Combined simulation and inversion of SP and resistivity logs for theestimation of connate-water resistivity and Archie’s cementation exponent

Jesús M. Salazar1, Gong Li Wang1, Carlos Torres-Verdín1, and Hee Jae Lee1

ABSTRACT

Knowledge of initial water saturation is necessary to estimateoriginal hydrocarbon in place. A reliable assessment of thispetrophysical property is possiblewhen rock-coremeasurementsofArchie’s parameters, such as saturation exponent n and cemen-tation exponent m, are available. In addition, chemical analysisof formation water is necessary to measure connate-water resis-tivityRw. Suchmeasurements are seldomavailable inmost appli-cations; if they are available, their reliability may be question-able.We describe a new inversion method to estimate Rw andAr-chie’s cementation exponent from the combined use of boreholespontaneous-potential �SP� and raw array-induction resistivitymeasurements acquired in water-bearing depth intervals. Com-bined inversion of resistivity and SPmeasurements is performedassuming a piston-like invasion profile. In so doing, the reservoir

is divided into petrophysical layers to account for vertical hetero-geneities. Inversion products are values of invaded and virginformation resistivity, radius of invasion, and static spontaneouspotential �SSP�. Connate-water resistivity is calculated by as-suming membrane and diffusion potentials as the main contribu-tors to the SSP.Archie’s or dual-water equations enable the esti-mation ofm. The newcombined estimationmethod has been suc-cessfully applied to a data set acquired in a clastic formation.Data were acquired in a high permeability and moderately high-salt-concentration reservoir. Values of Rw and m yielded by theinversion are consistent with those obtained with a traditional in-terpretationmethod, thereby confirming the reliability of the esti-mation. The method is an efficient, rigorous alternative to con-ventional interpretation techniques for performing petrophysicalanalysis of exploratory and appraisal wells wherein rock-coremeasurementsmay not be available.

INTRODUCTION

Initial water saturation in hydrocarbon reservoirs has an enor-mous impact on calculating and producing original oil in place.When laboratorymeasurements �core,water analysis, etc.� are avail-able, this variable is properly constrained. However, such measure-ments are not always available, and even if they are, their reliabilitymay be questionable. Therefore, a strong need exists for alternativemethods to estimate those parameters before calculating initial wa-ter saturation.Two of the main parameters needed to calculate water saturation

are connate-water resistivity Rw and the cementation exponent m,which can be obtained from connate-water analysis and special coreanalysis, respectively. Core measurements are often expensive be-cause they involve the cost of extracting the core sample and subse-quent laboratorywork.Moreover, measurements ofwater resistivity

are difficult because of the need to acquire connate-water sampleswhen wells are already in production and water injection/steamflooding has been applied to enhance production. Fluid samples tak-en byfluid-acquisition tools are often contaminatedwithmudfiltrateand/or hydrocarbon.In the early days of formation evaluation, spontaneous potential

�SP� and resistivity were the only borehole measurements availablefor interpretation to log analysts or petrophysicists �Doll, 1949�. Oneof the first physical models of SPwas developed using a resistor net-work �Segesman, 1962�, where dipole layers were simulated usingvoltage sources. Zhang and Wang �1997, 1999� developed a finite-element algorithm to simulate SP measurements using the vectorand scalar potential theory. Their algorithm successfully reproducedthe resistormodel developed by Segesman.In this paper, we simulate SP measurements using Zhang and

Wang’s algorithm and resistivity measurements with the numerical-

Manuscript received by the Editor 16 June 2007; revisedmanuscript received 27 September 2007; published online 18April 2008.1University of Texas atAustin, Department of Petroleum and Geosystems Engineering,Austin, Texas, U.S.A. E-mail: salazarjm@mail.utexas.edu; glwang@

mail.utexas.edu; cverdin@mail.utexas.edu; heejae@mail.utexas.edu.© 2008 Society of ExplorationGeophysicists.All rights reserved.

