Numerical Simulation of Mud-Filtrate Invasion in Deviated Wells...the mud-filtration process. We...

Numerical Simulation of Mud-FiltrateInvasion in Deviated Wells

Jianghui Wu, Carlos Torres-Verdı́n, SPE, Kamy Sepehrnoori, SPE, and Mojdeh Delshad, SPE, U. of Texas at Austin

SummaryUnderstanding the spatial distribution of fluids in the near-borehole region caused by mud-filtrate invasion is necessary forthe accurate petrophysical interpretation of wireline measurementsacquired in deviated wells. This paper provides a sorely neededpetrophysical and fluid-flow template that can be used to integrateseveral wireline measurements into a unique model of petrophysi-cal parameters in the near-borehole region of deviated wells.

We simulate numerically the physics of mud-filtrate invasion invertical, horizontal, and highly deviated overbalanced wells. Thenumerical algorithm is adapted from a general 3D multiphase-fluid-flow simulator that is widely used in large-scale reservoirapplications. Checks of numerical consistency and accuracy areperformed against one commercial reservoir simulator. Empha-sis is placed on describing the influence of mudcake buildup onthe mud-filtration process. We approach the latter problem byintroducing an effective-flow-rate function that describes the evo-lution in time of the rate of invasion of mud filtrate into rockformations. Parametric representations of the flow-rate functionare derived on the basis of previously published laboratory experi-ments of mud circulation.

A sensitivity analysis quantifies the influence of several geo-metrical and petrophysical parameters on the spatial distribution ofmud-filtrate invasion away from the borehole wall. These param-eters include relative permeability, capillary pressure, permeabilityanisotropy, dipping layers, and degree of hydraulic communicationbetween adjacent layers. Our simulations reveal the character ofinvasion profiles in complex geometrical environments takingplace under realistic petrophysical conditions. We show that stan-dard pistonlike descriptions of mud-filtrate invasion, commonlyused in well-log interpretation, can lead to inaccurate interpreta-tions of wireline measurements. An example is presented of theuse of our simulation technique by calculating the sensitivity ofborehole electromagnetic induction measurements to specific con-ditions of mud-filtrate invasion in a vertical well.

IntroductionMud-filtrate invasion takes place in permeable rock formationspenetrated by a well that is hydraulically overbalanced by mudcirculation. This condition is of interest in numerous oilfield ap-plications including drilling engineering, reservoir simulation, res-ervoir stimulation, and well-log interpretation. Electrical, electro-magnetic, acoustic, and nuclear logging instrument responses areall influenced by the spatial distribution of fluids in the vicinity ofthe borehole resulting from the invasion of mud filtrate.

The invasion of mud filtrate into permeable rock formations isresponsible for the development of mudcake on the borehole wall(solids deposition), as well as for the displacement of existingin-situ fluids laterally away from the borehole.1–5 Standard proce-dures used for the interpretation of well-logging measurementsoften conceive of the invasion of mud filtrate as a radial sequenceof pistonlike fluid-saturation fronts. There have been a handful oftechniques put forth to simulate (numerically and in the laboratory)the physics of mud-filtrate invasion. Drilling variables such as mud

density and chemistry, mud circulation pressure, and time of fil-tration may all significantly affect the spatial extent of mud-filtrateinvasion. In-situ rock formation properties such as porosity, abso-lute permeability, relative permeability, pore pressure, shale chem-istry, capillary pressure, and residual fluid saturations also playimportant roles in controlling both the dynamic formation of mud-cake and the time evolution of the invasion process.

Simple 1D models of mud-filtrate invasion exist based on theassumptions of a vertical well and a horizontal and infinitely thickrock formation.2 These models have been derived by the enforce-ment of mass-balance conditions and at best assume a homoge-neous and isotropic spatial distribution of porosity and permeabil-ity while neglecting capillary forces and relative permeability ef-fects. To date, there are no available numerical algorithms capableof simulating the physics of mud-filtrate invasion in rock forma-tions comprising multiple hydraulically connected beds, nor arethere algorithms that can simulate the geometrical complexity as-sociated with deviated or horizontal wells.

In this paper, we introduce a general numerical algorithm tosimulate the physics of mud-filtrate invasion in vertical and devi-ated boreholes. This algorithm is adapted from an existing 3Dmultiphase-fluid-flow simulator widely used to replicate the be-havior of large-scale hydrocarbon reservoirs. The simulator, de-veloped by the U. of Texas at Austin, is commercially referred toas UTCHEM.6–9 A detailed description of UTCHEM’s mul-tiphase, multicomponent, and multichemical-species fluid-flowmodel is presented in the UTCHEM technical documentation.10

One of the salient technical problems often considered in mud-filtrate-invasion studies is the description of mudcake buildup. Thedevelopment and thickening of mudcake on the borehole wallcauses the permeability of mudcake to decrease monotonically. Inturn, this causes the flow rate of mud filtrate to decrease toward alow steady-state value. Both chemical and fluid mechanical pro-cesses determine (a) the rate at which solids accumulate on theborehole wall and (b) the thickness and effective permeability ofthe mudcake. Complications may arise as a result of periodic re-trieval of drillpipe for bit changes. This can cause scraping ofmudcake, resulting in secondary mudcake buildup and, hence, sec-ondary mud-filtrate invasion.

