Uncertainty quantification for evaluating the impacts of fracture zone ...

$: Uncertainty quantification for evaluating the impacts of fracture zone ...$
254 © 2014 Society of Chemical Industry and John Wiley & Sons, Ltd | Greenhouse Gas Sci Technol. 5:254–267 (2015); DOI: 10.1002/ghg

Received June 17, 2014 ; revised June 12, 2014 ; accepted June 16, 2014 Published online at Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/ghg.1456

Modeling and Analysis

Uncertainty quantifi cation for evaluating the impacts of fracture zone on pressure build-up and ground surface uplift during geological CO 2 sequestration Jie Bao, Zhangshuan Hou, Yilin Fang, Huiying Ren , and Guang Lin , Pacifi c Northwest National Laboratory , Richland , WA 99352 , USA

Abstract: A series of numerical test cases refl ecting broad and realistic ranges of geological formation and fracture zone properties was developed to systematically evaluate the impacts of fracture zone on pressure build-up and ground surface uplift during CO 2 injection. Numerical test cases were conducted using a coupled hydro-geomechanical simulator, eSTOMP-RBSM (extreme-scale Subsurface Transport over Multiple Phases, rigid-body-spring model). For effi cient sensitivity analysis and reliable construc-tion of a reduced-order model, a quasi-Monte Carlo sampling method was applied to effectively sample a high-dimensional input parameter space to explore uncertainties associated with hydrologic proper-ties. The uncertainty quantifi cation results show that the impacts on geomechanical response from the fracture zone mainly depend on reservoir and fracture zone permeability. When the fracture zone per-meability is two to three orders of magnitude smaller than the reservoir permeability, the fracture zone can be considered as an impermeable block that resists fl uid transport in the reservoir, which causes pressure increase near the fracture zone. When the fracture zone permeability is close to the reservoir permeability, or higher than 10 −15 m 2 in this study, the fracture zone can be considered as a conduit that penetrates the caprock, connecting the fl uid fl ow between the reservoir and the upper rock.© 2014 Society of Chemical Industry and John Wiley & Sons, Ltd

Keywords: CO 2 geological sequestration ; fracture zone ; uncertainty quantifi cation ; numerical simulation

Correspondence to: Jie Bao, Experimental and Computational Engineering Group, Energy and Environment Directorate, Pacifi c Northwest National

Laboratory, Richland, WA 99352, USA. E-mail: [email protected]

Introduction

CO 2 sequestration in deep saline aquifers may be a promising mitigation method for the reduc-tion of CO 2 emitted to the atmosphere. 1 Im-

perfections in caprock, like fracture zone, comprise primary risk factors for leakage in engineered CO 2

injection systems. 2–4 Th rough numerical simulations, Jha and Juanes 5 and Rinaldi et al . 6,7 demonstrate the fault slip and CO 2 leakage in the fracture zone which is a few hundred meters away from the injection point. For practical sequestration sites, such as In Salah, Algeria site, the fracture zone contacts the injection well. Various analyses 8–11 have been done to study the

Modeling and Analysis: Uncertainty quantifi cation for CO2 sequestration J Bao et al.

255© 2014 Society of Chemical Industry and John Wiley & Sons, Ltd | Greenhouse Gas Sci Technol. 5:254–267 (2015); DOI: 10.1002/ghg

double-lobe uplift pattern caused by the fracture. To avoid the slip and the leakage, the CO 2 injection sites should be located at least a few kilometers, if possible, away from a detectable large pre-existing fracture zone to reduce the risk of fault slip and CO 2 leak-age. However, the injection-induced pore-pressure increase in a reservoir propagates much faster than the CO 2 plume. 12–14 Th e pressure changes may spread over 20 km away from the injection well, while the plume may remain within an area of 5 km away from the injection well. Pre-existing fracture zone impacts the pressure in the reservoir and the corresponding geomechanics deformation, even though it may not cause CO 2 leakage. Geomechanical responses, such as induced seismicity, fracturing, and preexisting fault reactivation upon CO 2 injection and sequestration, indicate the safety and sustainability of the injection activities. Top-surface/ground-surface displacements can partially refl ect underground geomechanical ac-tivities, and therefore provide critical information that can be used to infer potential hazards caused by CO 2 sequestration activities.

Various pioneer researches introduced the assump-tions, methods, and tools to study the impacts of fracture zone on CO 2 sequestration on an example test case or a specifi ed actual site. 5–8 In this study, we focus on a systematically study of the factors aff ecting the safety and sustainability of a CO 2 injection practice when a fracture zone is involved. Because of the complexity of the processes involved in CO 2 geological sequestration, especially those associated with the fracture zone, it is challenging to fully understand the impacts of the factors. In recent years, uncertainty quantifi cation (UQ) techniques have proven necessary for quantitative evaluations of uncertainty in the outputs of a system that result from diff erent sources of input uncertainty. In our previous work, 12 a procedure which includes quasi-Monte Carlo (QMC) sampling method, analysis of variance method (ANOVA) based on the generalized linear method (GLM) 15–18 were introduced to systematically evaluate the importance of the hydraulic and mechan-ical parameters in CO 2 injection system based on eSTOMP-RBSM 19 simulation predictions. It showed that reservoir permeability plays a critical role in pressure build-up and ground surface uplift during CO 2 sequestration, and it is consistent with another of our previous work, 20 which is based on a simplifi ed analytical model and an open source fi nite element solver Elmer. 13,21 Besides the reservoir permeability,

injection rate, and reservoir porosity show consider-able impacts on pressure build-up and ground surface uplift . In this study, we follow the similar UQ proce-dure as our previous work, and study the factors aff ecting the diff erence between the injection system with and without fracture zone. For convenience, the UQ methods and procedure are briefl y introduced in this manuscript in the following sections.

