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Estimation of Permeability andPermeability Anisotropy From

Straddle-Packer Formation-TesterMeasurements Based on the Physics of

Two-Phase Immiscible Flow and InvasionRenzo Angeles, Carlos Torres-Verdín, Hee-Jae Lee, and Faruk O. Alpak, University of Texas at Austin; and

James Sheng,* Baker Atlas

SummaryWe describe the successful application of a new method to esti-mate permeability and permeability anisotropy from transient mea-surements of pressure acquired with a wireline straddle-packerformation tester. Unlike standard algorithms used for the interpre-tation of formation-tester measurements, the method developed inthis paper incorporates the physics of two-phase immiscible flowas well as the processes of mudcake buildup and invasion.

An efficient 2D (cylindrical coordinates) implicit-pressure ex-plicit-saturation finite-difference algorithm is used to simulateboth the process of invasion and the pressure measurements ac-quired with the straddle-packer formation tester. Initial conditionsfor the simulation of formation-tester measurements are deter-mined by the spatial distributions of pressure and fluid satura-tion resulting from mud-filtrate invasion. Inversion is performedwith a Levenberg-Marquardt nonlinear minimization algorithm.Sensitivity analyses are conducted to assess nonuniqueness andthe impact of explicit assumptions made about fluid viscosity,capillary pressure, relative permeability, mudcake growth, andtime of invasion on the estimated values of permeability and per-meability anisotropy.

Applications of the inversion method to noisy synthetic mea-surements include homogeneous, anisotropic, single- and multi-layer formations for cases of low- and high-permeability rocks. Wealso study the effect of unaccounted impermeable bed boundarieson inverted formation properties. For cases where a priori infor-mation can be sufficiently constrained, our inversion methodologyprovides reliable and accurate estimates of permeability and per-meability anisotropy. In addition, we show that estimation errorsof permeability inversion procedures that neglect the physics oftwo-phase immiscible fluid flow and mud-filtrate invasion can beas high as 100%.

IntroductionModular and multiprobe formation testers have proved advanta-geous in the determination of permeability at intermediate-scalelengths because of the increased distance between the observationand sink probes (Pop et al. 1993; Badaam et al. 1998; Proett et al.2000). Moreover, the use of dual-packer or “straddle-packer” mod-ules over point-probe modules is known to improve the interpre-tation of pressure transient measurements when testing laminated,shaly, fractured, vuggy, unconsolidated, and low-permeability for-mations (Ayan et al. 2001). Several papers have been published todescribe interpretation techniques and applications of these newformation-testing approaches (Kuchuk 1998; Hurst et al. 2000;Onur et al. 2004).

The new method introduced in this paper interprets formation-tester measurements acquired with wireline straddle-packer tools.It incorporates the physics of two-phase, axisymmetric, immisciblefluid flow to simulate the measurements, and it is combined witha nonlinear minimization algorithm for history-matching purposes.Comparable inversion approaches have been documented in theopen technical literature (Proett et al. 2000; Xian et al. 2004;Jackson et al. 2003) but they assumed single-phase fluid flow.Recently, Zeybek et al. (2001) introduced a multiphase flowmethod to integrate formation-tester pressure and fractional flowmeasurements with the objective of refining relative permeabilityvalues estimated from openhole resistivity logs. The same authorsconsidered the manual inversion of radial invasion profiles, hori-zontal permeability, and permeability anisotropy but did not assessthe uncertainty of their estimations introduced by a priori assump-tions about multiphase flow parameters. By contrast, the develop-ments reported in this paper integrate the flow simulator with adynamically coupled mudcake growth and mud-filtrate invasionalgorithm (Wu et al. 2002), which improves the physical consis-tency and reliability of the quantitative estimation of both perme-ability and permeability anisotropy.

MethodThere are three main components in the workflow developed inthis paper:

1. Mud-filtrate invasion algorithm2. Two-phase axisymmetric simulator3. Nonlinear minimization algorithmTransient measurements of pressure and flow rate are com-

pared to the outputs of a two-phase axisymmetric simulator toyield new model parameters through nonlinear minimization. Theinvasion algorithm makes use of these parameters (permeabilityand permeability anisotropy), in addition to pressure overbalance,invasion geometry, mudcake properties, and other rock-formationproperties, to simulate the process of mud-filtrate invasion. Sub-sequently, the calculated spatial distributions of pressure and fluidsaturation resulting from mud-filtrate invasion are used as initialconditions for pressure-transient tests. To reduce the time requiredby the inversion, this last step could be approximated with aninvariant mud-filtrate invasion profile calculated only once duringthe minimization. However, such a strategy is not recommendedfor supercharged formations where updates of initial conditionsduring minimization can drastically impact inversion results. Forthe synthetic case examples considered in this paper, we constantlyupdate the initial conditions during minimization. The geometry ofthe formation model (e.g., multilayer formations, impermeable bedshoulders) is fixed when entered into the flow simulator. Conse-quently, field measurements, well-log measurements, and otherindependent sources of information are needed to define the geo-metrical properties of the rock formation model. To complete theestimation, the fluid-flow simulator yields pressure transients to becompared against actual measurements. This process repeats itselfuntil the quadratic norm of the residuals between simulations andmeasurements decreases to a predefined value. When the latter

* Currently with TOTAL.

Copyright © 2007 Society of Petroleum Engineers

This paper (SPE 95897) was first presented at the 2005 SPE Annual Technical Conferenceand Exhibition, Dallas, 9–12 October, and revised for publication. Original manuscript re-ceived for review 14 July 2005. Revised manuscript received 22 May 2007. Paper peerapproved 29 May 2007.

339September 2007 SPE Journal

condition is met, the inversion algorithm outputs the estimatedvalues of permeability and permeability anisotropy.

Systematic description of the inversion methodology on syn-thetic measurements requires the use of a “base case” model (de-scribed later), which includes petrophysical variables and geo-metrical properties that can reproduce arbitrary formation modelsand tool configurations. Fig. 1 shows the measurement configu-ration for the assumed straddle-packer wireline formation tester.Dimensions of the “base case” tool were chosen according totypical physical dimensions of wireline formation testers designedfor interval pressure transient tests (IPTTs): two vertical observa-tion probes and one packer flow area that acts as a sink. Thesimulator used in this paper was developed by the FormationEvaluation Research Program of the University of Texas at Austin.

Special considerations about skin factor, tool storage, and totalcompressibility are discussed in a separate section of the paper.Although not investigated here, skin damage can be readily imple-mented in the inversion method by using a similar approach to themultilayer formation example presented in this work. In addition,even though we ignore tool-storage effects, the latter can be stud-ied with time-variable flow rates of fluid production.