GEOPHYSICS,VOL. 73, NO. 3 �MAY-JUNE2008�; P. E107–E114, 7 FIGS., 3TABLES.10.1190/1.2890408

E107

mode matching �NMM� method �Chew et al., 1984; Zhang et al.,1999�. We use raw borehole-corrected array-induction resistivity�AIT, a mark of Schlumberger� measurements and consider the ef-fect of invasion and layer thickness on both resistivity and SP. Ourobjective is to calculateRw andm for a field data set acquired in awa-ter-bearing clastic formation based on the combined inversion of SPand raw array-induction measurements. The estimation method isbased on quantitative simulations and interpretation of the measure-ments with realistic physical assumptions, not on qualitative inter-pretations. Additionally, to benchmark the technique, values of Rw

obtainedwith thismethod are compared to those obtainedwith Pick-ett plots �Pickett, 1966�. Finally, we summarize the possible applica-tions of themethod.

RESISTIVITY MODELING AND INVERSION

The purpose of resistivity modeling �forward modeling� and in-version is to estimate the invaded-zone Rxo, virgin-zone Rt resistivi-ties, and the radius of invasion rinv from raw array-induction conduc-tivity �inverse of resistivity�measurements. Figure 1 shows the sub-surfacemodel assumed in the simulations. Initially, we assume a sin-gle-layer model and a piston-like radial profile of invasion. The sys-tem is bounded at the top and bottom by shale beds �shoulders�withresistivities equal to Rshtop and Rshbot, respectively.Also, the model isradially bounded by a borehole with fluid resistivity equal to Rm andthe virgin zone, whose resistivity is one of themain inputs necessaryto estimatewater saturation.

Forward model

Induction tools measure formation conductivity � by inducinglow-frequency electric currents into the formation surrounding theborehole. We adopt a 2D axial-symmetric model. In such a model,current loop sources reside at the center of the borehole, with a mag-netic moment pointing upward; the conductivity is invariant in theazimuthal direction. Sedimentary rocks are generally not magnetic;hence, the magnetic permeability � is assumed equal to the vacuummagnetic permeability �0. As a result, the electric field comprisesonly the azimuthal component E� and varies only in the meridianplane.

Assuming there is only one source current loop, with radius rs andvertical position zs, the governing equation in cylindrical coordi-nates �r,z� forE� is given by

� �

� r

1

r

�

� rr �

� 2

� z2� k2�E��r,z;rs,zs�

� �i��0I� �r � rs�� �z � zs� , �1�

where I is the electric-current intensity, � isDirac’s delta function,�is angular frequency, k2 � i��0� , i � ��1, and the time conven-tion e�i�t is assumed, where t is time. The boundary condition isE��r,z���� � 0, where� � ��r,z��0�r� ,� �z� .We assume a piecewise-constant spatial distribution of electrical

conductivity:

� �r,z� � � 0�r�H�z1 � z� � � M�r�H�z � zM�

� m�1

M�1

� m�r��H�z � zm� � H�z � zm�1�� ,

�2�

whereH�z� is the Heaviside function and � 0 and � M are the conduc-tivities of the first and last layers, respectively. There is a total ofM � 1 horizontal layerswith boundaries z1,z2, . . . ,zM; layer conduc-tivities are given by

� m�r� � �mudH�rw � r� � � t,mH�r � rinv,m�� � xo,m�H�r � rw� � H�r � rinv,m��,m � 0,1, . . . ,M ,

where rinv is the radial length of invasion, rw is the borehole radius,�mud is the borehole conductivity, and � xo and � t are the invaded-and virgin-zone conductivities, respectively.We use the NMMmethod to solve the above 2D simulation prob-

lem �Chewet al., 1984; Zhang et al., 1999�. This algorithm combinesa 1D finite-element solution in the radial direction with an analyticalsolution in the vertical direction. When augmented with amplitudeand slope basis functions �Zhang et al., 1999�, the NMM is hundredsof times more computer efficient than either 2D finite-element or fi-nite-difference method. �One function describes the amplitude, andthe remaining function describes the slope of the electric field at nod-al points.�

Inversion

Inversion of array-induction resistivity measurements is posed astheminimization of a quadratic cost function that comprises an addi-tive term which weighs the data residuals and one additive term in-troduced to stabilize the inversion in the presence of noisy andsparsely sampled measurements. We adopt the quadratic cost func-tion, given by

C�x� �1

2� e�x� 2 � 2 x 2 , �3�

where x includes the unknown model parameters �layer-by-layer values of Rxo, Rt, and rinv�, 2 is the regularization �stabiliza-tion� parameter, and e is the vector of data residuals given bye�x� � d�x� � do. In this expression, d�x� contains the indexed nu-merically simulated measurements and do contains the correspond-

Figure 1. Single-layer subsurface model assumed in the simulationand inversion of borehole-resistivity measurements. The model as-sumes a piston-like invasion front.