Recently, and on the basis of laboratory experiments of mudcirculation, Dewan and Chenevert1 reported a methodology to pre-dict the time evolution of mudcake buildup as well as the effectivepetrophysical properties of mudcake. Dewan and Chenevert’s de-scription is based entirely on six mud-filtrate parameters, all ofwhich can be determined from a standard mud-filtrate test. Asexpected, Dewan and Chenevert’s work predicts a monotonicallydecreasing rate of mud-filtrate invasion as a function of time.Moreover, Dewan and Chenevert emphasize the fact that the rateof mudcake thickening remains practically unaffected by the petro-physical properties of the invaded rock formation. Borrowing fromthese results, in our simulations of mud-filtrate invasion we havecompletely avoided the problem of modeling mudcake buildup.Instead, we have chosen to model the effect of mudcake buildup onthe invasion profile by defining an equivalent time-dependent flowrate of mud-filtrate invasion.

With an equivalent flow-rate function in hand, we have adaptedUTCHEM to perform the remaining task of simulating the inva-sion of mud filtrate into permeable formations. Results from thisexperience are presented here as an attempt to quantify in a sys-tematic manner the effects of a wide range of geometrical andpetrophysical parameters on the invasion phenomenon. We visu-

Copyright © 2004 Society of Petroleum Engineers

This paper (SPE 87919) was revised for publication from paper SPE 71739, first presentedat the 2001 SPE Annual Technical Conference and Exhibition, New Orleans, 30 Septem-ber–3 October. Original manuscript received for review 18 November 2001. Revised manu-script received 8 January 2004. Paper peer approved 11 February 2004.

143April 2004 SPE Reservoir Evaluation & Engineering

alize the numerical simulations of mud-filtrate invasion in theform of spatial cross sections of water saturation. Our study con-siders the influence of mud composition, in-situ fluid saturation,relative permeability, permeability anisotropy, capillary pressure,gravity segregation, and hydraulic communication between adja-cent layers. We also consider the geometrical influence imposedby horizontal and highly deviated wells. In all cases, the simulatedspatial distributions of mud-filtrate invasion substantially departfrom the standard notion of a radial sequence of pistonlike fluid-saturation fronts.

We believe that our 3D numerical simulator of mud-filtrateinvasion will be useful as a working template to integrate all theavailable wireline measurements into a consistent and effectiveinterpretation of petrophysical parameters in the vicinity of theborehole wall.11–13 This template also will be readily available forthe interpretation and inversion of multiphase-fluid-flow param-eters from pressure and flow-rate data measured with wirelineformation testers.14 We show an example of the use of such atemplate in computing the sensitivity of wireline-induction-instrument responses to petrophysical conditions of mud-filtrateinvasion in a vertical well.

Mudcake Parameters and Effective Flow Rate ofMud-Filtrate InvasionIn general, the mechanical and petrophysical properties associatedwith mudcake can be characterized with six parameters: referenceporosity, reference permeability, “pressure-up” compressibility in-dex, “pressure-down” compressibility index, zero-pressure shearstrength, and erosion friction factor.1,15–17 All these parameterscan in principle be determined from an appropriately designed testsequence of mud filtration. In addition, mud has two parameters:the filtrate viscosity and the solids content. All eight parametersare necessary to calculate the flow rate of mud-filtrate invasioninto permeable rock formations.

In the case of a static mud-filtrate invasion, the flow rate ofinvasion can be calculated from the total volume of mud-filtrateinvasion during a given period of time. The equations describingthe time evolution of both the fluid volume and the flow rate ofinvasion are given by1

V�t� = C�t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

and q�t� =dV�t�

dt=

C

2

1

�t, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

respectively, where t is time after the onset of invasion and C is aconstant determined by the eight parameters listed previously. Fig.1 is a geometrical description of the basic benchmark model con-sidered in our simulations of mud-filtrate invasion. Geometricaland fluid-flow properties associated with the various componentsof this model are summarized in Table 1. Mud properties aresummarized in Table 2. The model consists of one central layer ofthickness equal to 0.61 m and permeability equal to 300 md, bor-dered by two layers of thickness equal to 0.61 m and permeability

equal to 100 md. Impermeable slabs of infinite thickness terminatethe model above and below the three permeable horizontal layers.

A finite-difference invasion simulator, referred to here asINVADE, was developed on the basis of the solution of fluid-flowpartial-differential equations and boundary conditions for immis-cible radial flow and coupled mudcake growth. INVADE was builtas a specialized version of UTCHEM. In the process of invasion,the overbalance pressure is maintained as a boundary condition.The flow rate of mud filtrate across the wellbore is calculated fromthe petrophysical properties of the formation coupled with themodel of mudcake growth. An additional feature of INVADE isthat the simulations can take into account the process of salt mix-ing between mud filtrate and connate water. This feature becomesextremely important in the assessment of wireline electrical resis-tivity logs. Fig. 2 is a plot of the evolution of the flow-rate functionduring 2 days after the onset of invasion. In this figure, the bore-hole is assumed vertical, and flow-rate functions are calculated foreach of the horizontal layers shown in Fig. 1. The two curvesexhibit the expected behavior of a monotonically decreasing flowrate with increasing time caused by progressive mudcake thicken-ing and hardening. As shown in Fig. 3, the volume of mud-filtrateinvasion per formation thickness reaches 0.12 m3/m for each layer.The dash-dot curve shown in Fig. 2 describes an equivalent con-stant-flow-rate function responsible for the same accumulated vol-

Fig. 1—Geometrical description of the three-layer reservoirmodel and borehole environment considered in this paper. Inthe figure, Rw is the wellbore radius. Tables 1 and 2 summarizethe petrophysical and mud properties assumed for this model.