Numerical Simulation of the Process ofMud-Filtrate InvasionAn adaptation of INVADE, developed by Wu et al. (2002), is usedto simulate the process of mud-filtrate invasion. Simulations in-clude the dynamically coupled effects of mudcake growth andmultiphase, immiscible mud-filtrate invasion. In simple terms, theflow rate of mud-filtrate invasion depends on both mud and rockformation properties. This approach differs from the proceduredescribed by Gök et al. (2006), who considered stationary com-posite zones and assumed single-phase flow. By coupling the in-vasion algorithm with the flow simulator, our inversion method isnot restricted to discontinuous fluid saturation gradients and, moreimportantly, it does not assume that the invaded zone across thestraddle-packer interval is immobile nor stays under constant fluidsaturation during the test (i.e., our inversion approach remainsaccurate in cases of significant fluid cleanup). The INVADE al-gorithm assumes that the rock formation is drilled with a water-based mud (WBM).

Our base-case model replicates the conditions of an invadedzone through the injection of brine into the formation during 1.5days. This value, as well as other assumptions on rock formationand mud properties, was arbitrarily chosen to illustrate the methodproposed in this paper rather than to describe a specific situation.Additional assumptions include the values of brine salinity, equalto 5,000 ppm (1.75 lbm/STB) and formation water salinity, equalto 120,000 ppm (42.06 lbm/STB). Table 1 summarizes the prop-erties of the assumed mud (and mudcake). Fig. 2 describes the as-sumed water/oil relative permeability and capillary pressure curves.

Fig. 1—Configuration of the “base-case” straddle-packer, awireline formation tester consisting of two vertical observationprobes and a dual-packer module.

Fig. 2—Water/oil relative permeability and capillary pressurecurves assumed in the numerical simulations of both mud-filtrate invasion and formation-tester measurements.

340 September 2007 SPE Journal

To couple the outputs of the invasion algorithm with the nu-merical simulation of pressure transient measurements, the simu-lator calculates spatial distributions of pressure, salt concentration,and fluid saturation resulting from 1.5 days of invasion that areentered as initial conditions for the simulation of formation-testermeasurements.

Numerical Simulation of Two-Phase FlowThe simulation of formation-tester measurements is performedwith a water/oil two-phase, axisymmetric reservoir simulator de-veloped by the Formation Evaluation Program of the University ofTexas at Austin. Detailed information about this simulator is givenby Lee et al. (2004). The simulator uses finite differences and theIMPES (implicit-pressure explicit-saturation) algorithm to solvethe nonlinear system of equations of pressure and saturation. Vari-ous boundary conditions can be enforced by the algorithm. Space-and time-dependent variables such as temperature and salt con-centration are also included in the simulations.

Base-Case Model. The Base-Case Model describes standard mea-surement parameters and rock formation properties assumed formost of the studies considered in this paper. Fig. 3 shows thefinite-difference grid along with the physical dimensions of theassumed hydrocarbon-bearing rock formation. The vertical sepa-ration between grid nodes is nonuniform, ranging from 0.5 ft nearthe probes to 1 ft at the remaining grid nodes. In the radial direc-tion, the simulator enforces a logarithmic discretization, startingfrom an initial value of 0.049 ft near the wellbore to 122.2 ft at theouter radial boundary of the reservoir. Fig. 4 describes the finite-

difference grid used in this paper to assess the effects of imper-meable bed boundaries on formation-tester measurements.

There are three observation points for the measurement of pres-sure transients at distances of 5, 13, and 20 ft measured from thetop of the reservoir, respectively. The packer interval (sink) has alength of 2 ft. Upper, lower, and external reservoir boundaries areassumed impermeable (flow rate is zero). Table 2 summarizes thegeometrical dimensions of the reservoir model considered in thissection, whereas Table 3 describes the associated rock and fluidproperties. Initial conditions for formation-tester measurementsprior to the onset of mud-filtrate invasion are given in Table 4. Forthe base-case model, the drawdown sequence enforces a constantproduction flow rate at the packer of 21 STB/D during 60 minutes,after which the buildup sequence continues for 60 additional min-utes. Fig. 5 illustrates the assumed flow-rate sequence.

To validate the finite-difference grid used in this paper, weconducted the test shown in Fig. 6, where we compared packer-pressure measurements simulated for the base case model againstthe corresponding single-phase radial-flow analytical Ei solution.The rock system is anisotropic in order to emphasize radial flowconditions in the comparison exercise. Results indicate an excel-lent match between the numerical and analytical results.

Nonlinear Inversion AlgorithmGiven the complex nonlinear relationship between borehole pres-sure-gauge measurements and rock formation petrophysical prop-

Fig. 3—Graphical description of the finite-difference grid usedfor the numerical simulations associated with the base-casemodel. The reservoir extends from 0.354 ft (wellbore radius) to1,000 ft (drainage radius) horizontally and from 3,990 ft to 4,020ft vertically. The numbers located to the left of the tool sche-matic indicate the distance in feet from each probe and packerto the top of the reservoir. The total pay thickness is 30 ft.

Fig. 4—Graphical description of the finite-difference grid usedfor the study of impermeable bed boundaries. Numbers locatedto the left of the tool schematic indicate the distance in feet fromeach probe and packer to the top of the reservoir. The numericalsimulations assume negligible values of permeability and po-rosity outside the permeable layer.


erties, the inversion algorithm requires several sequential linearsteps to match the simulated pressure transients with measuredpressure transients. Similar types of applications can be found inthe open technical literature (Kuchuk 1998; Onur et al. 2004).

The inversion algorithm considered in this paper is based on thegeneral framework for constrained minimization described by Ha-bashy and Abubakar (2004). Specifically, we use a modified ver-sion of the Levenberg-Marquardt (LM) minimization methodimplemented by Alpak (2005). The LM minimization method iswidely used for nonlinear least-squares problems because of itsstability and good rate of convergence. Away from the minimum,the algorithm is similar to the steepest-descent method, whereas inthe neighborhood of the minimum, the algorithm is similar to theGauss-Newton method. Accordingly, model parameters are ob-tained by minimizing a cost function composed of the quadraticmisfit between measured and simulated samples of transient pres-sure plus an additive quadratic model functional. For this work, thecost function is written as

C�x� =1

2�� e�x�� 2 + �x � 2�, . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

where �, the Lagrange multiplier, is a scalar quantity (0<�<�) thatassigns relative weight to the two additive terms included in Eq. 1,and e(x), the vector of data residuals, is defined below. The firstadditive term of the cost function yields an estimate of the un-known model x that honors the measurements, whereas the secondadditive term prevents instability in the estimation due to non-uniqueness as well as insufficient and noisy measurements. Selec-tion of the Lagrange multiplier is based on the criteria

1

�= � max�m��m�, provided that

min�m��m�

max�m��m�� ,

where � is a constant equal to 10−8 and �m are the eigenvalues ofthe real and symmetric matrix JT(x)�J(x). The Jacobian matrix,J(x), is given by

J�x�=�Jmn =�em

�xn, m = 1, 2, 3, . . . , M; n = 1, 2, 3, . . . , N�,

where M is the number of measurements, and N is the number ofunknown model parameters. The weight of the misfit term in thecost function is progressively adjusted by the Lagrange multiplieras the inversion algorithm iterates toward the minimum of the costfunction. This approach guarantees a stable intermediate solutionat every iteration without over-biasing the final solution by thespecific choice of regularization term.

