Research Article Study on the Mechanism of Rock...
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Research Article Study on the Mechanism of Rock Stress Sensitivity Using a Random Pore Network Simulation Qi Chen, 1,2 Quanwen Liu, 2 and Zhengwu Tao 3 1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, China 2 Guangdong University of Petrochemical Technology, Guangdong, Maoming 525000, China 3 PetroChina Tarim Oilfield Company, Korla, Xinjiang 841000, China Correspondence should be addressed to Quanwen Liu; [email protected] Received 18 March 2016; Revised 27 April 2016; Accepted 10 May 2016 Academic Editor: Renal Backov Copyright © 2016 Qi Chen et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Studies of rock stress sensitivity are mainly focused on experimental and data processing methods, and the mechanism cannot be adequately explained using specific pore shape models. is study, based on a random pore network simulation, explains the rock stress sensitivity mechanism for the first time. Based on the network model theory, the hydraulic conductivity equation, the dimensionless radius equation, and the effective stress equation for partially saturated rock are used to generate a three-dimensional random pore network model based on the QT platform. e simulation results show that the influence of the effective stress on the dimensionless radius becomes more significant as the aspect ratio decreases, and the relationship between dimensionless radius and effective stress can be effectively interpreted through different combinations of pore shapes. Moreover, the mechanism behind permeability stress sensitivity can be explained by establishing the relationship between permeability and effective stress. 1. Introduction Rock stress sensitivity refers to the changes in rock petrophys- ical parameters caused by effective stress, including porosity stress sensitivity [1–3], permeability stress sensitivity [4, 5], and stress sensitivity of electrical resistivity [1, 6], among which permeability stress sensitivity is most frequently dis- cussed [4, 5]. e study of permeability stress sensitivity is mainly focused on experimental and data processing methods [5], which lack in-depth analysis of the micro-pore structure eigenvalue. Although some researchers have previously built theoret- ical models based on specific pore shapes, such as circular [7], oval [8], conical, hyperbolic triangular [9], and hyperbolic quadrilateral [10], these models, in which only specific pores are considered, are not sufficient to explain rock stress sensitivity. erefore, pore network models that consider various pore shapes are preferable to study the mechanisms applicable to the micro-pore structure [11]. Pore network models include pore space-based imaging network models [12] and regular pore network models [11]. Use of the pore space-based imaging network model is relatively computationally expensive [12], whereas regular pore network models set the pore size and distribution and adopt certain distribution functions by integrating per- colation theory, which results in the pore network model featuring the same complexity as that of a real rock [13]. As a result, this study, based on a QT platform, focuses on the dimensionless radius-randomized stress change by adopting a C++ program to generate randomized pore network models and then integrating the different pore types, proportions, and micro-pore networks. e permeability stress sensitivity mechanism is then explained by establishing the relationship between permeability and the dimensionless radius function. 2. Establishing the Randomized Pore Network Model 2.1. eoretical Basis of Network Models 2.1.1. Percolation eory. Percolation theory is central to the study of randomized pore network models, which is mainly Hindawi Publishing Corporation Journal of Chemistry Volume 2016, Article ID 5343786, 10 pages http://dx.doi.org/10.1155/2016/5343786
Transcript of Research Article Study on the Mechanism of Rock...
Research ArticleStudy on the Mechanism of Rock Stress Sensitivity Usinga Random Pore Network Simulation
Qi Chen12 Quanwen Liu2 and Zhengwu Tao3
1State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation Southwest Petroleum UniversityChengdu Sichuan 610500 China2Guangdong University of Petrochemical Technology Guangdong Maoming 525000 China3PetroChina Tarim Oilfield Company Korla Xinjiang 841000 China
Correspondence should be addressed to Quanwen Liu lqw005533163com
Received 18 March 2016 Revised 27 April 2016 Accepted 10 May 2016
Academic Editor Renal Backov
Copyright copy 2016 Qi Chen et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Studies of rock stress sensitivity are mainly focused on experimental and data processing methods and the mechanism cannotbe adequately explained using specific pore shape models This study based on a random pore network simulation explains therock stress sensitivity mechanism for the first time Based on the network model theory the hydraulic conductivity equation thedimensionless radius equation and the effective stress equation for partially saturated rock are used to generate a three-dimensionalrandom pore network model based on the QT platformThe simulation results show that the influence of the effective stress on thedimensionless radius becomes more significant as the aspect ratio decreases and the relationship between dimensionless radiusand effective stress can be effectively interpreted through different combinations of pore shapes Moreover the mechanism behindpermeability stress sensitivity can be explained by establishing the relationship between permeability and effective stress
1 Introduction
Rock stress sensitivity refers to the changes in rock petrophys-ical parameters caused by effective stress including porositystress sensitivity [1ndash3] permeability stress sensitivity [4 5]and stress sensitivity of electrical resistivity [1 6] amongwhich permeability stress sensitivity is most frequently dis-cussed [4 5] The study of permeability stress sensitivityis mainly focused on experimental and data processingmethods [5] which lack in-depth analysis of the micro-porestructure eigenvalue
Although some researchers have previously built theoret-ical models based on specific pore shapes such as circular [7]oval [8] conical hyperbolic triangular [9] and hyperbolicquadrilateral [10] these models in which only specific poresare considered are not sufficient to explain rock stresssensitivity Therefore pore network models that considervarious pore shapes are preferable to study the mechanismsapplicable to the micro-pore structure [11]
Pore network models include pore space-based imagingnetwork models [12] and regular pore network models [11]
Use of the pore space-based imaging network model isrelatively computationally expensive [12] whereas regularpore network models set the pore size and distributionand adopt certain distribution functions by integrating per-colation theory which results in the pore network modelfeaturing the same complexity as that of a real rock [13] Asa result this study based on a QT platform focuses on thedimensionless radius-randomized stress change by adoptingaC++ program to generate randomized pore networkmodelsand then integrating the different pore types proportionsand micro-pore networks The permeability stress sensitivitymechanism is then explained by establishing the relationshipbetween permeability and the dimensionless radius function
2 Establishing the RandomizedPore Network Model
21 Theoretical Basis of Network Models
211 Percolation Theory Percolation theory is central to thestudy of randomized pore network models which is mainly
Hindawi Publishing CorporationJournal of ChemistryVolume 2016 Article ID 5343786 10 pageshttpdxdoiorg10115520165343786
2 Journal of Chemistry
used to describe random structures and connectivity It isused to study the fluid distribution and flow in a random anddisorderly medium as in randomized pore network modelsusing statistical methods The randomized pore networkmodel employed in this study is designed as follows thelines in the model symbolize throats of a certain volumeand flow resistance with different distribution modes suchas a uniform distribution and the intersections of the linessymbolize the throats without volume and flow resistancethat only function as connections therefore the calculationof percolation parameters should be mainly focused on thelinearity calculation without considering the node Such atechnique based on the linear distribution of the networkstudies the impact of the microdistribution on the macro-properties of the porous media Since the percolation theorycomplies with the probability theory and the statistical accu-racy depends on the sample size the samples shouldmeet therequirements for a reliable statistical result and therefore thenetwork simulation based on percolation theory requires thatthe three-dimensional network model has nodes of at least20 times 20 times 20 [13] Using these conditions and the allowablecomputer performance the models can be applicable to abroader range of situations Within the calculation limits ofthe computer the more nodes the models have the moremacroattributes they will reflect
212 Similarity Principle between Water and Electricity Thenetwork simulation is based on the similarity principlebetween water and electricity By analyzing the flow in theporous medium the network structure of the circuit can beused to conduct a simulation analysis The current in thecircuit follows Ohmrsquos law
119868 =
1
119877
Δ119864 (1)
where 119868 is the current 119877 is the resistance and Δ119864 is thevoltage
By analyzing the network circuit we assume that thecircular cross section pipes filled with conduction fluid aremade of resistors
119868 =
1205871199032
120588119897
Δ119864 (2)
where 119903 is the pipe radius 120588 is the pipe electrical resistivityand 119897 is the resistor or pipe length
Assuming that flow is laminar the simplest cylindricalpipe is taken as an example and the Poiseuille equation isused
119902 =
1205871199034
8120583119897
Δ119901 (3)
where 119902 is the flow 120583 is the fluid viscosity and Δ119901 is thedifferential pressure at both ends of the pipe
The circuit network and fluid pore network are all madeup of pipes According to (2) and (3) Ohmrsquos law and thePoiseuille equation both explain the relationship between theflow of a current or fluid and the differential pressure at both
ends of the pipes (ie voltage and flow differential pressurevalues) the relationship between pipes and the differentialpressure of both ends of pipes (they are voltages and flowingpressure differential value) and the relationship among pipesThe flow relationship is similar to the similarity principlebetween water and electricity
Based on this the Poiseuille equation can be simplified toOhmrsquos law
119902 =
1
1198771015840Δ119901 (4)
where 11198771015840 = 12058711990348120583119897It can be seen from the similarity principle between water
and electricity above that the fluid in the porous network issimilar to the current in the circuit network therefore wecan conduct the fluid flow simulation in the pore networkmodel based on the current in the circuit network and usethe analyticalmethod for the current to analyze the simulatedfluid model
213 Kirchhoff Law Kirchhoff rsquos circuit laws are divided into(1) Kirchhoff rsquos current law whereby the current inflows ofany node in the circuit are equal to the current outflows and(2) Kirchhoff voltage law whereby the voltage in any circuitloop shall be zero after completing the loop in the direction ofcurrent flow For the pore networkmodel Kirchhoff rsquos currentlaw is adopted to build the equation set of the model
22 Key Points of the Program
221 The Equation Deduction of the Effective Stress forPartially Saturated Rock This study is focused on the quasi-static two-phase flow random pore network model Thefluid flow in the model is fully controlled by the capillarypressure assuming that the fluid is incompressible and theinfluence of a viscous force is ignored There are severalassumptions reflected in the physical model The oil-wateror gas-water interface is relatively static namely the fluiddistribution of the corresponding pore throat unit willchange once displacement occurs This assumption is thesame as the low-speed percolation of most porous mediaFurthermore in the network simulation we generally assumethat the displacement process is instantly completed andthen balanced namely the nonwetting fluid pressure (119901
119899)
is greater than or equal to the inlet capillary pressure of athroat (119901
119888) and therefore the displacement action will start
and the nonwetting fluid will displace the wetting fluid inthe throat The throats without displacement are filled withwetting fluid and the pressure is kept constant (zero) Undersuch circumstances the capillary pressure of the displacedthroat is as follows
119901119888= 119901119899 (5)
Throats with displacement contain a type of flowing fluidwhereas the wetting fluid at the corner is taken as a staticand nonflowing fluid its pressure value remains at 0 theflow inside the pipes can be deemed as single-phase flow andthe fluid pressure (119901
119891) equals the nonwetting displacement
Journal of Chemistry 3
pressure (119901119899) For throats that have not been displaced the
fluid inside is the wetting phase (water) and the pressure is0 There is no pressure difference between the nonwettingand wetting phases and capillary pressure does not existTherefore the effective stress expression is as follows
119901eff = 119901ov minus 120572119901119891 = 119901ov minus 120572119901119899 (6)
where 119901eff is the effective stress and 119901ov is the overburdenpressure
This theory was proposed by Bishop [14] who experi-mentally defined the general effective stress for a partiallysaturated porous medium as follows
1205901015840= 120590 minus 120572 [119901
119899minus 119878119908(119901119899minus 119901119908)] (7)
When wetting-phase saturation values are at the extremeendpoints (0 and 1) the product of wetting-phase saturationand capillary pressure (119878
119908119901119888) is 0 and when a relatively low
value is obtained for example 01 or less the peak value of119878119908119901119888will occur However this value is still very small so in
most cases it is negligible when compared with the effectivestress [14] Therefore the expression above can be simplifiedas
1205901015840= 120590 minus 120572119901
119899 (8)
This expression is the effective stress equation for two-phase fluids and is identical to (6) however the value of thecoefficient has been a topic for debate [15 16] If the valueof the coefficient is 0 the effective stress is the confiningpressure If the effective stress coefficient is 1 the effectivestress is the net stress To simplify this study the effectivestress coefficient is set as 0 under which circumstancesthe pore fluid pressure is assumed to impart no pressureon the effective stress or has much less impact than theconfining pressure Therefore this study on rock micro-porestructure is focused on the change in rock pore structureunder different confining pressures
222 Key Equations The key equation of this methodinvolves the hydraulic conductivity at the throats and thedimensionless radius equation The flow equation for ahyperbolic triangle throat [9] is
119902 = FT 1198884
120583119897
Δ119901 (9)
where FT is the dimensionless conductivity per unit length ofunit area which is solved by finite difference The hydraulicconductivity equation is
119867 =
119902
Δ119901
= FT 1198884
120583119897
(10)
Because there is no dimensionless radius equation forhyperbolic triangle throats in the literature here we present asimple deduction based on the formula from Gangi [17]
119903119901= 1 minus 119862
0(
119875
1198750
)
23
(11)
Hyperbolic triangles poreHyperbolic quadrilateral pore
Figure 1 Quadrilateral-triangles stress transformation
Moreover it is known from another Gangi formula that[17]
119875
1198750
=
3120587 (1 minus ]2)4119864
119875eff (12)
and considering that
1198620=
21198881119877
1199031
asymp 2 (13)
it can be deduced that the dimensionless radius equation forhyperbolic triangle throats is
119903119901= 1 minus [
3radic2120587 (1 minus ]2)2120576119864
119901eff]
23
(14)
In a similar manner the hydraulic conductivity equationand dimensionless radius equations for circular throats ovalthroats conical throats hyperbolic triangle throats and starthroats can be deduced shown in Table 1
In these equations 119903119901is the dimensionless radius which
is the ratio of the pore throat radius under effective stressto the pore throat radius under no effective stress FT isthe dimensionless conductivity per unit length of unit areawhich is solved by finite difference ] is the Poisson ratio119887 is the length of the semiminor axis 119888 is the length of thesemimajor axis 120576 is the aspect ratio 120576 = 119887119888 and 119864 is Youngrsquosmodulus
The dimensionless radius equations for hyperbolic trian-gle throats and star-shaped throats are similar only the coeffi-cient and index are different Therefore their conclusions aresimilar which is illustrated by Figure 1 and in the followingdiscussion
223 Coefficient Matrix Solution A key part of the networkmodel simulation is building operation expressions the basicidea of which is based on Kirchhoff rsquos law nodes and lineconductivity Noble and Daniels [18] presented the equationfor a network of119873 nodes
(119902) = 119860119870119860119879(119881) (15)
whereas (119902) = [11990211199022sdot sdot sdot 119902119899]1015840 (119881) = [119881
11198812sdot sdot sdot 119881119899]1015840
For a simple network the coefficient matrix can beobtained through a structural analysis however the simu-lation method of the model is based on statistical theory
4 Journal of Chemistry
Table 1 Key equation
Diagrammatic drawing Hydraulic conductivity equation Dimensionless radius equation
Circular throat
r
119867 = 0125
1205871199034
120583119897
119903119901= 1 minus
2 (1 minus ]2)119864
119901eff
Oval throat
ba
119867 = 025
12058712057631198884
120583119897(1205762+ 1)
119903119901= 1 minus
2 (1 minus ]2)120576119864
119901eff
Conical throat
bc
l 119867 = 0685
12058712057631198884
120583119897(1205762+ 1)
119903119901= 1 minus [
4 (1 minus ]2)3120576119864
119901eff]
12
Hyperbolictriangle throat 119867 = FT 119888
4
120583119897
119903119901= 1 minus [
3radic2120587 (1 minus ]2)2120576119864
119901eff]
23
Star throatdc
119867 = FT 1198884
120583119897
119903119901= 1 minus [
3radic2 (1 minus ]2)4120576119864
119901eff]
13
which dictates that the number of nodes and lines in thenetwork model needs to be large enough to ensure thereliability of the simulation As a result when the number ofnodes and connections is large enough the model cannot besolved by simple algebra and the Cholesky decompositionis required in order to solve the coefficient matrix 119860119870119860119879However the Cholesky decomposition affects the precisionof the coefficient matrix mainly due to the round-off errorand furthermore the program code is complicated [11] To thisend we use the iterative method to solve the matrix