144 April 2004 SPE Reservoir Evaluation & Engineering

ume of invading fluid as that of the variable function during the2-day interval. The equivalent constant flow rate is calculated withthe equation

qe =

�t1

t2

q�t�dt

t2 − t1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

where qe is the equivalent constant flow rate, q(t) is the flow rateof invasion, t is time, t1 is invasion start time, and t2 is invasionstop time. In this case, the equivalent constant flow rate is equal to0.06 m3/d/m.

In the remaining sections of this paper, we make use of theflow-rate functions shown in Fig. 2 to perform a sensitivity studyof mud-filtrate invasion. A slight variation of these functions willbe introduced for the case of horizontal wells. Additional numeri-cal simulations for specific mud properties and mud-circulationconditions will require the estimation of a different set of flow-ratefunctions from the corresponding six mud parameters introducedby Dewan and Chenevert.1

Checks of Accuracy and Internal Consistency ofthe Simulation AlgorithmUnless otherwise noted, our sensitivity studies are based on theassumption of (a) a freshwater-based mud, (b) a borehole radius of0.1 m, and (c) in-situ fluid saturation consisting of oil and irre-ducible water. Nominal relative permeability and capillary pres-sure curves used in our study to characterize the three permeablelayers of Fig. 1 are shown in Figs. 4 through 6. Notice that werestrict ourselves to water/oil relative permeability curves becauseof our assumption of water-based mud. In the subsections to fol-low, we consider a number of geometrical and petrophysical varia-tions to the basic model shown in Fig. 1, and we evaluate theirinfluence on the mud-filtrate-invasion process. Simulations areperformed that make use of the effective flow rates of mud-filtrateinvasion shown in Fig. 2. These flow rates are entered intoUTCHEM to perform the numerical simulations in a way similarto that of a standard water-injection process.

2D Grid Validation for Vertical-Well Models. We performedextensive numerical checks to verify the accuracy and internal

Fig. 3—Time evolution of the accumulated volume of mud fil-trate during the process of invasion. The two curves identifyinvasion volumes across the horizontal layers shown in Fig. 1.

Fig. 4—Nominal set of oil/water relative permeability curvesused in the sensitivity analysis of mud-filtrate invasion. The solidand dashed curves describe relative permeabilities as a functionof water saturation for water and oil fractions, respectively.

Fig. 5—Alternative set of oil/water relative permeability curves.The solid and dashed curves describe relative permeabilities asa function of water saturation for water and oil fractions, re-spectively (compare to Fig. 4).

Fig. 2—Time evolution of the flow rate of mud-filtrate invasion.Comparison between constant and monotonically decreasingrates of mud-filtrate invasion across the horizontal layersshown in Fig. 1.


consistency of the numerical simulations. The first test consisted ofperforming a study of the convergence of the simulations withrespect to a decrease in the step size used in the spatial-discretization scheme. Convergence of the ensuing numerical re-sults was observed, with a decrease in the radial step size used inthe finite-difference mesh. On the basis of these results, weadopted a radial logarithmic discretization scheme consisting of 60radial grid steps. UTCHEM treats the outer boundary of the radialgrid as a flow boundary. Therefore, the radial location of the outerboundary will not affect the simulation results provided that theinvasion front is within that boundary. In the numerical simula-tions considered in this paper, the outer boundary is located 6.1 maway from the borehole wall.

3D Grid Validation for Horizontal-Well Models. A special dis-cretization scheme was necessary to simulate mud-filtrate invasionin horizontal wells. The procedure we followed consisted of con-structing 3D Cartesian grids with relatively small spatial steps inthe vicinity of the borehole wall. In addition, nonhomogeneous

normal-flow boundary conditions were enforced on grid cells in-terfacing the borehole wall. This allowed us to impose an inte-grated normal-flow condition consistent with the prescribed flowrate for mud-filtrate invasion. We validated such an approach byperforming simulations of invasion into a homogeneous and iso-tropic 300-md formation and by disregarding the gravity effects ofthe flow from the borehole into the formation. Numerical simula-tions for this model were then compared against an equivalentvertical-borehole model calculated exclusively with a 2D grid. Fig.7 shows the results of such a comparison. The two curves in thatfigure describe water-saturation profiles simulated radially awayfrom the borehole wall for hypothetical vertical and horizontalwells, respectively, assuming the same rock formation properties.The corresponding radial profiles of water pressure are shown inFig. 8. Despite some discrepancies between the two simulationresults, especially in the radial profiles of water pressure (the maxi-mum saturation difference is 0.03, and the maximum pressuredifference is 0.2 psi), the 3D simulations performed with a hori-zontal well exhibit a high degree of similarity with the 2D simu-lations performed with the equivalent vertical well, thus givingcredence to our 3D spatial-discretization approach.