The inversion algorithm is based on the following definition ofvector of data residuals e(x):

e�x� =�e1�x�

···ej �x�

···eM�x�

� =��ps1�x� − pm1��pm1

···�psj �x� − pm j��pm j

···�psM�x� − pmM��pmM

� , . . . . . . . (2)

where pmj is the measured pressure and psj is the numericallysimulated pressure. The index j attached to the pressure measure-

ments is used to identify the corresponding time sample. In theabove expression, data residuals are normalized in order to putboth packer and probe pressure transients on equal footing. Analternative approach is to define the measurements as pressuredifferentials (i.e., pj�pº−pj) where pº is formation pressure priorto the time of measurement, or to use the logarithmic value of thepressure differentials, log(pj), in which case the normalization ofEq. 2 is no longer required. In this paper, total pressures were usedto obtain the inversion results reported in Tables 5 and 6 exceptfor those values identified with (“*”), where pressure differentialswere necessary to ensure stability of the inversion. Pressure dif-ferentials were also used to obtain the results reported in Table 7.Our experience shows that, in general, the use of pressure differ-entials substantially increases the stability of the minimization pro-cess. Further comparisons between the use of relative misfit errorsand pressure differentials to perform the inversion can be found inthe work by Angeles (2005).

The vector of model parameters included in Eq. 2, x, is given by

x = �x1

···xN

� = �log�y1�

···log�yN�

�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

Fig. 5—Time schedule of flow rate assumed for the simulationsof formation tester measurements in connection with the basecase model.

Fig. 6—Validation test of the finite-difference grid employed tosimulate the base-case model. Three different formation perme-ability values (10, 100, and 1000 mD) are the basis of compari-son between simulated packer-pressure measurements and thesingle-phase radial-flow analytical solution.


where N is the number of unknowns, and yi is permeability. For thepurposes of this paper, model parameters are either permeability orpermeability anisotropy ratio. When inverting for permeabilities,logarithmic values are used to define the model parameters yi, as

indicated in Eq. 3. On the other hand, when inverting for perme-ability anisotropy, actual (nontransformed) values are used instead.The logarithmic transformation of permeability enhances the con-vergence rate of the inversion algorithm (Angeles 2005). When the


inversion is implemented to estimate mobilities, the correspondingviscosities of the fluids involved are assumed constant.

Although not implemented in this paper, the quadratic costfunction defined by Eq. 1 could include a data weighting matrix,Wd, to de-emphasize the influence of noisy or biased pressuresamples. In such a case, the term �e(x)�2 becomes �Wd·e(x)�2, andif the measurement noise is stationary and uncorrelated, thenWd�diag{1/j}, where j is the standard deviation of the noisepresent in the jth pressure measurement.

Cramér-Rao Uncertainty Bounds. We use the Cramér-Rao(Cramer 1945) approach to estimate the uncertainty of the esti-mated parameters (permeability and permeability anisotropy). Ac-cordingly, a perturbation is performed on the parameters yieldedby the inversion to evaluate the expression

� ≈ 2�2I + �JT�x*�J�x*��−1, . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

where is the standard deviation of the uncorrelated, zero-meanGaussian noise used to contaminate the pressure data, � is theLagrange multiplier included in the quadratic cost function (Eq. 1),x* is the vector of inverted model parameters, and � is the esti-mator’s covariance matrix. The square root of the diagonal entries(variances) of the latter matrix provide the uncertainty values suchthat for 99.7% of the occurrences, the ith model parameter will fallwithin ±3√�ii of the estimated value (Habashy and Abubakar2004). In this paper, uncertainty bounds were calculated only forthe case of noisy measurements for inversion examples of multi-layer rock formations; nonetheless, uncertainty bounds could alsobe calculated for all noisy synthetic cases considered the paper.

Data Misfit and Impact Value. In addition to the estimated pa-rameters, there are two diagnostic outputs provided by the inver-sion process: data misfit and impact value. Data misfit is quantifiedwith the root mean square (RMS) difference between the input andsimulated transient pressure measurements at the end of the inver-sion. This value is given as a percentage of the quadratic norm ofthe input transient pressure measurements. For noise-free syntheticcases, the data misfit is expected to be 0.0%.

We introduce an “impact” value (IM) to quantify the relativeimportance of specific assumptions made in the inversion process(e.g., fluid viscosity, irreducible water saturation, and level ofnoise). The IM is defined as

IM = 100l

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

with

l =� 1

L − 1 �l=1

L

�x*l − xl� ��1�2

. . . . . . . . . . . . . . . . . . . . . . . . . . (6)

In these expressions, the subscript l identifies the specific inversionresult for a given inversion example, xl* is the estimated valueobtained from the inversion, xl

� is its corresponding target value (orthe actual value for synthetic examples), and max is the maximumvalue of l in the total inversion scheme for single-layer inversionexamples (as reported in Tables 5 and 6) or for multilayer inversionexamples (as reported in Table 7). For illustration, consider the case of 10psi noise in the “drawdown only” section of Table 5. Accordingly,


one has l = ��1�2�*��10.1 − 10� + �101.9 − 100�

+ �1200.9 − 1000� ��1�2

= 142.06.

For single-layer inversion examples, max was found to be 2,837.3;hence, the corresponding impact value is IM�100*(142.06)/2,837.3�5.01. Impact values range from 0 to 100, the closer to100 the largest the influence of a given assumption on the invertedparameters. Conversely, impact values close to 0 indicate the low-est possible influence on inverted parameters. The “impact” valueis provided here for qualitative interpretation of the sensitivitystudies and is influenced by the specific inversion examplesconsidered within a given set of inversion results. Similar diag-nostic procedures could be implemented in field applications bychanging the target value x l

� to the final matched value and bycalculating the variations of x l* for specific assumptions on for-mation properties.

Considerations on Damage Skin, Tool Storage,and CompressibilityThe synthetic cases considered in this paper do not explicitly in-clude the effects of skin, tool storage, or total fluid compressibilityon pressure-transient measurements. However, it becomes impor-tant to illustrate how these parameters can be readily incorporatedinto the inversion method described in this paper.

To include skin factor in the inversion, we suggest constructinga composite radial model where a damage or stimulated layeradjacent to the wellbore is assigned a radius (rs) and a permeability(ks), both of which could be regarded as unknown parameters inthe inversion. Subsequently, skin (s) could be estimated withHawkins’ formula (1956):

s = � k

ks− 1� ln

rs

rw, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

where rw is the wellbore radius. Alternative approaches are pos-sible such as those described by Pucknell and Clifford (1991).However, we anticipate that the increased number of unknownswill worsen the stability of the inversion, therefore requiring ad-ditional information to guarantee reliable estimations.

To include tool storage, the simulator (as with most reservoirsimulators) can take as input time-variable flow rates of fluidproduction. In cases of severe compression and decompression offluid across the flowline, data processing techniques (e.g., low-pass filtering) applied to the measured flow rates could help toimprove the estimation. Note, however, that these propositionshave not been explored by the authors.