The principal of the iterative method which graduallyapproaches the real solution is to take the assumed valueas the solution and perform continuous iterations until itmeets the convergence condition and obtains the solution ofthe equation For the convergent system set the deviationobtained from each displacement will decrease and its solu-tion becomes closer to the real solutionThe iterationmethodcan also automatically adjust the occasional calculation errorthat occurs in the iteration The method includes simpleiteration and super-relaxed iteration and in the followingsection the simplest two-dimensional square network modelis used as an example to introduce the solution process
The basis of the simple iteration also known as successiveiteration is to construct the fixed-point equation in order toobtain the approximate solutionThe simple iterationmethodis solved via the following steps
(1) Assign Initial Values for Nodes As the ultimate solution isnot associated with the initial given values the initial valuecan be randomly set yet the convergence rate depends onthe accuracy of the initial values We assume that the flowdirection is from left to right in the square network the leftend of the nodes is assignedwith voltage119881
1 and the right end
of the nodes is assignedwith voltage1198812 while other nodes are
assigned with a voltage of 0
(2) Establish the Equation Taking the 0 node in Figure 2 as anexample the following equation can be obtained according toKirchhoff rsquos current law
(119881119894119895minus 119881119894+1119895
) 119892(119894119895sim119894+1119895)
+ (119881119894119895minus 119881119894119895+1
) 119892(119894119895sim119894119895+1)
+ (119881119894119895minus 119881119894minus1119895
) 119892(119894119895sim119894minus1119895)
+ (119881119894119895minus 119881119894119895minus1
) 119892(119894119895sim119894119895minus1)
= 0
(16)
Journal of Chemistry 5
j
1(i + 1 j)
2(i j + 1)
3(i minus 1 j)
4(i j minus 1)
0(i j)
i
Figure 2 One node cell of square network model
This equation can be used for every node in the networkand in this way we obtain the system of linear equationswhose quantities are identical to the node quantities
(11988111minus 11988121) 119892(11sim21)
+ (11988111minus 11988112) 119892(11sim12)
+ (11988111minus 11988101) 119892(11sim01)
+ (11988111minus 11988110) 119892(11sim10)
= 0
(11988112minus 11988122) 119892(12sim22)
+ (11988112minus 11988113) 119892(12sim13)
+ (11988112minus 11988102) 119892(12sim02)
+ (11988112minus 11988111) 119892(12sim11)
= 0
(1198811119895minus 1198812119895) 119892(1119895sim2119895)
+ (1198811119895minus 1198811119895+1
) 119892(1119895sim1119895+1)
+ (1198811119895minus 1198810119895) 119892(1119895sim0119895)
+ (1198811119895minus 1198811119895minus1
) 119892(1119895sim1119895minus1)
= 0
(11988121minus 11988131) 119892(21sim31)
+ (11988121minus 11988122) 119892(21sim22)
+ (11988121minus 11988111) 119892(21sim11)
+ (11988121minus 11988120) 119892(21sim20)
= 0
(11988131minus 11988141) 119892(31sim41)
+ (11988131minus 11988132) 119892(31sim32)
+ (11988131minus 11988121) 119892(31sim21)
+ (11988131minus 11988130) 119892(31sim30)
= 0
(119881119894119895minus 119881119894+1119895
) 119892(119894119895sim119894+1119895)
+ (119881119894119895minus 119881119894119895+1
) 119892(119894119895sim119894119895+1)
+ (119881119894119895minus 119881119894minus1119895
) 119892(119894119895sim119894minus1119895)
+ (119881119894119895minus 119881119894119895minus1
) 119892(119894119895sim119894119895minus1)
= 0
(17)
To conduct the iteration solution process (17) is changedas follows
119881119899+1
119894119895=
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
(18)
(3) Iteration Solution The inputoutput current value in thenetwork is calculated after each iteration The iteration isonly deemed completed if the inputoutput currents are equalor the difference is within the error After the iteration isfinished the inputoutput current is solved the resistance iscalculated using Ohmrsquos law and then the other parametersof the network model such as electrical resistivity aresolved
As the convergence rate of the simple iteration is low weuse the super-relaxed iteration and the program obtains theadjacent node voltage values from the last step calculationduring the node calculation For example when we calculatethe points (119894 119895) in the square network the voltage of theleft point (119894 minus 1 119895) and the point below (119894 119895 minus 1) willbe replaced by the voltage values obtained from the laststep
119881119899+1
119894119895=
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
(19)
The method is called Gauss-Seidel iteration and itreplaces the new values obtained from the previous last step
in order to speed up the convergence ratesThe increment canbe written as
6 Journal of Chemistry
119881119899+1
119894119895= 119881119899
119894119895+ (
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
minus 119881119899
119894119895) (20)
To speed up convergence and introduce the relaxingfactor 119890 (obtaining the values 1-2) the above expression canbe written as follows
119881119899+1
119894119895= 119881119899
119894119895+ 119890(
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
minus 119881119899
119894119895) (21)
(4) Initial Radius Assignment Considering the randomnessof the rock pore radius and the strong heterogeneity of thereservoir rocks the network models used in our researchdetermine the pipe radius by random functions in order togenerate randomized and nonhomogeneous network mod-els This study selects radius as per the logarithmic meandistribution and uniform distribution of which the normal-ized standard deviations are 005 030 055 080 and 105and 005 030 and 055 respectively The radius is generatedthrough the following random numbers
119903 = 119890rand( )((int(lg(119903max)lowast50)minusint(lg(119903min)lowast50))50)
+ lg (119903min) (22)
where rand( ) is the random number 119903max is the maximumthroat radius and 119903min is the minimum throat radius
It can be seen from the above equation that the radius ofany throat in the network is between the maximum throatradius and minimum throat radius It should be particularlynoted that the hydraulic radii (ie twice the volume tosurface ratio of the pores 119903
119867= 2119881
119901119878119901 where 119881
119901is
the total pore volume and 119878119901is the total wetted surface
area) all remain constant (40 120583m) Under the conditionsof log uniform distribution and uniform distribution thehydraulic radius and normalized standard deviation shouldmeet the requirements in Table 2 When the hydraulic radiusis set at 40 120583m the normalized standard deviation and thelargestsmallest throat radius in Table 3 should be used
23 Program Implementation
231 InputOutput Parameters Based on the theory of thenetwork model and the key programming points the C++language is used to compile random network model pro-cedures on the QT platform Regarding the grid size (119909119910 and 119911 direction) the maximumminimum throat radiimaximumminimumaspect ratios formationwater viscositygrid unit length (pipe length) connectivity probability andthe proportion of pores of each shape aspect ratios areinputted through program interfaces
When running the program the first output is a 3D cuberandom network model whose size is set at 100 times 100 times
100 (Figure 3) followed by the relevant parameters of this
program (Table 4) Finally it produces the aspect ratios ofdifferent shaped throats and graphs of the dimensionlessradius and the effective stress
232 Program Result Analysis In order to analyze theinfluence of the aspect ratio on the calculation result theprogram is set according to a specific throat and the changein dimensionless radius due to effective stress is studiedby setting different aspect ratios See Figures 4ndash7 for thecomparison results of the program valuation
It is observed from Figures 4ndash7 that the smaller the aspectratio the more significant the influence of the effective stresson the dimensionless radius Taking star-shaped throats as anexample we can see in the dimensionless radius equation thatthe smaller the aspect ratio 120576 is the more the dimensionlessradiuswill be influenced by the stress changeThe star-shapedthroats experience deformation with the change in stress(Figure 8) and when the aspect ratio is smaller the throatapproaches a crack shape and the influence of the crack onstress is more evident
The relationship described above is between the dimen-sionless radius and the effective stress in a specific throatand does not consider the range in aspect ratio values orthe combination of the aspect ratio and the proportionTherefore taking the star-shaped pore as an example theplan and operation results shown in Figure 9 consider thespecific throat aspect ratio and proportion combinationshowever there will certainly be different plans for oval coneand trilateral throats
In addition to the different aspect ratios and proportioncombinations for one specific pore shape other relationshipsexist between dimensionless radius and effective stress withcombinations of two- three- or four-pore throat typesDue to limited space the combination of three differentthroat types is taken as an example and the relationshipbetween dimensionless radius and effective stress is effec-tively explained by a specific combination plan (Figure 10)
3 Explanation for Rock Stress Sensitivity
The analysis above indicates that the pore structure of arock changes with the degree of effective stress which
Journal of Chemistry 7
Table 2 (Log) uniform distribution
Log uniform distribution Uniform distribution
119903119867=
119903max + 119903min2
119903119867= 2
1199032
max + 119903max119903min + 1199032
min
3 (119903max + 119903min)
120590119903= radic
ln (119903max119903min) (119903max + 119903min)
2 (119903max minus 119903min)minus 1 120590
119903=
(119903max minus 119903min)
radic3 (119903max + 119903min)
Table 3 Value of 119904119903 119903max and 119903min
Log uniform distribution Uniform distribution119904119903
119903maxmm 119903minmm 119904119903
119903maxmm 119903minmm005 434563 365437 005 433557 364448030 592411 207589 030 557657 176288055 701570 984301 055 599655 145480080 760414 395856105 786472 135281
Table 4 Program output parameters
Output parameters ValuePore volume 269386119890 + 006
Pore surface area 135504Hydraulic radius 397605Numbers of tube bundle 2970000Total input electric current 0109619Total output electric current 00851105Average coordination number 59374
induces changes in the rock physical properties Therefore acrucial factor in studies of rock stress sensitivity particularlypermeability stress sensitivity is understanding the responseof the pore radius to effective stress namely
119903 = 119891 (119901eff) (23)
The relationship between the pore radius derived from themodel and the dimensionless radius 119903
119901is as follows
radius = InitRadius lowast 119903119901 (24)
which establishes the function for the pore radius (radius)and the effective stress (119901eff ) It can be seen from Table 1that the hydraulic power conductivity (ℎcon) is a function ofthe radius (radius) and the inputoutput flow is a functionof the hydraulic power (ℎcon) Furthermore the permeabilitycalculation function (which is based on Darcyrsquos Law) isrelated to the inflow or outflow from which we establish therelationship between permeability and effective stress (119901eff )Therefore our model by fitting data from the top of the rockcore through a combination of different pore shape models(see Figure 11) can effectively explain themechanisms behindrock permeability stress sensitivity
Figure 3 3D pore network model
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
Figure 4 Effect of aspect ratio on oval pores
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
Figure 5 Effect of aspect ratio on taper pores
4 Conclusion
(1) The effective stress equation for partially saturatedrock is derived and verified and the core operationexpressions of the network model are obtained usingiteration methods
8 Journal of Chemistry
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
Figure 6 Effect of aspect ratio on triangular pores
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
Figure 7 Effect of aspect ratio on star-shaped pores
(a) 120576 = 02 (b) 120576 = 0005
Figure 8 Star-shaped throat at different aspect ratios
(2) The network model simulation shows that a circularthroat is the special case for an ellipse and the smallerthe aspect ratio the greater the effect of stress on thedimensionless radius
(3) The relationship between the experimental dimen-sionless radius and the effective stress can beexplained through different pore shape combinationsbased on the network model
Journal of Chemistry 9
0
02
04
06
08
1
0 5 10 15 20 25 30Effective stress Peff (MPa)
The relation of dimensionless radius rpand effective stress Peff
Dim
ensio
nles
s rad
iusr
p
Figure 9 Star-shaped [10 (60) + 01 (20) + 005 (10) + 001 (10)]
Experimental dataCalculated curve
0
02
04
06
08
1
Dim
ensio
nles
s rad
iusr
p
0 5 10 15 20 25 30Effective stress Peff (MPa)
Figure 10 Combination of three pores types 20 cone [10 (60) + 01 (20) + 0005 (10) + 0001 (10)] + 30 triangle [10 (60) + 01(20) + 0005 (10) + 0001 (10)] + 50 star [10 (60) + 01 (20) + 0005 (10) + 0001 (10)]
0
02
04
06
08
1
0 20 40 60 80Effective stress Peff (MPa)
Matched curveExperimental curve
Dim
ensio
nles
s per
mea
bilit
yk
The relation of dimensionless permeability kand effective stress Peff
Figure 11 Comparison diagram of the program calculation curve and experiment data 50 oval [001 (60) + 0005 (30) + 00012 (10)]+ 50 star [01 (27) + 0005 (36) + 00012 (37)]
10 Journal of Chemistry
(4) The permeability stress sensitivity is effectivelyexplained using the network simulation
Competing Interests
The authors declare that they have no competing interests
References
[1] V M Dobrynin ldquoEffect of overburden pressure on some prop-erties of sandstonesrdquo Society of PetroleumEngineers Journal vol2 no 4 pp 360ndash366 1962
[2] H S Ali M A Al-Marhoun S A Abu-Khamsin et alldquoThe effect of overburden pressure on relative permeabilityrdquoin Proceedinga of the 5th SPE Middle East Oil Show ManamaBahram March 1987
[3] M E Shafiee and A Kantzas ldquoInvestigation on the effectof overburden pressure on vuggy carbonate oil reservoircore propertiesrdquo in Proceedings of the Canadian InternationalPetroleum Conference (CIPC rsquo09) pp 16ndash18 Calgary AlbertaJune 2009
[4] D Tiab and E C Donaldson Petrophysics Theory and Practiceof Measuring Reservoir Rock and Fluid Transport PropertiesElsevierGulf Professional Publishing 2012
[5] M Li Y Bernabe W-I Xiao Z-Y Chen and Z-Q LiuldquoEffective pressure law for permeability of E-bei sandstonesrdquoJournal of Geophysical Research B Solid Earth vol 114 no 7Article ID B07205 p 223 2009
[6] D G Longeron M J Argaud and J P Feraud ldquoEffect of over-burden pressure and the nature andmicroscopic distribution offluids on electrical properties of rock samplesrdquo SPE FormationEvaluation vol 6 pp 194ndash202 1989
[7] J C Jaeger and N Cook Fundamentals of Rock MechanicsChapman amp Hall 2nd edition 1976
[8] D A Seeburger and A Nur ldquoA pore space model for rock per-meability and bulk modulusrdquo Journal of Geophysical Researchvol 89 no 1 pp 527ndash536 1984
[9] D P Yale and A Nur ldquoNetwork modeling of flow storage anddeformation in porous rocksrdquo SEGTechnical ProgramExpandedAbstracts vol 4 no 1 p 437 1999
[10] R F Sigal ldquoThe pressure dependence of permeabilityrdquo Petro-physics vol 43 no 2 pp 92ndash102 2002
[11] T Zhengwu Numerical Experiments of Capillary PressureCurve Southwest Petroleum University Chengdu China 2013(Chinese)
[12] J T Fredrich ldquo3D imaging of porousmedia using laser scanningconfocal microscopy with application to microscale transportprocessesrdquo Physics and Chemistry of the Earth Part A SolidEarth and Geodesy vol 24 no 7 pp 551ndash561 1999
[13] C D Tsakiroglou and M Fleury ldquoPore network analysis ofresistivity index for water-wet porous mediardquo Transport inPorous Media vol 35 no 1 pp 89ndash128 1999
[14] AW Bishop ldquoThe principle of effective stressrdquoTekniskUkebladvol 106 no 39 pp 859ndash863 1960
[15] W G Gray and B A Schrefler ldquoThermodynamic approach toeffective stress in partially saturated porous mediardquo EuropeanJournal of Mechanics ASolids vol 20 no 4 pp 521ndash538 2001
[16] M Nuth and L Laloui ldquoEffective stress concept in unsaturatedsoils clarification and validation of a unified frameworkrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 32 no 7 pp 771ndash801 2008
[17] A F Gangi ldquoVariation of whole and fractured porous rockpermeability with confining pressurerdquo International Journal ofRock Mechanics andMining Sciences and vol 15 no 5 pp 249ndash257 1978
[18] B B Noble and J W DanielsApplied Linear Algebra AmericanMathematical Society 3rd edition 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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2 Journal of Chemistry
used to describe random structures and connectivity It isused to study the fluid distribution and flow in a random anddisorderly medium as in randomized pore network modelsusing statistical methods The randomized pore networkmodel employed in this study is designed as follows thelines in the model symbolize throats of a certain volumeand flow resistance with different distribution modes suchas a uniform distribution and the intersections of the linessymbolize the throats without volume and flow resistancethat only function as connections therefore the calculationof percolation parameters should be mainly focused on thelinearity calculation without considering the node Such atechnique based on the linear distribution of the networkstudies the impact of the microdistribution on the macro-properties of the porous media Since the percolation theorycomplies with the probability theory and the statistical accu-racy depends on the sample size the samples shouldmeet therequirements for a reliable statistical result and therefore thenetwork simulation based on percolation theory requires thatthe three-dimensional network model has nodes of at least20 times 20 times 20 [13] Using these conditions and the allowablecomputer performance the models can be applicable to abroader range of situations Within the calculation limits ofthe computer the more nodes the models have the moremacroattributes they will reflect
212 Similarity Principle between Water and Electricity Thenetwork simulation is based on the similarity principlebetween water and electricity By analyzing the flow in theporous medium the network structure of the circuit can beused to conduct a simulation analysis The current in thecircuit follows Ohmrsquos law
119868 =
1
119877
Δ119864 (1)
where 119868 is the current 119877 is the resistance and Δ119864 is thevoltage
By analyzing the network circuit we assume that thecircular cross section pipes filled with conduction fluid aremade of resistors
119868 =
1205871199032
120588119897
Δ119864 (2)
where 119903 is the pipe radius 120588 is the pipe electrical resistivityand 119897 is the resistor or pipe length