E108 Salazar et al.

ing indexed field measurements �raw array-induction conductivi-ties�. For the estimation of layer resistivities, we assume that layerboundaries are known a priori.We approach the minimization of the quadratic cost function giv-

en by equation 3 using the distorted Born iterative method �DBIM��Chew and Liu, 1994�. The computation of sensitivities �entries ofthe Jacobianmatrix� is crucial to solving the nonlinearminimizationproblem by iterative linear steps.To calculate the sensitivities, we consider a generic single-trans-

mitter, single-receiver induction system. Let rT be the radius and zT

be the vertical location of the transmitter. Likewise, let rR and zR bethe radius and vertical location of the receiver, respectively. Whenapplying a perturbation to the background conductivity � 0, the cor-responding perturbation of apparent conductivity � a is given by�Zhang, 1984�

�� a � �0

��

drdzg�r,z��� �r,z� , �4�

where

g�r,z� � Re��2��2�0

2I

rG�r,z;rR,zR�G�r,z;rT,zT�� ,

is the tool constant, G�r,z;rs,zs� is the Green’s functionthat satisfies equation 1 with the right-hand side replaced with�� �r � rs�� �z � zs�, and �� � � � � 0 is a perturbation of thebackground conductivity. Substituting equation 2 into equation 4yields

�� a�r,z��� xo,m

� �rw

rinv,m

�zm

zm�1

drdzg�r,z� ,

�� a�r,z��� t,m

� �rinv,m

�zm

zm�1

drdzg�r,z� ,

�� a�r,z�� rinv,m

� �� xo,m � � t,m� �zm

zm�1

dzg�rinv,m,z� , �5�

where � xo,m and � t,m are invaded- and virgin-zone layer conductivi-ties, respectively, and rinv,m is the radius of invasion for each layer.In the above equations, we assume the formation boundary zm is

known. Initially in theDBIM, the Jacobianmatrix, whose entries aregiven by equation 5, is recalculated after updating the conductivitydistribution �ChewandLiu, 1994�. Because the induction problem isquasi-linear �Zhang, 1984�, the Green’s function is approximatedwith that of a homogeneous medium penetrated by a borehole andfixed for all iteration steps. Consequently, at each step, the Jacobianmatrix is reset by evaluating only the three integrals given by equa-tion 5. This approach represents a highly cost-effective process be-cause the repeated evaluation of g�r,z� is unnecessary. For the inver-sion, we assume that�mud and rw are known and that the conductivityof the homogeneous medium is equal to 0.01 S/m. This strategygreatly reduces the computer cost required by calculating deriva-tives.

We invoke the multiplicative regularization technique describedby Habashy andAbubakar �2004� to calculate the regularization pa-rameter included in the quadratic cost function �equation 3�with therelationship

2 � e�x� 2

�,

where � is a constant that can be determined with numerical experi-ments and, in our case, is equal to 2.0. In thick formations �over hun-dreds of feet long� containing many layers, the inversion processacts as a depthwindow sliding over the data set layer by layer. This ispossible because induction is primarily a localized measurement,namely, apparent conductivity is mostly affected by layers close tothemeasurement point whereas the effect of layers far from themea-surement point is comparatively small.