To include total compressibility, one could add as unknownparameters both water (cw) and oil (co) compressibilities. Giventhat fluid saturations are output by the flow simulator, it is possibleto calculate averaged values of water and oil saturations (Sw andSo, respectively). Accordingly, the average total compressibility ct

is given by

ct = Swcw + Soco + cf , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)where cf is the formation rock compressibility.

Sensitivity of Straddle-Packer Formation TesterMeasurements to Rock-Formation PropertiesThis section evaluates the impact of specific assumptions madeabout rock-formation properties on the simulated pressure-transient measurements and the corresponding inversion results.For this purpose, the base-case model (described earlier) is used toestablish the standard measurement parameters and rock-formationproperties assumed for all the examples under consideration. Ex-plicit correlations between permeability, porosity, capillary pres-sure, and relative permeability are enforced by the inversion algo-rithm as described in the Appendix. The flow rate of mud filtratewas restricted to a maximum of 0.003 B/D/ft for the examplespresented in this section, except for the sensitivity analyses ofmud-filtrate invasion, where flow rates exceeded 0.2 B/D/ft.

Refer to Tables 5 through 7 for a summary of the results ob-tained in this section. With the exception of the cases identified as“Only Drawdown” and “Impermeable Bed Boundaries,” all theresults presented in these tables were obtained using the simulta-neous inversion of the two monitoring probes and the packer aswell as both drawdown and buildup pressure sequences.

Sensitivity to Variations of Permeability. Four different types ofrock formations are used to assess the effect of the spatial distributionof permeability and permeability anisotropy on inversion results:

1. Homogeneous and isotropic formations2. Homogeneous and anisotropic formations3. Two-layer and isotropic formations4. Finely laminated formations

Homogeneous and Isotropic Formation. Fig. 7 describes thepressure responses simulated at two of the three sampling points ofthe formation tester: monitoring probe 1 and packer. It is observedthat the simulated pressure-transient measurements for high-permeability rocks exhibit relatively lower sensitivity to variationsof rock permeability than those associated with low-permeabilityrocks (i.e., small variations of permeability cause small variations


of pressure only when permeabilities are high). For instance, avariation from 100 to 200 mD modifies the late-time packer pres-sure response by 5%. On the other hand, an increase from 900 to1,000 mD only modifies it by 0.01%. This effect is further en-hanced when the measurements are artificially contaminated withzero-mean additive Gaussian noise.

Another interesting observation from Fig. 7 is the range ofvariation of pressure measured at both packer and probes. Probe-pressure variations are given in a few fractions of psi, whereaspacker responses vary in the range of psi or tens of psi. For ex-ample, a pressure drawdown of 7.7 psi is simulated for a 10-mDrock formation at the probe. This represents 6.7% of the drawdownpressure (114.8 psi) simulated at the packer for the same formation.

Fig. 8 describes the flow regimes observed in the base-casemodel. At early times, the log-log plot exhibits the –1/2 slopecharacteristic of spherical flow (before approximately 0.001 hours)but subsequently stabilizes to a constant value (i.e., a radial flowregime is reached after approximately 0.01 hours of buildup forboth monitoring and packer-pressure measurements). Table 5shows the inverted values of permeability starting with an initialguess of 40 mD toward their respective target values. In this case,the inversion algorithm is driven by noise-free synthetic pressuretransient data generated with the two-phase flow simulator at bothprobes and the packer. A total of 2,592 time pressure samples fromthe complete test interval (including drawdown and buildup) wereused by the inversion algorithm. The time sampling used to acquire

these pressure samples is the same as the one enforced by thenumerical simulator. Each iteration requires approximately 3.5minutes of CPU time on a 1.6-GHz Windows-based PC. The in-version reached convergence within nine iterations. Final invertedpermeabilities are the same as the target values.

Effect of Additive Zero-Mean Random Gaussian Noise. Forthis exercise, packer and probe measurements were equally con-taminated with zero-mean additive random Gaussian noise of stan-dard deviations equal to 1 and 10 psi. Pressure responses associ-ated with high-permeability formations are the most affected bythe presence of noise, especially at the observation probes, wherepressure variations are smaller in amplitude than those at thepacker flow area. As permeability increases, the error increases to20% for the case of 10-psi Gaussian noise (Table 5). An explana-tion for this is that high-permeability formations entail smallerpressure differentials and hence are more susceptible to the pres-ence of noise than low-permeability formations. For instance, avariation of 1 psi of the measured transient pressure for a forma-tion of 1,000-mD represents a contamination of 60% of the mea-surements, and this causes an error equal to 1% in the estimatedpermeability. Correspondingly, a variation of 1 psi represents acontamination of 8% for 100-mD pressure measurements, and thiscauses a zero error on the estimated permeability. Fig. 9 describesthe misfit between the estimated and measured pressure values.

Effect of Buildup and Drawdown Measurements. In order toassess the relative information content of each stage of the pressuretransient test, two inversion schemes were designed: one usingonly the drawdown pressure measurements, and another using bothdrawdown and buildup pressure measurements. Both cases as-sumed noise-free and noisy synthetic data (10 psi standard devia-tion of zero-mean additive Gaussian noise). As expected, both teststages (buildup+drawdown) contribute to decrease the nonunique-ness of the inversion with respect to the case of only one stage(drawdown in this case), especially when the data are contami-nated with relatively large amounts of noise (i.e., 10 psi standarddeviation zero-mean additive Gaussian noise). Although notshown here, the convergence rate significantly improves whenusing both measurement stages (Angeles 2005).

Homogeneous and Anisotropic Formation. Three values ofanisotropy ratio (defined as the ratio between horizontal and ver-tical permeability, kh/kv) were considered for the base-case rockformation model: 1, 10, and 100. Fig. 10 shows the corresponding

Fig. 7—Noise-free simulated pressure transient measurementsat the observation probe and packer located at 5 and 20 ft,respectively, from the top of the reservoir.

Fig. 8—Log-log plot for the packer and probe 2 buildup pres-sure measurements simulated for the base-case model. Simu-lated measurements for the monitoring probe 1 are not shownhere but are also used for inversion. In addition, the plot com-pares single- and two-phase flow synthetic measurements. Thepermeability of the formation is 100 mD.


simulated pressure measurements. From the figures, one can ob-serve that an increase of anisotropy ratio causes an increase of themagnitude of the pressure differential at the packer flow area.Conversely, the pressure differential decreases at the vertical ob-servation probes. This effect is emphasized for the case of high-permeability formations, where the corresponding pressure mea-surements are much smaller in magnitude than those associatedwith low-permeability formations. It then follows that the pressuremeasurements acquired at the probes might not contribute signifi-cantly to reduce nonuniqueness of the inversion if the formationpermeability is high (above 500 mD for the examples shown in thispaper). In the latter case, alternative pressure-testing strategiescould be used to reduce nonuniqueness of the inversion (e.g., byacquiring pressure measurements at two different depths). This isan important technical issue because pressure measurements areoften corrupted by noise, thereby decreasing their reliability toestimate rock formation properties.

