SPE 95897

Efficient and Accurate Estimation of Permeability and Permeability Anisotropy from Straddle-Packer Formation Tester Measurements Using the Physics of Two-Phase Immiscible Flow and Invasion Renzo Angeles, Hee Jae Lee, Faruk O. Alpak, and Carlos Torres-Verdín, SPE, The University of Texas at Austin, and James Sheng, SPE, Baker Atlas

Introduction Copyright 2005, Society of Petroleum Engineers

Modular and multi-probe formation testers have proved advantageous in the determination of permeability at intermediate scale lengths due to the increased distance between the observation and sink probes

This paper was prepared for presentation at the 2005 SPE Annual Technical Conference and Exhibition held in Dallas, Texas, U.S.A., 9 – 12 October 2005. This paper was selected for presentation by an SPE Program Committee following review of information contained in a proposal submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to a proposal of not more than 300 words; illustrations may not be copied. The proposal must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

1-3. Moreover, the use of dual-packer or “straddle-packer” modules over point-probe modules is known to improve the interpretation of pressure transient measurements when testing laminated, shaly, fractured, vuggy, unconsolidated, and low-permeability formations 4. Several papers have been published to describe interpretation techniques and applications of these new formation testing approaches

5-7. Abstract The new methodology introduced in this paper interprets formation tester measurements acquired with wireline straddle-packer tools. It incorporates the physics of two-phase, axi-symmetric, immiscible fluid flow to simulate the measurements, and is combined with a rigorous nonlinear minimization algorithm for history matching purposes. Comparable inversion approaches have been recently introduced in the open technical literature

This paper describes the successful application of a novel methodology to estimate permeability and permeability anisotropy from transient measurements of pressure acquired with a wireline straddle-packer formation tester. Unlike standard algorithms used for the interpretation of formation tester measurements, the methodology developed in this paper rigorously incorporates the physics of two-phase immiscible flow as well as the process of mudcake buildup and invasion. The effects of supercharge, skin, and convective transport of salt are also explicitly included in the estimation algorithm.

3,8,9 but they assume single-phase fluid flow. In addition, the developments considered in this paper integrate the flow simulator with a dynamically-coupled mudcake growth and mud-filtrate invasion algorithm

An efficient 2D (cylindrical coordinates) implicit-pressure explicit-saturation finite-difference algorithm is used to simulate both the process of invasion and the pressure measurements acquired with the straddle-packer formation tester. Initial conditions for the simulation of formation tester measurements are determined by the spatial distributions of pressure and fluid saturation resulting from mud-filtrate invasion. Inversion is performed with a rigorous Levenberg-Marquardt nonlinear minimization algorithm. Sensitivity analyses are conducted to assess non-uniqueness and the impact of assumptions on rock-fluid and mud properties such as fluid viscosity, capillary pressure, relative permeability, mudcake growth, and time of invasion.

10 which improves the physical consistency and reliability of the quantitative estimation of both permeability and permeability anisotropy. Methodology There are three main components in the workflow used by this paper:

(1) Mud-filtrate invasion algorithm, (2) Two-phase axi-symmetric simulator, and (3) Nonlinear minimization algorithm.

Transient measurements of pressure are compared to the simulated outputs of a two-phase axi-symmetric simulator to yield new model parameters via nonlinear minimization. The invasion algorithm makes use of these parameters (permeability and permeability anisotropy) in addition to pressure overbalance, invasion geometry, mudcake properties, and other rock-formation properties, to simulate the process of mud-filtrate invasion. Subsequently, the calculated spatial distributions of pressure and fluid saturation resulting from mud-filtrate invasion are used as initial conditions for the

Applications of the estimation algorithm to noisy synthetic measurements include homogeneous, anisotropic, layered, and multi-layered formations, for both low- and high-porosity rocks. We also study the effect of unaccounted impermeable bed boundaries on the inversion results. For cases where a-priori information can be sufficiently constrained, our inversion methodology provides reliable and accurate estimates of permeability and permeability anisotropy.

2 SPE 95897

pressure transient tests. For the synthetic case examples considered in this paper, the geometry of the formation model (e.g. multilayer formations, impermeable bed shoulders) is input to the flow simulator; however, for field measurements, well-log measurements are needed to define the geometrical properties of the rock formation model. Transient flow rate measured during the test is also input to the simulator. To close the loop, the simulator yields pressure transients to be compared against actual measurements. The process repeats itself until the quadratic norm of the residuals between simulations and measurements decreases to a prescribed value. When the latter condition is met, the inversion algorithm yields the final estimates of permeability and permeability anisotropy.

The systematic description of the inversion methodology on synthetic measurements requires the use of a ‘base case’ model (described later), which includes petrophysical variables and geometrical properties that can reproduce arbitrary formation models and tool configurations. Figure 1 shows the measurement configuration for the assumed straddle-packer wireline formation tester. Dimensions of the ‘base case’ tool were chosen according to typical physical dimensions of wireline formation testers designed for interval pressure transient tests (IPTTs): two vertical observation probes and one packer flow area that acts as a sink. The simulator used in this paper was developed by the Formation Evaluation Research Program of The University of Texas at Austin. Numerical Simulation of the Process of Mud-Filtrate Invasion An adaptation of INVADE, developed by Wu et al. 10, is used to simulate the process of mud-filtrate invasion. Simulations include the dynamically coupled effects of mudcake growth and multiphase, immiscible mud-filtrate invasion. In simple terms, the flow rate of mud-filtrate invasion depends on both mud and rock formation properties. The INVADE algorithm assumes that the rock formation is drilled with a water-based mud (WBM).

The ‘base case’ model replicates the conditions of an invaded zone through the injection of brine into the formation during 1.5 days. This value, as well as other assumptions on rock formation and mud properties, was arbitrarily chosen to illustrate the methodology proposed in this paper rather than to describe a general situation. Additional assumptions include the values of brine salinity, equal to 5,000 ppm (1.7525 lb/STB) and formation water salinity, equal to 120,000 ppm (42.0608 lb/STB). Table 1 summarizes the properties of the assumed mudcake. Figure 2 describes the assumed relative permeability and capillary pressure curves.