Assuming that flow is laminar the simplest cylindricalpipe is taken as an example and the Poiseuille equation isused
119902 =
1205871199034
8120583119897
Δ119901 (3)
where 119902 is the flow 120583 is the fluid viscosity and Δ119901 is thedifferential pressure at both ends of the pipe
The circuit network and fluid pore network are all madeup of pipes According to (2) and (3) Ohmrsquos law and thePoiseuille equation both explain the relationship between theflow of a current or fluid and the differential pressure at both
ends of the pipes (ie voltage and flow differential pressurevalues) the relationship between pipes and the differentialpressure of both ends of pipes (they are voltages and flowingpressure differential value) and the relationship among pipesThe flow relationship is similar to the similarity principlebetween water and electricity
Based on this the Poiseuille equation can be simplified toOhmrsquos law
119902 =
1
1198771015840Δ119901 (4)
where 11198771015840 = 12058711990348120583119897It can be seen from the similarity principle between water
and electricity above that the fluid in the porous network issimilar to the current in the circuit network therefore wecan conduct the fluid flow simulation in the pore networkmodel based on the current in the circuit network and usethe analyticalmethod for the current to analyze the simulatedfluid model
213 Kirchhoff Law Kirchhoff rsquos circuit laws are divided into(1) Kirchhoff rsquos current law whereby the current inflows ofany node in the circuit are equal to the current outflows and(2) Kirchhoff voltage law whereby the voltage in any circuitloop shall be zero after completing the loop in the direction ofcurrent flow For the pore networkmodel Kirchhoff rsquos currentlaw is adopted to build the equation set of the model
22 Key Points of the Program
221 The Equation Deduction of the Effective Stress forPartially Saturated Rock This study is focused on the quasi-static two-phase flow random pore network model Thefluid flow in the model is fully controlled by the capillarypressure assuming that the fluid is incompressible and theinfluence of a viscous force is ignored There are severalassumptions reflected in the physical model The oil-wateror gas-water interface is relatively static namely the fluiddistribution of the corresponding pore throat unit willchange once displacement occurs This assumption is thesame as the low-speed percolation of most porous mediaFurthermore in the network simulation we generally assumethat the displacement process is instantly completed andthen balanced namely the nonwetting fluid pressure (119901
119899)
is greater than or equal to the inlet capillary pressure of athroat (119901
119888) and therefore the displacement action will start
and the nonwetting fluid will displace the wetting fluid inthe throat The throats without displacement are filled withwetting fluid and the pressure is kept constant (zero) Undersuch circumstances the capillary pressure of the displacedthroat is as follows
119901119888= 119901119899 (5)
Throats with displacement contain a type of flowing fluidwhereas the wetting fluid at the corner is taken as a staticand nonflowing fluid its pressure value remains at 0 theflow inside the pipes can be deemed as single-phase flow andthe fluid pressure (119901
119891) equals the nonwetting displacement
Journal of Chemistry 3
pressure (119901119899) For throats that have not been displaced the
fluid inside is the wetting phase (water) and the pressure is0 There is no pressure difference between the nonwettingand wetting phases and capillary pressure does not existTherefore the effective stress expression is as follows
119901eff = 119901ov minus 120572119901119891 = 119901ov minus 120572119901119899 (6)
where 119901eff is the effective stress and 119901ov is the overburdenpressure
This theory was proposed by Bishop [14] who experi-mentally defined the general effective stress for a partiallysaturated porous medium as follows
1205901015840= 120590 minus 120572 [119901
119899minus 119878119908(119901119899minus 119901119908)] (7)
When wetting-phase saturation values are at the extremeendpoints (0 and 1) the product of wetting-phase saturationand capillary pressure (119878
119908119901119888) is 0 and when a relatively low
value is obtained for example 01 or less the peak value of119878119908119901119888will occur However this value is still very small so in
most cases it is negligible when compared with the effectivestress [14] Therefore the expression above can be simplifiedas
1205901015840= 120590 minus 120572119901
119899 (8)
This expression is the effective stress equation for two-phase fluids and is identical to (6) however the value of thecoefficient has been a topic for debate [15 16] If the valueof the coefficient is 0 the effective stress is the confiningpressure If the effective stress coefficient is 1 the effectivestress is the net stress To simplify this study the effectivestress coefficient is set as 0 under which circumstancesthe pore fluid pressure is assumed to impart no pressureon the effective stress or has much less impact than theconfining pressure Therefore this study on rock micro-porestructure is focused on the change in rock pore structureunder different confining pressures
222 Key Equations The key equation of this methodinvolves the hydraulic conductivity at the throats and thedimensionless radius equation The flow equation for ahyperbolic triangle throat [9] is
119902 = FT 1198884
120583119897
Δ119901 (9)
where FT is the dimensionless conductivity per unit length ofunit area which is solved by finite difference The hydraulicconductivity equation is
119867 =
119902
Δ119901
= FT 1198884
120583119897
(10)
Because there is no dimensionless radius equation forhyperbolic triangle throats in the literature here we present asimple deduction based on the formula from Gangi [17]
119903119901= 1 minus 119862
0(
119875
1198750
)
23
(11)
Hyperbolic triangles poreHyperbolic quadrilateral pore
Figure 1 Quadrilateral-triangles stress transformation
Moreover it is known from another Gangi formula that[17]
119875
1198750
=
3120587 (1 minus ]2)4119864
119875eff (12)
and considering that
1198620=
21198881119877
1199031
asymp 2 (13)
it can be deduced that the dimensionless radius equation forhyperbolic triangle throats is
119903119901= 1 minus [
3radic2120587 (1 minus ]2)2120576119864
119901eff]
23
(14)
In a similar manner the hydraulic conductivity equationand dimensionless radius equations for circular throats ovalthroats conical throats hyperbolic triangle throats and starthroats can be deduced shown in Table 1
In these equations 119903119901is the dimensionless radius which
is the ratio of the pore throat radius under effective stressto the pore throat radius under no effective stress FT isthe dimensionless conductivity per unit length of unit areawhich is solved by finite difference ] is the Poisson ratio119887 is the length of the semiminor axis 119888 is the length of thesemimajor axis 120576 is the aspect ratio 120576 = 119887119888 and 119864 is Youngrsquosmodulus
The dimensionless radius equations for hyperbolic trian-gle throats and star-shaped throats are similar only the coeffi-cient and index are different Therefore their conclusions aresimilar which is illustrated by Figure 1 and in the followingdiscussion
223 Coefficient Matrix Solution A key part of the networkmodel simulation is building operation expressions the basicidea of which is based on Kirchhoff rsquos law nodes and lineconductivity Noble and Daniels [18] presented the equationfor a network of119873 nodes
(119902) = 119860119870119860119879(119881) (15)
whereas (119902) = [11990211199022sdot sdot sdot 119902119899]1015840 (119881) = [119881
11198812sdot sdot sdot 119881119899]1015840
For a simple network the coefficient matrix can beobtained through a structural analysis however the simu-lation method of the model is based on statistical theory
4 Journal of Chemistry
Table 1 Key equation
Diagrammatic drawing Hydraulic conductivity equation Dimensionless radius equation
Circular throat
r
119867 = 0125
1205871199034
120583119897
119903119901= 1 minus
2 (1 minus ]2)119864
119901eff
Oval throat
ba
119867 = 025
12058712057631198884
120583119897(1205762+ 1)
119903119901= 1 minus
2 (1 minus ]2)120576119864
119901eff
Conical throat
bc
l 119867 = 0685
12058712057631198884
120583119897(1205762+ 1)
119903119901= 1 minus [
4 (1 minus ]2)3120576119864
119901eff]
12
Hyperbolictriangle throat 119867 = FT 119888
4
120583119897
119903119901= 1 minus [
3radic2120587 (1 minus ]2)2120576119864
119901eff]
23
Star throatdc
119867 = FT 1198884
120583119897
119903119901= 1 minus [
3radic2 (1 minus ]2)4120576119864
119901eff]
13
which dictates that the number of nodes and lines in thenetwork model needs to be large enough to ensure thereliability of the simulation As a result when the number ofnodes and connections is large enough the model cannot besolved by simple algebra and the Cholesky decompositionis required in order to solve the coefficient matrix 119860119870119860119879However the Cholesky decomposition affects the precisionof the coefficient matrix mainly due to the round-off errorand furthermore the program code is complicated [11] To thisend we use the iterative method to solve the matrix
The principal of the iterative method which graduallyapproaches the real solution is to take the assumed valueas the solution and perform continuous iterations until itmeets the convergence condition and obtains the solution ofthe equation For the convergent system set the deviationobtained from each displacement will decrease and its solu-tion becomes closer to the real solutionThe iterationmethodcan also automatically adjust the occasional calculation errorthat occurs in the iteration The method includes simpleiteration and super-relaxed iteration and in the followingsection the simplest two-dimensional square network modelis used as an example to introduce the solution process
The basis of the simple iteration also known as successiveiteration is to construct the fixed-point equation in order toobtain the approximate solutionThe simple iterationmethodis solved via the following steps
(1) Assign Initial Values for Nodes As the ultimate solution isnot associated with the initial given values the initial valuecan be randomly set yet the convergence rate depends onthe accuracy of the initial values We assume that the flowdirection is from left to right in the square network the leftend of the nodes is assignedwith voltage119881
1 and the right end
of the nodes is assignedwith voltage1198812 while other nodes are
assigned with a voltage of 0
(2) Establish the Equation Taking the 0 node in Figure 2 as anexample the following equation can be obtained according toKirchhoff rsquos current law
(119881119894119895minus 119881119894+1119895
) 119892(119894119895sim119894+1119895)
+ (119881119894119895minus 119881119894119895+1
) 119892(119894119895sim119894119895+1)
+ (119881119894119895minus 119881119894minus1119895
) 119892(119894119895sim119894minus1119895)
+ (119881119894119895minus 119881119894119895minus1
) 119892(119894119895sim119894119895minus1)
= 0
(16)
Journal of Chemistry 5
j
1(i + 1 j)
2(i j + 1)
3(i minus 1 j)
4(i j minus 1)
0(i j)
i
Figure 2 One node cell of square network model
This equation can be used for every node in the networkand in this way we obtain the system of linear equationswhose quantities are identical to the node quantities
(11988111minus 11988121) 119892(11sim21)
+ (11988111minus 11988112) 119892(11sim12)
+ (11988111minus 11988101) 119892(11sim01)
+ (11988111minus 11988110) 119892(11sim10)
= 0
(11988112minus 11988122) 119892(12sim22)
+ (11988112minus 11988113) 119892(12sim13)
+ (11988112minus 11988102) 119892(12sim02)
+ (11988112minus 11988111) 119892(12sim11)
= 0
(1198811119895minus 1198812119895) 119892(1119895sim2119895)
+ (1198811119895minus 1198811119895+1
) 119892(1119895sim1119895+1)
+ (1198811119895minus 1198810119895) 119892(1119895sim0119895)
+ (1198811119895minus 1198811119895minus1
) 119892(1119895sim1119895minus1)
= 0
(11988121minus 11988131) 119892(21sim31)
+ (11988121minus 11988122) 119892(21sim22)
+ (11988121minus 11988111) 119892(21sim11)
+ (11988121minus 11988120) 119892(21sim20)
= 0
(11988131minus 11988141) 119892(31sim41)
+ (11988131minus 11988132) 119892(31sim32)
+ (11988131minus 11988121) 119892(31sim21)
+ (11988131minus 11988130) 119892(31sim30)
= 0
(119881119894119895minus 119881119894+1119895
) 119892(119894119895sim119894+1119895)
+ (119881119894119895minus 119881119894119895+1
) 119892(119894119895sim119894119895+1)
+ (119881119894119895minus 119881119894minus1119895
) 119892(119894119895sim119894minus1119895)
+ (119881119894119895minus 119881119894119895minus1
) 119892(119894119895sim119894119895minus1)
= 0
(17)
To conduct the iteration solution process (17) is changedas follows
119881119899+1
119894119895=
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
(18)
(3) Iteration Solution The inputoutput current value in thenetwork is calculated after each iteration The iteration isonly deemed completed if the inputoutput currents are equalor the difference is within the error After the iteration isfinished the inputoutput current is solved the resistance iscalculated using Ohmrsquos law and then the other parametersof the network model such as electrical resistivity aresolved
As the convergence rate of the simple iteration is low weuse the super-relaxed iteration and the program obtains theadjacent node voltage values from the last step calculationduring the node calculation For example when we calculatethe points (119894 119895) in the square network the voltage of theleft point (119894 minus 1 119895) and the point below (119894 119895 minus 1) willbe replaced by the voltage values obtained from the laststep
119881119899+1
119894119895=
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
(19)
The method is called Gauss-Seidel iteration and itreplaces the new values obtained from the previous last step
in order to speed up the convergence ratesThe increment canbe written as
6 Journal of Chemistry
119881119899+1
119894119895= 119881119899
119894119895+ (
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
minus 119881119899
119894119895) (20)
To speed up convergence and introduce the relaxingfactor 119890 (obtaining the values 1-2) the above expression canbe written as follows
119881119899+1
119894119895= 119881119899
119894119895+ 119890(
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
minus 119881119899
119894119895) (21)
(4) Initial Radius Assignment Considering the randomnessof the rock pore radius and the strong heterogeneity of thereservoir rocks the network models used in our researchdetermine the pipe radius by random functions in order togenerate randomized and nonhomogeneous network mod-els This study selects radius as per the logarithmic meandistribution and uniform distribution of which the normal-ized standard deviations are 005 030 055 080 and 105and 005 030 and 055 respectively The radius is generatedthrough the following random numbers
119903 = 119890rand( )((int(lg(119903max)lowast50)minusint(lg(119903min)lowast50))50)
+ lg (119903min) (22)
where rand( ) is the random number 119903max is the maximumthroat radius and 119903min is the minimum throat radius
It can be seen from the above equation that the radius ofany throat in the network is between the maximum throatradius and minimum throat radius It should be particularlynoted that the hydraulic radii (ie twice the volume tosurface ratio of the pores 119903
119867= 2119881
119901119878119901 where 119881
119901is
the total pore volume and 119878119901is the total wetted surface
area) all remain constant (40 120583m) Under the conditionsof log uniform distribution and uniform distribution thehydraulic radius and normalized standard deviation shouldmeet the requirements in Table 2 When the hydraulic radiusis set at 40 120583m the normalized standard deviation and thelargestsmallest throat radius in Table 3 should be used
23 Program Implementation
231 InputOutput Parameters Based on the theory of thenetwork model and the key programming points the C++language is used to compile random network model pro-cedures on the QT platform Regarding the grid size (119909119910 and 119911 direction) the maximumminimum throat radiimaximumminimumaspect ratios formationwater viscositygrid unit length (pipe length) connectivity probability andthe proportion of pores of each shape aspect ratios areinputted through program interfaces
When running the program the first output is a 3D cuberandom network model whose size is set at 100 times 100 times
100 (Figure 3) followed by the relevant parameters of this
program (Table 4) Finally it produces the aspect ratios ofdifferent shaped throats and graphs of the dimensionlessradius and the effective stress
232 Program Result Analysis In order to analyze theinfluence of the aspect ratio on the calculation result theprogram is set according to a specific throat and the changein dimensionless radius due to effective stress is studiedby setting different aspect ratios See Figures 4ndash7 for thecomparison results of the program valuation
It is observed from Figures 4ndash7 that the smaller the aspectratio the more significant the influence of the effective stresson the dimensionless radius Taking star-shaped throats as anexample we can see in the dimensionless radius equation thatthe smaller the aspect ratio 120576 is the more the dimensionlessradiuswill be influenced by the stress changeThe star-shapedthroats experience deformation with the change in stress(Figure 8) and when the aspect ratio is smaller the throatapproaches a crack shape and the influence of the crack onstress is more evident
The relationship described above is between the dimen-sionless radius and the effective stress in a specific throatand does not consider the range in aspect ratio values orthe combination of the aspect ratio and the proportionTherefore taking the star-shaped pore as an example theplan and operation results shown in Figure 9 consider thespecific throat aspect ratio and proportion combinationshowever there will certainly be different plans for oval coneand trilateral throats
In addition to the different aspect ratios and proportioncombinations for one specific pore shape other relationshipsexist between dimensionless radius and effective stress withcombinations of two- three- or four-pore throat typesDue to limited space the combination of three differentthroat types is taken as an example and the relationshipbetween dimensionless radius and effective stress is effec-tively explained by a specific combination plan (Figure 10)
3 Explanation for Rock Stress Sensitivity
The analysis above indicates that the pore structure of arock changes with the degree of effective stress which
Journal of Chemistry 7
Table 2 (Log) uniform distribution
Log uniform distribution Uniform distribution
119903119867=
119903max + 119903min2
119903119867= 2
1199032
max + 119903max119903min + 1199032
min
3 (119903max + 119903min)
120590119903= radic
ln (119903max119903min) (119903max + 119903min)
2 (119903max minus 119903min)minus 1 120590
119903=
(119903max minus 119903min)
radic3 (119903max + 119903min)