Unlike the strategy adopted for one-parameter inversion, thealgorithm was allowed to vary the unknown model parameterswithin narrower prescribed bounds, thereby reducing the problemof nonuniqueness discussed previously. For instance, if the in-version algorithm was used to estimate a horizontal permeabilityof 100 mD, the minimization would be constrained to find a so-lution between the upper and lower bounds of 1 and 300 mD,respectively, compared to the upper and lower bounds of 0.1 and10,000 mD, respectively, usually enforced for the case of one-parameter inversions.

The inversion algorithm required between 5 to 25 iterations toachieve convergence toward the estimated parameters. Comparedto the typical range of 5 to 10 iterations normally used for one-parameter inversions, the required number of iterations is rela-tively high. The algorithm used 552 time-pressure samples ac-quired with the same time sampling interval used by the simulator.Also, it was observed that several combinations of permeabilityand permeability anisotropy could lead to local minima. This ob-servation indicates that there are several equivalent solutions to theinverse problem that honor the measurements.

Different inversion techniques were implemented to obtain theresults described previously. Using a priori knowledge, the initialguess parameters were given values close to their actual values.In practical applications, such information could be derivedfrom well-log, core, or production data. When this a priori infor-mation was not adequate, the algorithm would continuously restart

the search with different initial guesses to warrant stable conver-gence while enforcing the same bounds to explore several localminima. The final result was chosen as the one that entailed thelowest data misfit.

However, two problems remain for the case of low-perme-ability formations: first, more local minima exist than for the caseof high-permeability formations (for the same range of anisotropyratios, a much larger combination of formation permeabilities hon-ors similar pressure measurements), thereby biasing the inversiontoward values close to the initial guess, and second, estimatedvalues would converge toward the correct values, but the rate ofconvergence was slow. To circumvent these two problems, theinversion algorithm was modified to use pressure differentials(p�pº−pmj) rather than raw pressure measurements. This strat-egy proved efficient to reduce nonuniqueness in the inversion re-sults. The corresponding inversion result is identified with an as-terisk (“*”) in Table 5.

The impact of noise contamination on the input pressure-transient measurements is also shown in Table 5. Because of thesevere nonuniqueness of the inverse problem, the algorithm usedseveral “restart” values to yield the final estimates. Fig. 11 is alog-log plot that compares the inverted pressure measurementsagainst the original noisy measurements generated with zero-meanadditive random Gaussian noise of standard deviation equal to 1psi. For this example, the inversion algorithm yielded values ofpermeability and permeability anisotropy equal to 100.8 mD and110, respectively, compared to target values of 100 mD and 100,respectively. Results are deemed satisfactory.

Multilayer FormationsTwo-Layer Formation. Fig. 12 describes the examples of two-layer formations considered in this section. We position two of theobservation probes within the top layer and locate the packerwithin the bottom layer. Both layers are assumed homogeneousand isotropic. Also, we assume that layer boundaries as well asdistances between the packer and probes are known a priori. Mea-surements were contaminated with zero-mean additive randomGaussian noise of standard deviation equal to 0.1 psi. Simulatedpressure differentials (p) were entered to the inversion algorithminstead of raw pressure measurements to mitigate the problem ofnonuniqueness. We also calculate Cramer-Rao uncertainty bounds(using the “+/–” operator) to quantify the level of confidence of theestimated properties. Table 7 describes the results of this inversionexercise. Even though both Cases A and B were initialized with aguess of 300 mD, we observe that Case A led to convergence in thefirst attempt, as opposed to Case B, wherein the inversion stag-nated at a local minimum. Such a behavior prompted us to “restart”the inversion several times while pursuing the global minimum.The same strategy was successfully applied to cases of noise-contaminated measurements.

Another important observation from this inversion exercise isthat the Cramer-Rao bounds decrease when inverting properties ofbottom layers. This behavior indicates that the best estimates cor-respond to zones closer to the packer, where pressure transients aremore sensitive to rock-formation properties.

Finely Laminated Formation. A different inversion methodologyis adopted for the case of finely laminated rock formations. Fig. 13shows the example of a formation model composed of seven ho-mogeneous and isotropic layers. Testing of this formation model isperformed within the lowest pay zone, where the packers are lo-cated, while the vertical observation probes sample pressureswithin the medium and top pay zones. Notice that the verticalseparation of the numerical grid nodes is 0.5 ft near the samplingpoints, while the thickness of the formation layers varies from 1.5ft (top) to 2 ft (medium and bottom). Only the three pay-zonepermeabilities are assumed unknown in the estimation. Flow-rateschedules and formation properties are the same as those assumedfor the base-case formation model.

In this example, the inverse problem is severely nonunique, andtherefore a priori information is necessary to estimate the locationof layer boundaries and the initial guess permeabilities. Moreover,

Fig. 9—Comparison of input pressure measurements and pres-sure measurements simulated with the permeability estimatedfrom inversion. Input pressure measurements were simulatedfor a formation with homogeneous and isotropic permeabilityequal to 100 mD, and they were contaminated with additivezero-mean random Gaussian noise of standard deviation equalto 10 psi. The inverted permeability is equal to 101.6 mD.


the “restart” strategy proves to be insufficient if it is not combinedwith the enforcement of physical bound constraints on the esti-mated parameters, also assumed to be known a priori. The resultsshown in Table 7 were obtained with an initial guess of 120, 250,and 150 mD for the permeabilities of top, center, and bottom payzones, respectively. Table 8 describes the bound constraints en-forced for this example. Note that in field applications, one wouldpreferentially position the packer within more than one pay zoneand acquire additional sets of pressure data to increase the confi-dence on these bound constraints. We observe that the Cramer-Raobounds indicate more confidence on the estimated bottom pay-zone permeability (located across the packer) than on the perme-abilities of the remaining pay zones.

Sensitivity to Variations of Fluid ViscosityGiven the two-phase nature of the fluid-flow phenomenon as-sumed in this paper, sensitivity analyses were performed to evalu-

ate the impact of assumptions made on the viscosity of the fluidsinvolved (oil and water for the base-case formation model). Pres-sure transient measurements were simulated for values of oil vis-cosity equal to +500% and –20% of the original viscosity. Thesame relative variations are applied for the appraisal of waterviscosity. It was observed that the impact of oil viscosity is muchmore significant than that of water viscosity. Pressures changedramatically when oil viscosity is perturbed from its original value.On the other hand, water viscosity affects only the initial pressuremeasurements during drawdown. This behavior can be explainedby recalling the plots of water saturation, relative permeability, andcapillary pressure. Specifically, as the drawdown stage begins,water saturation decreases from almost 0.63 to 0.45 (i.e., withina region where the relative permeability of water is not as highas that of oil). Thus, the effect of water is noticeable only at thestart of drawdown. An increase of oil viscosity decreases themobility, and the corresponding effect on pressure is similar to

Fig. 10—Simulated pressure transient measurements for three values of permeability anisotropy (=kh/kv): 1, 10, and 100 of a rockformation with horizontal permeability (kh) equal to 100 mD.


that of single-phase flow. Flow-rate schedule and remaining for-mation properties were the same as those of the base-case forma-tion model.