To couple the outputs of the invasion algorithm with the numerical simulation of pressure transient measurements, the simulator calculates the spatial distributions of pressure, salt concentration, and fluid saturation resulting from 1.5 days of invasion that are input as initial conditions for the simulation of formation tester measurements. Numerical Simulation of Two-Phase Flow The simulation of formation tester measurements is performed with a water-oil two-phase, axi-symmetric reservoir simulator

developed by the Formation Evaluation Program of The University of Texas at Austin. Detailed information about this simulator is given by Lee et al. 11. The simulator uses finite differences and the IMPES (Implicit-Pressure Explicit-Saturation) algorithm to solve the nonlinear system of equations of pressure and saturation. Various boundary conditions can be enforced by the algorithm. Space- and time-dependent variables such as temperature and salt concentration are also included in the simulations. Base Case Model The Base Case Model describes standard measurement parameters and rock formation properties assumed for most of the studies considered in this paper. Figure 3 shows the finite-difference grid along with the physical dimensions of the assumed hydrocarbon-bearing rock formation. The vertical separation between grid nodes is non-uniform, ranging from 0.5 ft near the probes to 1 ft at the remaining grid nodes. In the radial direction, however, the simulator enforces a logarithmic discretization, starting from an initial value of 0.049 ft near the wellbore to 122.2 ft at the outer radial boundary of the reservoir. Figure 4 describes the finite-difference grid used in this paper to assess the effects of impermeable bed boundaries on formation tester measurements.

There are three observation points for the measurement of pressure transients at distances of 5, 13, and 20 ft from the top of the reservoir, respectively. The packer interval (sink) has a length of 2 ft. Upper, lower, and external reservoir boundaries are assumed impermeable (flow rate is zero). Table 2 summarizes the geometrical dimensions of the reservoir model considered in this section, whereas Table 3 describes the associated rock and fluid properties. Initial conditions for formation tester measurements prior to the onset of mud-filtrate invasion are given in Table 4.

The production flow rate during the test is kept constant at 21 STB/D, as shown in Figure 5. In addition, the tests consist of 60 minutes for both drawdown and buildup sequences. Nonlinear Inversion Algorithm Given the complex nonlinear relationship between borehole pressure-gauge measurements and rock formation petrophysical properties, the inversion algorithm requires several sequential linear steps to match the simulated pressure transients with the actual measurements. Similar types of applications can be found in the recent literature 5,7.

The inversion algorithm considered in this paper is based on a modified version of the Levenberg-Marquardt minimization method, developed by Alpak 12. Accordingly, model parameters are obtained by minimizing a cost function composed of the quadratic misfit between measured and simulated samples of transient pressure plus an additive quadratic penalty model functional. This cost function is written as

{ } 22 21( ) ( ) ( ) .2 d xC μ χ p

⎡ ⎤= ⋅ − + ⋅ −⎢ ⎥⎣ ⎦x W e x W x x

(1)

The first additive term of the cost function produces an

estimate of the model x from the misfit of the data within a

SPE 95897 3

where N is the number of unknowns, and yR includes all a-priori information on the model parameters such as those derived from independent measurements. For the purposes of this paper, model parameters are either permeability or permeability anisotropy ratio. When inverting for permeabilities, logarithmic values are used to define the model parameters y, as shown in equation (4). On the other hand, when inverting for permeability anisotropy ratios, raw values are used instead. This transformation increases the convergence rate of the inversion algorithm

prescribed value χ 2 (determined from a-priori estimates of noise in the pressure measurements). The second term is included to regularize the minimization; it provides a safeguard against cases when measurements are redundant or lack sensitivity to certain model parameters thereby causing non-uniqueness. It also suppresses any possible magnification of errors in the parameter estimation due to measurement noise. The scalar parameter μ, also known as Lagrange multiplier (0< μ <∞), determines the relative importance of the two additive terms in equation (1). Wx

T Wx is the inverse of the model covariance matrix representing the degree of confidence in the prescribed model xp. On the other hand, Wd

T

Wd is the inverse of the data covariance matrix, which describes the estimated uncertainties (due to noise contamination) in the available measurements. If the measurement noise is stationary and uncorrelated, then Wd = diag{σj} where σj is the standard deviation of the noise for the jth measurement.

13. If values of mobility are required, the viscosities of the fluids involved are assumed constant by the inversion.

Another feature of the inversion algorithm is the backtracking line search method, which is used along the descent direction to guarantee a reduction of the cost function from one iteration to the next 12.

Cramer-Rao Uncertainty Bounds We use the Cramer-Rao12 approach to estimate the uncertainty of the estimated parameters (permeability, permeability anisotropy). Accordingly, a perturbation is performed of the parameters yielded by the inversion to calculate the Jacobian (J) and to evaluate the expression

The choice of Lagrange multiplier is adaptively linked to the condition number of the Hessian matrix of the cost function. Thus, the weight of the misfit term in the cost function is progressively increased with respect to the regularization term as the inversion algorithm iterates toward the minimum of the cost function. This approach enhances the stability of the solution obtained at each iteration without over biasing the final solution by the specific choice of the regularization term.

1**

22 )()(

−

⎥⎦

⎤⎢⎣

⎡+=Σ xJxJI T

μσσ , (5)

where σ is the standard deviation of the Gaussian noise used to contaminate the pressure data, μ is the Lagrange multiplier, x* is the vector of inverted model parameters, and Σ is defined as the estimator’s covariance matrix. The square root of the diagonal terms (variances) of the latter matrix provide the uncertainty values such that for 99.7% of the occurrences, the ith model parameter will fall within

The inversion algorithm is based on the following vector of residuals e(x):

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

−

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

mMmMsM

mjmjsj

mms

M

j

ppp

ppp

ppp

e

e

e

/))((

/))((

/))((

)(

)(

)( 1111

x

x

x

x

x

x

e(x) =

, (2) iiΣ± 3 of the estimated value12. The uncertainty bounds were calculated only for the case of noisy measurements for layered and multi-layered rock formations. whereupon

∑=

−=M

j mj

sj

pp

1

22 1

)()(

xxe Sensitivity of Straddle-Packer Formation Tester

Measurements to Rock Formation Properties . (3)

This section evaluates the impact of specific assumptions made about rock formation properties on the simulated pressure transient measurements and the corresponding inversion results. The base case model (described earlier) is used to establish the standard measurement parameters and rock formation properties assumed for all the examples under consideration. Correlations between permeability, porosity, capillary pressure, and relative permeability are enforced by the inversion algorithm as described in the Appendix. The flow rate of mud filtrate was restricted to a maximum of 0.003 bbl/day/ft for the examples presented in this section, except in the case of mud-filtrate invasion sensitivity analyses, where flow rates exceeded 0.2 bbl/day/ft.

In the above expressions, M is the number of measurements, pmj is the measured pressure, psj is the simulated pressure, and x is the vector of unknown parameters. The index j attached to the pressure measurements is used to identify the corresponding time sample of pressure. An alternative approach is to define the measurements as differential pressure, i.e., ∆pj = phydrostatic – pj, where phydrostatic is the hydrostatic pressure prior to the time of measurement, or to use the logarithmic value of pressure, log(pj). These alternative methods to define the measurements are discussed by Angeles 13.