With UQ, a complex system can be reduced to a ‘smaller’ or ‘simpler’ system but still provide reason-able descriptions of major aspects of the original system or process, and the simplifi ed model is oft en called as reduced-order model (ROM). 22 Usually, both UQ analysis and ROM construction are considered computationally expensive because it requires accu-mulating a large number of system responses to input excitations. 23 Considering the high dimensionality of the input parameter space and the number of numeri-cal model evaluations required to achieve reliable relationships between input parameters/factors and various observed responses, an effi cient numerical simulator is essential to provide eff ective predicted responses to diff erent input parameter setups. In this study, the fracture zone is modeled as a small region with diff erent permeability from the surrounding area and lower mechanical stiff ness. We focus on studying the pressure build-up and surface displacement near the hypothetical fracture zone, which is 10 km away from the injection well. In addition, in order to study the impacts of existence of fracture zones, this study evaluate the diff erences between the cases with and without fracture zones. Th e manuscript is organized as follows: a hydro-geomechanical model for CO 2 sequestration simulation is briefl y introduced. Con-fi gurations and parameterization of the test cases, for the study of impacts from reservoir and fracture zone hydraulic properties, are then discussed. Finally, the sensitivity analysis, ROM development, and response surfaces are summarized.

A hydro-geomechanical model for CO 2 geological sequestration Similar to our previous work, 12 the sensitivity analysis of pressure and ground surface uplift responses to preexisting fracture zone during CO 2 injection was based on simulation results of the engineering simula-tor eSTOMP-RBSM, which couples the geologic media multi-phase fl ow transport solver STOMP and a rigid-body-spring model according to Kawai’s

J Bao et al. Modeling and Analysis: Uncertainty quantifi cation for CO2 sequestration


work. 24 Th e code eSTOMP-RBSM was designed for large scale parallel computation by applying the Global Arrays (GA) toolkit, which provides an effi cient and portable ‘shared-memory’ programming interface for distributed-memory computers. 25,26 Th e partial diff erential equations are solved by the parallel linear system solver: Portable, Extensible Toolkit for Scientifi c Computation (PETSc). 27,28 Details of the development of the coupled eSTOMP-RBSM can be found in Fang et al. 19

In our previous study, 29 the injection-induced porosity and permeability change play a considerable role of aff ecting the pressure and ground surface uplift response, especially for the cases with large injection rate, or low permeability reservoir. Th ere-fore, in this study, the injection-induced porosity change is applied in the whole domain, which in-cludes reservoir, caprock, base rock, upper rock, and fracture zone. Th e injection-induced porosity change can be modeled either by empirical functions fi tted to laboratory data 30,31 or fi eld observation, 32 or by theoretical relationship, such as the model in Liu and Rutqvist’s work. 33 In this study, we adopted the commonly used method, which was introduced in Rutqvist and Tsang’s work 34 and Rinaldi et al. ’s work: 7

( )exp(5 10 ) ,r M r08θ θ θ σ θ= − × ⋅ ′ +−

(1)

k k exp 22.2 1 ,0

0

θθ

= −⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

(2)

where θ is porosity, θ 0 is porosity at zero stress, θ r is residual porosity at high stress, and Mσ ′ is the mean eff ective stress (in Pa). In Eqn ( 2) , k is permeability, and k 0 is the zero-stress permeability.

Geometry confi guration and parameterization Geometry confi guration As shown in Fig. 1 , a vertical fracture zone is located near the injection well. Th e distance between the fracture zone and the injection well is Fa , and it is fi xed at 10 km east of the injection well. Th e depth of the upper edge of the fracture zone, F d , is fi xed at 0 m. Th e length of the fracture zone, F l , is set at 30 km. Th e thickness of the fracture zone, F t , is 50 m. Th e entire investigated domain is 80 km × 80 km × 3 km, and the injection well is located at the center of

the domain. Th e aquifer layer is simplifi ed as a 200-m thick horizontal layer located between depths of −1300 m and −1500 m, and the lower edge of the fracture zone is fi xed at the depth of −3000 m. Th e caprock is from −1200 m to −1300 m. Supercritical CO 2 is injected into aquifer continuously at a con-stant rate. As mentioned in the introduction section, fracture zone is modeled as a small region with diff erent properties from the host area. Th e perme-ability and porosity in the fracture zone can be higher or lower than those in the host domain, while the elastic stiff ness is assumed to be smaller than in the surrounding area. We focus on the cases that the fracture zone is 10 km away from the injection well, so the fault reactivation, plastic eff ect, and slip are neglected in this study. Figure 2 shows the results for a specifi c test example – a baseline model of our study – and compares the diff erence between the cases with and without the fracture zone. Th e main properties adopted for this selected test are listed in Table 1 , which is one selected test among the 256 tested cases for the UQ study for this research. In the baseline model test, because the fracture zone is 10 km away from the injection well, the CO 2 plume is not obvi-ously aff ected by the fracture zone, as shown in Figs 2 (c) and 2 (d). However, the pressure evolution is much faster than the CO 2 plume propagation; there is a noticeable pressure increase in the fracture zone,