Sensitivity to Variations of Capillary Pressureand Relative PermeabilityTable 5 also shows the effects of variations of the assumed water/oil capillary pressure and relative permeability curves on inversionresults. For this purpose, the study makes use of modified Brooks-Corey parameters as well as base-case model properties. Sensitiv-ity analyses consider variations of water/oil capillary pressure andrelative permeability in two ways: by changing the pore-size dis-tribution index, and by changing the irreducible water saturation.

Sensitivity to Variations of Pore-Size Distribution Index. Threevalues of pore-size distribution index, �, were considered: 0.5(very wide range), 2 (wide range), and 4 (medium range). Theexpressions used in this exercise are the drainage equations asso-ciated with the modified Brooks-Corey model, namely:

S*w =Sw − Swr

1 − Swr − Sor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

S*o =1 − Sw − Sor

1 − Swr − Sor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)

Pc,dr = Pce�S*w�−1

� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

krw,dr = �S*w��2+3��

� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)

kro,dr = �S*o�2�1 − �S*w�

�2+3��

� �, . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)

where S is fluid saturation, Pc,dr is drainage capillary pressure, kr,dr

is drainage relative permeability, and Pce is capillary entry pres-sure. In these equations, subscripts are used to identify water (w),oil (o), and irreducible saturations (r). The superscript * indicatesnormalized saturations as defined by Eqs. 9 and 10.

Table 5 shows the corresponding inversion results. By consid-ering that the base case value is �≈2, it can be concluded that thisvariable has marginal impact on the final inverted properties.

Sensitivity to Variations of Irreducible Water Saturation.Variations of the assumed irreducible water saturation were testedfor values of 0.15, 0.25, and 0.35 (used in the base-case model). As

done before, modified Brooks-Corey drainage equations were usedto define the capillary pressure and relative permeability curves.Unlike the effect of variations of �, variations of Swi dramaticallyinfluence the simulated noise-free pressure measurements com-pared to those of the base-case model.

Sensitivity to Variations of Production Flow RateFor the base-case model, the formation tester withdraws formationfluids at a rate of 21 B/D from the packer-straddle section of theborehole. Variations of 4 B/D to the latter production rate areenforced by the simulator to obtain the inversion results summa-rized in Table 5. Note that the estimated permeability increases or

Fig. 12—Two-layer and isotropic formations considered forcase examples A and B, respectively. Capillary pressures, rela-tive permeabilities, and rates of mud-filtrate invasion are as-sumed the same for the two layers. The two observation probesare located within the top layer, whereas the packer flow area ispositioned within the bottom layer. Numbers located to the leftof the tool schematic indicate the distance in feet from eachprobe and packer to the top of the reservoir. Reservoir dimen-sions and remaining formation properties are the same as thoseof the base-case model.

Fig. 13—Geometrical description of a finely laminated formationmodel. The packer flow area is located within the bottom layer,while the two observation probes sample pressure from theupper layers. These layers are separated by low-permeabilityintermediate layers whose boundaries and permeabilities areknown a priori. Numbers located to the left of the tool schematicindicate the distance in feet from each probe and packer to thetop of the reservoir.

Fig. 11—Log-log plot comparing the buildup “measured” noisysynthetic data against simulation results for one of the inver-sion cases with anisotropy (dotted lines). Although used in theinversion, pressure measurements acquired with the monitor-ing probe 1 are not shown here. The agreement between the twosets of measurements is excellent despite the high level of noise.


decreases in proportion to the decrease or increase of the ratevariation, respectively.

Sensitivity to Presence of ImpermeableBed BoundariesPressure-transient measurements remain sensitive to the presenceof impermeable bed boundaries in the vicinity of measurementpoints. The purpose of this section is to assess the influence ofunaccounted impermeable bed boundaries on the inversion resultsfor different values of layer thickness. Fig. 4 shows the finite-difference grid used to perform the simulations. The packer islocated in the middle of the formation as the layer thickness, h,is decreased to prescribed values. There are impermeable beds

adjacent to the formation with porosity and permeability equal to0.0001% and 0.000001 mD, respectively. Fig. 14 describes thesimulated pressure transients, and Table 6 describes the corre-sponding inversion results. For these cases, the inversion is per-formed using both drawdown and buildup pressure measurementsacquired only with one monitoring probe (probe 2) in addition topressure measurements acquired with the packer.

In general, the pressure differential during drawdown measuredby both the observation probe and the packer increases in thepresence of impermeable bed boundaries, thereby biasing the in-version results toward permeability values lower than that of thebase case. The closer the packer is located to an unaccountedimpermeable bed, the lower the corresponding estimate of perme-ability yielded by the inversion.

Sensitivity to Mud-Filtrate Invasion ParametersThis section evaluates the impact of some important assumptionsmade about parameters associated with the process of mud-filtrateinvasion on the permeabilities yielded by the inversion. Specifi-cally, three invasion parameters are given consideration:

1. Presence of an invasion zone.2. Time of invasion.3. Overbalance pressure.

In all these cases, the formation test is initialized using the spatialdistributions of pressure and fluid saturation derived from a pre-vious mud-filtrate invasion simulation (i.e., we are assuming thatno invasion occurs while pressure transients are acquired by theformation tester). This assumption may not be accurate if the for-mation test is performed concomitant to drilling.

Sensitivity to Presence of an Invasion Zone. Fig. 15 illustratesthe supercharging effect on pressure transients simulated for a1-mD rock formation assuming mud and formation properties asspecified in Table 1. The original formation pressure is masked bythe large sandface pressure observed on the wellbore side of themud-filtrate invaded zone. Fig. 16 compares the pressure-transient

Fig. 14—Effect of impermeable bed boundaries on the simu-lated pressure measurements for a formation of permeabilityequal to 100 mD. In the figures, �h designates the thickness ofthe permeable rock shouldered by impermeable beds (top andbottom bed boundaries).

Fig. 15—Pressure supercharging effect extending more than 2in. into the formation simulated 1.5 days after the onset of mud-filtrate invasion in an isotropic formation with permeabilityequal to 1 mD. At the sample depth (grid number 29 in thez-axis), the pressure difference between virgin formation andsandface pressure is as high as 10 psi.


measurements simulated at the two probes and the packer when theinvaded zone is included in the formation model. Relatively largevariations are observed in the simulated pressure transient mea-surements when the process of invasion is not considered by thesimulations.

Sensitivity to Time of Invasion. Simulations of pressure tran-sients were performed for three different times of invasion: 0.5,1.0, and 1.5 days. The effect of unaccounted time of invasion is toincrease the inverted value of permeability when the actual time ofinvasion is shorter than assumed by the inversion.

Sensitivity to Mud Overbalance Pressure. Three different valuesof mud pressure were entered to the flow simulator to generatenoise-free synthetic pressure transient measurements: 3,300,3,400, and 3,500 psi, corresponding to overbalance pressures of300, 400, and 500 psi, respectively. Incorrect assumptions aboutmud overbalance pressure (differences of approximately 100 psi)lead to errors of 2% on the estimated permeability.