The vector of model parameters, x, is given by the difference between the vector of the actual model parameters, y, and a reference model (or background model), yR , i.e., Refer to Tables 5, 6, and 7 for a summary of the results

obtained in this section.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

RN

R

NN y

y

y

y

x

x 111

)log(

)log( = log(y) - yR , (4) x = Data Misfit and Impact Value

In addition to the estimated parameters, there are two diagnostic outputs from the inversion process: (a) Data Misfit

4 SPE 95897

and (b) Impact Value. Data misfit is quantified with the root mean square (RMS) difference between the input and simulated transient pressure measurements at the end of the inversion. This value is given as a percentage of the quadratic norm of the input transient pressure measurements. For the noise free synthetic cases the data misfit is expected to be 0.0%.

The “Impact” value is used to quantify the relative importance of certain assumptions made in the inversion process, e.g. fluid viscosity, irreducible water saturation, level of noise, etc., to obtain estimates of permeability and permeability anisotropy. It ranges from 0 to 100. Impact values close to 100 indicate the largest possible influence of a given assumption on the inverted parameters. Conversely, impact values close to 0 indicate the lowest possible influence of the corresponding assumption on the estimated values of permeability and permeability anisotropy. The “Impact” value is determined from the percentage of the standard deviation of each output parameter (for each case) with respect to the target (actual) value of that parameter and with respect to the total inversion scheme presented in this study. The “Impact” value is provided here for qualitative interpretation of the sensitivity studies and it is influenced by the specific inversion cases considered within a given table of inversion results. Sensitivity to Variations of Permeability Four different types of rock formation configurations are used to assess the effect of the spatial distribution of permeability and permeability anisotropy on the inversion results:

(a) Homogeneous and Isotropic Formation, (b) Homogeneous and Anisotropic Formation, (c) Layered and Isotropic Formation, and (d) Multi-Layered Formation.

Homogeneous and Isotropic Formation Figure 8 describes the pressure responses simulated at the three sampling points of the formation tester: the two monitor probes and the packer. It is observed that the simulated pressure measurements for high-permeability rocks exhibit relatively lower sensitivity to variations of rock permeability than those associated with low permeability rocks, i.e. small variations of permeability cause small variations of pressure only when permeabilities are high. For instance, a variation from 100 to 200 mD modifies the late-time packer pressure response by 5 %. On the other hand, an increase from 900 to 1000 mD, only modifies it by 0.01 %. This effect is further increased when the measurements are artificially contaminated with zero-mean additive Gaussian noise.

Another interesting observation from Figure 8 is that the pressure drop at the probes away from the sink is much lower than that at the sink. Probe pressure variations are given in a few fractions of psi whereas packer responses vary in the range of psi or tens of psi. For example, a pressure drawdown of 7.7 psi is simulated for a 10 mD rock formation at the probe. This represents 6.7 % of the drawdown pressure (114 .8 psi) simulated at the packer for the same formation.

Table 5 shows the inverted values of permeability starting with an initial guess of 40 mD toward their respective target values. In this case, the inversion algorithm uses noise-free synthetic pressure transient data generated with the two-phase

flow simulator at both probes and the packer. A total of 2592 time pressure samples from the complete test interval (including drawdown and buildup) were used by the inversion algorithm. The time sampling used to acquire these pressure samples is the same as the one enforced by the numerical simulator. One iteration requires approximately 3.5 minutes of CPU time on a 1.6GHz Windows-based PC. The inversion reached convergence within 9 iterations. Final inverted permeabilities are the same as the target values. Effect of Additive Zero-Mean Random Gaussian Noise For this exercise, measurements were contaminated with zero-mean additive random Gaussian noise of standard deviations equal to 1 and 10 psi. The pressure responses associated with high-permeability formations are the most affected by the presence of noise, especially at the observation probes where pressure variations are smaller in amplitude than those at the packer flow area. As permeability increases, the error increases up to 20% for the case of 10 psi Gaussian noise. An explanation for this is that high permeability formations entail smaller pressure differentials and hence are more susceptible to the presence of noise than low permeability formations. For instance, a variation of 1 psi of the measured transient pressure for a formation of 1000 mD represents a contamination of 60% of the measurements, and this causes an error equal to 1% in the estimated permeability. Correspondingly, a variation of 1 psi represents a contamination of 8% for 100 mD pressure measurements, and this causes a zero error of the estimated permeability. Figure 9 describes the misfit between the estimated and measured pressure values. Effect of Buildup and Drawdown Measurements In order to assess the relative information content of each stage of the pressure transient test, two inversion schemes were designed: one using only the drawdown pressure measurements and another one using both drawdown and buildup pressure measurements. Both cases assumed noise-free and noisy synthetic data (10 psi-standard deviation of zero-mean additive Gaussian noise). As expected, both test stages (buildup + drawdown) contribute more to the non-uniqueness of the inversion than only one stage (drawdown in this case), especially when the data are contaminated with relatively large amounts of noise (i.e. 10 psi standard deviation zero-mean additive Gaussian noise). Although not shown in the plots, the convergence rate significantly improves when using both measurement stages. Specifically, the minimum of the cost function is further decreased when using both test stages, thereby improving the reliability of the inversion Homogeneous and Anisotropic Formation Three values of anisotropy ratio (defined as the ratio between horizontal and vertical permeability, Kh/Kv) were considered for the base-case rock formation model: 1, 10, and 100. Figure 10 shows the corresponding simulated pressure measurements. From the figures, one can observe that an increase of anisotropy ratio causes an increase of the magnitude of the pressure differential at the packer flow area. Conversely, the pressure differential decreases at the vertical observation probes. This effect is emphasized for the case of high permeability formations, where the corresponding

SPE 95897 5

pressure measurements are much smaller in magnitude than those associated with low permeability rocks. It then follows that the pressure measurements acquired at the probes might not contribute significantly to reduce non-uniqueness of the inversion if the formation permeability is high (above 500 mD for the examples shown in this paper). In the latter case, alternative pressure testing strategies could be used to reduce non-uniqueness of the inversion (e.g. by acquiring pressure measurements at two different depths). This is an important issue because pressure measurements are often corrupted by noise, thereby decreasing their reliability to estimate rock formation properties.

Unlike the scheme adopted for one-parameter inversion, the algorithm was allowed to vary the unknown model parameters within narrower prescribed bounds thereby reducing the problem of non-uniqueness discussed above. For instance, if the inversion algorithm was used to estimate a horizontal permeability of 100 mD, the minimization would be constrained to find a solution between the upper and lower bounds of 1 and 300 mD, respectively, compared to the upper and lower bounds of 0.1 and 10000 mD, respectively, usually enforced for the case of one-parameter inversions.

The inversion algorithm required between 5 to 25 nonlinear iterations to achieve convergence toward the estimated parameters. Compared to the typical range of 5 to 10 iterations normally used for one-parameter inversions, the required number of iterations is relatively high. The algorithm used 552 time pressure samples acquired with the same time sampling interval used by the simulator. Also, it was observed that several combinations of permeability and permeability anisotropy could lead to local minima. This observation indicates that there are several equivalent solutions to the inverse problem.