Figure 1. Schematic of computational geometry confi guration.



and the geomechanical deformation near the fracture zone is aff ected considerable, as marked on Figs 2 (b) and 2 (f). Figure 3 (a) shows the comparison of pres-sure change profi le as a function of distance from the injection well. Th e fracture zone causes almost no

pressure diff erences in reservoir. Figure 3 (b) shows the ground surface uplift as a function of distance from injection well; the fracture zone increases the uplift near the fracture zone, which is 10 km away from the injection well.

Figure 2. Comparison between the cases with and without fracture zone for a specifi c test example at the selected cross section (x = 0 m) after 10 years of injection: (a) pressure change distribution without fracture zone; (b) pressure change distribution with fracture zone; (c) gas saturation fi eld without fracture zone; (d) gas saturation fi eld with fracture zone; (e) vertical displacement fi eld without fracture zone; (f) vertical displacement fi eld with fracture zone.



Parameterization and sampling Th e properties of the reservoir and the fracture zone are perturbed from the baseline test case as discussed above. UQ framework is able to systematically evalu-ate the impacts of the uncertainty of fracture zone and surrounding zones on the geomechanical re-sponses during CO 2 injection. Our previous stud-ies 12,20 show that the injection rate, initial reservoir permeability, and porosity have greater impact the geomechanical response during CO 2 injection and sequestration than other parameters, such as the hydraulic properties of caprock. To systematically evaluate the fracture zone impacts on pressure and ground surface response, the porosity and permeabil-ity of the fracture zone are selected as input param-eters as well. Table 2 lists the range of each input parameter. Besides the fi ve input parameters, the values of other properties that were fi xed during the sensitivity analysis are the same as the base line test case as listed in Table 1 .

The tested problem is three-dimensional, as shown in Fig. 1 , and the modeling domain contains over 1 million grid nodes. The simulation is time consuming, so the efficiency of the sampling

method for setting up the test cases is critical to the UQ analysis. QMC methods have become one of the most popular sampling method because of its faster convergence and effective sampling of high-dimensional parametric spaces without

Figure 3. Comparison between the cases with and without a fracture zone for a specifi c test example after 10 years of injection: (a) pressure change profi le as a function of distance from injection well in aquifer (z = −1400 m); (b) ground surface vertical displacement as a function of distance from the injection well.

Table 1. Parameters for baseline model test.

Injection rate (MMT year −1 ) 1.48

Log 10 initial aquifer permeability (log 10 (m 2 ))

–12.9

Initial aquifer porosity 0.115

Log 10 initial fracture zone permeability (log 10 (m 2 ))

–14.1

Initial fracture zone porosity 0.071

Log 10 initial upper permeability (1og 10 (m 2 ))

–16

Initial upper porosity 0.1

Log 10 initial caprock permeability (1og 10 (m 2 ))

–20

Initial caprock porosity 0.01

Log 10 initial base permeability (1og 10 (m 2 ))

–20

Initial base porosity 0.01

Upper, caprock, aquifer, and base Young’s Modulus (GPa)

25 53,54

Fracture zone Young’s modulus (GPa) 2.5

Upper, caprock, aquifer, base and fracture zone Poisson ratio

0.25 55

Table 2. Selected uncertain input parameters and their sampling ranges.

Injection rate (MMT year −1 ) 0.01 to 1.5 12

Log 10 initial aquifer permeability (log 10 (m 2 ))

−15 to −12 35

Initial aquifer porosity 0.05 to 0.3 35

Log 10 initial fracture zone permeability (log 10 (m 2 ))

−16 to −14 8

Initial fracture zone porosity 0.05 to 0.1 8



clumping and gaps. 35,36 Research by Singhee and Rutenbar 37 demonstrated that QMC is better than the close completion sampling method Latin hypercube sampling (LHS). 38,39 In our study, the QMC samples were generated from existing scram-bled Sobol sequences 40 ranging from 0 to 1, and the samples then were scaled to the actual input parameter ranges as listed in Table 2 . In this study, 256 test cases were generated based on QMC sampling. The paired scatter plots of the five independent parameters are shown in Fig. 4 . There are some patterns in the sampling points that do not look random, as shown in Fig. 4 , because Sobol sequences is not a pure random number list. The newly generated sampling points depends on the

distribution of the existing points, which enable the sampling points to fill in the gaps of the pa-rameter space without clumping. This would yield more reliable response surface than using pure random sampling points.

Response variables To separate the impacts of fracture zone on pressure build up and ground surface uplift from the total injection induced pressure and ground surface response, another 256 cases were tested with the same confi gurations but without the fracture zone. Th e relative diff erences of output responses between the cases with and without a fracture zone are defi ned as:

Figure 4. Paired scatter plots of the fi ve input parameters: injection rate, reservoir permeability, reser-voir porosity, fracture zone permeability, and fracture zone porosity.