SP MODEL

The SP has four main components �Hallemburg, 1971�: the diffu-sion potential, the membrane or Nernst potential �both diffusion andmembrane are known as the electrochemical potentials�, the electro-kinetic or streaming potential, and the oxidation/reduction �redox�potential. Electrokinetic and redox potentials are negligible in bore-hole applications compared with the electrochemical potentials.Here,we assume the total potentialmeasured by a borehole SPtool issolely the sum of themembrane and diffusion potentials �Wyllie andSouthwick, 1954�.In a permeable zone at borehole conditions, the maximum differ-

ential potential �in absolute value� is known as the static SP �SSP�.The SSPat borehole conditions is measuredwith respect to the shalebaseline �Pirson, 1963� and, inmillivolts, is given by

SSP� �70.7� �460� TF

537�log�Rmfe

Rwe� , �6�

where Rwe designates the equivalent water resistivity, Rmfe is theequivalent mud-filtrate resistivity, and TF is the formation tempera-ture in degrees Fahrenheit.We use equation 6 to estimate Rwe via thecombined inversion of SPand resistivitymeasurements.

Modeling and inversion of SP

TheSSPis an electromotive force because of electric dipole layersdistributed along the borehole wall, invasion fronts, and formationboundaries. In this study, vector potential theory �Zhang andWang,1997, 1999� is used to compute SP in awater-based,mud-filled bore-hole.Accordingly, electric dipole layers that could extend to infinityare replaced with magnetic current rings located at the intersectionpoints of the borehole wall, invasion fronts, and formation bound-aries.In cylindrical coordinates �r,z�, the governing equation is given

by

1

2�� �

� r

1

r�

�V

� r�

�

� z

1

r�

�V

� z�

� ��SSP� �r � rs�� �z � zs� , �7�

where V is the current potential �Zhang andWang, 1997, 1999�, � isthe electrical conductivity of the formation, �SSP is the total potentialdifference across dipole layers, and rs and zs are the radius and verti-

SP and resistivity inversion E109

cal position, respectively, of a source in the meridian plane repre-senting a magnetic current ring in 3D space.We use the 2D finite-el-ement method to solve equation 7 with a frontal solver to expeditesolving the resulting linear system of equations.Results from resistivity inversion are used directly in the inver-

sion of SPmeasurements. In addition, we assume that mud resistivi-ty, borehole radius, and layer boundary location are a priori informa-tion determined from other measurements. Therefore, in the SP in-version, we only invert for the magnitudes of SSP. From equation 7,we realize �1� the SP response is linear with respect to the sourcemagnitude and �2� when there are many sources, the total SP re-sponse is the superposition of SP responses originating separatelyfrom all sources. Thus, one canwrite the equation for SP �SPn� as

SPn � m�1

M

�SSP,mSPnm, n � 1,2, . . . ,N , �8�

whereN is the total number of SPmeasurements,M is the total num-ber of SP sources, �SSP,m is the magnitude of the potential differencefor the mth SP source, and SPnm is the response to a unit SP sourcewith radius rs,m and vertical position zs,m, obtained by solving equa-tion 7.Equation 8 can bewritten inmatrix notation as

C�y � b , �9�

whereC� � �SPnm�N�M, y � ��SSP,m�M�1, and b � �SPn�N�1. Becausethe coefficientmatrixC� is rectangular, we use the generalized singu-lar-value-decomposition �SVD� method to solve equation 9. Ac-cordingly, the solution of the eigenvalue problem is given by

y � V����1U�Tb ,

where U��RN�M and V��RM�M are two unitary matrices formed byeigenvectors,���RM�M is a diagonalmatrix that consists of singularvalues obtained from the SVD, and the superscript T designates thetranspose of a matrix. We assume that SP sources are far from eachother; therefore, no truncation of the spectrum of singular values isnecessary, whereuponmatrixC� is invertible.

Figure 2 shows a multilayer subsurface model �heterogeneousformation� that includes three invaded layers with different radii ofinvasion and resistivities for each cylindrical and vertical layer. Theresistivity of each radial block �Rxo and Rt�, the radius of invasion,and the SSPat each layer boundary are the properties estimated withthe combined inversion of SPand resistivitymeasurements.