Table 3 Value of 119904119903 119903max and 119903min
Log uniform distribution Uniform distribution119904119903
119903maxmm 119903minmm 119904119903
119903maxmm 119903minmm005 434563 365437 005 433557 364448030 592411 207589 030 557657 176288055 701570 984301 055 599655 145480080 760414 395856105 786472 135281
Table 4 Program output parameters
Output parameters ValuePore volume 269386119890 + 006
Pore surface area 135504Hydraulic radius 397605Numbers of tube bundle 2970000Total input electric current 0109619Total output electric current 00851105Average coordination number 59374
induces changes in the rock physical properties Therefore acrucial factor in studies of rock stress sensitivity particularlypermeability stress sensitivity is understanding the responseof the pore radius to effective stress namely
119903 = 119891 (119901eff) (23)
The relationship between the pore radius derived from themodel and the dimensionless radius 119903
119901is as follows
radius = InitRadius lowast 119903119901 (24)
which establishes the function for the pore radius (radius)and the effective stress (119901eff ) It can be seen from Table 1that the hydraulic power conductivity (ℎcon) is a function ofthe radius (radius) and the inputoutput flow is a functionof the hydraulic power (ℎcon) Furthermore the permeabilitycalculation function (which is based on Darcyrsquos Law) isrelated to the inflow or outflow from which we establish therelationship between permeability and effective stress (119901eff )Therefore our model by fitting data from the top of the rockcore through a combination of different pore shape models(see Figure 11) can effectively explain themechanisms behindrock permeability stress sensitivity
Figure 3 3D pore network model
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
Figure 4 Effect of aspect ratio on oval pores
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
Figure 5 Effect of aspect ratio on taper pores
4 Conclusion
(1) The effective stress equation for partially saturatedrock is derived and verified and the core operationexpressions of the network model are obtained usingiteration methods
8 Journal of Chemistry
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
Figure 6 Effect of aspect ratio on triangular pores
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
Figure 7 Effect of aspect ratio on star-shaped pores
(a) 120576 = 02 (b) 120576 = 0005
Figure 8 Star-shaped throat at different aspect ratios
(2) The network model simulation shows that a circularthroat is the special case for an ellipse and the smallerthe aspect ratio the greater the effect of stress on thedimensionless radius
(3) The relationship between the experimental dimen-sionless radius and the effective stress can beexplained through different pore shape combinationsbased on the network model
Journal of Chemistry 9
0
02
04
06
08
1
0 5 10 15 20 25 30Effective stress Peff (MPa)
The relation of dimensionless radius rpand effective stress Peff
Dim
ensio
nles
s rad
iusr
p
Figure 9 Star-shaped [10 (60) + 01 (20) + 005 (10) + 001 (10)]
Experimental dataCalculated curve
0
02
04
06
08
1
Dim
ensio
nles
s rad
iusr
p
0 5 10 15 20 25 30Effective stress Peff (MPa)
Figure 10 Combination of three pores types 20 cone [10 (60) + 01 (20) + 0005 (10) + 0001 (10)] + 30 triangle [10 (60) + 01(20) + 0005 (10) + 0001 (10)] + 50 star [10 (60) + 01 (20) + 0005 (10) + 0001 (10)]
0
02
04
06
08
1
0 20 40 60 80Effective stress Peff (MPa)
Matched curveExperimental curve
Dim
ensio
nles
s per
mea
bilit
yk
The relation of dimensionless permeability kand effective stress Peff
Figure 11 Comparison diagram of the program calculation curve and experiment data 50 oval [001 (60) + 0005 (30) + 00012 (10)]+ 50 star [01 (27) + 0005 (36) + 00012 (37)]
10 Journal of Chemistry
(4) The permeability stress sensitivity is effectivelyexplained using the network simulation
Competing Interests
The authors declare that they have no competing interests
References
[1] V M Dobrynin ldquoEffect of overburden pressure on some prop-erties of sandstonesrdquo Society of PetroleumEngineers Journal vol2 no 4 pp 360ndash366 1962
[2] H S Ali M A Al-Marhoun S A Abu-Khamsin et alldquoThe effect of overburden pressure on relative permeabilityrdquoin Proceedinga of the 5th SPE Middle East Oil Show ManamaBahram March 1987
[3] M E Shafiee and A Kantzas ldquoInvestigation on the effectof overburden pressure on vuggy carbonate oil reservoircore propertiesrdquo in Proceedings of the Canadian InternationalPetroleum Conference (CIPC rsquo09) pp 16ndash18 Calgary AlbertaJune 2009
[4] D Tiab and E C Donaldson Petrophysics Theory and Practiceof Measuring Reservoir Rock and Fluid Transport PropertiesElsevierGulf Professional Publishing 2012
[5] M Li Y Bernabe W-I Xiao Z-Y Chen and Z-Q LiuldquoEffective pressure law for permeability of E-bei sandstonesrdquoJournal of Geophysical Research B Solid Earth vol 114 no 7Article ID B07205 p 223 2009
[6] D G Longeron M J Argaud and J P Feraud ldquoEffect of over-burden pressure and the nature andmicroscopic distribution offluids on electrical properties of rock samplesrdquo SPE FormationEvaluation vol 6 pp 194ndash202 1989
[7] J C Jaeger and N Cook Fundamentals of Rock MechanicsChapman amp Hall 2nd edition 1976
[8] D A Seeburger and A Nur ldquoA pore space model for rock per-meability and bulk modulusrdquo Journal of Geophysical Researchvol 89 no 1 pp 527ndash536 1984
[9] D P Yale and A Nur ldquoNetwork modeling of flow storage anddeformation in porous rocksrdquo SEGTechnical ProgramExpandedAbstracts vol 4 no 1 p 437 1999
[10] R F Sigal ldquoThe pressure dependence of permeabilityrdquo Petro-physics vol 43 no 2 pp 92ndash102 2002
[11] T Zhengwu Numerical Experiments of Capillary PressureCurve Southwest Petroleum University Chengdu China 2013(Chinese)
[12] J T Fredrich ldquo3D imaging of porousmedia using laser scanningconfocal microscopy with application to microscale transportprocessesrdquo Physics and Chemistry of the Earth Part A SolidEarth and Geodesy vol 24 no 7 pp 551ndash561 1999
[13] C D Tsakiroglou and M Fleury ldquoPore network analysis ofresistivity index for water-wet porous mediardquo Transport inPorous Media vol 35 no 1 pp 89ndash128 1999
[14] AW Bishop ldquoThe principle of effective stressrdquoTekniskUkebladvol 106 no 39 pp 859ndash863 1960
[15] W G Gray and B A Schrefler ldquoThermodynamic approach toeffective stress in partially saturated porous mediardquo EuropeanJournal of Mechanics ASolids vol 20 no 4 pp 521ndash538 2001
[16] M Nuth and L Laloui ldquoEffective stress concept in unsaturatedsoils clarification and validation of a unified frameworkrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 32 no 7 pp 771ndash801 2008
[17] A F Gangi ldquoVariation of whole and fractured porous rockpermeability with confining pressurerdquo International Journal ofRock Mechanics andMining Sciences and vol 15 no 5 pp 249ndash257 1978
[18] B B Noble and J W DanielsApplied Linear Algebra AmericanMathematical Society 3rd edition 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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CatalystsJournal of
Journal of Chemistry 3
pressure (119901119899) For throats that have not been displaced the
fluid inside is the wetting phase (water) and the pressure is0 There is no pressure difference between the nonwettingand wetting phases and capillary pressure does not existTherefore the effective stress expression is as follows
119901eff = 119901ov minus 120572119901119891 = 119901ov minus 120572119901119899 (6)
where 119901eff is the effective stress and 119901ov is the overburdenpressure
This theory was proposed by Bishop [14] who experi-mentally defined the general effective stress for a partiallysaturated porous medium as follows
1205901015840= 120590 minus 120572 [119901
119899minus 119878119908(119901119899minus 119901119908)] (7)
When wetting-phase saturation values are at the extremeendpoints (0 and 1) the product of wetting-phase saturationand capillary pressure (119878
119908119901119888) is 0 and when a relatively low
value is obtained for example 01 or less the peak value of119878119908119901119888will occur However this value is still very small so in
most cases it is negligible when compared with the effectivestress [14] Therefore the expression above can be simplifiedas
1205901015840= 120590 minus 120572119901
119899 (8)
This expression is the effective stress equation for two-phase fluids and is identical to (6) however the value of thecoefficient has been a topic for debate [15 16] If the valueof the coefficient is 0 the effective stress is the confiningpressure If the effective stress coefficient is 1 the effectivestress is the net stress To simplify this study the effectivestress coefficient is set as 0 under which circumstancesthe pore fluid pressure is assumed to impart no pressureon the effective stress or has much less impact than theconfining pressure Therefore this study on rock micro-porestructure is focused on the change in rock pore structureunder different confining pressures
222 Key Equations The key equation of this methodinvolves the hydraulic conductivity at the throats and thedimensionless radius equation The flow equation for ahyperbolic triangle throat [9] is
119902 = FT 1198884
120583119897
Δ119901 (9)
where FT is the dimensionless conductivity per unit length ofunit area which is solved by finite difference The hydraulicconductivity equation is
119867 =
119902
Δ119901
= FT 1198884
120583119897
(10)
Because there is no dimensionless radius equation forhyperbolic triangle throats in the literature here we present asimple deduction based on the formula from Gangi [17]
119903119901= 1 minus 119862
0(
119875
1198750
)
23
(11)
Hyperbolic triangles poreHyperbolic quadrilateral pore
Figure 1 Quadrilateral-triangles stress transformation
Moreover it is known from another Gangi formula that[17]
119875
1198750
=
3120587 (1 minus ]2)4119864
119875eff (12)
and considering that
1198620=
21198881119877
1199031
asymp 2 (13)
it can be deduced that the dimensionless radius equation forhyperbolic triangle throats is
119903119901= 1 minus [
3radic2120587 (1 minus ]2)2120576119864
119901eff]
23
(14)
In a similar manner the hydraulic conductivity equationand dimensionless radius equations for circular throats ovalthroats conical throats hyperbolic triangle throats and starthroats can be deduced shown in Table 1
In these equations 119903119901is the dimensionless radius which
is the ratio of the pore throat radius under effective stressto the pore throat radius under no effective stress FT isthe dimensionless conductivity per unit length of unit areawhich is solved by finite difference ] is the Poisson ratio119887 is the length of the semiminor axis 119888 is the length of thesemimajor axis 120576 is the aspect ratio 120576 = 119887119888 and 119864 is Youngrsquosmodulus
The dimensionless radius equations for hyperbolic trian-gle throats and star-shaped throats are similar only the coeffi-cient and index are different Therefore their conclusions aresimilar which is illustrated by Figure 1 and in the followingdiscussion
223 Coefficient Matrix Solution A key part of the networkmodel simulation is building operation expressions the basicidea of which is based on Kirchhoff rsquos law nodes and lineconductivity Noble and Daniels [18] presented the equationfor a network of119873 nodes
(119902) = 119860119870119860119879(119881) (15)
whereas (119902) = [11990211199022sdot sdot sdot 119902119899]1015840 (119881) = [119881
11198812sdot sdot sdot 119881119899]1015840
For a simple network the coefficient matrix can beobtained through a structural analysis however the simu-lation method of the model is based on statistical theory
4 Journal of Chemistry
Table 1 Key equation
Diagrammatic drawing Hydraulic conductivity equation Dimensionless radius equation
Circular throat
r
119867 = 0125
1205871199034
120583119897
119903119901= 1 minus
2 (1 minus ]2)119864
119901eff
Oval throat
ba
119867 = 025
12058712057631198884
120583119897(1205762+ 1)
119903119901= 1 minus
2 (1 minus ]2)120576119864
119901eff
Conical throat
bc
l 119867 = 0685
12058712057631198884
120583119897(1205762+ 1)
119903119901= 1 minus [
4 (1 minus ]2)3120576119864
119901eff]
12
Hyperbolictriangle throat 119867 = FT 119888
4
120583119897
119903119901= 1 minus [
3radic2120587 (1 minus ]2)2120576119864
119901eff]
23
Star throatdc
119867 = FT 1198884
120583119897
119903119901= 1 minus [
3radic2 (1 minus ]2)4120576119864
119901eff]
13
which dictates that the number of nodes and lines in thenetwork model needs to be large enough to ensure thereliability of the simulation As a result when the number ofnodes and connections is large enough the model cannot besolved by simple algebra and the Cholesky decompositionis required in order to solve the coefficient matrix 119860119870119860119879However the Cholesky decomposition affects the precisionof the coefficient matrix mainly due to the round-off errorand furthermore the program code is complicated [11] To thisend we use the iterative method to solve the matrix
The principal of the iterative method which graduallyapproaches the real solution is to take the assumed valueas the solution and perform continuous iterations until itmeets the convergence condition and obtains the solution ofthe equation For the convergent system set the deviationobtained from each displacement will decrease and its solu-tion becomes closer to the real solutionThe iterationmethodcan also automatically adjust the occasional calculation errorthat occurs in the iteration The method includes simpleiteration and super-relaxed iteration and in the followingsection the simplest two-dimensional square network modelis used as an example to introduce the solution process
The basis of the simple iteration also known as successiveiteration is to construct the fixed-point equation in order toobtain the approximate solutionThe simple iterationmethodis solved via the following steps
(1) Assign Initial Values for Nodes As the ultimate solution isnot associated with the initial given values the initial valuecan be randomly set yet the convergence rate depends onthe accuracy of the initial values We assume that the flowdirection is from left to right in the square network the leftend of the nodes is assignedwith voltage119881
1 and the right end
of the nodes is assignedwith voltage1198812 while other nodes are
assigned with a voltage of 0
(2) Establish the Equation Taking the 0 node in Figure 2 as anexample the following equation can be obtained according toKirchhoff rsquos current law
(119881119894119895minus 119881119894+1119895
) 119892(119894119895sim119894+1119895)
+ (119881119894119895minus 119881119894119895+1
) 119892(119894119895sim119894119895+1)
+ (119881119894119895minus 119881119894minus1119895
) 119892(119894119895sim119894minus1119895)
+ (119881119894119895minus 119881119894119895minus1
) 119892(119894119895sim119894119895minus1)
= 0
(16)
Journal of Chemistry 5
j
1(i + 1 j)
2(i j + 1)
3(i minus 1 j)
4(i j minus 1)
0(i j)
i
Figure 2 One node cell of square network model
This equation can be used for every node in the networkand in this way we obtain the system of linear equationswhose quantities are identical to the node quantities
(11988111minus 11988121) 119892(11sim21)
+ (11988111minus 11988112) 119892(11sim12)
+ (11988111minus 11988101) 119892(11sim01)
+ (11988111minus 11988110) 119892(11sim10)
= 0
(11988112minus 11988122) 119892(12sim22)
+ (11988112minus 11988113) 119892(12sim13)
+ (11988112minus 11988102) 119892(12sim02)
+ (11988112minus 11988111) 119892(12sim11)
= 0
(1198811119895minus 1198812119895) 119892(1119895sim2119895)
+ (1198811119895minus 1198811119895+1
) 119892(1119895sim1119895+1)
+ (1198811119895minus 1198810119895) 119892(1119895sim0119895)
+ (1198811119895minus 1198811119895minus1
) 119892(1119895sim1119895minus1)
= 0
(11988121minus 11988131) 119892(21sim31)
+ (11988121minus 11988122) 119892(21sim22)
+ (11988121minus 11988111) 119892(21sim11)
+ (11988121minus 11988120) 119892(21sim20)
= 0
(11988131minus 11988141) 119892(31sim41)
+ (11988131minus 11988132) 119892(31sim32)
+ (11988131minus 11988121) 119892(31sim21)
+ (11988131minus 11988130) 119892(31sim30)
= 0
(119881119894119895minus 119881119894+1119895
) 119892(119894119895sim119894+1119895)
+ (119881119894119895minus 119881119894119895+1
) 119892(119894119895sim119894119895+1)
+ (119881119894119895minus 119881119894minus1119895
) 119892(119894119895sim119894minus1119895)
+ (119881119894119895minus 119881119894119895minus1
) 119892(119894119895sim119894119895minus1)
= 0
(17)
To conduct the iteration solution process (17) is changedas follows
119881119899+1
119894119895=
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
(18)
(3) Iteration Solution The inputoutput current value in thenetwork is calculated after each iteration The iteration isonly deemed completed if the inputoutput currents are equalor the difference is within the error After the iteration isfinished the inputoutput current is solved the resistance iscalculated using Ohmrsquos law and then the other parametersof the network model such as electrical resistivity aresolved
As the convergence rate of the simple iteration is low weuse the super-relaxed iteration and the program obtains theadjacent node voltage values from the last step calculationduring the node calculation For example when we calculatethe points (119894 119895) in the square network the voltage of theleft point (119894 minus 1 119895) and the point below (119894 119895 minus 1) willbe replaced by the voltage values obtained from the laststep
119881119899+1
119894119895=
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
(19)
The method is called Gauss-Seidel iteration and itreplaces the new values obtained from the previous last step
in order to speed up the convergence ratesThe increment canbe written as
6 Journal of Chemistry
119881119899+1
119894119895= 119881119899
119894119895+ (
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
minus 119881119899
119894119895) (20)
To speed up convergence and introduce the relaxingfactor 119890 (obtaining the values 1-2) the above expression canbe written as follows
119881119899+1
119894119895= 119881119899
119894119895+ 119890(
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
minus 119881119899
119894119895) (21)
(4) Initial Radius Assignment Considering the randomnessof the rock pore radius and the strong heterogeneity of thereservoir rocks the network models used in our researchdetermine the pipe radius by random functions in order togenerate randomized and nonhomogeneous network mod-els This study selects radius as per the logarithmic meandistribution and uniform distribution of which the normal-ized standard deviations are 005 030 055 080 and 105and 005 030 and 055 respectively The radius is generatedthrough the following random numbers
119903 = 119890rand( )((int(lg(119903max)lowast50)minusint(lg(119903min)lowast50))50)
+ lg (119903min) (22)
where rand( ) is the random number 119903max is the maximumthroat radius and 119903min is the minimum throat radius
It can be seen from the above equation that the radius ofany throat in the network is between the maximum throatradius and minimum throat radius It should be particularlynoted that the hydraulic radii (ie twice the volume tosurface ratio of the pores 119903