Numerical Simulations Performed With ECLIPSE-100*. Anadditional assessment of the numerical accuracy and internal con-sistency of our 3D spatial discretization was performed with thecommercial simulator ECLIPSE-100. This study required the con-struction of 2D and 3D Cartesian finite-difference grids for inputinto ECLIPSE-100, similar to the UTCHEM grids described pre-viously. Saturation and pressure simulations obtained withECLIPSE-100 for the cases of a vertical and a horizontal well areshown in Figs. 9 and 10, respectively. As with the numericalsimulations performed with UTCHEM, the radial profiles of waterpressure exhibit only minor discrepancies, whereas the water-saturation profiles tend to separate with increasing values of dis-tance away from the borehole wall. We observe a maximum satu-ration difference of 0.03 and a maximum pressure difference of 0.4psi. More importantly, the ECLIPSE-100 results overlap extremelywell with the corresponding UTCHEM results, hence giving fur-ther credence to our numerical simulation approach. We remarkalso that the availability of a 3D simulation model provides theadded flexibility to simulate spatial variations of the flow rate ofmud filtrate along the borehole wall.3 Such an option is exploredin a subsequent section of this paper.

* Mark of Geoquest, Schlumberger, Abingdon, U.K.

Fig. 6—Nominal capillary pressure curve used in the sensitivityanalysis of mud-filtrate invasion.

Fig. 7—Comparison of the numerical simulations of water satu-ration performed with UTCHEM for equivalent horizontal andvertical wells. The curves describe radial profiles of water satu-ration away from the borehole wall assuming a homogeneousformation and a time of invasion of 2 days.

Fig. 8—Comparison of the numerical simulations of water pres-sure performed with UTCHEM for equivalent horizontal and ver-tical wells. The curves describe radial profiles of water satura-tion away from the borehole wall assuming a homogeneousformation and a time of invasion of 2 days.


Numerical Sensitivity Studies ofMud-Filtrate InvasionThe simulation results presented in this section consider severalvariations about the basic benchmark geometrical model describedin Fig. 1. We assume a mud composed of fresh water and rockformations saturated with oil and irreducible water (see Table 1).Results of the simulations of mud-filtrate invasion are shown in theform of spatial cross sections of water saturation away from theborehole wall. Table 3 is a summary of the test cases consideredin the numerical simulation of the process of mud-filtrate invasion.

Sensitivity Studies With a Vertical-Well Model. Case 1: Con-stant Rate of Mud-Filtrate Invasion. Fig. 11 shows the corre-sponding cross section (radial vs. vertical directions) of water satu-ration simulated under the assumption of a constant-flow-ratefunction with 2 days’ invasion (see Fig. 2). Moreover, the simu-lations were performed using the relative permeability and capil-lary pressure curves shown in Figs. 4 and 6, respectively. Hydrau-lic communication in the vertical direction (vertical crossflow) wasallowed between adjacent layers.

Case 2: Monotonically Decreasing Rate of Mud-Filtrate In-vasion. This test case differs from Case 1 in that a variable rate ofmud-filtrate invasion was enforced in the simulation of invasion.

Fig. 12 shows the corresponding cross section of water saturation.The distribution of water saturation exhibits a smoother spatialdecline away from the borehole wall than is shown in Fig. 11.Unless otherwise noted, we adopt the use of a variable rate ofmud-filtrate invasion for the simulations described in subsequenttest cases.

Case 3: Enhanced Vertical Permeability. We consider thevertical permeability one-tenth of the base horizontal permeability(kV � 0.1 kH). A small value of kV results in a decreased rate ofcrossflow between adjacent layers. This effect can be identifiedfrom the cross section shown in Fig. 13, especially when comparedto the cross section in Fig. 12.

Case 4: Sensitivity to a Perturbation in the Water/Oil RelativePermeability Curves. We maintain the same model conditionsdescribed for Case 2 except that a change is made in the oil/waterrelative permeability curves entered into the simulations. The al-ternative set of oil/water relative permeability curves is shown inFig. 5. Fig. 14 displays the ensuing cross section of water satura-tion. A marked enhancement in the radial extent of invasion isobserved as a result of the change in relative permeability curves.

Fig. 9—Comparison of the numerical simulations of water satu-ration performed with ECLIPSE-100 for equivalent horizontaland vertical wells. The curves describe radial profiles of watersaturation away from the borehole wall assuming a homoge-neous formation and a time of invasion of 2 days.

Fig. 10—Comparison of the numerical simulations of waterpressure performed with ECLIPSE-100 for equivalent horizontaland vertical wells. The curves describe radial profiles of watersaturation away from the borehole wall assuming a homoge-neous formation and a time of invasion of 2 days.

Fig. 11—Case 1: spatial cross section of water saturation. Ex-ample of a vertical well and a time-constant rate of mud-filtrateinvasion. The time of invasion is 2 days.


Case 5: Sensitivity to Null Capillary Pressure. We return toCase 2 to study the influence of capillary pressure on the timeevolution of the invasion process. Fig. 15 shows the cross sectionof water saturation obtained assuming that the rock’s capillarypressure is zero for all values of water saturation. This cross sec-tion exhibits a relatively sharp transition between the water and oilphases saturating the permeable formation and is reminiscent ofthe pistonlike saturation profiles often assumed in well-log inter-pretation. Moreover, the radial extent of the invasion process isrelatively shallow, and there is almost no evidence of crossflowbetween adjacent layers.