Different inversion techniques were implemented to obtain the results described above 13. Using a-priori knowledge, the initial guess parameters were given values close to their actual values. In practical applications, such information could be derived from well-log, core, or production data. When this a-priori information was not adequate, the algorithm would continuously re-start the search with different initial guesses to warrant stable convergence while enforcing the same bounds to explore several local minima. The final result was chosen as the one that entailed the lowest data misfit.

However, two problems remain for the case of low permeability formations: (1) more local minima exist than for the case of high permeability formations, thereby biasing the inversion toward values close to the initial guess, and (2) estimated values would converge toward the correct values but the rate of convergence was slow. To circumvent these two problems, the inversion algorithm was modified to use pressure differentials (∆P = Phydrostatic-Pmeasured) rather than raw pressure measurements. The corresponding inversion result is identified with an asterisk (“*”) in Table 5. It was found that, in general, the use of pressure differentials yielded inverted parameters that were in good agreement with the target values. The impact of noise contamination on the input pressure transient measurements is also shown in Table 5. Due to the severe non-uniqueness of the inverse problem, the algorithm made use of several ‘re-start’ values to yield the final estimates. Results are deemed satisfactory.

Layered and Isotropic Formation Figure 11 describes the examples of layered formations considered for this section of the paper. Two observation probes are located in the top layer while the packer section is set against the bottom layer. The permeability of the top layer is different from the permeability of the bottom layer. Rock absolute permeabilities are homogeneous and isotropic within a given layer. Case A assumes permeabilities of 200 and 400 mD for the top and bottom layer, respectively, whereas Case B assumes permeabilities equal to 400 and 100 mD for the top and bottom layer, respectively. Distances between the packer probes and the boundary between the layers are known a priori, i.e., the finite-difference grid used to simulate the measurements is fixed and designed to consider the location of the packer and probes with respect to layer boundaries.

Simulated pressure measurements for the case examples considered in this section are shown in Figure 12. Pressure differentials (∆P) were input to the inversion algorithm instead of raw pressure measurements to mitigate the problem of non-uniqueness. Table 7 summarizes the inversion results and, additionally, it includes the Cramer-Rao uncertainty bounds of the inversion. Although both cases A and B started with an initial guess of 300 mD, case A led to convergence in the first attempt. Case B found 181.3 and 20.3 mD for top and bottom layers, respectively. The result shown was obtained using the “re-start” strategy. It can be clearly seen that the best estimates correspond to the zones close to the packer, where the pressure transients are more sensitive to the rock formation.

Multi-Layered Formation A different inversion methodology is adopted for the case of multi-layered or finely laminated rock formations. Figure 13 shows the example of a layered formation model composed of homogeneous and isotropic layers. Testing of this formation model is performed within the lowest layer, where the packers are located. The vertical observation probes monitor the pressure response which will be mainly affected by the low permeability layers included in the formation model. Notice that the vertical separation of the numerical grid nodes is 0.5 ft near the sampling points and the thickness of the formation layers varies from 1.5 ft (top layer) to 2 ft (medium and bottom layers, respectively).

Figure 14 shows the numerically simulated formation tester measurements. Flow-rate schedules and formation properties are the same as those assumed for the base case formation model. In this example, the inverse problem is non-unique and hence a priori information is necessary to estimate the location of layer boundaries and the initial guess permeabilities. The results shown in Table 7 were obtained with an initial guess of 120, 250, and 150 mD for top, center, and bottom layers, respectively. Table 8 describes the bound constraints imposed for this example.

Sensitivity to Variations of Fluid Viscosity

Given the two-phase nature of the fluid-flow phenomenon assumed in this paper, sensitivity analyses were performed to evaluate the impact of the assumptions made on the viscosity of the fluids involved (oil and water for the base case formation model). Pressure transient measurements were

6 SPE 95897

simulated for values of oil viscosity equal to +500% and -20% of the original viscosity. The same variations are applied for the case of water viscosity variations. It was observed that the impact of oil viscosity is much more significant than that of water viscosity. Pressures change dramatically when oil viscosity is perturbed from its original value. On the other hand, water viscosity affects only the initial pressure measurements during drawdown. This can be explained by recalling the plots of water saturation, relative permeability, and capillary pressure. Specifically, as the drawdown stage begins, water saturation decreases from almost 0.63 to 0.45, i.e. within a region where the relative permeability of water is not as high as that of oil. Thus, the effect of water is noticeable only at the start of drawdown. An increase of oil viscosity decreases the mobility and the corresponding effect on pressure is similar to that of single-phase flow. Note that what is estimated from the inversion algorithm represents the mobility in the uninvaded zone. For the inversion, raw pressure measurements from both drawdown and buildup stages were input to the inversion algorithm using the same time sampling as the one used by the flow simulator. Flow-rate schedule and remaining formation properties were the same as those of the base case formation model.

Sensitivity to Variations of Capillary Pressure and Relative Permeability Table 5 also shows the effects of variations of the assumed water-oil capillary pressure and relative permeability curves on the inversion results. For this purpose, the study makes use of modified Brooks-Corey parameters as well as the base case model properties. Sensitivity analyses consider variations of water-oil capillary pressure and relative permeability in two ways: (a) by changing the pore size distribution index, and (b) by changing the irreducible water saturation. Sensitivity to Variations of Pore-Size Distribution Index Three values of pore-size distribution index, λ, were considered: 0.5 (very wide range), 2 (wide range), and 4 (medium range). The equations used in this exercise are the drainage equations associated with the modified Brooks-Corey model, namely:

orwr

wrww SS

SSS

−−−

=1

*

orwr

orwo SS

SSS

−−−−

=11*

( ) λ1

*,

−= wcedrc SPP

( ) λλ )32(

*,

+

= wdrrw Sk

, (6) , (7) , (8) , and (9)

( ) ( ) ⎥⎦

⎤⎢⎣

⎡−=

+λ

λ )2(*2*

, 1 wodrro SSk

. (10)

where S is fluid saturation, Pc,dr is drainage capillary pressure, kr,dr is drainage relative permeability, Pce is capillary entry

pressure, and λ is the pore-size distribution index. In the above equations, subscripts are used to designate water (w), oil (o), and irreducible saturations (r). The superscript * indicates normalized saturations as defined by equations (6) and (7).

It was observed that, although relatively small, the most significant changes of pressure measurements arise during the drawdown stage. Inversion results shown in Table 5 confirm the latter statement. For these case examples, raw pressure measurements from the two observation probes and the packer were input to the inversion algorithm including both drawdown and buildup stages.

Sensitivity to Variations of Irreducible Water Saturation Variations of the assumed irreducible water saturation were tested for values of 0.15, 0.25, and 0.35 (used in the base case model). As done before, modified Brooks-Corey drainage equations were used to define the capillary pressure and relative permeability curves. Unlike the effect of variations of λ (pore-size distribution index), variations of Swi dramatically influence the simulated noise-free pressure measurements compared to those of the base case model.