Inversion Results for the Case of Single-PhaseFluid FlowWe compare inversion results obtained with the approach devel-oped in this paper against inversion results obtained when the flow

regime is inaccurately assumed to be single phase. Fig. 17 showsthe pressure-transient measurements simulated at the two probesand the packer for the case of a formation with homogeneous andisotropic permeability equal to 10 mD. Single-phase flow wassimulated by equating to zero the capillary pressure and waterrelative permeability curves, and by assigning a value of 1 to theoil relative permeability curve. We note a relatively large pressuredifferential simulated for both types of synthetic measurementsassuming the same production flow rate. Unlike single-phase flow,two-phase fluid flow constrains the displacement of oil by thespecific value of water saturation, thereby entailing a larger pres-sure differential than for the case of single-phase flow. Fig. 8,described earlier, shows a similar comparison using a log-log plot,where formation permeability is equal to 100 mD. Table 6 sum-marizes the inversion results obtained from this sensitivity analy-sis. It is found that inaccurate assumptions about single-phase flowentail inverted permeabilities lower than the actual values.

DiscussionThe inversion algorithm includes different components that add tothe complexity of the estimation problem but, at the same time,contribute to improving the reliability and physical consistency of

Fig. 16—Pressure-transient measurements simulated to assessthe effect of presence of a mud-filtrate invaded zone for thecase of a formation with isotropic and homogeneous perme-ability equal to 100 mD. The two plots shown above describenoise-free pressure measurements simulated at one of the ob-servation probes and the packer.

Fig. 17—Pressure measurements simulated to assess the effectof inaccurate single-phase flow assumptions on the invertedvalues of permeability for the case of a formation with isotropicand homogeneous permeability equal to 10 mD. The two plotsshown above describe noise-free pressure measurementssimulated at one observation probe and the packer. Remainingformation properties are the same as those of the base-casemodel.


the results. To emphasize such an important property of the inver-sion method developed in this paper, Fig. 18 summarizes the for-mation properties and test parameters that exhibit the largest im-pact on the simulated formation tester measurements. Incorrectassumptions about viscosity are by far the most important in theanalysis (which, incidentally, is also the case for single-phase in-versions). Presence of a mud-filtrate invaded zone, relative-permeability and capillary-pressure curves, knowledge of im-permeable bed boundaries, and single-phase flow assumptions arethe second-largest causes of data misfit in the analysis. Moreover,Fig. 19 suggests that the estimation error caused by additionaluncertainties associated with two-phase flow analysis is smallerthan either the error associated with the use of conventional single-phase flow techniques or the error associated with neglecting pres-ence of mud-filtrate invasion. In other words, the two-phase char-acter of the flow phenomenon under consideration remains crucialto accurately interpret pressure measurements acquired with a for-mation tester.

Likewise, presence of permeability anisotropy causes a rela-tively large pressure drawdown at the packer while it decreases theamplitude of pressure transients measured at the probes. This is nota desirable situation for the case of high-permeability rock forma-tions, where the amplitude of pressure transients significantly de-creases and hence leads to unreliable inversions in the presence ofnoise. This fact, coupled with nonuniqueness when inverting morethan one unknown parameter, requires the use of alternative strat-egies to constrain the solution and to redefine the model and inputmeasurements. Similar situations arise for the cases of unknownpetrophysical properties associated with multilayer formations. In-version exercises emphasize the importance of good initial guesses(obtained from auxiliary measurements such as rock-core andwell-log data) as well as physical bound constraints imposed onthe unknown properties.

Variations of fluid viscosity reveal fluid-flow characteristicscompletely different from those of single-phase flow. For the base-case model, the deleterious impact of incorrect assumptions madeon oil viscosity was significant compared to that of incorrect val-ues of water viscosity. This can be explained by the fact that thesaturation region for relative permeability and capillary pressurefluctuates between values of water saturation of 0.45 and 0.63,where most of the displaced fluid is oil. As inferred from thecorresponding inversion results, the effect of inaccurate assump-tions about fluid viscosity is similar to that caused by inaccurateassumptions about the character of the flow regime.

A similar behavior was observed for the case of inaccurateassumptions about water/oil relative-permeability and capillary-pressure curves. It was found that, in general, the pore-size distri-

bution index (�) associated with a modified Brooks-Corey modelmarginally affected the inversion results. Significant biases in theinversion results were observed only for the cases of high-permeability rock formations. Conversely, inaccurate assumptionson irreducible water saturation (Swi) have a measurable impact onthe estimated values of permeability because of the increase ofeffective permeability to water in relation to effective permeabilityto oil.

The sensitivity analysis to presence of impermeable beds wasintended to assess the effects of inaccurate assumptions about thedistance to upper and lower impermeable beds on the invertedvalues of permeability. Presence of unaccounted impermeablebeds near the sampling points caused a relatively large increase ofpressure drawdown measurements at the packer and monitoringprobes. Consequently, the inversion algorithm yielded values ofpermeability lower than the target values.

In field applications, presence of multiple local minima couldbe overcome in a similar manner as described in this paper with theuse of “restart” sequences. In addition, initial values of unknownparameters could be obtained from independent sources of infor-mation (e.g., well logs, single-phase transient analysis, and rock-core data) and used to enforce bound constraints on the inversion.

ConclusionsThe following is a summary of the most important conclusionsstemming from this paper:1. We showed that significant variations of pressure can be caused

by inaccurate assumptions made about two-phase rock-fluidproperties (such as relative permeability, capillary pressure, andoil viscosity), depending on specific conditions of invasion andvalues of permeability. Traditional approaches used for the in-terpretation of dual-packer pressure measurements are based onthe assumption of single-phase flow. To estimate permeability,these approaches include a “correction factor” to account fortwo-phase flow effects. The algorithm developed in this paperexplicitly considers the two-phase nature of fluid flow duringthe acquisition of formation-tester measurements. Moreover, wehave shown that neglecting the physics of two-phase flow andmud-filtrate invasion can result in permeability estimation errorsas high as 100%.

2. By explicitly including the processes of dynamic mudcakegrowth and mud-filtrate invasion into the inversion algorithm,the methodology presented here offers the advantage of not

Fig. 18—Pie chart describing the relative impact of several for-mation properties and formation-test parameters on the reli-ability of inverted values of permeability and permeabilityanisotropy. The size of each slice is proportional to the stan-dard deviation of the inverted properties compared to the cor-responding target (true) property values.

Fig. 19—Relative error in the estimation of three formation per-meabilities (10, 100, and 1,000 mD). Either the assumption ofsingle-phase flow or the omission of mud-filtrate invasion in theanalysis leads to a much higher error than that introduced byerroneous two-phase flow assumptions. A 0.2 perturbation ofirreducible water saturation (Swi) for the Brooks-Corey modelsis probably too large, although still important in the two-phaseflow analysis for the base-case model.


being restricted to assume stationary and/or piston-like mud-filtrate invaded layers. Once initial pressures and fluid satura-tions (calculated from the simulation of mud-filtrate invasion)are explicitly included as initial conditions for the simulation offormation-tester measurements, there is no need to include ar-bitrary fluid interfaces. Fluid saturation is allowed to change inthe vicinity of the packer during the test. This flexibility inthe simulation improves the estimation of permeability, espe-cially in cases wherein significant fluid cleanup is observedduring the test.