119867= 2119881
119901119878119901 where 119881
119901is
the total pore volume and 119878119901is the total wetted surface
area) all remain constant (40 120583m) Under the conditionsof log uniform distribution and uniform distribution thehydraulic radius and normalized standard deviation shouldmeet the requirements in Table 2 When the hydraulic radiusis set at 40 120583m the normalized standard deviation and thelargestsmallest throat radius in Table 3 should be used
23 Program Implementation
231 InputOutput Parameters Based on the theory of thenetwork model and the key programming points the C++language is used to compile random network model pro-cedures on the QT platform Regarding the grid size (119909119910 and 119911 direction) the maximumminimum throat radiimaximumminimumaspect ratios formationwater viscositygrid unit length (pipe length) connectivity probability andthe proportion of pores of each shape aspect ratios areinputted through program interfaces
When running the program the first output is a 3D cuberandom network model whose size is set at 100 times 100 times
100 (Figure 3) followed by the relevant parameters of this
program (Table 4) Finally it produces the aspect ratios ofdifferent shaped throats and graphs of the dimensionlessradius and the effective stress
232 Program Result Analysis In order to analyze theinfluence of the aspect ratio on the calculation result theprogram is set according to a specific throat and the changein dimensionless radius due to effective stress is studiedby setting different aspect ratios See Figures 4ndash7 for thecomparison results of the program valuation
It is observed from Figures 4ndash7 that the smaller the aspectratio the more significant the influence of the effective stresson the dimensionless radius Taking star-shaped throats as anexample we can see in the dimensionless radius equation thatthe smaller the aspect ratio 120576 is the more the dimensionlessradiuswill be influenced by the stress changeThe star-shapedthroats experience deformation with the change in stress(Figure 8) and when the aspect ratio is smaller the throatapproaches a crack shape and the influence of the crack onstress is more evident
The relationship described above is between the dimen-sionless radius and the effective stress in a specific throatand does not consider the range in aspect ratio values orthe combination of the aspect ratio and the proportionTherefore taking the star-shaped pore as an example theplan and operation results shown in Figure 9 consider thespecific throat aspect ratio and proportion combinationshowever there will certainly be different plans for oval coneand trilateral throats
In addition to the different aspect ratios and proportioncombinations for one specific pore shape other relationshipsexist between dimensionless radius and effective stress withcombinations of two- three- or four-pore throat typesDue to limited space the combination of three differentthroat types is taken as an example and the relationshipbetween dimensionless radius and effective stress is effec-tively explained by a specific combination plan (Figure 10)
3 Explanation for Rock Stress Sensitivity
The analysis above indicates that the pore structure of arock changes with the degree of effective stress which
Journal of Chemistry 7
Table 2 (Log) uniform distribution
Log uniform distribution Uniform distribution
119903119867=
119903max + 119903min2
119903119867= 2
1199032
max + 119903max119903min + 1199032
min
3 (119903max + 119903min)
120590119903= radic
ln (119903max119903min) (119903max + 119903min)
2 (119903max minus 119903min)minus 1 120590
119903=
(119903max minus 119903min)
radic3 (119903max + 119903min)
Table 3 Value of 119904119903 119903max and 119903min
Log uniform distribution Uniform distribution119904119903
119903maxmm 119903minmm 119904119903
119903maxmm 119903minmm005 434563 365437 005 433557 364448030 592411 207589 030 557657 176288055 701570 984301 055 599655 145480080 760414 395856105 786472 135281
Table 4 Program output parameters
Output parameters ValuePore volume 269386119890 + 006
Pore surface area 135504Hydraulic radius 397605Numbers of tube bundle 2970000Total input electric current 0109619Total output electric current 00851105Average coordination number 59374
induces changes in the rock physical properties Therefore acrucial factor in studies of rock stress sensitivity particularlypermeability stress sensitivity is understanding the responseof the pore radius to effective stress namely
119903 = 119891 (119901eff) (23)
The relationship between the pore radius derived from themodel and the dimensionless radius 119903
119901is as follows
radius = InitRadius lowast 119903119901 (24)
which establishes the function for the pore radius (radius)and the effective stress (119901eff ) It can be seen from Table 1that the hydraulic power conductivity (ℎcon) is a function ofthe radius (radius) and the inputoutput flow is a functionof the hydraulic power (ℎcon) Furthermore the permeabilitycalculation function (which is based on Darcyrsquos Law) isrelated to the inflow or outflow from which we establish therelationship between permeability and effective stress (119901eff )Therefore our model by fitting data from the top of the rockcore through a combination of different pore shape models(see Figure 11) can effectively explain themechanisms behindrock permeability stress sensitivity
Figure 3 3D pore network model
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
Figure 4 Effect of aspect ratio on oval pores
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
Figure 5 Effect of aspect ratio on taper pores
4 Conclusion
(1) The effective stress equation for partially saturatedrock is derived and verified and the core operationexpressions of the network model are obtained usingiteration methods
8 Journal of Chemistry
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
Figure 6 Effect of aspect ratio on triangular pores
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
Figure 7 Effect of aspect ratio on star-shaped pores
(a) 120576 = 02 (b) 120576 = 0005
Figure 8 Star-shaped throat at different aspect ratios
(2) The network model simulation shows that a circularthroat is the special case for an ellipse and the smallerthe aspect ratio the greater the effect of stress on thedimensionless radius
(3) The relationship between the experimental dimen-sionless radius and the effective stress can beexplained through different pore shape combinationsbased on the network model
Journal of Chemistry 9
0
02
04
06
08
1
0 5 10 15 20 25 30Effective stress Peff (MPa)
The relation of dimensionless radius rpand effective stress Peff
Dim
ensio
nles
s rad
iusr
p
Figure 9 Star-shaped [10 (60) + 01 (20) + 005 (10) + 001 (10)]
Experimental dataCalculated curve
0
02
04
06
08
1
Dim
ensio
nles
s rad
iusr
p
0 5 10 15 20 25 30Effective stress Peff (MPa)
Figure 10 Combination of three pores types 20 cone [10 (60) + 01 (20) + 0005 (10) + 0001 (10)] + 30 triangle [10 (60) + 01(20) + 0005 (10) + 0001 (10)] + 50 star [10 (60) + 01 (20) + 0005 (10) + 0001 (10)]
0
02
04
06
08
1
0 20 40 60 80Effective stress Peff (MPa)
Matched curveExperimental curve
Dim
ensio
nles
s per
mea
bilit
yk
The relation of dimensionless permeability kand effective stress Peff
Figure 11 Comparison diagram of the program calculation curve and experiment data 50 oval [001 (60) + 0005 (30) + 00012 (10)]+ 50 star [01 (27) + 0005 (36) + 00012 (37)]
10 Journal of Chemistry
(4) The permeability stress sensitivity is effectivelyexplained using the network simulation
Competing Interests
The authors declare that they have no competing interests
References
[1] V M Dobrynin ldquoEffect of overburden pressure on some prop-erties of sandstonesrdquo Society of PetroleumEngineers Journal vol2 no 4 pp 360ndash366 1962
[2] H S Ali M A Al-Marhoun S A Abu-Khamsin et alldquoThe effect of overburden pressure on relative permeabilityrdquoin Proceedinga of the 5th SPE Middle East Oil Show ManamaBahram March 1987
[3] M E Shafiee and A Kantzas ldquoInvestigation on the effectof overburden pressure on vuggy carbonate oil reservoircore propertiesrdquo in Proceedings of the Canadian InternationalPetroleum Conference (CIPC rsquo09) pp 16ndash18 Calgary AlbertaJune 2009
[4] D Tiab and E C Donaldson Petrophysics Theory and Practiceof Measuring Reservoir Rock and Fluid Transport PropertiesElsevierGulf Professional Publishing 2012
[5] M Li Y Bernabe W-I Xiao Z-Y Chen and Z-Q LiuldquoEffective pressure law for permeability of E-bei sandstonesrdquoJournal of Geophysical Research B Solid Earth vol 114 no 7Article ID B07205 p 223 2009
[6] D G Longeron M J Argaud and J P Feraud ldquoEffect of over-burden pressure and the nature andmicroscopic distribution offluids on electrical properties of rock samplesrdquo SPE FormationEvaluation vol 6 pp 194ndash202 1989
[7] J C Jaeger and N Cook Fundamentals of Rock MechanicsChapman amp Hall 2nd edition 1976
[8] D A Seeburger and A Nur ldquoA pore space model for rock per-meability and bulk modulusrdquo Journal of Geophysical Researchvol 89 no 1 pp 527ndash536 1984
[9] D P Yale and A Nur ldquoNetwork modeling of flow storage anddeformation in porous rocksrdquo SEGTechnical ProgramExpandedAbstracts vol 4 no 1 p 437 1999
[10] R F Sigal ldquoThe pressure dependence of permeabilityrdquo Petro-physics vol 43 no 2 pp 92ndash102 2002
[11] T Zhengwu Numerical Experiments of Capillary PressureCurve Southwest Petroleum University Chengdu China 2013(Chinese)
[12] J T Fredrich ldquo3D imaging of porousmedia using laser scanningconfocal microscopy with application to microscale transportprocessesrdquo Physics and Chemistry of the Earth Part A SolidEarth and Geodesy vol 24 no 7 pp 551ndash561 1999
[13] C D Tsakiroglou and M Fleury ldquoPore network analysis ofresistivity index for water-wet porous mediardquo Transport inPorous Media vol 35 no 1 pp 89ndash128 1999
[14] AW Bishop ldquoThe principle of effective stressrdquoTekniskUkebladvol 106 no 39 pp 859ndash863 1960
[15] W G Gray and B A Schrefler ldquoThermodynamic approach toeffective stress in partially saturated porous mediardquo EuropeanJournal of Mechanics ASolids vol 20 no 4 pp 521ndash538 2001
[16] M Nuth and L Laloui ldquoEffective stress concept in unsaturatedsoils clarification and validation of a unified frameworkrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 32 no 7 pp 771ndash801 2008
[17] A F Gangi ldquoVariation of whole and fractured porous rockpermeability with confining pressurerdquo International Journal ofRock Mechanics andMining Sciences and vol 15 no 5 pp 249ndash257 1978
[18] B B Noble and J W DanielsApplied Linear Algebra AmericanMathematical Society 3rd edition 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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CatalystsJournal of
4 Journal of Chemistry
Table 1 Key equation
Diagrammatic drawing Hydraulic conductivity equation Dimensionless radius equation
Circular throat
r
119867 = 0125
1205871199034
120583119897
119903119901= 1 minus
2 (1 minus ]2)119864
119901eff
Oval throat
ba
119867 = 025
12058712057631198884
120583119897(1205762+ 1)
119903119901= 1 minus
2 (1 minus ]2)120576119864
119901eff
Conical throat
bc
l 119867 = 0685
12058712057631198884
120583119897(1205762+ 1)
119903119901= 1 minus [
4 (1 minus ]2)3120576119864
119901eff]
12
Hyperbolictriangle throat 119867 = FT 119888
4
120583119897
119903119901= 1 minus [
3radic2120587 (1 minus ]2)2120576119864
119901eff]
23
Star throatdc
119867 = FT 1198884
120583119897
119903119901= 1 minus [
3radic2 (1 minus ]2)4120576119864
119901eff]
13
which dictates that the number of nodes and lines in thenetwork model needs to be large enough to ensure thereliability of the simulation As a result when the number ofnodes and connections is large enough the model cannot besolved by simple algebra and the Cholesky decompositionis required in order to solve the coefficient matrix 119860119870119860119879However the Cholesky decomposition affects the precisionof the coefficient matrix mainly due to the round-off errorand furthermore the program code is complicated [11] To thisend we use the iterative method to solve the matrix
The principal of the iterative method which graduallyapproaches the real solution is to take the assumed valueas the solution and perform continuous iterations until itmeets the convergence condition and obtains the solution ofthe equation For the convergent system set the deviationobtained from each displacement will decrease and its solu-tion becomes closer to the real solutionThe iterationmethodcan also automatically adjust the occasional calculation errorthat occurs in the iteration The method includes simpleiteration and super-relaxed iteration and in the followingsection the simplest two-dimensional square network modelis used as an example to introduce the solution process
The basis of the simple iteration also known as successiveiteration is to construct the fixed-point equation in order toobtain the approximate solutionThe simple iterationmethodis solved via the following steps
(1) Assign Initial Values for Nodes As the ultimate solution isnot associated with the initial given values the initial valuecan be randomly set yet the convergence rate depends onthe accuracy of the initial values We assume that the flowdirection is from left to right in the square network the leftend of the nodes is assignedwith voltage119881
1 and the right end
of the nodes is assignedwith voltage1198812 while other nodes are
assigned with a voltage of 0
(2) Establish the Equation Taking the 0 node in Figure 2 as anexample the following equation can be obtained according toKirchhoff rsquos current law
(119881119894119895minus 119881119894+1119895
) 119892(119894119895sim119894+1119895)
+ (119881119894119895minus 119881119894119895+1
) 119892(119894119895sim119894119895+1)
+ (119881119894119895minus 119881119894minus1119895
) 119892(119894119895sim119894minus1119895)
+ (119881119894119895minus 119881119894119895minus1
) 119892(119894119895sim119894119895minus1)
= 0
(16)
Journal of Chemistry 5
j
1(i + 1 j)
2(i j + 1)
3(i minus 1 j)
4(i j minus 1)
0(i j)
i
Figure 2 One node cell of square network model
This equation can be used for every node in the networkand in this way we obtain the system of linear equationswhose quantities are identical to the node quantities
(11988111minus 11988121) 119892(11sim21)
+ (11988111minus 11988112) 119892(11sim12)
+ (11988111minus 11988101) 119892(11sim01)
+ (11988111minus 11988110) 119892(11sim10)
= 0
(11988112minus 11988122) 119892(12sim22)
+ (11988112minus 11988113) 119892(12sim13)
+ (11988112minus 11988102) 119892(12sim02)
+ (11988112minus 11988111) 119892(12sim11)
= 0
(1198811119895minus 1198812119895) 119892(1119895sim2119895)
+ (1198811119895minus 1198811119895+1
) 119892(1119895sim1119895+1)
+ (1198811119895minus 1198810119895) 119892(1119895sim0119895)
+ (1198811119895minus 1198811119895minus1
) 119892(1119895sim1119895minus1)
= 0
(11988121minus 11988131) 119892(21sim31)
+ (11988121minus 11988122) 119892(21sim22)
+ (11988121minus 11988111) 119892(21sim11)
+ (11988121minus 11988120) 119892(21sim20)
= 0
(11988131minus 11988141) 119892(31sim41)
+ (11988131minus 11988132) 119892(31sim32)
+ (11988131minus 11988121) 119892(31sim21)
+ (11988131minus 11988130) 119892(31sim30)
= 0
(119881119894119895minus 119881119894+1119895
) 119892(119894119895sim119894+1119895)
+ (119881119894119895minus 119881119894119895+1
) 119892(119894119895sim119894119895+1)
+ (119881119894119895minus 119881119894minus1119895
) 119892(119894119895sim119894minus1119895)
+ (119881119894119895minus 119881119894119895minus1
) 119892(119894119895sim119894119895minus1)
= 0
(17)
To conduct the iteration solution process (17) is changedas follows
119881119899+1
119894119895=
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
(18)
(3) Iteration Solution The inputoutput current value in thenetwork is calculated after each iteration The iteration isonly deemed completed if the inputoutput currents are equalor the difference is within the error After the iteration isfinished the inputoutput current is solved the resistance iscalculated using Ohmrsquos law and then the other parametersof the network model such as electrical resistivity aresolved
As the convergence rate of the simple iteration is low weuse the super-relaxed iteration and the program obtains theadjacent node voltage values from the last step calculationduring the node calculation For example when we calculatethe points (119894 119895) in the square network the voltage of theleft point (119894 minus 1 119895) and the point below (119894 119895 minus 1) willbe replaced by the voltage values obtained from the laststep
119881119899+1
119894119895=
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
(19)
The method is called Gauss-Seidel iteration and itreplaces the new values obtained from the previous last step
in order to speed up the convergence ratesThe increment canbe written as
6 Journal of Chemistry
119881119899+1
119894119895= 119881119899
119894119895+ (
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
minus 119881119899
119894119895) (20)
To speed up convergence and introduce the relaxingfactor 119890 (obtaining the values 1-2) the above expression canbe written as follows
119881119899+1
119894119895= 119881119899
119894119895+ 119890(
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
minus 119881119899
119894119895) (21)
(4) Initial Radius Assignment Considering the randomnessof the rock pore radius and the strong heterogeneity of thereservoir rocks the network models used in our researchdetermine the pipe radius by random functions in order togenerate randomized and nonhomogeneous network mod-els This study selects radius as per the logarithmic meandistribution and uniform distribution of which the normal-ized standard deviations are 005 030 055 080 and 105and 005 030 and 055 respectively The radius is generatedthrough the following random numbers
119903 = 119890rand( )((int(lg(119903max)lowast50)minusint(lg(119903min)lowast50))50)
+ lg (119903min) (22)
where rand( ) is the random number 119903max is the maximumthroat radius and 119903min is the minimum throat radius
It can be seen from the above equation that the radius ofany throat in the network is between the maximum throatradius and minimum throat radius It should be particularlynoted that the hydraulic radii (ie twice the volume tosurface ratio of the pores 119903
119867= 2119881
119901119878119901 where 119881
119901is
the total pore volume and 119878119901is the total wetted surface
area) all remain constant (40 120583m) Under the conditionsof log uniform distribution and uniform distribution thehydraulic radius and normalized standard deviation shouldmeet the requirements in Table 2 When the hydraulic radiusis set at 40 120583m the normalized standard deviation and thelargestsmallest throat radius in Table 3 should be used
23 Program Implementation