Sensitivity to Variations of Production Flow Rate

Figure 15 illustrates the assumed time variations of flow rate for the inversion examples under consideration. For the base case model, the formation tester withdraws formation fluids at a rate of 21 bbl/day from the packer-straddled section of the borehole. Variations of 4 bbl/day to the latter production rate are enforced by the simulator to obtain the inversion results summarized in Table 5. Note that the estimated permeability is increased or decreased in proportion to the decrease or increase of the rate variation, respectively. The inversion algorithm uses raw pressure transient measurements obtained at the two probes and at the packer for drawdown and buildup stages.

Sensitivity to Presence of Impermeable Bed Boundaries Pressure transient measurements remain sensitive to the presence of impermeable bed boundaries in the vicinity of measurement points. The purpose of this section is to assess the influence of unaccounted impermeable bed boundaries on the inversion results for different values of layer thickness. Figure 6 shows the finite-difference grid used to perform the simulations. The packer is located 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 to 0.0001% and 0.000001 mD, respectively. Figure 16 describes the simulated pressure transient measurements.

In general, the pressure differential during drawdown measured by both the observation probe and the packer increases in the presence of impermeable bed boundaries, i.e. the pressure differential during drawdown increases for those cases where the impermeable bed boundaries are located in the vicinity of the packer, thereby biasing the inversion results toward permeability values lower than that of the base case.

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Sensitivity to Mud-filtrate Invasion Parameters This section evaluates the impact of some important assumptions made about parameters associated with the process of mud-filtrate invasion on the permeabilities yielded by the inversion. Specifically, three invasion parameters are given consideration:

(a) Presence of an invasion zone, (b) Time of invasion, and (c) Overbalance pressure.

Sensitivity to the Presence of an Invasion Zone Figure 17 compares the pressure transient measurements simulated at the two probes and the packer when the invaded zone is included in the formation model. The simulation that considers presence of mud-filtrate invasion assumes properties of mud specified in Table 9. Relatively large variations are observed in the simulated pressure transient measurements when the process of invasion is not considered by the simulations. Sensitivity to the Time of Invasion Using the same mud properties described in Table 9, simulations of pressure transients were performed for three different times of invasion: 0.5, 1.0, and 1.5 days. Figure 18 illustrates the pressure responses at one observation probe and the packer. The effect of the unaccounted time of invasion is to increase the inverted value of permeability if the actual time of invasion is shorter than assumed by the inversion.

Sensitivity to the Mud Overbalance Pressure Three different values of mud pressure were input to the flow simulator to generate noise-free synthetic pressure transient measurements: 3300, 3400, and 3500 psi, corresponding to overbalance pressures of 300, 400, and 500 psi, respectively. Incorrect assumptions about mud overbalance pressure (differences around 100 psi) lead to errors of 2% on the estimated permeability.

Inversion Results for the Case of Single-Phase Fluid Flow We compare inversion results obtained with the approach developed in this paper against inversion results obtained when the flow regime is inaccurately assumed to be single phase. Figure 19 shows the pressure transient measurements simulated at the two probes and the packer for the case of a formation with homogeneous and isotropic permeability equal to 10 mD. Single-phase flow was simulated by equating to zero the capillary pressure and water relative permeability curves, and by assigning a value of one to the oil relative permeability curve. Notice the relatively large pressure differential simulated for both types of synthetic measurements assuming the same production flow rate. Unlike single-phase flow, two-phase fluid flow constrains the displacement of oil by the specific value of water saturation, thereby entailing a larger pressure differential than for the case of single-phase flow. Table 6 summarizes the inversion results obtained from this sensitivity analysis. It follows that

inaccurate assumptions about single-phase flow entail permeabilities lower than the actual values. Discussion The inversion algorithm includes different components that add to the complexity of the estimation problem but, at the same time, contribute to improve the reliability of the results. The two-phase character of the flow phenomenon under consideration remains crucial to accurately interpret measurements acquired with a formation tester. Whereas standard pressure-transient techniques assume single-phase fluid flow, water saturation can experience significant variations during drawdown and buildup. This condition also affects the relative permeability and capillary pressure and, as a result, the inverted petrophysical parameters are often significantly conditioned by the limiting assumption of single-phase fluid flow.

Likewise, presence of permeability anisotropy causes a relatively large pressure drawdown at the packer while it decreases the amplitude of pressure transients measured at the probes. This is not a desirable situation for the case of high-permeability rock formations, where the amplitude of pressure transients significantly decreases and hence leads to unreliable inversions in the presence of noise. This fact, coupled with non-uniqueness when inverting more than one unknown parameter, requires the use of alternative strategies to constrain the solution and to re-define the model and input measurements. Similar situations arise for the cases of unknown petrophysical properties associated with layered and multi-layered formations. Inversion exercises emphasize the importance of good initial guesses (obtained from auxiliary measurements such as rock-core and well-log data) as well as physical bound constraints imposed on the unknown properties.

Variations of fluid viscosity reveal fluid-flow characteristics completely different from those of single-phase flow. For the base case model, the deleterious impact of incorrect assumptions made on oil viscosity was significant compared to that of incorrect values of water viscosity. This can be explained by the fact that the saturation region for relative permeability and capillary pressure fluctuates between values of water saturation of 0.45 and 0.63, where most of the displaced fluid is oil. As inferred from the inversion results, the effect of inaccurate assumptions about fluid viscosity is similar to that due to inaccurate assumptions about the character of the flow regime.

A similar behavior was observed for the case of inaccurate assumptions about water-oil relative permeability and capillary pressure curves. It was found that, in general, the pore-size distribution index (λ) associated with a modified Brooks-Corey model marginally affected the inversion results. Significant biases in the inversion results were observed only for the cases of high-permeability rock formations. Conversely, inaccurate assumptions on irreducible water saturation (Swi) have a measurable impact on the estimated values of permeability because of the increase of effective permeability to water in relation to effective permeability to oil.

8 SPE 95897

The sensitivity analysis to presence of impermeable beds was intended to assess the effects of inaccurate assumptions about the distance to upper and lower impermeable beds on the inverted values of permeability. Presence of impermeable beds near the sampling points caused a relatively increase of pressure drawdown measurements at the packer and monitoring probes. Consequently, the inversion algorithm yielded values of permeability lower than the target values. Conclusions The following is a summary of the most important conclusions stemming from this paper: 1. It was shown that considerable variations of pressure can

be caused by inaccurate assumptions made about two-phase rock-fluid properties (such as relative permeability, capillary pressure, and oil viscosity), depending on specific conditions of invasion and values of permeability. Traditional approaches used for the interpretation of dual-packer pressure measurements are based on the assumption of single-phase flow. To estimate permeability, these approaches include a “correction factor” to account for two-phase flow effects. The algorithm developed in this paper explicitly considers the two-phase nature of fluid flow during the acquisition of formation tester measurements.