Comments on assumptions and limitations of thesimulation of SP measurements

The SP sources assumed in our algorithm are exactly the sameones used to construct the Schlumberger’s SP-3 chart �Schlum-berger, 1991�. That is to say, they are dipole layers that could extendto infinity from the borehole wall or from invasion fronts. Under thetheory of current potential, these dipole layers are equivalent tomany infinitesimal rings centered on the borehole axis. Physically,they are static magnetic-current rings that drive the flow of electriccurrent in both the borehole and the formation. Such infinitesimalcurrent rings and dipole layers aremathematically equivalent to eachother. However, infinitesimal current rings aremore amenable to nu-merical simulation.The dipole-layer assumption for SP sources commonly has been

used in log interpretation. Such an approach will continue to be useduntil a better assumption is introduced in the SPtheory,which shouldbe more physically complete and numerically easier to implementthan the current one.In modern induction tools, the reliability of SPmeasurements can

be compromised as a result of the short distance that exists betweenthe SP electrode and the grounded metal. However, the effect of thatdistance becomes appreciable only when formations are highly re-sistive �more than 100 �m�. The field case considered in this articlecorresponds to a very conductive formation, in which the resistivityis generally less than 20 �m, whereby the effect of short distancebetween the SPelectrode and the groundedmetal is negligible. Con-sidering that the favorable measurement condition for inductiontools is generally low-resistivity formations, it is reasonable to as-sume that the effect of short distance between the SP electrode andthe grounded metal is always negligible as long as this condition issatisfied.

ESTIMATION OF RW AND m

Field SP and raw AIT measurements as well borehole and mudproperties are used as input to the combined SP-resistivity inversion.Resistivity-inverted results are input to the SP inversion. To estimateconnate-water resistivity, the inversion is carried out in a wet-sandinterval. Once the SSP is calculated at each layer boundary from theinverted SP, the corresponding equivalent water resistivity is calcu-latedwith themaximumnegative SSPusing equation 6. Subsequent-ly, an empirical correlation from log interpretation charts �Schlum-berger, 1991; Bigelow, 1992� is used to estimateRw, namely,

Rw �Rwe � 0.131 · 10

� 1

log�TF/19.9��2�

� 0.5� Rwe � 10� 0.0426

log�TF/50.8�� . �10�

In ourmethod, we assume that formationwater is a solution of so-dium chloride �NaCl�. Therefore, equation 10 is not valid for alltypes of formation water or waters with very high salt concentration

Figure 2. Multilayer subsurface model assumed in the combinedsimulation and inversion of resistivity and SPmeasurements.

E110 Salazar et al.

�above 220,000 ppm�.Additional transformations are needed for thecase of water with ion components other than those of NaCl �Big-elow, 1992�.To estimate the cementation exponent, we initially assume a clean

�shale-free� 100% water-saturated clastic rock. Archie’s equation�Archie, 1942� is used to compute the cementation exponentwithoutspecific adjustments for the presence of shale as follows:

m �1

log��� log�Rw

Rt� , �11�

where � is the nonshale porosity taken as an average in the intervalof analysis and the tortuosity factor a �not shown in equation 11� isassumed equal to one. On the other hand, when the presence of clayis considered, we use the dual-water model for shaly sands �Clavieret al., 1984� because several of its governing parameters can be cal-culated fromwell logs �Dewan, 1983�:

m �1

log��� log�Rw

Rt�1� Sb�1�

Rw

Rb���1� , �12�

where Sb and Rb are bound-water saturation and bound-water resis-tivity �a function of shale resistivity�, respectively. Equation 12 isvalid in wet sands and reduces to equation 11 in clean sands �whereSb � 0�. Figure 3 summarizes the steps used to estimate connate-water resistivity and cementation exponent via the combined inver-sion of resistivity and SPmeasurements.

FIELD CASE STUDY

We select a formation with a water-bearing depth zone to test thedeveloped method on field measurements. The case under analysiscorresponds to a north Louisiana �U. S.� fairly clean clastic forma-

tion.The reservoir is 100%water saturatedwithmoderately high saltconcentration �more than 100,000 ppm�. Table 1 summarizes thethickness and average petrophysical properties of the formation.Figure 4 displays the well logs and estimated porosity and perme-ability for the formation under analysis. Well logs include gammaray �GR�, SP, andAIT resistivity �2-ft vertical resolution�. Porosity iscalculated from density-neutron logs via a nonlinear dual-mineral�shale-quartz� model, and permeability is calculated from porosityand irreducible water saturation using a modified Timur-Tixierequation �Balan et al., 1995�. Water saturation is not shown here be-cause the formation is completelywater saturated �Sw � 100%�.