231 InputOutput Parameters Based on the theory of thenetwork model and the key programming points the C++language is used to compile random network model pro-cedures on the QT platform Regarding the grid size (119909119910 and 119911 direction) the maximumminimum throat radiimaximumminimumaspect ratios formationwater viscositygrid unit length (pipe length) connectivity probability andthe proportion of pores of each shape aspect ratios areinputted through program interfaces
When running the program the first output is a 3D cuberandom network model whose size is set at 100 times 100 times
100 (Figure 3) followed by the relevant parameters of this
program (Table 4) Finally it produces the aspect ratios ofdifferent shaped throats and graphs of the dimensionlessradius and the effective stress
232 Program Result Analysis In order to analyze theinfluence of the aspect ratio on the calculation result theprogram is set according to a specific throat and the changein dimensionless radius due to effective stress is studiedby setting different aspect ratios See Figures 4ndash7 for thecomparison results of the program valuation
It is observed from Figures 4ndash7 that the smaller the aspectratio the more significant the influence of the effective stresson the dimensionless radius Taking star-shaped throats as anexample we can see in the dimensionless radius equation thatthe smaller the aspect ratio 120576 is the more the dimensionlessradiuswill be influenced by the stress changeThe star-shapedthroats experience deformation with the change in stress(Figure 8) and when the aspect ratio is smaller the throatapproaches a crack shape and the influence of the crack onstress is more evident
The relationship described above is between the dimen-sionless radius and the effective stress in a specific throatand does not consider the range in aspect ratio values orthe combination of the aspect ratio and the proportionTherefore taking the star-shaped pore as an example theplan and operation results shown in Figure 9 consider thespecific throat aspect ratio and proportion combinationshowever there will certainly be different plans for oval coneand trilateral throats
In addition to the different aspect ratios and proportioncombinations for one specific pore shape other relationshipsexist between dimensionless radius and effective stress withcombinations of two- three- or four-pore throat typesDue to limited space the combination of three differentthroat types is taken as an example and the relationshipbetween dimensionless radius and effective stress is effec-tively explained by a specific combination plan (Figure 10)
3 Explanation for Rock Stress Sensitivity
The analysis above indicates that the pore structure of arock changes with the degree of effective stress which
Journal of Chemistry 7
Table 2 (Log) uniform distribution
Log uniform distribution Uniform distribution
119903119867=
119903max + 119903min2
119903119867= 2
1199032
max + 119903max119903min + 1199032
min
3 (119903max + 119903min)
120590119903= radic
ln (119903max119903min) (119903max + 119903min)
2 (119903max minus 119903min)minus 1 120590
119903=
(119903max minus 119903min)
radic3 (119903max + 119903min)
Table 3 Value of 119904119903 119903max and 119903min
Log uniform distribution Uniform distribution119904119903
119903maxmm 119903minmm 119904119903
119903maxmm 119903minmm005 434563 365437 005 433557 364448030 592411 207589 030 557657 176288055 701570 984301 055 599655 145480080 760414 395856105 786472 135281
Table 4 Program output parameters
Output parameters ValuePore volume 269386119890 + 006
Pore surface area 135504Hydraulic radius 397605Numbers of tube bundle 2970000Total input electric current 0109619Total output electric current 00851105Average coordination number 59374
induces changes in the rock physical properties Therefore acrucial factor in studies of rock stress sensitivity particularlypermeability stress sensitivity is understanding the responseof the pore radius to effective stress namely
119903 = 119891 (119901eff) (23)
The relationship between the pore radius derived from themodel and the dimensionless radius 119903
119901is as follows
radius = InitRadius lowast 119903119901 (24)
which establishes the function for the pore radius (radius)and the effective stress (119901eff ) It can be seen from Table 1that the hydraulic power conductivity (ℎcon) is a function ofthe radius (radius) and the inputoutput flow is a functionof the hydraulic power (ℎcon) Furthermore the permeabilitycalculation function (which is based on Darcyrsquos Law) isrelated to the inflow or outflow from which we establish therelationship between permeability and effective stress (119901eff )Therefore our model by fitting data from the top of the rockcore through a combination of different pore shape models(see Figure 11) can effectively explain themechanisms behindrock permeability stress sensitivity
Figure 3 3D pore network model
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
Figure 4 Effect of aspect ratio on oval pores
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
Figure 5 Effect of aspect ratio on taper pores
4 Conclusion
(1) The effective stress equation for partially saturatedrock is derived and verified and the core operationexpressions of the network model are obtained usingiteration methods
8 Journal of Chemistry
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
Figure 6 Effect of aspect ratio on triangular pores
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
Figure 7 Effect of aspect ratio on star-shaped pores
(a) 120576 = 02 (b) 120576 = 0005
Figure 8 Star-shaped throat at different aspect ratios
(2) The network model simulation shows that a circularthroat is the special case for an ellipse and the smallerthe aspect ratio the greater the effect of stress on thedimensionless radius
(3) The relationship between the experimental dimen-sionless radius and the effective stress can beexplained through different pore shape combinationsbased on the network model
Journal of Chemistry 9
0
02
04
06
08
1
0 5 10 15 20 25 30Effective stress Peff (MPa)
The relation of dimensionless radius rpand effective stress Peff
Dim
ensio
nles
s rad
iusr
p
Figure 9 Star-shaped [10 (60) + 01 (20) + 005 (10) + 001 (10)]
Experimental dataCalculated curve
0
02
04
06
08
1
Dim
ensio
nles
s rad
iusr
p
0 5 10 15 20 25 30Effective stress Peff (MPa)
Figure 10 Combination of three pores types 20 cone [10 (60) + 01 (20) + 0005 (10) + 0001 (10)] + 30 triangle [10 (60) + 01(20) + 0005 (10) + 0001 (10)] + 50 star [10 (60) + 01 (20) + 0005 (10) + 0001 (10)]
0
02
04
06
08
1
0 20 40 60 80Effective stress Peff (MPa)
Matched curveExperimental curve
Dim
ensio
nles
s per
mea
bilit
yk
The relation of dimensionless permeability kand effective stress Peff
Figure 11 Comparison diagram of the program calculation curve and experiment data 50 oval [001 (60) + 0005 (30) + 00012 (10)]+ 50 star [01 (27) + 0005 (36) + 00012 (37)]
10 Journal of Chemistry
(4) The permeability stress sensitivity is effectivelyexplained using the network simulation
Competing Interests
The authors declare that they have no competing interests
References
[1] V M Dobrynin ldquoEffect of overburden pressure on some prop-erties of sandstonesrdquo Society of PetroleumEngineers Journal vol2 no 4 pp 360ndash366 1962
[2] H S Ali M A Al-Marhoun S A Abu-Khamsin et alldquoThe effect of overburden pressure on relative permeabilityrdquoin Proceedinga of the 5th SPE Middle East Oil Show ManamaBahram March 1987
[3] M E Shafiee and A Kantzas ldquoInvestigation on the effectof overburden pressure on vuggy carbonate oil reservoircore propertiesrdquo in Proceedings of the Canadian InternationalPetroleum Conference (CIPC rsquo09) pp 16ndash18 Calgary AlbertaJune 2009
[4] D Tiab and E C Donaldson Petrophysics Theory and Practiceof Measuring Reservoir Rock and Fluid Transport PropertiesElsevierGulf Professional Publishing 2012
[5] M Li Y Bernabe W-I Xiao Z-Y Chen and Z-Q LiuldquoEffective pressure law for permeability of E-bei sandstonesrdquoJournal of Geophysical Research B Solid Earth vol 114 no 7Article ID B07205 p 223 2009
[6] D G Longeron M J Argaud and J P Feraud ldquoEffect of over-burden pressure and the nature andmicroscopic distribution offluids on electrical properties of rock samplesrdquo SPE FormationEvaluation vol 6 pp 194ndash202 1989
[7] J C Jaeger and N Cook Fundamentals of Rock MechanicsChapman amp Hall 2nd edition 1976
[8] D A Seeburger and A Nur ldquoA pore space model for rock per-meability and bulk modulusrdquo Journal of Geophysical Researchvol 89 no 1 pp 527ndash536 1984
[9] D P Yale and A Nur ldquoNetwork modeling of flow storage anddeformation in porous rocksrdquo SEGTechnical ProgramExpandedAbstracts vol 4 no 1 p 437 1999
[10] R F Sigal ldquoThe pressure dependence of permeabilityrdquo Petro-physics vol 43 no 2 pp 92ndash102 2002
[11] T Zhengwu Numerical Experiments of Capillary PressureCurve Southwest Petroleum University Chengdu China 2013(Chinese)
[12] J T Fredrich ldquo3D imaging of porousmedia using laser scanningconfocal microscopy with application to microscale transportprocessesrdquo Physics and Chemistry of the Earth Part A SolidEarth and Geodesy vol 24 no 7 pp 551ndash561 1999
[13] C D Tsakiroglou and M Fleury ldquoPore network analysis ofresistivity index for water-wet porous mediardquo Transport inPorous Media vol 35 no 1 pp 89ndash128 1999
[14] AW Bishop ldquoThe principle of effective stressrdquoTekniskUkebladvol 106 no 39 pp 859ndash863 1960
[15] W G Gray and B A Schrefler ldquoThermodynamic approach toeffective stress in partially saturated porous mediardquo EuropeanJournal of Mechanics ASolids vol 20 no 4 pp 521ndash538 2001
[16] M Nuth and L Laloui ldquoEffective stress concept in unsaturatedsoils clarification and validation of a unified frameworkrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 32 no 7 pp 771ndash801 2008
[17] A F Gangi ldquoVariation of whole and fractured porous rockpermeability with confining pressurerdquo International Journal ofRock Mechanics andMining Sciences and vol 15 no 5 pp 249ndash257 1978
[18] B B Noble and J W DanielsApplied Linear Algebra AmericanMathematical Society 3rd edition 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Journal of Chemistry 5
j
1(i + 1 j)
2(i j + 1)
3(i minus 1 j)
4(i j minus 1)
0(i j)
i
Figure 2 One node cell of square network model
This equation can be used for every node in the networkand in this way we obtain the system of linear equationswhose quantities are identical to the node quantities
(11988111minus 11988121) 119892(11sim21)
+ (11988111minus 11988112) 119892(11sim12)
+ (11988111minus 11988101) 119892(11sim01)
+ (11988111minus 11988110) 119892(11sim10)
= 0
(11988112minus 11988122) 119892(12sim22)
+ (11988112minus 11988113) 119892(12sim13)
+ (11988112minus 11988102) 119892(12sim02)
+ (11988112minus 11988111) 119892(12sim11)
= 0
(1198811119895minus 1198812119895) 119892(1119895sim2119895)
+ (1198811119895minus 1198811119895+1
) 119892(1119895sim1119895+1)
+ (1198811119895minus 1198810119895) 119892(1119895sim0119895)
+ (1198811119895minus 1198811119895minus1
) 119892(1119895sim1119895minus1)
= 0
(11988121minus 11988131) 119892(21sim31)
+ (11988121minus 11988122) 119892(21sim22)
+ (11988121minus 11988111) 119892(21sim11)
+ (11988121minus 11988120) 119892(21sim20)
= 0
(11988131minus 11988141) 119892(31sim41)
+ (11988131minus 11988132) 119892(31sim32)
+ (11988131minus 11988121) 119892(31sim21)
+ (11988131minus 11988130) 119892(31sim30)
= 0
(119881119894119895minus 119881119894+1119895
) 119892(119894119895sim119894+1119895)
+ (119881119894119895minus 119881119894119895+1
) 119892(119894119895sim119894119895+1)
+ (119881119894119895minus 119881119894minus1119895
) 119892(119894119895sim119894minus1119895)
+ (119881119894119895minus 119881119894119895minus1
) 119892(119894119895sim119894119895minus1)
= 0
(17)
To conduct the iteration solution process (17) is changedas follows
119881119899+1
119894119895=
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
(18)
(3) Iteration Solution The inputoutput current value in thenetwork is calculated after each iteration The iteration isonly deemed completed if the inputoutput currents are equalor the difference is within the error After the iteration isfinished the inputoutput current is solved the resistance iscalculated using Ohmrsquos law and then the other parametersof the network model such as electrical resistivity aresolved
As the convergence rate of the simple iteration is low weuse the super-relaxed iteration and the program obtains theadjacent node voltage values from the last step calculationduring the node calculation For example when we calculatethe points (119894 119895) in the square network the voltage of theleft point (119894 minus 1 119895) and the point below (119894 119895 minus 1) willbe replaced by the voltage values obtained from the laststep
119881119899+1
119894119895=
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
(19)
The method is called Gauss-Seidel iteration and itreplaces the new values obtained from the previous last step
in order to speed up the convergence ratesThe increment canbe written as
6 Journal of Chemistry
119881119899+1
119894119895= 119881119899
119894119895+ (
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
minus 119881119899
119894119895) (20)
To speed up convergence and introduce the relaxingfactor 119890 (obtaining the values 1-2) the above expression canbe written as follows
119881119899+1
119894119895= 119881119899
119894119895+ 119890(
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
minus 119881119899
119894119895) (21)
(4) Initial Radius Assignment Considering the randomnessof the rock pore radius and the strong heterogeneity of thereservoir rocks the network models used in our researchdetermine the pipe radius by random functions in order togenerate randomized and nonhomogeneous network mod-els This study selects radius as per the logarithmic meandistribution and uniform distribution of which the normal-ized standard deviations are 005 030 055 080 and 105and 005 030 and 055 respectively The radius is generatedthrough the following random numbers
119903 = 119890rand( )((int(lg(119903max)lowast50)minusint(lg(119903min)lowast50))50)
+ lg (119903min) (22)
where rand( ) is the random number 119903max is the maximumthroat radius and 119903min is the minimum throat radius
It can be seen from the above equation that the radius ofany throat in the network is between the maximum throatradius and minimum throat radius It should be particularlynoted that the hydraulic radii (ie twice the volume tosurface ratio of the pores 119903
119867= 2119881
119901119878119901 where 119881
119901is
the total pore volume and 119878119901is the total wetted surface
area) all remain constant (40 120583m) Under the conditionsof log uniform distribution and uniform distribution thehydraulic radius and normalized standard deviation shouldmeet the requirements in Table 2 When the hydraulic radiusis set at 40 120583m the normalized standard deviation and thelargestsmallest throat radius in Table 3 should be used
23 Program Implementation
231 InputOutput Parameters Based on the theory of thenetwork model and the key programming points the C++language is used to compile random network model pro-cedures on the QT platform Regarding the grid size (119909119910 and 119911 direction) the maximumminimum throat radiimaximumminimumaspect ratios formationwater viscositygrid unit length (pipe length) connectivity probability andthe proportion of pores of each shape aspect ratios areinputted through program interfaces
When running the program the first output is a 3D cuberandom network model whose size is set at 100 times 100 times
100 (Figure 3) followed by the relevant parameters of this
program (Table 4) Finally it produces the aspect ratios ofdifferent shaped throats and graphs of the dimensionlessradius and the effective stress
232 Program Result Analysis In order to analyze theinfluence of the aspect ratio on the calculation result theprogram is set according to a specific throat and the changein dimensionless radius due to effective stress is studiedby setting different aspect ratios See Figures 4ndash7 for thecomparison results of the program valuation
It is observed from Figures 4ndash7 that the smaller the aspectratio the more significant the influence of the effective stresson the dimensionless radius Taking star-shaped throats as anexample we can see in the dimensionless radius equation thatthe smaller the aspect ratio 120576 is the more the dimensionlessradiuswill be influenced by the stress changeThe star-shapedthroats experience deformation with the change in stress(Figure 8) and when the aspect ratio is smaller the throatapproaches a crack shape and the influence of the crack onstress is more evident
The relationship described above is between the dimen-sionless radius and the effective stress in a specific throatand does not consider the range in aspect ratio values orthe combination of the aspect ratio and the proportionTherefore taking the star-shaped pore as an example theplan and operation results shown in Figure 9 consider thespecific throat aspect ratio and proportion combinationshowever there will certainly be different plans for oval coneand trilateral throats
In addition to the different aspect ratios and proportioncombinations for one specific pore shape other relationshipsexist between dimensionless radius and effective stress withcombinations of two- three- or four-pore throat typesDue to limited space the combination of three differentthroat types is taken as an example and the relationshipbetween dimensionless radius and effective stress is effec-tively explained by a specific combination plan (Figure 10)
3 Explanation for Rock Stress Sensitivity
The analysis above indicates that the pore structure of arock changes with the degree of effective stress which
Journal of Chemistry 7
Table 2 (Log) uniform distribution
Log uniform distribution Uniform distribution
119903119867=
119903max + 119903min2
119903119867= 2
1199032
max + 119903max119903min + 1199032
min
3 (119903max + 119903min)
120590119903= radic
ln (119903max119903min) (119903max + 119903min)
2 (119903max minus 119903min)minus 1 120590
119903=
(119903max minus 119903min)
radic3 (119903max + 119903min)
Table 3 Value of 119904119903 119903max and 119903min
Log uniform distribution Uniform distribution119904119903
119903maxmm 119903minmm 119904119903
119903maxmm 119903minmm005 434563 365437 005 433557 364448030 592411 207589 030 557657 176288055 701570 984301 055 599655 145480080 760414 395856105 786472 135281
Table 4 Program output parameters
Output parameters ValuePore volume 269386119890 + 006
Pore surface area 135504Hydraulic radius 397605Numbers of tube bundle 2970000Total input electric current 0109619Total output electric current 00851105Average coordination number 59374
induces changes in the rock physical properties Therefore acrucial factor in studies of rock stress sensitivity particularlypermeability stress sensitivity is understanding the responseof the pore radius to effective stress namely
119903 = 119891 (119901eff) (23)
The relationship between the pore radius derived from themodel and the dimensionless radius 119903
119901is as follows
radius = InitRadius lowast 119903119901 (24)