Sensitivity Studies With a Horizontal-Well Model. Case 6:Homogeneous and Isotropic Formation. As explained earlier,the numerical simulation of mud-filtrate invasion in horizontal anddeviated wells requires a 3D spatial-discretization scheme. Fig. 16is a cross section of water saturation resulting from the simulationof filtrate invasion in a horizontal well, assuming density for waterand oil equal to 1 and 0.85 g/cm3, respectively, and a time ofinvasion of 2 days. This cross section is taken to coincide with aplane perpendicular to the borehole axis. Because of symmetry,

only one half of such a plane is shown in Fig. 16. The invaded rockformation is assumed to be homogeneous and isotropic, but itspermeability is not high enough to exhibit gravity-segregation ef-fects on the distribution of fluids around the borehole wall. Weremark that the cross section shown in Fig. 16 exhibits a highdegree of azimuthal symmetry around the borehole axis.

Case 7: Angle-Dependent Rate of Mud-Filtrate Invasion. De-pending on the rate of mud circulation, and because of gravityforces, mudcake has a known tendency to thicken more rapidly onthe “low” side of a horizontal wellbore than elsewhere on its wall.It is therefore expected that the rate of mud filtration will vary asa function of the angle around the borehole wall. We have made anattempt to model this behavior by introducing the angle-dependentrate of filtrate invasion shown in Fig. 17. The borehole is dividedinto 28 angular segments, with the flow rate for each angularsegment given by

q��,t� =1

N� q�t� + sin ��

2� � �

90°− 1�� q� , . . . . . . . . . . . . (4)

Fig. 13—Case 3: spatial cross section of water saturation. Ex-ample of a vertical well and a rock formation exhibiting a re-duced vertical permeability. The variables kr and kz indicateradial and vertical permeabilities, respectively, and the time ofinvasion is 2 days.

Fig. 14—Case 4: spatial cross section of water saturation. Ex-ample of a vertical well and of the use of the alternative set ofoil/water relative permeability curves shown in Fig. 5. The timeof invasion is 2 days.

Fig. 15—Case 5: spatial cross section of water saturation. Ex-ample of a vertical well and of a zero capillary pressure. Thetime of invasion is 2 days.

Fig. 12—Case 2: spatial cross section of water saturation. Ex-ample of a vertical well and a monotonically decreasing rate ofmud-filtrate invasion. The time of invasion is 2 days.


where � is the angle (degrees) between the borehole segment andthe downward direction, t is time, N is the total number of angularsegments, q(t) is the flow rate of invasion, q(�,t) is the flow rate forthe angular segment with angle equal to �, and q� is the flow-ratedifference between the 180° and 90° segments. This dependenceon vertical location is, of course, to be superimposed to the as-sumed time behavior shown in Fig. 2.

The corresponding simulation of mud-filtrate invasion yields across section of water saturation slightly different from that shownfor Case 6. A plot of the difference between the distributions ofwater saturation with and without the inclusion of an angle-dependent flow rate is shown in Fig. 18. As expected, we observeboth an excess and a deficit of water saturation above and belowthe borehole, respectively, as a consequence of the angle-dependent rate of invasion described in Eq. 4.

Case 8: A High-Permeability Layer Above the Well. A hori-zontal slab of permeability equal to 3,000 md is positioned just

above the horizontal well, over the penetrated rock formation. Thecontrast of permeability between the drilled formation and the slabis 1:10. Fig. 19 shows the corresponding cross section of watersaturation resulting from the invasion process. This cross sectionshows a clear tendency of the mud filtrate to flow toward thehigh-permeability slab despite the force of gravity.

Sensitivity Studies With a Deviated-Well Model. Case 9: BasicThree-Layer Model. We return to the basic three-layer modelshown in Fig. 1. In this case, the well is assumed to penetrate thelayered formations at an angle of 45°. Fig. 20 is a cross sectiontaken along a vertical plane through the axis of the borehole thatshows the simulated distribution of water saturation resulting frommud-filtrate invasion. This cross section clearly illustrates the in-fluence of the most-permeable layer as well as that of the existingdegree of hydraulic communication (crossflow) between adja-cent layers.

Case 10: Enhanced Horizontal Permeability. A slight varia-tion of Case 9 is pursued here to quantify the sensitivity of the

Fig. 17—Angle-dependent rate of mud-filtrate invasion in a hori-zontal well. This flow-rate curve is used to study the effect ofvariable-size mudcake buildup around the wall of a horizontalborehole. Note that a maximum flow rate occurs at the highestvertical point along the borehole wall.

Fig. 18—Case 7: spatial cross section of the difference in watersaturation between the cases of an angle-dependent and anangle-independent rate of mud-filtrate invasion. Example of ahorizontal well and a homogeneous and isotropic oil-bearingformation invaded with a water-based mud. The density of waterand oil are equal to 1 and 0.85 g/cm3, respectively, and the timeof invasion is 2 days.

Fig. 16—Case 6: spatial cross section of water saturation. Ex-ample of a horizontal well and a homogeneous and isotropicoil-bearing formation invaded with a water-based mud. The den-sity of water and oil are equal to 1 and 0.85 g/cm3, respectively,and the time of invasion is 2 days.

Fig. 19—Case 8: spatial cross section of water saturation. Ex-ample of a horizontal well and a high-permeability layer abovethe well. The time of invasion is 2 days.


invasion process to a 10:1 enhancement in the horizontal perme-ability of the central layer. We observe a highly asymmetric spatialdistribution of water saturation that develops as a result ofthe enhancement in horizontal permeability. The simulation re-sults, shown in the cross section of Fig. 21, significantly departfrom those of the isotropic case displayed in Fig. 20. We alsoobserve a substantial reduction of the effect of crossflow betweenadjacent layers.