= −P

P P PP

For pressure ( ):( )

,rd wf wof

wof

(3)

uu u u

uFor vertical displacement ( ) :

( ),rd wf wof

wof

= −

(4)

where the subscript rd means relative diff erence, wf stands for with fracture zone, and wof is without fracture zone. Four output response variables were studied to evaluate the impacts of presence of a fracture zone, including the relative diff erence in the ground surface uplift at the injection well at point A ( D uA ) shown in Fig. 1 , the relative diff erence in the ground surface uplift on the top of the fracture zone at point B ( D uB ) in Fig. 1 , the relative diff erence in pressure change at the injection point (z = −1400 m) at point C ( D pC ) in Fig. 1 , and the relative diff erence in pressure change at the lower edge of the fracture zone at point D ( D pD ) in Fig. 1 .

Sensitivity analysis and response surface Qualitative sensitivity analysis Figure 5 shows the boxplots 41–43 for the relationship between the output response and the input param-eters, which can qualitatively evaluate the importance of the input parameters. For the relative displacement diff erence at point A ( D uA ), the reservoir permeability is much more important than the other four input parameters. With the increase in reservoir permeabil-ity, D uA increases. Larger reservoir permeability leads to smaller pressure in reservoir and smaller ground surface uplift , so the value of P wof or u wof are usually very small. Th erefore, any small infl uence from the fracture zone would be noticed and enlarged by the system, which leads to larger D uA . For the relative displacement diff erence at point B ( D uB ), with the increase in reservoir permeability, D uB increases gradually, due to the similar reason as D uA . Other

Figure 5. Boxplots for the relationships between input parameters and output responses.



input parameters do not show obvious eff ects on the ground relative surface uplift diff erences. Similar to D uA , for the relative pressure diff erence at point C ( D pC ), D pc increases with reservoir permeability, and does not depend on other input parameters, due to the same reason as D uA . With increase in reservoir permeability and decrease in fracture zone permeabil-ity, the relative pressure diff erence at point D ( D pD ) increases, because larger reservoir permeability make the system sensitive to the fracture zone, as discussed above, and the smaller fracture zone permeability make the fracture zone has higher contrast to the host domain. In summary, for an injection site with small reservoir permeability, the impact from a fracture zone, which is a few kilometers away from the injec-tion well, is very limited. For a site with large reservoir permeability, the fracture zone may cause relatively big diff erences from the case without the fracture zone, and the magnitude of the relative diff erence partially depends on the fracture zone permeability.

Quantitative sensitivity analysis Th e qualitative analysis above reveals the general relationship between the response variable and input parameters, but it cannot measure the importance of the input parameters quantitatively, and the results would be easily biased by people’s experiences and preferences. Th erefore, a quantitative sensitivity analysis is necessary for a more precise evaluation. Th e sensitivity analysis can base on the original system outputs, or a regression model that can fi t the original system outputs. Analysis based on original system can avoid inaccuracy and biases from regres-sion models, but it requires large amount sampling points with necessary number of replicated value of input parameters, 44,45 such as replicated Latin Hyper-cube sampling (rLHS) method. 46 For a high dimen-sional input parameters space, the number of sam-pling points is usually more than 10 000 for getting a converged results, 47,48 and it is unaff ordable for the most studies. Th erefore, sensitivity analysis based on a regression model is more effi cient, especially for high dimensional input parameters space, because it does not require replicated value of each input parameters. Th e choice of the regression method directly aff ects the reliability and conclusions of sensitivity analysis. In this study, we employed the analysis of variance method (ANOVA) based on the generalized linear regression model (GLM), 15–18 which uses the value of

the residual of deviance in the GLM. A GLM is fi tted with the following starting model:

Y c c x ,i j i j ij

J

0 ,1

∑ ε= + +=

(5)

where x i,j represents the i th realization of the j th parameter (as listed in Table 2 ), which can be original or transformed fi rst-order, two-way interaction, or higher-order terms; c j is the fi tted coeffi cient for the j th parameter, which can be calculated by least squares method 49,50 based on all the realizations, which are the 256 simulation tests as introduced in Section 3. Y i represents the i th realization of the response variable, such as D uA , D uB , D pc , and D pD in this study ε is model-fi tting residuals. ANOVA is a collection of statistical models used to analyze the regression models. In this study, the residual of deviance changes of the GLM regression model (Eqn ( 5) ) by adding the terms c j x i,j is defi ned as a quantitative measurement of the importance of the j th parameter x i,j .

Response surface Figure 6 shows the sensitivity scores of input param-eters, which is consistent with the qualitative analysis based on the boxplots. Th e sensitivity scores stand for the percentage of contribution from the input param-eters to the output response. For D uA , D uB and D pC , reservoir permeability contributes about 75% variabil-ity out of the total variance of the output responses. For D pD , fracture zone permeability contributes about 70%, and reservoir permeability contributes around 30% to the variance of D pD . According to the sensitiv-ity scores, the top three important input parameters of each output response are kept for the ROM, and further used for generating the response surface. Equation ( 5) can regress the relationship between output response and the corresponding top three important input parameters, and it is consistent with the qualitative analysis based on the boxplots, as shown on the response surfaces in Figure 7 . Th e model fi tnesses R 2 51,52 are 0.68, 0.83, 0.69, and 0.93 for D uA , D uB , D pC , and D pD , respectively. Th e response surface can not only show the relationship between input and output parameters, but also the sensitivity of inputs. If the response surface is more perpendicu-lar to a certain axis, the output depends more on the input parameters represented by that axis. D uA mainly depends on reservoir permeability, and generally increases with reservoir permeability increase. When