which establishes the function for the pore radius (radius)and the effective stress (119901eff ) It can be seen from Table 1that the hydraulic power conductivity (ℎcon) is a function ofthe radius (radius) and the inputoutput flow is a functionof the hydraulic power (ℎcon) Furthermore the permeabilitycalculation function (which is based on Darcyrsquos Law) isrelated to the inflow or outflow from which we establish therelationship between permeability and effective stress (119901eff )Therefore our model by fitting data from the top of the rockcore through a combination of different pore shape models(see Figure 11) can effectively explain themechanisms behindrock permeability stress sensitivity
Figure 3 3D pore network model
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
Figure 4 Effect of aspect ratio on oval pores
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
Figure 5 Effect of aspect ratio on taper pores
4 Conclusion
(1) The effective stress equation for partially saturatedrock is derived and verified and the core operationexpressions of the network model are obtained usingiteration methods
8 Journal of Chemistry
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
Figure 6 Effect of aspect ratio on triangular pores
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
Figure 7 Effect of aspect ratio on star-shaped pores
(a) 120576 = 02 (b) 120576 = 0005
Figure 8 Star-shaped throat at different aspect ratios
(2) The network model simulation shows that a circularthroat is the special case for an ellipse and the smallerthe aspect ratio the greater the effect of stress on thedimensionless radius
(3) The relationship between the experimental dimen-sionless radius and the effective stress can beexplained through different pore shape combinationsbased on the network model
Journal of Chemistry 9
0
02
04
06
08
1
0 5 10 15 20 25 30Effective stress Peff (MPa)
The relation of dimensionless radius rpand effective stress Peff
Dim
ensio
nles
s rad
iusr
p
Figure 9 Star-shaped [10 (60) + 01 (20) + 005 (10) + 001 (10)]
Experimental dataCalculated curve
0
02
04
06
08
1
Dim
ensio
nles
s rad
iusr
p
0 5 10 15 20 25 30Effective stress Peff (MPa)
Figure 10 Combination of three pores types 20 cone [10 (60) + 01 (20) + 0005 (10) + 0001 (10)] + 30 triangle [10 (60) + 01(20) + 0005 (10) + 0001 (10)] + 50 star [10 (60) + 01 (20) + 0005 (10) + 0001 (10)]
0
02
04
06
08
1
0 20 40 60 80Effective stress Peff (MPa)
Matched curveExperimental curve
Dim
ensio
nles
s per
mea
bilit
yk
The relation of dimensionless permeability kand effective stress Peff
Figure 11 Comparison diagram of the program calculation curve and experiment data 50 oval [001 (60) + 0005 (30) + 00012 (10)]+ 50 star [01 (27) + 0005 (36) + 00012 (37)]
10 Journal of Chemistry
(4) The permeability stress sensitivity is effectivelyexplained using the network simulation
Competing Interests
The authors declare that they have no competing interests
References
[1] V M Dobrynin ldquoEffect of overburden pressure on some prop-erties of sandstonesrdquo Society of PetroleumEngineers Journal vol2 no 4 pp 360ndash366 1962
[2] H S Ali M A Al-Marhoun S A Abu-Khamsin et alldquoThe effect of overburden pressure on relative permeabilityrdquoin Proceedinga of the 5th SPE Middle East Oil Show ManamaBahram March 1987
[3] M E Shafiee and A Kantzas ldquoInvestigation on the effectof overburden pressure on vuggy carbonate oil reservoircore propertiesrdquo in Proceedings of the Canadian InternationalPetroleum Conference (CIPC rsquo09) pp 16ndash18 Calgary AlbertaJune 2009
[4] D Tiab and E C Donaldson Petrophysics Theory and Practiceof Measuring Reservoir Rock and Fluid Transport PropertiesElsevierGulf Professional Publishing 2012
[5] M Li Y Bernabe W-I Xiao Z-Y Chen and Z-Q LiuldquoEffective pressure law for permeability of E-bei sandstonesrdquoJournal of Geophysical Research B Solid Earth vol 114 no 7Article ID B07205 p 223 2009
[6] D G Longeron M J Argaud and J P Feraud ldquoEffect of over-burden pressure and the nature andmicroscopic distribution offluids on electrical properties of rock samplesrdquo SPE FormationEvaluation vol 6 pp 194ndash202 1989
[7] J C Jaeger and N Cook Fundamentals of Rock MechanicsChapman amp Hall 2nd edition 1976
[8] D A Seeburger and A Nur ldquoA pore space model for rock per-meability and bulk modulusrdquo Journal of Geophysical Researchvol 89 no 1 pp 527ndash536 1984
[9] D P Yale and A Nur ldquoNetwork modeling of flow storage anddeformation in porous rocksrdquo SEGTechnical ProgramExpandedAbstracts vol 4 no 1 p 437 1999
[10] R F Sigal ldquoThe pressure dependence of permeabilityrdquo Petro-physics vol 43 no 2 pp 92ndash102 2002
[11] T Zhengwu Numerical Experiments of Capillary PressureCurve Southwest Petroleum University Chengdu China 2013(Chinese)
[12] J T Fredrich ldquo3D imaging of porousmedia using laser scanningconfocal microscopy with application to microscale transportprocessesrdquo Physics and Chemistry of the Earth Part A SolidEarth and Geodesy vol 24 no 7 pp 551ndash561 1999
[13] C D Tsakiroglou and M Fleury ldquoPore network analysis ofresistivity index for water-wet porous mediardquo Transport inPorous Media vol 35 no 1 pp 89ndash128 1999
[14] AW Bishop ldquoThe principle of effective stressrdquoTekniskUkebladvol 106 no 39 pp 859ndash863 1960
[15] W G Gray and B A Schrefler ldquoThermodynamic approach toeffective stress in partially saturated porous mediardquo EuropeanJournal of Mechanics ASolids vol 20 no 4 pp 521ndash538 2001
[16] M Nuth and L Laloui ldquoEffective stress concept in unsaturatedsoils clarification and validation of a unified frameworkrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 32 no 7 pp 771ndash801 2008
[17] A F Gangi ldquoVariation of whole and fractured porous rockpermeability with confining pressurerdquo International Journal ofRock Mechanics andMining Sciences and vol 15 no 5 pp 249ndash257 1978
[18] B B Noble and J W DanielsApplied Linear Algebra AmericanMathematical Society 3rd edition 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
6 Journal of Chemistry
119881119899+1
119894119895= 119881119899
119894119895+ (
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
minus 119881119899
119894119895) (20)
To speed up convergence and introduce the relaxingfactor 119890 (obtaining the values 1-2) the above expression canbe written as follows
119881119899+1
119894119895= 119881119899
119894119895+ 119890(
119881119899
119894+1119895119892(119894119895sim119894+1119895)
+ 119881119899
119894119895+1119892(119894119895sim119894119895+1)
+ 119881119899+1
119894minus1119895119892(119894119895sim119894minus1119895)
+ 119881119899+1
119894119895minus1119892(119894119895sim119894119895minus1)
119892(119894119895sim119894+1119895)
+ 119892(119894119895sim119894119895+1)
+ 119892(119894119895sim119894minus1119895)
+ 119892(119894119895sim119894119895minus1)
minus 119881119899
119894119895) (21)
(4) Initial Radius Assignment Considering the randomnessof the rock pore radius and the strong heterogeneity of thereservoir rocks the network models used in our researchdetermine the pipe radius by random functions in order togenerate randomized and nonhomogeneous network mod-els This study selects radius as per the logarithmic meandistribution and uniform distribution of which the normal-ized standard deviations are 005 030 055 080 and 105and 005 030 and 055 respectively The radius is generatedthrough the following random numbers
119903 = 119890rand( )((int(lg(119903max)lowast50)minusint(lg(119903min)lowast50))50)
+ lg (119903min) (22)
where rand( ) is the random number 119903max is the maximumthroat radius and 119903min is the minimum throat radius
It can be seen from the above equation that the radius ofany throat in the network is between the maximum throatradius and minimum throat radius It should be particularlynoted that the hydraulic radii (ie twice the volume tosurface ratio of the pores 119903
119867= 2119881
119901119878119901 where 119881
119901is
the total pore volume and 119878119901is the total wetted surface
area) all remain constant (40 120583m) Under the conditionsof log uniform distribution and uniform distribution thehydraulic radius and normalized standard deviation shouldmeet the requirements in Table 2 When the hydraulic radiusis set at 40 120583m the normalized standard deviation and thelargestsmallest throat radius in Table 3 should be used
23 Program Implementation
231 InputOutput Parameters Based on the theory of thenetwork model and the key programming points the C++language is used to compile random network model pro-cedures on the QT platform Regarding the grid size (119909119910 and 119911 direction) the maximumminimum throat radiimaximumminimumaspect ratios formationwater viscositygrid unit length (pipe length) connectivity probability andthe proportion of pores of each shape aspect ratios areinputted through program interfaces
When running the program the first output is a 3D cuberandom network model whose size is set at 100 times 100 times
100 (Figure 3) followed by the relevant parameters of this
program (Table 4) Finally it produces the aspect ratios ofdifferent shaped throats and graphs of the dimensionlessradius and the effective stress
232 Program Result Analysis In order to analyze theinfluence of the aspect ratio on the calculation result theprogram is set according to a specific throat and the changein dimensionless radius due to effective stress is studiedby setting different aspect ratios See Figures 4ndash7 for thecomparison results of the program valuation
It is observed from Figures 4ndash7 that the smaller the aspectratio the more significant the influence of the effective stresson the dimensionless radius Taking star-shaped throats as anexample we can see in the dimensionless radius equation thatthe smaller the aspect ratio 120576 is the more the dimensionlessradiuswill be influenced by the stress changeThe star-shapedthroats experience deformation with the change in stress(Figure 8) and when the aspect ratio is smaller the throatapproaches a crack shape and the influence of the crack onstress is more evident
The relationship described above is between the dimen-sionless radius and the effective stress in a specific throatand does not consider the range in aspect ratio values orthe combination of the aspect ratio and the proportionTherefore taking the star-shaped pore as an example theplan and operation results shown in Figure 9 consider thespecific throat aspect ratio and proportion combinationshowever there will certainly be different plans for oval coneand trilateral throats
In addition to the different aspect ratios and proportioncombinations for one specific pore shape other relationshipsexist between dimensionless radius and effective stress withcombinations of two- three- or four-pore throat typesDue to limited space the combination of three differentthroat types is taken as an example and the relationshipbetween dimensionless radius and effective stress is effec-tively explained by a specific combination plan (Figure 10)
3 Explanation for Rock Stress Sensitivity
The analysis above indicates that the pore structure of arock changes with the degree of effective stress which
Journal of Chemistry 7
Table 2 (Log) uniform distribution
Log uniform distribution Uniform distribution
119903119867=
119903max + 119903min2
119903119867= 2
1199032
max + 119903max119903min + 1199032
min
3 (119903max + 119903min)
120590119903= radic
ln (119903max119903min) (119903max + 119903min)
2 (119903max minus 119903min)minus 1 120590
119903=
(119903max minus 119903min)
radic3 (119903max + 119903min)
Table 3 Value of 119904119903 119903max and 119903min
Log uniform distribution Uniform distribution119904119903
119903maxmm 119903minmm 119904119903
119903maxmm 119903minmm005 434563 365437 005 433557 364448030 592411 207589 030 557657 176288055 701570 984301 055 599655 145480080 760414 395856105 786472 135281
Table 4 Program output parameters
Output parameters ValuePore volume 269386119890 + 006
Pore surface area 135504Hydraulic radius 397605Numbers of tube bundle 2970000Total input electric current 0109619Total output electric current 00851105Average coordination number 59374
induces changes in the rock physical properties Therefore acrucial factor in studies of rock stress sensitivity particularlypermeability stress sensitivity is understanding the responseof the pore radius to effective stress namely
119903 = 119891 (119901eff) (23)
The relationship between the pore radius derived from themodel and the dimensionless radius 119903
119901is as follows
radius = InitRadius lowast 119903119901 (24)
which establishes the function for the pore radius (radius)and the effective stress (119901eff ) It can be seen from Table 1that the hydraulic power conductivity (ℎcon) is a function ofthe radius (radius) and the inputoutput flow is a functionof the hydraulic power (ℎcon) Furthermore the permeabilitycalculation function (which is based on Darcyrsquos Law) isrelated to the inflow or outflow from which we establish therelationship between permeability and effective stress (119901eff )Therefore our model by fitting data from the top of the rockcore through a combination of different pore shape models(see Figure 11) can effectively explain themechanisms behindrock permeability stress sensitivity
Figure 3 3D pore network model
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
Figure 4 Effect of aspect ratio on oval pores
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
Figure 5 Effect of aspect ratio on taper pores
4 Conclusion
(1) The effective stress equation for partially saturatedrock is derived and verified and the core operationexpressions of the network model are obtained usingiteration methods
8 Journal of Chemistry
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
Figure 6 Effect of aspect ratio on triangular pores
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
Figure 7 Effect of aspect ratio on star-shaped pores
(a) 120576 = 02 (b) 120576 = 0005
Figure 8 Star-shaped throat at different aspect ratios
(2) The network model simulation shows that a circularthroat is the special case for an ellipse and the smallerthe aspect ratio the greater the effect of stress on thedimensionless radius
(3) The relationship between the experimental dimen-sionless radius and the effective stress can beexplained through different pore shape combinationsbased on the network model
Journal of Chemistry 9
0
02
04
06
08
1
0 5 10 15 20 25 30Effective stress Peff (MPa)
The relation of dimensionless radius rpand effective stress Peff
Dim
ensio
nles
s rad
iusr
p
Figure 9 Star-shaped [10 (60) + 01 (20) + 005 (10) + 001 (10)]
Experimental dataCalculated curve
0
02
04
06
08
1
Dim
ensio
nles
s rad
iusr
p
0 5 10 15 20 25 30Effective stress Peff (MPa)
Figure 10 Combination of three pores types 20 cone [10 (60) + 01 (20) + 0005 (10) + 0001 (10)] + 30 triangle [10 (60) + 01(20) + 0005 (10) + 0001 (10)] + 50 star [10 (60) + 01 (20) + 0005 (10) + 0001 (10)]
0
02
04
06
08
1
0 20 40 60 80Effective stress Peff (MPa)
Matched curveExperimental curve
Dim
ensio
nles
s per
mea
bilit
yk
The relation of dimensionless permeability kand effective stress Peff
Figure 11 Comparison diagram of the program calculation curve and experiment data 50 oval [001 (60) + 0005 (30) + 00012 (10)]+ 50 star [01 (27) + 0005 (36) + 00012 (37)]
10 Journal of Chemistry
(4) The permeability stress sensitivity is effectivelyexplained using the network simulation
Competing Interests
The authors declare that they have no competing interests
References
[1] V M Dobrynin ldquoEffect of overburden pressure on some prop-erties of sandstonesrdquo Society of PetroleumEngineers Journal vol2 no 4 pp 360ndash366 1962
[2] H S Ali M A Al-Marhoun S A Abu-Khamsin et alldquoThe effect of overburden pressure on relative permeabilityrdquoin Proceedinga of the 5th SPE Middle East Oil Show ManamaBahram March 1987
[3] M E Shafiee and A Kantzas ldquoInvestigation on the effectof overburden pressure on vuggy carbonate oil reservoircore propertiesrdquo in Proceedings of the Canadian InternationalPetroleum Conference (CIPC rsquo09) pp 16ndash18 Calgary AlbertaJune 2009
[4] D Tiab and E C Donaldson Petrophysics Theory and Practiceof Measuring Reservoir Rock and Fluid Transport PropertiesElsevierGulf Professional Publishing 2012
[5] M Li Y Bernabe W-I Xiao Z-Y Chen and Z-Q LiuldquoEffective pressure law for permeability of E-bei sandstonesrdquoJournal of Geophysical Research B Solid Earth vol 114 no 7Article ID B07205 p 223 2009
[6] D G Longeron M J Argaud and J P Feraud ldquoEffect of over-burden pressure and the nature andmicroscopic distribution offluids on electrical properties of rock samplesrdquo SPE FormationEvaluation vol 6 pp 194ndash202 1989
[7] J C Jaeger and N Cook Fundamentals of Rock MechanicsChapman amp Hall 2nd edition 1976
[8] D A Seeburger and A Nur ldquoA pore space model for rock per-meability and bulk modulusrdquo Journal of Geophysical Researchvol 89 no 1 pp 527ndash536 1984
[9] D P Yale and A Nur ldquoNetwork modeling of flow storage anddeformation in porous rocksrdquo SEGTechnical ProgramExpandedAbstracts vol 4 no 1 p 437 1999
[10] R F Sigal ldquoThe pressure dependence of permeabilityrdquo Petro-physics vol 43 no 2 pp 92ndash102 2002
[11] T Zhengwu Numerical Experiments of Capillary PressureCurve Southwest Petroleum University Chengdu China 2013(Chinese)
[12] J T Fredrich ldquo3D imaging of porousmedia using laser scanningconfocal microscopy with application to microscale transportprocessesrdquo Physics and Chemistry of the Earth Part A SolidEarth and Geodesy vol 24 no 7 pp 551ndash561 1999
[13] C D Tsakiroglou and M Fleury ldquoPore network analysis ofresistivity index for water-wet porous mediardquo Transport inPorous Media vol 35 no 1 pp 89ndash128 1999
[14] AW Bishop ldquoThe principle of effective stressrdquoTekniskUkebladvol 106 no 39 pp 859ndash863 1960
[15] W G Gray and B A Schrefler ldquoThermodynamic approach toeffective stress in partially saturated porous mediardquo EuropeanJournal of Mechanics ASolids vol 20 no 4 pp 521ndash538 2001
[16] M Nuth and L Laloui ldquoEffective stress concept in unsaturatedsoils clarification and validation of a unified frameworkrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 32 no 7 pp 771ndash801 2008
[17] A F Gangi ldquoVariation of whole and fractured porous rockpermeability with confining pressurerdquo International Journal ofRock Mechanics andMining Sciences and vol 15 no 5 pp 249ndash257 1978
[18] B B Noble and J W DanielsApplied Linear Algebra AmericanMathematical Society 3rd edition 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Journal of Chemistry 7
Table 2 (Log) uniform distribution
Log uniform distribution Uniform distribution
119903119867=
119903max + 119903min2
119903119867= 2
1199032
max + 119903max119903min + 1199032
min
3 (119903max + 119903min)
120590119903= radic
ln (119903max119903min) (119903max + 119903min)
2 (119903max minus 119903min)minus 1 120590
119903=
(119903max minus 119903min)
radic3 (119903max + 119903min)
Table 3 Value of 119904119903 119903max and 119903min
Log uniform distribution Uniform distribution119904119903
119903maxmm 119903minmm 119904119903
119903maxmm 119903minmm005 434563 365437 005 433557 364448030 592411 207589 030 557657 176288055 701570 984301 055 599655 145480080 760414 395856105 786472 135281