Combined inversion

We use raw borehole-correctedAITmeasurements to perform thecombined inversion method. The formation under analysis was di-vided into three petrophysical layers or flow subunits. Layer selec-tion was based upon main resistivity changes and petrophysicalproperties �porosity-permeability�. For resistivity inversion, inputs

Table 1. Summary of average petrophysical propertiesassumed for the case study.

Property Units Values

Thickness m 17.68

Interconnected porosity fraction 0.30

Water saturation fraction 1.0

Volumetric shale concentration fraction 0.02–0.10

Permeability md�m2�

100–2000��10�14 – 2 � 10�12�

Figure 3. Flow chart describing the algorithmic steps included in thecombined simulation and inversion of resistivity and SP measure-ments to estimate Rw and m and to calculate water saturation in hy-drocarbon-bearing zones.

Figure 4. Well logs and petrophysical properties for the case study.Separating AIT resistivity curves indicates invasion of mud filtrateinto the water-bearing sand. The porosity track shows shale-correct-ed density and neutron porosity �D

sh and �Nsh, effective initial-guess

porosity �eo, and nonshale porosity �e obtained with a nonlinear

model that relates densitywith resistivity.

SP and resistivity inversion E111

are layer boundaries, borehole radius and conductivity, and rawAITconductivity data. Inversion results are values of Rxo, Rt, and rinv foreach layer, which are additional inputs to the SP inversion. Once theinversion is finished, we obtain SSPvalues for each layer boundary.Table 2 describes the inversion results for resistivity and radius of

invasion. Figure 5 shows the results of resistivity inversion and com-pares field and simulated SPmeasurements at the conclusion of theinversion. Figure 6 compares field to simulated apparent array-in-duction resistivitymeasurements calculatedwith the inverted valuesof rinv,Rxo, andRt.

Assessment of m and Rw

Equivalent water resistivity is obtained from the inverted SSPwith equation 6. Subsequently, connate-water resistivity is comput-ed at reservoir temperature using equation 10.Archie’s �equation 11for clean sands� and dual-water �equation 12 for shaly sands�modelsyield m using the updated values of Rw and Rt obtained from inver-sion and porosity from logs. Table 3 describes the inverted values ofRw andm.When we compare the values of m obtained with Archie’s equa-

tion to the one obtainedwith the dual-water equation,we see a reduc-tion of the cementation exponent. The resulting value ofm, decreas-ing with increasing shaliness in the dual-water model, depends onthe ratio between bound-water resistivity and true formation resis-tivity �Rb/Rt�. Clavier et al. �1984, their Figure 8� cite no clear ten-dency of the variations ofm as a function of shale content; measure-ments are scattered between m � 1.5 and m � 2.5 for a volumetricshale concentration between 0 and 0.50. Therefore, values ofmwithbehavior opposite to that observed in this paper are feasible.

Appraisal of Rw

To cross-validate the inverted value of connate-water resistivity,we constructed a traditional Pickett plot �Pickett, 1966� for log�Rt�versus log���, which yields a straight line whose slope is equal to�m. The ordinate at log��� � 0 yields log�Rw� �assuming a � 1�.Figure 7 displays a Pickett plot for the formation under analysis.

The value of Rw obtained from the Pickett plot is 0.037 �m at 61°C.Such a result represents a difference of 12%with respect to the value

Table 2. Summary of inversion results for the case study.

Thickness�m�

Rxo

��m�Rt

��m�rinv�m�

SSP�mV�

Shoulder 3.7586 3.7586 0 56.4128

3.66 1.8938 0.5425 0.35 �6.5244

5.79 1.1665 0.4849 0.38 �1.7396

8.23 1.0916 0.3387 0.54 �49.3705

Shoulder 4.2012 4.2012 0 —

Table 3. Calculated values of connate-water resistivity andArchie’s cementation exponent.

Property Units Value

Rw �m 0.042

Temperature °C 61

NaCl ppm 105,000

m — Archie — 1.74

m — dual water — 1.69

Volumetric shale concentration Fraction 0.10

a) b)

Figure 5. Results of the combined inversion of resistivity and SP. �a�Simulated SPobtained from the inverted SSPfromfield SPmeasure-ments. �b�Comparison of inverted Rxo and Rt to fieldAIT �2-ft verti-cal resolution�measurements.