the fracture zone properties and surrounding zones can be used to estimate the output response, such as the relative diff erences at the investigated points A, B, C, and D. From the analysis above, the fracture zone and reservoir permeability play an important role in the relative diff erences, so the permeability ratio is defi ned as:

kkk

log ,ratiof

r10=

⎛⎝⎜

⎞⎠⎟ (6)

where k f and k r are fracture zone permeability and reservoir permeability respectively, with units of m 2 in Eqn ( 6) . Figure 8 shows boxplots for the relation-ship between the output response and the permeabil-ity ratio. Th e relative diff erences at points A and C decrease when k ratio increases. It means that a less-permeable fracture zone can increase the pressure at the injection point and ground surface uplift at the injection well. Th e relative diff erence at point B does not depend on the permeability ratio. Th e relative pressure diff erence at point D shows strong dependence on the permeability ratio. If the second order GLM regression model is used to fi t the output response based only on the permeability ratio, the model can be expressed as:

D C k C k C ,ratio ratio1 22

0= + + (7)

reservoir permeability is smaller than 10 −14.5 m 2 , D uA slightly increases with decrease of reservoir perme-ability. D uB , increases gradually with reservoir perme-ability, and most response surfaces are perpendicular to the axis for reservoir permeability. When reservoir permeability is larger than 10 −13 m 2 , D uB is highly dependent on fracture zone permeability, and in-creases with fracture zone permeability. D pC is very similar to D uA , because the investigated point C is under the point A, and the ground surface behavior at point A is the refl ection of the pressure evolution behavior at point C. D pc mainly depends on reservoir permeability, and generally increases with reservoir permeability increase. When reservoir permeability is smaller than 10 −14.5 m 2 , D pc slightly increases with decrease of reservoir permeability. For D pD , the contrast of the reservoir and fracture zone perme-abilities aff ects the pressure distribution, so both permeabilities shows signifi cant importance. When reservoir permeability increases and fracture zone permeability decreases, the relative diff erence of pressure at point D ( D pD ) increases.

Reduced order model for permeability ratio Th e fracture zone can be considered as a discontinuity of underground properties, so the contrast between

Figure 6. Sensitivity scores for the input parameters: (a) relative displacement difference at point A; (b) relative displacement difference at point B; (c) relative pressure difference at point C; (d) relative pressure difference at point D.



D pD , the model fi tness R 2 is 0.91, which means that Eqn ( 7) is a good ROM for estimation of D pD , and D pD strongly depends on the permeability ratio. Figure 9 shows the correlation between the values of D pD predicted by Eqn ( 7) and the actual values of D pD from the 256 simulation test cases. With increase in permeability ratio, D pD decreases. When fracture zone permeability is two to three orders of magnitude smaller than reservoir permeability, the fracture zone is an impermeable zone blocking fl uid transport in the reservoir, which causes pressure increase near the fracture zone. When fracture zone permeability is

where D is the output response, which can be D uA , D uB , D pC , or D pD . Th e values of coeffi cients ( C 1 , C 2 , and C 0 ) were calculated using the least squares method. Th e model fi tnesses R 2 for D uA and D pC are around 0.5, and R 2 for D uB is smaller than 0.1. Even if a higher order of GLM is used for the regression, R 2 cannot be increased. Th is means that about 50% of the variation of D uA and D pC can be explained by the permeability ratio, so they partially depend on the permeability ratio. D uB cannot be represented by a ROM that is a function of k ratio only, which means that it does not depend on the permeability ratio. For

Figure 7. Response surfaces of the relative differences between the cases with and without a fault at investigated points: (a) relative displacement difference at point A; (b) relative displacement difference at point B; (c) relative pressure difference at point C; (d) relative pressure difference at point D.



runs using eSTOMP-RBSM. It was shown that the impacts on pressure and ground surface response from the fracture zone depend more on the reservoir and fracture zone permeability than other hydraulic properties. Generally, for an injection site with small reservoir permeability, the impact from a fracture zone, which is around 10 km away from the injection well, is very limited. For a site with large reservoir permeability, the fracture zone may cause relatively big diff erences from the case without the fracture zone, and the magnitude of the relative diff erence partially depends on the fracture zone permeability. When the fracture zone permeability is two to three orders of magnitude smaller than the reservoir permeability, the fracture zone can be considered as an impermeable block that resists fl uid transport in the reservoir, which causes pressure increase near the fracture zone. When the fracture zone permeability is close to the reservoir permeability or higher than 10 −15 m 2 , the fracture zone can be considered as a conduit that penetrates the caprock and connects the fl uid transport between the reservoir and upper rock. Th is is not surprising, but the study can provide a

close to reservoir permeability, the fracture zone serves as a conduit that penetrates the caprock, which facilitates the fl uid fl ow between the reservoir and upper rock, so the pressure decreases near the fracture zone in the reservoir. Th is also explains why D uB does not depend on permeability ratio. Th e pressure increase at the fracture zone in the reservoir can increase the ground surface uplift , but fl uid penetrat-ing into upper rock increases the ground surface uplift as well. 13 When fracture zone is more permeable than reservoir, the fracture zone decrease the pressure in reservoir near the fracture zone signifi cantly, causing the relative diff erence of pressure at point D ( D pD ) close to −1.