Table 4 Program output parameters
Output parameters ValuePore volume 269386119890 + 006
Pore surface area 135504Hydraulic radius 397605Numbers of tube bundle 2970000Total input electric current 0109619Total output electric current 00851105Average coordination number 59374
induces changes in the rock physical properties Therefore acrucial factor in studies of rock stress sensitivity particularlypermeability stress sensitivity is understanding the responseof the pore radius to effective stress namely
119903 = 119891 (119901eff) (23)
The relationship between the pore radius derived from themodel and the dimensionless radius 119903
119901is as follows
radius = InitRadius lowast 119903119901 (24)
which establishes the function for the pore radius (radius)and the effective stress (119901eff ) It can be seen from Table 1that the hydraulic power conductivity (ℎcon) is a function ofthe radius (radius) and the inputoutput flow is a functionof the hydraulic power (ℎcon) Furthermore the permeabilitycalculation function (which is based on Darcyrsquos Law) isrelated to the inflow or outflow from which we establish therelationship between permeability and effective stress (119901eff )Therefore our model by fitting data from the top of the rockcore through a combination of different pore shape models(see Figure 11) can effectively explain themechanisms behindrock permeability stress sensitivity
Figure 3 3D pore network model
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
Figure 4 Effect of aspect ratio on oval pores
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
Figure 5 Effect of aspect ratio on taper pores
4 Conclusion
(1) The effective stress equation for partially saturatedrock is derived and verified and the core operationexpressions of the network model are obtained usingiteration methods
8 Journal of Chemistry
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
Figure 6 Effect of aspect ratio on triangular pores
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
Figure 7 Effect of aspect ratio on star-shaped pores
(a) 120576 = 02 (b) 120576 = 0005
Figure 8 Star-shaped throat at different aspect ratios
(2) The network model simulation shows that a circularthroat is the special case for an ellipse and the smallerthe aspect ratio the greater the effect of stress on thedimensionless radius
(3) The relationship between the experimental dimen-sionless radius and the effective stress can beexplained through different pore shape combinationsbased on the network model
Journal of Chemistry 9
0
02
04
06
08
1
0 5 10 15 20 25 30Effective stress Peff (MPa)
The relation of dimensionless radius rpand effective stress Peff
Dim
ensio
nles
s rad
iusr
p
Figure 9 Star-shaped [10 (60) + 01 (20) + 005 (10) + 001 (10)]
Experimental dataCalculated curve
0
02
04
06
08
1
Dim
ensio
nles
s rad
iusr
p
0 5 10 15 20 25 30Effective stress Peff (MPa)
Figure 10 Combination of three pores types 20 cone [10 (60) + 01 (20) + 0005 (10) + 0001 (10)] + 30 triangle [10 (60) + 01(20) + 0005 (10) + 0001 (10)] + 50 star [10 (60) + 01 (20) + 0005 (10) + 0001 (10)]
0
02
04
06
08
1
0 20 40 60 80Effective stress Peff (MPa)
Matched curveExperimental curve
Dim
ensio
nles
s per
mea
bilit
yk
The relation of dimensionless permeability kand effective stress Peff
Figure 11 Comparison diagram of the program calculation curve and experiment data 50 oval [001 (60) + 0005 (30) + 00012 (10)]+ 50 star [01 (27) + 0005 (36) + 00012 (37)]
10 Journal of Chemistry
(4) The permeability stress sensitivity is effectivelyexplained using the network simulation
Competing Interests
The authors declare that they have no competing interests
References
[1] V M Dobrynin ldquoEffect of overburden pressure on some prop-erties of sandstonesrdquo Society of PetroleumEngineers Journal vol2 no 4 pp 360ndash366 1962
[2] H S Ali M A Al-Marhoun S A Abu-Khamsin et alldquoThe effect of overburden pressure on relative permeabilityrdquoin Proceedinga of the 5th SPE Middle East Oil Show ManamaBahram March 1987
[3] M E Shafiee and A Kantzas ldquoInvestigation on the effectof overburden pressure on vuggy carbonate oil reservoircore propertiesrdquo in Proceedings of the Canadian InternationalPetroleum Conference (CIPC rsquo09) pp 16ndash18 Calgary AlbertaJune 2009
[4] D Tiab and E C Donaldson Petrophysics Theory and Practiceof Measuring Reservoir Rock and Fluid Transport PropertiesElsevierGulf Professional Publishing 2012
[5] M Li Y Bernabe W-I Xiao Z-Y Chen and Z-Q LiuldquoEffective pressure law for permeability of E-bei sandstonesrdquoJournal of Geophysical Research B Solid Earth vol 114 no 7Article ID B07205 p 223 2009
[6] D G Longeron M J Argaud and J P Feraud ldquoEffect of over-burden pressure and the nature andmicroscopic distribution offluids on electrical properties of rock samplesrdquo SPE FormationEvaluation vol 6 pp 194ndash202 1989
[7] J C Jaeger and N Cook Fundamentals of Rock MechanicsChapman amp Hall 2nd edition 1976
[8] D A Seeburger and A Nur ldquoA pore space model for rock per-meability and bulk modulusrdquo Journal of Geophysical Researchvol 89 no 1 pp 527ndash536 1984
[9] D P Yale and A Nur ldquoNetwork modeling of flow storage anddeformation in porous rocksrdquo SEGTechnical ProgramExpandedAbstracts vol 4 no 1 p 437 1999
[10] R F Sigal ldquoThe pressure dependence of permeabilityrdquo Petro-physics vol 43 no 2 pp 92ndash102 2002
[11] T Zhengwu Numerical Experiments of Capillary PressureCurve Southwest Petroleum University Chengdu China 2013(Chinese)
[12] J T Fredrich ldquo3D imaging of porousmedia using laser scanningconfocal microscopy with application to microscale transportprocessesrdquo Physics and Chemistry of the Earth Part A SolidEarth and Geodesy vol 24 no 7 pp 551ndash561 1999
[13] C D Tsakiroglou and M Fleury ldquoPore network analysis ofresistivity index for water-wet porous mediardquo Transport inPorous Media vol 35 no 1 pp 89ndash128 1999
[14] AW Bishop ldquoThe principle of effective stressrdquoTekniskUkebladvol 106 no 39 pp 859ndash863 1960
[15] W G Gray and B A Schrefler ldquoThermodynamic approach toeffective stress in partially saturated porous mediardquo EuropeanJournal of Mechanics ASolids vol 20 no 4 pp 521ndash538 2001
[16] M Nuth and L Laloui ldquoEffective stress concept in unsaturatedsoils clarification and validation of a unified frameworkrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 32 no 7 pp 771ndash801 2008
[17] A F Gangi ldquoVariation of whole and fractured porous rockpermeability with confining pressurerdquo International Journal ofRock Mechanics andMining Sciences and vol 15 no 5 pp 249ndash257 1978
[18] B B Noble and J W DanielsApplied Linear Algebra AmericanMathematical Society 3rd edition 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
8 Journal of Chemistry
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
Figure 6 Effect of aspect ratio on triangular pores
1
08
06
04
02
0
Dim
ensio
nles
s rad
iusr
p
0 8 16 24 32
Effective stress Peff (MPa)
1205761 = 1
1205762 = 01
1205763 = 005
1205764 = 001
Figure 7 Effect of aspect ratio on star-shaped pores
(a) 120576 = 02 (b) 120576 = 0005
Figure 8 Star-shaped throat at different aspect ratios
(2) The network model simulation shows that a circularthroat is the special case for an ellipse and the smallerthe aspect ratio the greater the effect of stress on thedimensionless radius
(3) The relationship between the experimental dimen-sionless radius and the effective stress can beexplained through different pore shape combinationsbased on the network model
Journal of Chemistry 9
0
02
04
06
08
1
0 5 10 15 20 25 30Effective stress Peff (MPa)
The relation of dimensionless radius rpand effective stress Peff
Dim
ensio
nles
s rad
iusr
p
Figure 9 Star-shaped [10 (60) + 01 (20) + 005 (10) + 001 (10)]
Experimental dataCalculated curve
0
02
04
06
08
1
Dim
ensio
nles
s rad
iusr
p
0 5 10 15 20 25 30Effective stress Peff (MPa)
Figure 10 Combination of three pores types 20 cone [10 (60) + 01 (20) + 0005 (10) + 0001 (10)] + 30 triangle [10 (60) + 01(20) + 0005 (10) + 0001 (10)] + 50 star [10 (60) + 01 (20) + 0005 (10) + 0001 (10)]
0
02
04
06
08
1
0 20 40 60 80Effective stress Peff (MPa)
Matched curveExperimental curve
Dim
ensio
nles
s per
mea
bilit
yk
The relation of dimensionless permeability kand effective stress Peff
Figure 11 Comparison diagram of the program calculation curve and experiment data 50 oval [001 (60) + 0005 (30) + 00012 (10)]+ 50 star [01 (27) + 0005 (36) + 00012 (37)]
10 Journal of Chemistry
(4) The permeability stress sensitivity is effectivelyexplained using the network simulation
Competing Interests
The authors declare that they have no competing interests
References
[1] V M Dobrynin ldquoEffect of overburden pressure on some prop-erties of sandstonesrdquo Society of PetroleumEngineers Journal vol2 no 4 pp 360ndash366 1962
[2] H S Ali M A Al-Marhoun S A Abu-Khamsin et alldquoThe effect of overburden pressure on relative permeabilityrdquoin Proceedinga of the 5th SPE Middle East Oil Show ManamaBahram March 1987
[3] M E Shafiee and A Kantzas ldquoInvestigation on the effectof overburden pressure on vuggy carbonate oil reservoircore propertiesrdquo in Proceedings of the Canadian InternationalPetroleum Conference (CIPC rsquo09) pp 16ndash18 Calgary AlbertaJune 2009
[4] D Tiab and E C Donaldson Petrophysics Theory and Practiceof Measuring Reservoir Rock and Fluid Transport PropertiesElsevierGulf Professional Publishing 2012
[5] M Li Y Bernabe W-I Xiao Z-Y Chen and Z-Q LiuldquoEffective pressure law for permeability of E-bei sandstonesrdquoJournal of Geophysical Research B Solid Earth vol 114 no 7Article ID B07205 p 223 2009
[6] D G Longeron M J Argaud and J P Feraud ldquoEffect of over-burden pressure and the nature andmicroscopic distribution offluids on electrical properties of rock samplesrdquo SPE FormationEvaluation vol 6 pp 194ndash202 1989
[7] J C Jaeger and N Cook Fundamentals of Rock MechanicsChapman amp Hall 2nd edition 1976
[8] D A Seeburger and A Nur ldquoA pore space model for rock per-meability and bulk modulusrdquo Journal of Geophysical Researchvol 89 no 1 pp 527ndash536 1984
[9] D P Yale and A Nur ldquoNetwork modeling of flow storage anddeformation in porous rocksrdquo SEGTechnical ProgramExpandedAbstracts vol 4 no 1 p 437 1999
[10] R F Sigal ldquoThe pressure dependence of permeabilityrdquo Petro-physics vol 43 no 2 pp 92ndash102 2002
[11] T Zhengwu Numerical Experiments of Capillary PressureCurve Southwest Petroleum University Chengdu China 2013(Chinese)
[12] J T Fredrich ldquo3D imaging of porousmedia using laser scanningconfocal microscopy with application to microscale transportprocessesrdquo Physics and Chemistry of the Earth Part A SolidEarth and Geodesy vol 24 no 7 pp 551ndash561 1999
[13] C D Tsakiroglou and M Fleury ldquoPore network analysis ofresistivity index for water-wet porous mediardquo Transport inPorous Media vol 35 no 1 pp 89ndash128 1999
[14] AW Bishop ldquoThe principle of effective stressrdquoTekniskUkebladvol 106 no 39 pp 859ndash863 1960
[15] W G Gray and B A Schrefler ldquoThermodynamic approach toeffective stress in partially saturated porous mediardquo EuropeanJournal of Mechanics ASolids vol 20 no 4 pp 521ndash538 2001
[16] M Nuth and L Laloui ldquoEffective stress concept in unsaturatedsoils clarification and validation of a unified frameworkrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 32 no 7 pp 771ndash801 2008
[17] A F Gangi ldquoVariation of whole and fractured porous rockpermeability with confining pressurerdquo International Journal ofRock Mechanics andMining Sciences and vol 15 no 5 pp 249ndash257 1978
[18] B B Noble and J W DanielsApplied Linear Algebra AmericanMathematical Society 3rd edition 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Journal of Chemistry 9
0
02
04
06
08
1
0 5 10 15 20 25 30Effective stress Peff (MPa)
The relation of dimensionless radius rpand effective stress Peff
Dim
ensio
nles
s rad
iusr
p
Figure 9 Star-shaped [10 (60) + 01 (20) + 005 (10) + 001 (10)]
Experimental dataCalculated curve
0
02
04
06
08
1
Dim
ensio
nles
s rad
iusr
p
0 5 10 15 20 25 30Effective stress Peff (MPa)
Figure 10 Combination of three pores types 20 cone [10 (60) + 01 (20) + 0005 (10) + 0001 (10)] + 30 triangle [10 (60) + 01(20) + 0005 (10) + 0001 (10)] + 50 star [10 (60) + 01 (20) + 0005 (10) + 0001 (10)]
0
02
04
06
08
1
0 20 40 60 80Effective stress Peff (MPa)
Matched curveExperimental curve
Dim
ensio
nles
s per
mea
bilit
yk
The relation of dimensionless permeability kand effective stress Peff
Figure 11 Comparison diagram of the program calculation curve and experiment data 50 oval [001 (60) + 0005 (30) + 00012 (10)]+ 50 star [01 (27) + 0005 (36) + 00012 (37)]
10 Journal of Chemistry
(4) The permeability stress sensitivity is effectivelyexplained using the network simulation
Competing Interests
The authors declare that they have no competing interests
References
[1] V M Dobrynin ldquoEffect of overburden pressure on some prop-erties of sandstonesrdquo Society of PetroleumEngineers Journal vol2 no 4 pp 360ndash366 1962
[2] H S Ali M A Al-Marhoun S A Abu-Khamsin et alldquoThe effect of overburden pressure on relative permeabilityrdquoin Proceedinga of the 5th SPE Middle East Oil Show ManamaBahram March 1987
[3] M E Shafiee and A Kantzas ldquoInvestigation on the effectof overburden pressure on vuggy carbonate oil reservoircore propertiesrdquo in Proceedings of the Canadian InternationalPetroleum Conference (CIPC rsquo09) pp 16ndash18 Calgary AlbertaJune 2009
[4] D Tiab and E C Donaldson Petrophysics Theory and Practiceof Measuring Reservoir Rock and Fluid Transport PropertiesElsevierGulf Professional Publishing 2012
[5] M Li Y Bernabe W-I Xiao Z-Y Chen and Z-Q LiuldquoEffective pressure law for permeability of E-bei sandstonesrdquoJournal of Geophysical Research B Solid Earth vol 114 no 7Article ID B07205 p 223 2009
[6] D G Longeron M J Argaud and J P Feraud ldquoEffect of over-burden pressure and the nature andmicroscopic distribution offluids on electrical properties of rock samplesrdquo SPE FormationEvaluation vol 6 pp 194ndash202 1989
[7] J C Jaeger and N Cook Fundamentals of Rock MechanicsChapman amp Hall 2nd edition 1976
[8] D A Seeburger and A Nur ldquoA pore space model for rock per-meability and bulk modulusrdquo Journal of Geophysical Researchvol 89 no 1 pp 527ndash536 1984
[9] D P Yale and A Nur ldquoNetwork modeling of flow storage anddeformation in porous rocksrdquo SEGTechnical ProgramExpandedAbstracts vol 4 no 1 p 437 1999
[10] R F Sigal ldquoThe pressure dependence of permeabilityrdquo Petro-physics vol 43 no 2 pp 92ndash102 2002
[11] T Zhengwu Numerical Experiments of Capillary PressureCurve Southwest Petroleum University Chengdu China 2013(Chinese)
[12] J T Fredrich ldquo3D imaging of porousmedia using laser scanningconfocal microscopy with application to microscale transportprocessesrdquo Physics and Chemistry of the Earth Part A SolidEarth and Geodesy vol 24 no 7 pp 551ndash561 1999
[13] C D Tsakiroglou and M Fleury ldquoPore network analysis ofresistivity index for water-wet porous mediardquo Transport inPorous Media vol 35 no 1 pp 89ndash128 1999
[14] AW Bishop ldquoThe principle of effective stressrdquoTekniskUkebladvol 106 no 39 pp 859ndash863 1960
[15] W G Gray and B A Schrefler ldquoThermodynamic approach toeffective stress in partially saturated porous mediardquo EuropeanJournal of Mechanics ASolids vol 20 no 4 pp 521ndash538 2001
[16] M Nuth and L Laloui ldquoEffective stress concept in unsaturatedsoils clarification and validation of a unified frameworkrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 32 no 7 pp 771ndash801 2008
[17] A F Gangi ldquoVariation of whole and fractured porous rockpermeability with confining pressurerdquo International Journal ofRock Mechanics andMining Sciences and vol 15 no 5 pp 249ndash257 1978
[18] B B Noble and J W DanielsApplied Linear Algebra AmericanMathematical Society 3rd edition 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
10 Journal of Chemistry
(4) The permeability stress sensitivity is effectivelyexplained using the network simulation
Competing Interests
The authors declare that they have no competing interests
References
[1] V M Dobrynin ldquoEffect of overburden pressure on some prop-erties of sandstonesrdquo Society of PetroleumEngineers Journal vol2 no 4 pp 360ndash366 1962
[2] H S Ali M A Al-Marhoun S A Abu-Khamsin et alldquoThe effect of overburden pressure on relative permeabilityrdquoin Proceedinga of the 5th SPE Middle East Oil Show ManamaBahram March 1987
[3] M E Shafiee and A Kantzas ldquoInvestigation on the effectof overburden pressure on vuggy carbonate oil reservoircore propertiesrdquo in Proceedings of the Canadian InternationalPetroleum Conference (CIPC rsquo09) pp 16ndash18 Calgary AlbertaJune 2009
[4] D Tiab and E C Donaldson Petrophysics Theory and Practiceof Measuring Reservoir Rock and Fluid Transport PropertiesElsevierGulf Professional Publishing 2012
[5] M Li Y Bernabe W-I Xiao Z-Y Chen and Z-Q LiuldquoEffective pressure law for permeability of E-bei sandstonesrdquoJournal of Geophysical Research B Solid Earth vol 114 no 7Article ID B07205 p 223 2009
[6] D G Longeron M J Argaud and J P Feraud ldquoEffect of over-burden pressure and the nature andmicroscopic distribution offluids on electrical properties of rock samplesrdquo SPE FormationEvaluation vol 6 pp 194ndash202 1989
[7] J C Jaeger and N Cook Fundamentals of Rock MechanicsChapman amp Hall 2nd edition 1976
[8] D A Seeburger and A Nur ldquoA pore space model for rock per-meability and bulk modulusrdquo Journal of Geophysical Researchvol 89 no 1 pp 527ndash536 1984
[9] D P Yale and A Nur ldquoNetwork modeling of flow storage anddeformation in porous rocksrdquo SEGTechnical ProgramExpandedAbstracts vol 4 no 1 p 437 1999
[10] R F Sigal ldquoThe pressure dependence of permeabilityrdquo Petro-physics vol 43 no 2 pp 92ndash102 2002
[11] T Zhengwu Numerical Experiments of Capillary PressureCurve Southwest Petroleum University Chengdu China 2013(Chinese)
[12] J T Fredrich ldquo3D imaging of porousmedia using laser scanningconfocal microscopy with application to microscale transportprocessesrdquo Physics and Chemistry of the Earth Part A SolidEarth and Geodesy vol 24 no 7 pp 551ndash561 1999
[13] C D Tsakiroglou and M Fleury ldquoPore network analysis ofresistivity index for water-wet porous mediardquo Transport inPorous Media vol 35 no 1 pp 89ndash128 1999
[14] AW Bishop ldquoThe principle of effective stressrdquoTekniskUkebladvol 106 no 39 pp 859ndash863 1960
[15] W G Gray and B A Schrefler ldquoThermodynamic approach toeffective stress in partially saturated porous mediardquo EuropeanJournal of Mechanics ASolids vol 20 no 4 pp 521ndash538 2001
[16] M Nuth and L Laloui ldquoEffective stress concept in unsaturatedsoils clarification and validation of a unified frameworkrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 32 no 7 pp 771ndash801 2008
[17] A F Gangi ldquoVariation of whole and fractured porous rockpermeability with confining pressurerdquo International Journal ofRock Mechanics andMining Sciences and vol 15 no 5 pp 249ndash257 1978
[18] B B Noble and J W DanielsApplied Linear Algebra AmericanMathematical Society 3rd edition 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