2. Standard formation-tester interpretation techniques assume a constant invaded zone that often leads to biased estimations of permeability. Reliable estimations of permeability are only possible when the processes of dynamic mudcake growth and mud-filtrate invasion are explicitly taken into account by the inversion.

3. The combination of numerical near-wellbore simulations with a computationally efficient inversion algorithm based on the Levenberg-Marquardt minimization method provides a systematic way to reduce non-uniqueness in the presence of noisy pressure measurements. It was shown that a priori information on the unknown parameters is necessary to reduce non-uniqueness.

4. Synthetic pressure measurements were considered to appraise the reliability of the novel inversion methodology proposed in this paper for the interpretation of wireline formation tests performed with dual-packer modules. Although this approach still needs to be tested with field data, it provided reliable inversion results for the various types of formation models considered in this paper.

Nomenclature C(x) : cost function, [ ] e(x) : vector of residuals, [ ] ∆h : formation thickness, [ft] k : absolute permeability, [mD] kh : horizontal permeability, [mD] kv : vertical permeability, [mD] kh /kv : permeability anisotropy ratio, [ ] kro : oil-phase relative permeability, [ ]

krw : water-phase relative permeability, [ ] krw

0 : krw end point, [ ] kro

0 : kro end point, [ ] M : number of measurements, [ ] N : number of unknowns, [ ] Phydrostatic : hydrostatic pressure, [psi] Pmeasured : measured pressure, [psi] pmj : measured pressure data, [psi] psj : simulated pressure data, [psi] ∆P : pressure differential, [psi] Pc : capillary pressure, [psi] Pce : capillary entry pressure, [psi] Pd : particle diameter for the formation rock, [m] q : fluid flow rate [bbl/day] rw : wellbore radius, [ft ] re : external radius, [ft] Sw : water phase saturation, [ ] Swi : irreducible water phase saturation, [ ] Sw

* : normalized water phase saturation, [ ] So : oil phase saturation, [ ] Sor : residual oil saturation, [ ] So

* : normalized oil phase saturation, [ ] x : estimate of the model parameter, [ ] xp : Prescribed model parameter, [ ] Wx

T Wx : inverse of model covariance matrix, [ ] Wd

T Wd : inverse of data covariance matrix, [ ] Greek Symbols μ : Lagrange multiplier φ : effective porosity, [fraction] λ : pore size distribution index, [ ] χ : a-priori estimates of noise, [ ]

Acknowledgements The authors thank Dr. Kamy Sepehernoori for his assistance during the development of the two-phase fluid flow algorithm. The work reported in this paper was funded by the University of Texas at Austin’s Research Consortium on Formation Evaluation, jointly sponsored by Baker Atlas, BP, ConocoPhillips, ENI E&P, ExxonMobil, Halliburton, Mexican Institute for Petroleum, Occidental Petroleum Corporation, Petrobras, Precision Energy Services, Schlumberger, Shell International E&P, Statoil, and Total.

References 1. Pop, J.J., Badry, R.A., Morris, C.W., Wilkinson, D.J.,

Tottrup, P., and Jonas, J.K., “Vertical Interference Testing With a Wireline-Conveyed Straddle-Packer Tool, ” paper SPE 26481, presented at the 1993 SPE Annual Technical Conference and Exhibition, Houston, October 3-6, 1993.

2. Badaam, H., Al-Matroushi S., Young, N., Ayan, C., Mihcakan, M., and Kuchuk, F.J., “Estimation of Formation Properties Using Multiprobe Formation Tester in Layered Reservoirs,” paper SPE 49141, presented at the 1998 SPE Annual Technical Conference and Exhibition, New Orleans, September 27-30, 1998.

SPE 95897 9

3. Proett, M.A., Chin, W.C., and Mandal B., “Advanced Dual-Probe Formation Tester with Transient, Harmonic, and Pulsed Time-Delay Testing Methods Determines Permeability, Skin, and Anisotropy, ” paper SPE 64650, presented at the SPE International Oil and Gas Conference and Exhibition in China held in Beijing, November 7-10, 2000.

14. Balan, B., Mohaghegh, S., and Ameri, S., “State-Of-The-Art in Permeability Determination From Well Log Data: Part 1- A Comparative Study, Model Development, ” paper SPE 30978, presented at the SPE Eastern Regional Conference & Exhibition, Morgantown, West Virginia, September 17-21, 1995.

4. Ayan, C., Hafez, H., Hurst, S., Kuchuk, F., O’Callaghan, A., Peffer, J., Pop, J., and Zeybek, M., “Characterizing Permeability with Formation Testers,” Oilfield Review, Autumn 2001, p. 2-23.

Appendix Capillary Pressure and Relative Permeability 5. Kuchuk, F.J., “Interval Pressure Transient Testing with

MDT Packer-Probe Module in Horizontal Wells, ” paper SPE 39523, presented at the SPE India Oil and Gas Conference and Exhibition, New Delhi, February 17-19, 1998.

For inversion purposes, these curves are assumed a priori information and are automatically modified in response to variations of rock permeability. Relative permeabilities are assumed constant for a given type of rock regardless of permeability. Capillary pressures, however, usually exhibit a strong correlation with permeability and are therefore adjusted by the inversion algorithm to enforce physical consistency between the two parameters

6. Hurst, S.M., McCoy, T.F., and Hows, M.P., “Using the Cased-Hole Formation Tester Tool for Pressure Transient Analysis, ” paper SPE 63078, presented at the 2000 SPE Annual Technical Conference and Exhibition, Dallas, October 1-4, 2000. For the case of porosity, the algorithm relates the average

values of irreducible water saturation (Swirr) and porosity (poro_base) to a ‘base’ permeability (perm_base). These values are usually known a-priori from well logs; however, in this paper they are assumed equal to those of the base case model. The correlation model used to enforce a quantitative relationship between irreducible water saturation, porosity and permeability is the one proposed by Coates and Denoo

7. Onur, M., Hegeman, P.S., and Kuchuk, F.J., “Pressure-Transient Analysis of Dual Packer-Probe Wireline Formation Testers in Slanted Wells, ” paper SPE 90250, presented at the SPE Annual Technical Conference and Exhibition, Houston, September 26-29, 2004.

8. Xian, C., Carniege, C., Al Raisi, M.R., Petricola, M., and Chen, J., “An Integrated Efficient Approach To Perform IPTT Interpretation,” paper SPE 88561, presented at the SPE Asia Pacific Oil and Gas Conference and Exhibition, Perth, October 18-20, 2004.