Conclusion and discussion In this study, we explored fi ve important hydraulic parameters of a reservoir and fracture zone, and ranked their impacts on pressure build up and ground surface uplift during CO 2 injection when a fracture zone is located 10 km away from the injection well, through 256 realizations of the numerical test case

Figure 8. Boxplots for the relationships between output responses and permeability ratio k log f

rratio

k

k10( )( )= .

Figure 9. (a) Correlation between predicted and actual relative pressure difference at point D; (b) relationship between the relative pressure difference at point D and permeability ratio.



12. Bao J , Hou Z , Fang Y , Ren H and Lin G , Uncertainty quantifi cation for evaluating impacts of caprock and reservoir properties on pressure buildup and ground surface displace-ment during geological CO2 sequestration . Greenhouse Gas Sci Technol 3 : 338 – 358 ( 2013 ).

13. Bao J , Xu Z and Fang Y , A fi nite element model for simulation of CO2 sequestration . Environ Geotech 1(3) : 152 – 160 ( 2013 ).

14. Rutqvist J , Birkholzer J , Cappa F and Tsang CF , Estimating maximum sustainable injection pressure during geological sequestration of CO2 using coupled fl uid fl ow and geome-chanical fault-slip analysis . Energ Convers Manage 48 : 1798 – 1807 ( 2007 ).

15. Box GEP , Some theorems on quadratic forms applied in the study of analysis of variance problems, I. Effect of inequality of variance in the one-way classifi cation . Ann Math Stat 25 ( 2 ): 290 – 302 ( 1954 ).

16. Anscombe FJ , The validity of comparative experiments . J Roy Stat Soc 111 : 181 – 211 ( 1948 ).

17. McCullagh P and Nelder J , Generalized Linear Models, Second Edition . Chapman and Hall, Boca Raton, FL ( 1989 ).

18. Chambers JM and Hastie TJ . Statistical Models in S. Chapman and Hall/CRC, Boca Raton, FL ( 1992 ).

19. Fang Y , Nguyen BN , Carroll K , Xu Z , Yabusaki SB , Scheibe TD et al ., Development of a coupled thermo-hydro-mechanical model in discontinuous media for carbon sequestration . Int J Rock Mechanics Mining Sci 62 : 138 – 147 ( 2013 ).

20. Bao J , Xu Z , Lin G and Fang Y , Evaluating the impact of aquifer layer properties on geo-mechanical response during CO2 geological sequestration . Comput Geosci 54 : 28 - 37 ( 2013 ).

21. Ruokolainen J , Malinen M , Raback P , Zwinger T , Pursula A , Bycklin M , ElmerSolver Manual, CSC-IT Center for Science. [Online]. Available at: http://www.nic.funet.fi /pub/sci/physics/elmer/doc/ElmerSolverManual.pdf ( 2014 ).

22. Pan W , Bao J , Lo C , Lai K , Agarwal K , Koeppel BJ et al., A general approach to develop reduced order models for simulation of solid oxide fuel cell stacks . J Power Source 232: 139 – 151 ( 2013 ).

23. Amsallem D and Farhat C , Interpolation method for the adaptation of reduced-order models to parameter changes and its application to aeroelasticity . AIAA J 46 : 1803 – 1813 ( 2008 ).

24. Kawai T , New element models in discrete structural analysis . J Soc Naval Arch Japan 141 : 174 – 180 ( 1977 ).

25. Nieplocha J , Palmer B , Tipparaju V , Krishnan M , Trease H and Apra E , Advances, Applications and performance of the global arrays shared memory programming toolkit . Int J High Perform Comput Appl 20 ( 2 ): 203 – 231 ( 2006 ).

26. Krishnan M , Palmer B , Vishnu A , Krishnamoorthy S , Daily J and Chavarria D . The Global Arrays User Manual: Pacifi c Northwest National Laboratory, Richland, WA, USA ( 2012 ).

27. Balay S , Adams MF , Brown J , Brune P , buschelman K , Eijkhout V et al ., PETSc users manual: Argonne National Laboratory . Available at: http://www.mcs.anl.gov/petsc (2013) .

28. Balay S , Adams MF , Brown J , Brune P , Buschelman K , Eijkhout V et al ., PETSc web page . Available at: www.mcs.anl.gov/petsc ( 2014) .

29. Bao J , Xu Z and Fang Y , Uncertainty quantifi cation for the reliability of the analytical analysis for the simplifi ed model of CO2 geological sequestration . Greenhouse Gas Sci Technol ( 2014 ). DOI: 10.1002/ghg.1436

30. Mohiuddin MA , Korvin G , Abdulraheem A , Awal MR , Khan K , Khan MS et al., Stress-dependent porosity and permeability of a suite of samples from Saudi Arabian sandstone and

quantitative measurement of such impact as a func-tion of permeability contrast; therefore, the approach can be used to help evaluate the risk of an uncertain fracture zone in practice.