3. The combination of numerical near-wellbore simulations with acomputationally efficient inversion algorithm based on the Lev-enberg-Marquardt minimization method provides a systematicway to reduce nonuniqueness in the presence of noisy pressuremeasurements. We showed that a priori information on theunknown parameters is necessary to reduce nonuniqueness.

4. Synthetic pressure measurements were considered to appraisethe reliability of the new inversion method proposed in thispaper for the interpretation of wireline formation-tester mea-surements acquired with dual-packer modules. Although thisapproach still needs to be tested with field data, it providedreliable inversion results for the various synthetic formationmodels considered in this paper.

Nomenclature

C(x)� cost functione(x) � vector of residualsh � formation thickness, ftIM � impact value

k � absolute permeability, mDkh � horizontal permeability, mD

kh /kv � permeability anisotropy ratiokro � oil-phase relative permeabilityk0

ro � kro endpoint, mDkrw � water-phase relative permeabilityk0

rw � krw endpoint, mDkv � vertical permeability, mDM � number of measurementsnr � grid number in the radial directionnz � grid number in the vertical directionN � number of unknownspº � pressure prior to formation test, psi

pmj � measured pressure data, psipsj � simulated pressure data, psip � pressure differential, psiPc � capillary pressure, psi

Pce � capillary entry pressure, psiPd � particle diameter for the formation rock, mq � fluid flow rate, B/Dre � external radius, ftrw � wellbore radius, ftSo � oil-phase saturation

Sor � residual oil saturationS*o � normalized oil-phase saturationSw � water-phase saturation

Swi, Swr � irreducible water-phase saturationS*w � normalized water-phase saturation

x � estimate of the model parameterWd � data weight matrix

� � pore size distribution index� � Lagrange multiplier

�o � oil-phase viscosity, cp�w � water-phase viscosity, cp

� standard deviation� � estimator’s covariance matrix � effective porosity, fraction

AcknowledgmentsThe authors thank Kamy Sepehrnoori for his assistance during thedevelopment of the two-phase fluid-flow algorithm. Our gratitudeis also extended to three anonymous reviewers whose constructivetechnical and editorial feedback improved the quality of the firstmanuscript. The work reported in this paper was funded by theUniversity of Texas at Austin’s Research Consortium on Forma-tion Evaluation, jointly sponsored by Anadarko, Aramco, BakerAtlas, BP, British Gas, ConocoPhillips, Chevron, ENI E&P, Exx-onMobil, Halliburton, Hydro, Marathon, the Mexican Institute forPetroleum, Occidental Petroleum Corporation, Petrobras, Schlum-berger, Shell International E&P, Statoil, Total, and Weatherford.

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Appendix—Petrophysical Correlations forPhysically Consistent InversionProperties such as permeability are changed as the nonlinear in-version method approaches the minimum of the quadratic costfunction. This variation of rock-formation properties cannot beperformed without the use of physically consistent correlationsbetween relevant petrophysical properties. The purpose of thisAppendix is to describe the petrophysical correlations enforced bythe inversion algorithm to honor transient pressure measurements.

Relative permeabilities are assumed constant for a given typeof rock regardless of permeability. Capillary pressures and poros-ity, however, usually exhibit a strong correlation with permeabilityand therefore need to be automatically adjusted by the inversion.

For the case of porosity, the algorithm first uses average valuesof irreducible water saturation (Swi) and porosity ( b) at the pres-sure sensor depth (usually known a priori from well logs) to obtaina “base” permeability (kb). The correlation model used to enforcea quantitative relationship between irreducible water saturation,porosity, and permeability is the one proposed by Coates andDenoo (1981), namely,

kb = �100 b

2�1 − Swi�

Swi�2

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-1)

where kb is given in mD.Subsequently, the algorithm makes use of the Blake-Kozeny

model to determine a “base” particle diameter (pdb). In metric units,

pdb =150�1 − b�2kb

b3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-2)

Using this expression, porosity ( ) can be consistently scaled fora given value of permeability (k) using the recursive formula

= �150k�1 − �2

pdb2 �1�3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-3)

For capillary pressure, a similar procedure is employed by com-bining the Leverett J-function (Leverett 1941) and a modifiedBrooks-Corey (1964) reference model. Here, the pore-size distri-bution coefficient (�) is assumed constant for a given rock.

Renzo Angeles is a graduate research assistant and a PhDcandidate at the University of Texas at Austin. From 2000 to2003, he worked as a field engineer for Schlumberger, receiv-ing training in Peru, Colombia, Ecuador, the United States, andCanada. He is a recipient of the 2003 Presidential EndowedOsmar Abib Scholarship and the 2005–06 Chevron Scholarship.His interests involve pressure transient analysis, numerical simu-lation of reservoirs, inversion and optimization techniques, andformation characterization. His PhD research focuses on thequantitative analysis and inversion of formation tester mea-surements acquired in highly deviated wells using two- andthree-phase flow analysis including the effects of mud-filtrateinvasion. Carlos Torres-Verdín has been with the Departmentof Petroleum and Geosystems Engineering of University ofTexas at Austin, where he currently holds the position of asso-ciate professor, since 1999. From 1997 to 1999, he was ReservoirSpecialist and Technology Champion with YPF (Buenos Aires,Argentina). Since 1999, he has conducted research on bore-hole geophysics, well logging, formation evaluation, and inte-grated reservoir characterization. He is corecipient of the 2003,2004, and 2005 Best Paper Award by Petrophysics, and he isthe recipient of SPWLA’s 2006 Distinguished Technical Achieve-ment Award. He holds a PhD degree in engineering geo-science from the University of California, Berkeley. Hee-Jae Leeis currently a PhD candidate in the Petroleum and GeosystemsEngineering Department of the University of Texas at Austin. Heis working in the Formation Evaluation Research Group super-vised by Torres-Verdín. His main research area is in geome-chanical fluid-flow coupling, near-wellbore simulation, and isthe main code developer of the University of Texas’s FormationEvaluation Tool Box (UTFET). He holds BSc and MSc degrees inpetroleum engineering from Hanyang University, Korea. FarukO. Alpak is a research reservoir engineer with Shell Interna-tional E&P at Bellaire Technology Center, Houston. His researchinterests include parallel reservoir-simulation techniques, com-putational fluid dynamics, uncertainty analysis, inverse prob-lems, numerical optimization, and electromagnetic wavepropagation. Alpak holds PhD and MSc degrees in petroleumengineering from University of Texas at Austin. James Sheng iscurrently working for Total E&P in Houston as Research Adviser.Previously, he was Lead Scientist with Baker Hughes. He alsoworked as a reservoir engineer with several major and nationaloil companies. His work experience includes reservoir simula-tion and numerical modeling, well testing, wireline testing andsampling, heavy-oil recovery, production forecast, and en-hanced oil recovery. He holds a PhD degree from the Univer-sity of Alberta. He received several professional awards includ-ing the Outstanding Technical Editor Award for SPEREE (2005)and the Best Paper Award in JCPT (1998). He is currently servingas a Review Chairperson for SPEREE.


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