Figure 6. Comparison of measured and simulated array-inductionlogs �2-ft vertical resolution� yielded by inverting rinv, Rxo, and Rt.The relative separation of the simulated curves agreeswell with fielddata. Because the objective of the inversion was to estimate m andRw, only three petrophysical layers were considered for the inver-sion, which in turn reduced the accuracy of thematch between simu-lations and fieldmeasurements.

E112 Salazar et al.

obtained from inversion.The relatively small difference between thePickett plot calculation and the inversion result lends credence to thereliability of our combined inversionmethod.

Applications of inversion results

Ideally, in a reservoir penetrating a water zone, the objective is touse the results of inversion �Rw andm� to estimatewater saturation inthe upper hydrocarbon zone within the same geologic formation.The method, parameters, and simulation approach used to computewater saturation, porosity, and initial guess of permeability are de-scribed by Salazar et al. �2006�. Either resistivity matching or inver-sion �Salazar et al., 2005; Salazar et al., 2006� using the physics ofmud-filtrate invasion �Alpak et al., 2003;Wu et al., 2005� is an alter-nativemethod for estimating permeability.Values obtained from inversion provide a starting point to esti-

mate water saturation and layer-by-layer permeability accurately.Correct assessment ofwater saturation leads to reliable estimation ofin-place hydrocarbon saturation. Moreover, accurate estimation ofpermeability is necessary to select perforation intervals and layersfor fluid injection and to forecast production.

CONCLUSIONS

The combined inversion of resistivity and SP measurements is areliable method to estimate Rw and m in water-bearing formations.Results obtained from the inversion were consistent with those ob-tained from Pickett plots. The difference between the estimates ob-tainedwith the twomethodswas roughly 10%.The combined inversion method is highly recommended in zones

where NaCl is the most abundant solution component and whereconnate salt concentration does not change in short depth intervals.This method of inversion works very well in high-permeability,thick formations, and when connate water exhibits medium to highconcentrations ofNaCl.In depth zones where the most abundant salt components are dif-

ferent from NaCl �e.g., calcium chloride or potassium chloride�,equation 6 is not valid. In addition, when the salt concentration ofmud filtrate and connate water is similar, the deflection of the SP

curve is marginal and, therefore, the inversion method is not recom-mended. In low-permeability formations �k�5 md�, the electroki-netic components of the SPmay become important for the total con-tribution to SSPand hence could be needed in the formulation beforeapplying our inversionmethod.In the absence of water zones, resistivity inversion can still be

used to estimate Rxo and Rt for the refined calculation of initial watersaturation. This calculation method can be useful in the petrophysi-cal assessment of exploratory and appraisal wells that are normallydevoid of core samples and laboratorymeasurements.The fact that service companies do not charge to acquire SPmea-

surements contributes to the fact that neither service companies norclients give SP the attention it deserves. Commonly, SP measure-ments consist of extremely smooth, low-quality curves thatmake in-terpretation more challenging and hence may have an erroneous im-pact on applying the inversionmethod described in this article.

ACKNOWLEDGMENTS

We are obliged to ConocoPhillips for permission to publish thefield measurements described in this article. A note of special grati-tude goes toTomBarber, BarbaraAnderson, and two anonymous re-viewers, whose constructive comments and suggestions improvedthe original manuscript. The work reported in this paper was fundedby the University of Texas atAustin’s Research Consortium on For-mation Evaluation, jointly sponsored by Anadarko, Aramco, BakerAtlas, BHPBilliton, BP, British Gas, ConocoPhillips, Chevron, ENIE&P, ExxonMobil, Halliburton Energy Services, Hydro, MarathonOil Corporation,Mexican Institute for Petroleum, Occidental Petro-leum Corporation, Petrobras, Schlumberger, Shell InternationalE&P, Statoil, TOTAL, andWeatherford.

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Archie, G. E., 1942, The electrical resistivity log as an aid in determiningsome reservoir characteristics: Petroleum Transactions, American Insti-tute ofMining,Metallurgical, and PetroleumEngineers, 146, 54–62.

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