14, namely,

22

)1(_

100_ ⎥⎦

⎤⎢⎣

⎡ −=

wirr

wirr

SSbaseporo

baseperm9. Jackson, R.R., Banerjee, R., and Thambynayagam, R.K.M., “An Integrated Approach to Interval Pressure Transient Test Analysis Using Analytical and Numerical Methods, ” paper SPE 81515, presented at the SPE 13th Middle East Oil Show & Conference, Bahrain, April 5-8, 2003.

, (A.1)

where perm_base is given in milidarcies. Subsequently, the algorithm makes use of the Blake-Kozeny model to determine a ‘base’ particle diameter (pd_base) which allows the calculation of a scaled value of porosity (Φ) for the given value of permeability (k). In metric units,

10. Wu, J., Torres-Verdin, C., Proett, M., Sepehrnoori, K., and Belanger, D., “Inversion of Multi-Phase Petrophysical Properties Using Pumpout Sampling Data Acquired with a Wireline Formation Tester,” paper SPE 77345, presented at the Annual Technical Conference and Exhibition, San Antonio, September 29-October 2, 2002.

3

2

__)_1(150_

baseporobasepermbaseporobasepd −

= , and (A.2)

11. Lee, H., Wu, Z., and Torres-Verdín, C., “Development of a Two-Dimensional, Axi-Symmetric Single Well Code for Two-Phase Immiscible Fluid Flow, Salt Mixing, and Temperature Equilibration in the Near Wellbore Region with applications to the Simulation of Mud-Filtrate Invasion And Formation Tester Data,” Fourth Annual Report, Formation Evaluation Program at The University of Texas at Austin, August 19, 2004, Appendix II, p.3.

3/1

2

2

_

)1( 150 ⎥⎦

⎤⎢⎣

⎡ Φ−=Φ

basepdk

. (A.3)

For capillary pressure, a combination of the Leverett J-

function and a Modified Brooks-Corey reference model are used to scale the capillary entry pressure according to the values of porosity and permeability. The pore-size distribution coefficient (λ) is also assumed constant for a given rock. 12. Alpak, F.O., “Algorithms for Numerical Modeling and

Inversion of Multi-phase Fluid Flow and Electromagnetic Measurements,” Ph.D. dissertation, The University of Texas at Austin, Austin, 2005.

13. Angeles, R., “Inversion of Permeability and Permeability Anisotropy from Straddle-Packer Formation Tester Measurements using the Physics of Two-Phase Immiscible Flow and Invasion,” MSc. Thesis, The University of Texas at Austin, Austin, 2005.

10 SPE 95897

Variable Units Value

Mudcake Permeability mD 0.003 Mudcake Porosity fraction 0.400 Mud solid fraction fraction 0.280 Mudcake maximum thickness in 0.300 Mudcake compressibility exponent fraction 0.900 Mud hydrostatic pressure psi 3400 Mudcake rub-off time days 4.000

Table 1: Summary of mud properties assumed in the numerical simulations of mud-filtrate invasion. The initial formation pressure is 3000 psi for the base case, which entails an overbalance pressure of 400 psi.

Variable Units Value Wellbore Radius (rw) ft 0.354 External Radius (re) ft 1000 Reservoir Thickness ft 30 Datum Depth ft 4020 Water/Oil Contact Depth ft 6000 Number of nodes – radial direction -- 61 Number of nodes – vertical direction -- 42

Table 2: Summary of geometrical dimensions assumed for the base-case formation model.

Variable Units Value Porosity fraction 0.18 Water Density @ STP g/cc 1.001 Water Viscosity cp 1.2742 Water Compressibility 1/psi 2.55E-06 Water Formation Volume Factor -- 0.9964 Oil API @ STP -- 42 Oil Viscosity cp 0.355 Oil Compressibility 1/psi 1.90E-05 Production Flow Rate STB/day 21

Table 3: Summary of input rock and fluid properties for the base case formation model.

Variable Units Value

Initial Pressure @ datum psi 3000 Initial Water Saturation (all grid nodes) fraction 0.35 Initial F. Water Salinity (all grid nodes) ppm 120000 Skin Factor -- 0

Table 4: Summary of the assumed initial conditions for formation testing prior to the onset of mud-filtrate invasion (base case model). Tables 5, 6 and 7: see subsequent pages

Actual Permeability Lower Bound Upper Bound (mD) (mD) (mD)

Top Layer: 150 100 200 Medium Layer: 300 200 400 Bottom Layer: 100 20 200

Table 8: Upper and lower physical bound constraints enforced by the inversion of noise-free synthetic pressure measurements for the case of a multi-layer formation model.

Variable Units Value Mudcake Permeability mD 0.03 Mudcake Porosity fraction 0.8 Mud solid fraction fraction 0.16 Mudcake maximum thickness in 0.6 Mudcake compressibility exponent fraction 0.9 Mudcake exponent multiplier fraction 0 Mud hydrostatic pressure psi 3500 Mudcake rub-off time days 4

Table 9: Summary of mud properties assumed for the sensitivity studies of mud-filtrate invasion. The initial formation pressure is 3000 psi for the base case, which causes an overbalance pressure of 500 psi.