Acknowledgements Th is research has been accomplished and funded through the Pacifi c Northwest National Laboratory (PNNL) Carbon Sequestration Initiative, which is part of the PNNL Laboratory Directed Research and Development Program. A portion of this research was performed using the resources of the PNNL Institu-tional Computing program. PNNL is operated by Battelle for the US Department of Energy under Contract DE-AC05-76RL01830.

References 1. Vilarrasa V , Olivella S and Carrera J , Geomechanical stability

of the caprock during CO 2 sequestration in deep saline aquifers . Energ Procedia 4 : 5306 – 5313 ( 2011 ).

2. IPCC , Underground geologic storage in Carbon Dioxide Capture and Storage , ed by Metz B , Davidson O , Coninck Hd , Loos M and Meyer L , Cambridge University Press , Cambridge, UK ( 2005 ).

3. Chang KW , Bryant SL , eds, Dynamics of CO2 Plumes Encountering a Fault in a Reservoir , Sixth Annual Conference on Carbon Capture and Sequestration , Pittsburgh, PA, USA ( 2007 ).

4. Pasala SM , Forster CB , Deo M and Evans JP , Simulation of the impact of faults on CO2 injection intosandstone reser-voirs . Geofl uids 13 : 344 – 358 ( 2013 ).

5. Jha B and Juanes R , Coupled multiphase fl ow and porome-chanics: Computational model of pore-pressure effects on fault slip and earthquake triggering . Water Resour Res 50(5) : 3776 – 3808 ( 2014 ).

6. Rinaldi AP , Rutqvist J and Cappa F , Geomechanical effects on Co2 leakage during large-scale underground injection . Int J Greenhouse Gas Control 20 : 117 – 131 ( 2014 ).

7. Rinaldi AP , Jeanne P , Rutqvist J , Cappa F and Guglielmi Y , Effects of fault-zone architecture on earthquake magnitude and gas leakage related to Co2 injection in a multi-layered sedimen-tary system . Greenhouse Gas Sci Technol 4 : 99 – 120 ( 2014 ).

8. Rinaldi AP and Rutqvist J , Modeling of deep fracture zone opening and transient ground surface uplift at KB-502 CO2 injection well, In Salah Algeria . Int J Greenhouse Gas Control 12 : 155 – 167 ( 2013 ).

9. Rucci A , Vasco DW and Novali F , Monitoring the geologic storage of carbon dioxide using multicompoinent SAR interferometry . Geophys J Int 193 : 197 – 208 ( 2013 ).

10. Rutqvist J , Vasco DW and LMyer L , Coupled reservoir-geo-mechanical analysis of CO2 injection and ground deforma-tions at In Salah, Algeria . Int J Greenhouse Gas Control 4 : 225 – 230 ( 2010 ).

11. Shi JQ , Sinayuc C , Durucan S and Korre A , Assessment of carbon dioxide plume behavior within the storage reservoir and the lower caprock around the KB-502 injection well at In Salah . Int J Greenhouse Gas Control 7 : 115 – 126 ( 2012 ).



50. Björck Å , Numerical Methods for Least Squares Problems, First Edition. SIAM: Society for Industrial and Applied Mathematics, Philadelphia, PA, USA ( 1996 ).

51. Steel RGD and Torrie JH , Principles and Procedures of Statistics. McGraw-Hill , New York ( 1960 ).

52. Cameron AC and Windmeijer FAG , An R-squared measure of goodness of fi t for some common nonlinear regression models . J Econ 77 : 329 – 342 ( 1997 ).

53. Hart DJ and Wang HF , Laboratory measurements of a complete set of poroelastic moduli for Berea sandstone and Indiana limestone . J Geophys Res 100 : 17741 – 17751 ( 1995 ).

54. Palmstrom A and Singh R , The deformation modulus of rock masses - comparisons between in situtests and indirect estimates . Tunnell Underground Space Technol 16 ( 3 ): 115 – 131 ( 2001 ).

55. Gercek H , Poisson’s ratio values for rocks . Int J Rock Mechanic Mining Sci 44 : 1 – 13 ( 2007 ).

limestone reservoirs . 2000 Symposium of Core Analysts, Abu Dhabi, UAE (18–22 October 2000) .

31. Han G and Dusseault MB , Description of fl uid fl ow around a wellbore with stress-dependent porosity and permeability . J Petrol Sci Eng 40 ( 1/2 ): 1 – 16 ( 2003 ).

32. Davies JP and Davies DK , Stress-dependent permeability: Characterization and modeling . Soc Petrol Eng J 6 ( 2 ): 224 – 235 ( 2001 ).

33. Liu H and Rutqvist J , A new coal permeability model: Internal swelling stress and fracture-matrix interaction . Transp Porous Med 82 : 157 – 171 ( 2010 ).

34. Rutqvist J and Tsang C-F , A study of caprock hydromechani-cal changes associated with CO2 injection into a brine formation . Environ Geol 42 : 296 – 305 ( 2002 ).

35. Hou Z , Rockhold ML and Murray CJ , Evaluating the impact of caprock and reservoir properties on potential risk of CO2 leakage after injection . Environ Earth Sci 66 ( 8 ): 2403 – 2415 ( 2012 ).