Case 11: Multilayer Model. A more general deviated-wellmodel was constructed with a sequence of six horizontal layers.Different values of permeability, in the range of 100 to 800 md,were assigned to each of the layers to produce a highly structuredmodel of mud-filtrate invasion. Fig. 22 shows the correspondingcross section of water saturation simulated with the same set ofrelative permeability and capillary pressure curves for all thelayers. A uniform rate of mud-filtrate invasion also was assumedin the simulations irrespective of the permeabilities of the indi-vidual layers. Hydraulic communication between adjacent layerswas enforced while performing the numerical simulations. Thecross section in Fig. 22 sheds considerable light on the range ofvertical variability one could expect in practical cases of fluidinvasion into a vertically heterogeneous hydrocarbon reservoir.Most wireline-interpretation techniques dealing with deviatedwells fail to incorporate this level of geometrical complexity andhence can be subject to sizable errors in their estimations of in-situpetrophysical parameters.

Sensitivity of Wireline-Logging InstrumentResponse to the Spatial Distribution ofWater SaturationAssuming clay-free porous rock formations, Archie’s equationscan be readily used to transform the distributions of water satura-tion described previously into equivalent distributions of electricalresistivity. According to Archie, the bulk resistivity of a rock, Rt,is related to the resistivity of the brine in the rock, Rw, its porosity,�, and its water saturation, Sw, by

Rt =a � Rw

�m � Swn

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

where a is the tortuosity/cementation factor and m and n are thecementation and saturation exponents, respectively. The resistivityof connate water included in this formula can be calculated fromthe water’s salinity and temperature.

We have made a simplistic attempt to perform one such trans-formation. Our aim was to quantify whether a commercial induc-tion-logging instrument could capture some of the features of themud-filtration models described in the previous sections. Similaranalyses could be performed in conjunction with other wirelinetools (e.g., sonic, nuclear, formation tester, etc.); however, theexample of an induction tool seems ideal for the basic sensitivitystudy pursued in this paper. In related publications, Semmelbeck

and Holditch,11 Phelps,4 and Ramakrishnan and Wilkinson12 havepresented an in-depth analysis of the relationship borne by 1Dmud-filtrate-invasion models and array-induction measurements.

The spatial distributions for water saturation and salt concen-tration are obtained directly from the INVADE simulation results.In turn, salt concentrations are converted into equivalent values ofconnate water resistivity, Rw, using the formula18

Rw�r�� = �0.0123 +3647.5

Cw�r��0.955� 82

1.8T + 39, . . . . . . . . . . . . . . . . . (6)

where T is temperature measured in degrees Centigrade, Cw is saltconcentration measured in ppm, and r� is the location of the obser-vation point.

Figs. 23 and 24 are spatial cross sections of the simulateddistributions of salt concentration and formation resistivity, respec-tively, for Case 9. As described in Tables 1 and 2, the salt con-centrations in the mud filtrate and in the formation are assumedequal to 43,900 ppm and 102,500 ppm, respectively. The processof salt mixing causes the electrical resistivity of connate water toexperience considerable spatial variations in the vicinity of theborehole wall.

For completeness, Figs. 25 and 26 show the simulated crosssections of salt concentration and formation resistivity, respec-tively, for Case 10. We remark that the spatial distribution ofsalt concentration has a considerable influence on the spatial dis-tribution of electrical resistivity. This effect is often neglectedin the estimation of in-situ hydrocarbon saturation by way ofArchie’s equations. Parenthetically, a case study reported by Borahet al.19 makes use of the time-lapse physics of 1D mud-filtrate

Fig. 21—Case 10: spatial cross section of water saturation. Ex-ample of a deviated well and of an enhanced horizontal perme-ability. The angle of deviation is 45°, and the time of invasion is2 days.

Fig. 22—Case 11: spatial cross section of water saturation. Ex-ample of a deviated well intersected by six horizontal layers.Hydraulic communication exists between adjacent layers. Theangle of deviation is 45°, and the time of invasion is 2 days.

Fig. 20—Case 9: spatial cross section of water saturation. Ex-ample of a deviated well. The angle of deviation is 45°, and thetime of invasion is 2 days.


invasion to jointly interpret induction and nuclear logs acquired ina vertical well.

Sensitivity Studies of Time-Lapse Behavior. We study the time-lapse behavior of the process of invasion using as an example thecase of one vertical well and one permeable layer shouldered byimpermeable shale layers. Petrophysical and fluid-flow parametersassociated with the various components of this time-lapse case aresummarized in Table 4. A value of 43,900 ppm was assumed forthe salt concentration in the mud. This value of salt concentrationis equivalent to an electrical resistivity of 0.1 ohm-m at formationtemperature. The thickness of the formation was assumed to be 1.8m, and the salt concentration of connate water was made equal to102,500 ppm (equivalent to an electrical resistivity of 80 ohm-m).Figs. 27 and 28 are radial profiles away from the borehole wall ofthe cross sections of water saturation and salt concentration, re-spectively, simulated as a function of time after the onset of inva-sion, from 0.5 to 2 days. The radial profiles are taken through thecenter of the permeable layer. Fig. 29 shows the radial profile ofelectrical resistivity calculated from the radial profiles of watersaturation and salt concentration displayed in Figs. 27 and 28,respectively. At initial conditions, the resistivity of the formation is80 ohm-m. After 2 days of invasion, the invasion front has ad-vanced to 0.95 m away from the borehole wall, and the resistivityfor the invaded zone is approximately 2 to 3 ohm-m.

The spatial cross sections of electrical resistivity were used tosimulate the response of a dual-induction borehole-logging instru-ment (DIT-B*). Plots of the numerically simulated wireline dual-induction logs are displayed in Fig. 30 as functions of depth (mea-sured along the axis of the well) and for various times of invasion.

Fig. 23—Case 9: spatial cross section of salt concentration.Example of a deviated well. The angle of deviation is 45°, andthe time of invasion is 2 days.

Fig. 24—Case 9: spatial cross section of formation resistivity.Example of a deviated well. The angle of deviation is 45°, andthe time of invasion is 2 days. Electrical resistivities are dis-played with a logarithmic scale.

Fig. 25—Case 10: spatial cross section of salt concentration.Example of a deviated well and of an enhanced horizontal per-meability. The angle of deviation is 45°, and the time of invasionis 2 days.

Fig. 26—Case 10: spatial cross section of formation resistivity.Example of a deviated well and an enhanced horizontal perme-ability. The angle of deviation is 45°, and the time of invasionis 2 days. Electrical resistivities are displayed with a logarith-mic scale.


The deep induction log (ILD)* is sensitive to a deeper zone ofinvestigation than the medium induction log (ILM)* and henceshows a higher resistivity reading for the same invasion time. Astime progresses, the resistivity readings for both ILM and ILDdecrease because of the advancing column of mud filtrate. After0.5 days of invasion, the resistivity readings for ILM and ILD are7.3 and 8.9 ohm-m, respectively. At 2 days of invasion, the resis-tivity readings for ILM and ILD decrease to 3.5 and 4.4 ohm-m,respectively. This exercise shows that induction-instrument read-ings can experience measurable variations before the process ofinvasion comes to a halt. It is also evident from Fig. 30 thataccurate assessment of water saturation from deep induction logsfor the example under consideration would require a sizable cor-rection for the effect of invasion.

Pistonlike saturation/resistivity radial profiles of invasion areoften assumed in the interpretation of borehole induction data. Thesimulation examples described in this paper show that the combi-nation of water-saturation and salt-concentration profiles seldomleads to a pistonlike radial profile of electrical resistivity. Accurateinterpretation of borehole induction data requires consideration ofthe time-dependent nature of mud-filtrate invasion and salt mixing.In an effort to illustrate this significant point, Fig. 31 shows anexample of formation resistivities assuming pistonlike invasionprofiles for both water saturation and salt concentration as a func-tion of time of invasion. The electrical resistivity is 2.0 ohm-m inthe near-wellbore region and 80 ohm-m in the virgin zone. Simu-lation of dual-induction instrument readings was performed for theradial resistivity profiles shown in Fig. 31, and the results were

* Mark of Schlumberger Wireline Services, Houston.

Fig. 28—Time-lapse case: radial profiles of salt concentration.Example of a vertical well and a single permeable layer. Thecurves indicate salt concentration in the radial direction acrossthe center of the permeable layer at various times after the on-set of mud-filtrate invasion. The corresponding radial distribu-tions of water saturation are shown in Fig. 27.

Fig. 29—Time-lapse case: radial profiles of electrical resistivity.Electrical resistivities were calculated by use of Archie’s lawfrom the corresponding radial profiles of water saturation andsalt concentration described in Figs. 27 and 28, respectively.

Fig. 30—Time-lapse case: numerically simulated wireline logsof dual induction (medium, ILM, and deep sensing, ILD) as afunction of depth. The simulations were performed from thespatial cross sections of formation resistivity described graphi-cally in Fig. 29. Several curves are used to identify the simulatedinduction logs for different times after the onset of invasion.

Fig. 27—Time-lapse case: radial profiles of water saturation.Example of a vertical well and a single permeable layer. Thecurves indicate water saturation in the radial direction acrossthe center of the permeable layer at various times after the on-set of mud-filtrate invasion.


compared to the dual-induction readings shown in Fig. 30 (corre-sponding to radial invasion profiles simulated with INVADE).This comparison is shown in Fig. 32 in the form of maximumresistivity readings as a function of time of invasion for both ILMand ILD logs. Pistonlike invasion profiles entail higher resistivityreadings than those associated with invasion profiles simulatedwith INVADE. In this particular case, the assumption of a piston-like invasion profile leads to resistivity errors as high as 52%and, consequently, to significant errors in the estimation of in-situwater saturation.

ConclusionsWe have introduced a procedure to simulate numerically complexcases of mud-filtrate invasion in overbalanced vertical, horizontal,and highly deviated wells. This procedure consisted of (a) devel-oping an effective time-dependent flow-rate function that couldcapture the effect of mudcake buildup and (b) adapting an existing3D multiphase reservoir simulator to enforce boundary, initial, andsource conditions specific to the physics of mud-filtrate invasion.Tests of accuracy and internal consistency lent excellent credenceto our 3D simulations of mud-filtrate invasion in horizontal wells.