36. Tarantola A , Inverse Problem Theory and Model Parameter Estimation . Society of Indus Applied Mathamatics , Philadelphia, PA, USA ( 2005 ).

37. Singhee A and Rutenbar RA , Why quasi-Monte Carlo is better than Monte Carlo or Latin hypercube sampling for statistical circuit analysis . IEEE T Comput Aid D . 29 ( 11 ): 1763 – 1776 ( 2010 ).

38. McKay MD , Beckman RJ and Conover WJ , A comparison of three methods for selecting values of input variables in the analysis of output from a computer code . Am Stat Assoc 21 ( 2 ): 239 – 245 ( 1979 ).

39. Iman RL , Helton JC and Campbell JE , An approach to sensitivity analysis of computer models, Part 1. Introduction, input variable selection and preliminary variable assessment . J Qual Technol 13 ( 3 ): 174 – 183 ( 1981 ).

40. Bratley P and Fox BL , Algorithm 659: Implementing sobol’s quasirandom sequence generator . ACM T Math Software 14 : 88 – 100 ( 1988 ).

41. Tukey JW , Exploratory Data Analysis . Addison-Wesley , Boston, MA, USA ( 1977 ).

42. Benjamini Y , Opening the box of a boxplot . Am Stat 42 : 257 – 262 ( 1988 ).

43. Rousseeuw PJ , Ruts I and Tukey JW , The bagplot: A bivariate boxplot . Am Stat 53 : 382 – 387 ( 1999 ).

44. Tong C , Toward a more robust variance-based global sensitivity analysis of model outputs. Contract No: UCRL-TR-235561 . Lawrence Livermore National Laboratory Livermore, CA, USA ( 2007 ).

45. Nossent J , Elsen R and Bauwens W , Sobol’ sensitivity analysis of a complex environmental model . Environ Modell Soft 26 : 1515 – 1525 ( 2011 ).

46. Pleming JB and Manteufel RD , Replicated Latin Hypercube Sampling. 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference , Austin, TX (18–21 April 2005 ).

47. Lee LA , Carslaw KS , Pringle KJ and Mann GW , Mapping the uncertainty in global CCN using emulation . Atmos Chem Physic Discuss 12 : 14089 – 14114 ( 2012 ).

48. Lee LA , Carslaw KS , Pringle KJ , Mann GW and Spracklen DV , Emulation of a complex global aerosol model to quantify sensitivity to uncertain parameters . Atmos Chem Physic Discuss 11 : 122253 – 112273 ( 2011 ).

49. Rao CR , Toutenburg H , Fieger A , Heumann C , Nittner T and Scheid S , Linear Models: Least Squares and Alternatives , Springer series in statistics. Springer, New York, NY, USA ( 1999 ).

Jie Bao

Dr. Jie Bao got his Ph.D. from Me-chanical Engineering and Materials Science Department at University of Pittsburgh in 2010. Currently, he is an engineer in Experimental and Compu-tational Engineering Group at Pacifi c Northwest National Laboratory. He has expertise in numerical simulation for

turbulent fl ow, multi-phase fl ow, non-Newtonian fl uid, and porous media fl ow. His research focuses on lattice Boltzmann method, fi nite element method, massive parallel computation, GPU computing, UQ, and solving problems related to nuclear waste tank management, carbon sequestration, and different kinds of batteries.

Zhangshuan Hou

Dr. Zhangshuan Hou has expertise in Applied Statistics, Hydrogeology, Geo-physics, and Atmospheric Sciences. His research focuses on EDA, UQ, and inversion approaches, solving problems related to carbon sequestration, oil/gas exploration, environmental remediation, climate change, and power systems.

Yilin Fang

Dr. Yilin Fang got her Ph.D. from Civil and Environmental Engineering Depart-ment at the Pennsylvania State Uni-versity in 2003. Currently, she is a staff scientist in Hydrology Group at Pacifi c Northwest National Laboratory. Her research focuses on subsurface multi-phase fl ow and reactive transport, geo-

mechanics, high performance parallel computing, carbon sequestration, and biogeochemistry in land surface model.



Huiying Ren

Ms. Huiying Ren is a research associ-ate at Hydrology Group, Earth Sys-tems Science Division, Pacifi c North-west National Laboratory (PNNL). Her research fi elds at PNNL are analysis of acoustic, marine mammal pas-sive acoustic system and Uncertainty Quantifi cation (UQ) of CO2 seques-

tration. She received her B. Eng. and B. A. Degrees from Dalian University of Technology, Dalian, China in 2008. She joined Wright State University since then and received her M.S degree in Mechanical and Materials Department from Wright State University in 2010. Her master study was fl uid dynamic that focused on experi-mental study of turbulent fl ow and macro air vehicles (MAV) using particle image velocimetry (PIV) devices.

Guang Lin

Dr. Guang Lin got his Ph.D. from Divi-sion of Applied Mathematics at Brown University in 2007. Currently, he is a staff scientist in Computational Math-ematics Group at Pacifi c Northwest Na-tional Laboratory. His research focuses on uncertainty quantifi cation and multi-scale modeling of complex systems.

Uncertainty quantification for evaluating the impacts of fracture zone ...

Documents

Transcript of Uncertainty quantification for evaluating the impacts of fracture zone ...