A systematic sensitivity analysis was carried out to quantify theinfluence of several petrophysical and geometrical parameters onthe spatial distribution of mud-filtrate invasion away from theborehole wall. The sensitivity studies were summarized with crosssections of the spatial distribution of water saturation resultingfrom invasion. All our sensitivity studies adopted the referencemodel of a freshwater-based mud and porous rock formations satu-rated with both oil and irreducible water. The sensitivity studiesshowed that the mud-filtrate-invasion process is affected by thegeometry of the permeable beds and by petrophysical parameterssuch as relative permeability, permeability anisotropy, capillarypressure, gravity segregation, effective porosity, and hydrauliccommunication (crossflow) between adjacent layers.

The simulations reported in this paper shed new light on theinterpretation of wireline measurements in terms of petrophysicalparameters in the vicinity of the borehole wall. It was found thatthe spatial distribution of salt concentration resulting from theprocess of mud-filtrate invasion could significantly distort the spa-tial distribution of electrical resistivity. In turn, unaccounted spa-tial variations of salt concentration could cause substantial biasesin the estimation of in-situ hydrocarbon saturation.

We have found that a customary notion that regards invasion asa radial sequence of pistonlike invasion is the exception rather thanthe norm. It is envisioned that 3D petrophysical templates based onaccurate and comprehensive fluid-flow models of filtrate invasion

will provide modern quantitative ways to integrate a large varietyof wireline measurements into estimates of rock formation prop-erties. A basic example of the application of such a template wasshown here for the interpretation of wireline measurements ofelectromagnetic induction.

Nomenclaturea � Archie’s tortuosity/cementation factorC � static filtration constant

Cw � salt concentration measured in ppmk � formation permeability

kH � horizontal permeabilitykro � oil relative permeabilitykrw � water relative permeabilitykV � vertical permeabilitym � Archie’s cementation exponentN � total number of angular borehole segmentsn � Archie’s saturation exponent

q(�,t) � flow rate for a segment with an angle of �q� � flow-rate difference between the 180° segment and the

90° segmentqe � equivalent constant flow rate

q(t) � flow rate of mud filtrateR � radial distance away from the wellbore, such that R�0

at the wellboreRt � bulk resistivity of a medium

Rw � electrical resistivity of connate waterr� � observation point

Sw � water saturationt � time

t1 � starting time for invasiont2 � stop time for invasionT � temperature measured in degrees CentigradeV � total filtrate volumeX � location in the X-directionZ � vertical distance� � angle (degree) between borehole segment and

downward direction� � porosity

AcknowledgmentsWe would like to express our deepest appreciation to Baker Atlas,Halliburton, and Schlumberger for funding this work through the

Fig. 32—Graphical comparison of dual-induction resistivityreadings calculated from INVADE simulation results (Fig. 29)against those calculated from the pistonlike saturation profilesshown in Fig. 31. The plots show maximum dual-induction re-sistivity readings, ILM and ILD, as a function of time of invasion.

Fig. 31—Time-lapse case: radial profiles of electrical resistivityassuming pistonlike invasion. Electrical resistivities were cal-culated by use of Archie’s law assuming a pistonlike fluid-saturation front in the process of mud-filtrate invasion.


U. of Texas at Austin’s Center of Excellence in Formation Evalu-ation. Our gratitude goes to Guozhong Gao for simulating theinduction-resistivity logs shown in this paper. We are obliged toDavid Kennedy and Alberto Mezzatesta for their thorough reviewof the first version of the manuscript and for their constructivetechnical and editorial criticism. Their feedback has significantlyimproved the quality of the final paper.

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SI Metric Conversion Factorscp × 1.0* E – 03 � Pa�s°F (°F – 32)/1.8 � °Cft × 3.048* E – 01 � m

ft3 × 2.831 685 E – 02 � m3

in. × 2.54* E + 00 � cmpsi × 6.894 757 E + 00 � kPa

*Conversion factor is exact.

Jianghui Wu is a graduate student research assistant and PhDcandidate in the Dept. of Petroleum and Geosystems Engi-neering at the U. of Texas at Austin. Formerly, he was a reservoirengineer working for CNPC. His research interests include for-mation evaluation, reservoir engineering, and simulation. Heholds BS and MS degrees from the U. of Petroleum in China.Carlos Torres-Verdı́n has been an assistant professor in theDept. of Petroleum and Geosystems Engineering at the U. ofTexas at Austin, where he conducts research in formationevaluation and integrated reservoir characterization, since1999. From 1991 to 1997, he held the position of Research Sci-entist with Schlumberger-Doll Research; from 1997 to 1999, hewas Reservoir Specialist and Technology Champion with YPF(Buenos Aires). Torres-Verdı́n holds a PhD degree in engineer-ing geoscience from the U. of California, Berkeley. He is cur-rently a member of the SPEJ Editorial Board. Kamy Sepehrnooriis the Bank of America Centennial Professor in the Dept. ofPetroleum and Geosystems Engineering at the U. of Texas atAustin. His teaching and research interests include computa-tional methods, reservoir simulation, parallel computations, ap-plied mathematics, and enhanced oil recovery. Sepehrnooriholds a PhD degree in petroleum engineering from the U. ofTexas at Austin. Mojdeh Delshad is a research engineer withthe Center for Petroleum and Geosystems Engineering at theU. of Texas at Austin. Her research interests are in petrophysicalproperty modeling, enhanced oil recovery, reservoir engineer-ing, simulation, and groundwater modeling and remediation.Delshad holds MS and PhD degrees in petroleum engineeringfrom the U. of Texas at Austin. She is a member of the SPEEditorial Review Committee.


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