SPE 95897 11

Estimated Permeability Input Permeability Impact 10 mD 100 mD 1000 mD Homogeneous & Isotropic (1) only drawdown - noise-free 10.0 [0.00%] 100.0 [0.00%] 1000.0 [0.00%] 0.00 - 10 psi std. Gaussian noise 10.1 [0.72%] 101.9 [0.34%] 1200.9 [0.33%] 5.01 (2) drawdown + buildup - noise-free 9.9 [1.61%] 100.0 [0.00%] 1000.0 [0.00%] 0.00 - 1 psi std. Gaussian noise 10.0 [0.54%] 100.0 [0.03%] 1010.4 [0.03%] 0.26 - 10 psi std. Gaussian noise 11.8 [2.93%] 101.6 [0.34%] 1198.4 [0.33%] 4.94 Homogeneous & Anisotropic (1) anisotropy ratio (Kh/Kv) = 1 - noise-free 10.0 / 1.0 [0.00%] 99.9 / 1.0 [0.00%] 1017.4 / 1.1 [0.00%] 0.43 - 1 psi std. Gaussian noise 10.0 / 1.0 [2.45%] 100.7 / 1.0 [1.22%] 1030.0 / 1.2 [0.94%] 0.75 (2) anisotropy ratio (Kh/Kv) = 10 - noise-free 10.0 / 10.0 [0.12%] 101.4 / 11.0 [0.03%] 979.8 / 8.8 [0.02%] 0.50 - 1 psi std. Gaussian noise 10.0 / 10.8 [2.23%] 101.5 / 9.3 [0.81%] 1027.2 / 9.1 [0.69%] 0.68 (3) anisotropy ratio (Kh/Kv) = 100 - noise-free 10.0 / 99.8 [0.10%] 99.4 / 95.0 [0.00%] 1000.0 / 110.0 [0.00%] 0.01 - 1 psi std. Gaussian noise 10.0 / 107.1 [3.01%] 100.8 / 110.0 [0.68%] 1000.0 / 91.8 [0.33%] 0.02 Effect of Fluid Viscosity (1) effect of oil viscosity - viso = 0.0750 cp 50.2 [0.00%] 497.2 [0.00%] 4992.7 [0.00%] 100.00 - viso = 0.3550 cp 10.0 [0.00%] 100.0 [0.00%] 1000.1 [0.00%] 0.00 - viso = 1.7750 cp 2.0 [0.23%] 20.1 [0.02%] 201.3 [0.00%] 20.01 (2) effect of water viscosity - visw = 0.2548 cp 9.8 [0.05%] 98.8 [0.00%] 987.2 [0.00%] 0.32 - visw = 1.2742 cp 10.0 [0.00%] 100.0 [0.00%] 1000.0 [0.00%] 0.00 - visw = 6.3710 cp 10.1 [0.01%] 100.4 [0.00%] 1000.3 [0.01%] 0.01 Effect of Rel. Permeability and Pc (1) effect of pore-size distribution index - λ = 0.5 10.1 [0.00%] 99.7 [0.00%] 987.2 [0.00%] 0.32 - λ = 2.0 10.0 [0.00%] 99.0 [0.00%] 981.0 [0.00%] 0.47 - λ = 4.0 10.0 [0.00%] 98.6 [0.00%] 976.9 [0.00%] 0.58 (2) effect of irreducible water saturation - Swi = 0.15 4.7 [0.01%] 45.8 [0.01%] 462.5 [0.00%] 13.46 - Swi = 0.25 6.2 [0.01%] 66.4 [0.00%] 675.0 [0.00%] 8.14 - Swi = 0.35 10.0 [0.00%] 99.0 [0.00%] 998.3 [0.00%] 0.05 Effect of Production Flow Rate (1) effect of production flow rate - q = 17 bbl/day 12.4 [0.02%] 123.7 [0.00%] 1238.5 [0.00%] 5.97 - q = 21 bbl/day 10.0 [0.00%] 100.0 [0.00%] 1000.1 [0.00%] 0.00 - q = 25 bbl/day 8.4 [0.04%] 83.9 [0.00%] 839.5 [0.00%] 4.02

Table 5: Summary of sensitivity analyses performed to assess the effect of key parameters assumed in the inversion processes. Numbers within brackets are RMS data misfit errors for the corresponding inverted parameters. Impact values quantify the relative importance of a given parameter on the inversion results.

12 SPE 95897

Estimated Permeability Input Permeability Impact 10 mD 100 mD 1000 mD Effect of Impermeable Bed Boundaries (1) Formation thickness (∆h) = 30 ft - noise-free 10.0 [0.00%] 100.0 [0.00%] 1000.1 [0.00%] 0.00 - 1 psi std. Gaussian noise 10.0 [0.03%] 100.8 [0.01%] 989.5 [0.04%] 0.26 (2) Formation thickness (∆h) = 10 ft - noise-free 8.5 [0.34%] 81.0 [0.05%] 774.8 [0.00%] 5.63 - 1 psi std. Gaussian noise 8.5 [0.34%] 81.6 [0.05%] 748.2 [0.03%] 6.29 (3) formation thickness (∆h) = 2 ft - noise-free 4.0 [1.40%] 37.6 [0.21%] 330.2 [0.02%] 16.77 - 1 psi std. Gaussian noise 4.0 [1.40%] 37.7 [0.21%] 321.5 [0.04%] 16.98 Effect of Mud-filtrate Invasion (1) effect of presence of invasion zone 19.2 [1.52%] 41.6 [0.48%] 46.5 [0.50%] 23.81 (2) effect of time of mud invasion - time = 0.5 days 42.4 [0.18%] 180.5 [0.01%] 1256.0 [0.00%] 6.74 - time = 1.0 days 20.7 [0.18%] 142.8 [0.00%] 1110.8 [0.00%] 2.97 - time = 1.5 days 10.0 [0.00%] 100.0 [0.00%] 1000.1 [0.00%] 0.00 (3) effect of mud overbalance pressure - pressure = 300 psi 10.4 [0.02%] 102.2 [0.00%] 1007.1 [0.00%] 0.19 - pressure = 400 psi 10.0 [0.00%] 100.0 [0.00%] 1000.1 [0.00%] 0.00 - pressure = 500 psi 9.7 [0.02%] 98.2 [0.00%] 994.3 [0.00%] 0.15 Effect of Single-Phase Flow (1) effect of single-phase flow assumptions 5.9 [0.13%] 59.3 [0.03%] 597.4 [0.00%] 10.09

Table 6: Summary of sensitivity analyses performed to assess the effect of key parameters assumed in the inversion processes. Numbers within brackets are RMS data misfit errors for the corresponding inverted parameters. Impact values quantify the relative importance of a given parameter on the inversion results. Estimated Permeability Input Permeability Impact Top Layer Medium Layer Bottom Layer Layered & Isotropic (1) case A <200> -- <400> - noise-free 200.3 -- 400.5 [0.03%] 1.7 - 0.1 psi std. Gaussian noise 213.4 +/- 2.6 -- 400.2 +/- 0.6 [19.6%] 39.5 (2) case B <400> -- <100> - noise-free 399.5 -- 100.0 [0.04%] 1.5 - 0.1 psi std. Gaussian noise 433.9 +/- 1.5 -- 99.5 +/- 0.1 [18.9%] 100.0 Multi-layered (1) case A <150> <300> <100> - noise-free 153.8 297.8 100.0 [0.04%] 9.2 - 0.1 psi std. Gaussian noise 169.5 +/- 1.4 296.9 +/- 0.8 100.0 +/- 0.1 [1.28%] 41.2

Table 7: Summary of sensitivity analysis to assess inversion results in layered and multi-layered formations. Cramer-Rao uncertainty bounds12 are identified with a “+/-“ operator. Numbers within brackets [ ] are RMS data misfit errors for the corresponding inverted parameters. Numbers within < > describe target values for the inversion. Impact values quantify the relative importance of a given parameter on the inversion results.

Figure 1: Configuration of the ‘base-case’ straddle-

packer wireline formation tester consisting of two vertical observation probes and a dual-packer module.

Figure 3: Graphical description of the finite-difference grid used for the numerical simulations associated with the base case model. The reservoir extends from 0.354 ft (wellbore radius) to 1000 ft (drainage radius) horizontally, and from 3990 ft to 4020 ft vertically. The numbers in blue located to the left of the tool schematic indicate the distance in feet from each probe and packer to the top of the reservoir. The total pay thickness is 30 ft.

Figure 5: Graphical description of the finite-difference grid used for the study of impermeable bed boundaries. Numbers in blue located to the left of the tool schematic indicate the distance in feet from each probe and packer to the top of the reservoir. The simulator assumes negligible values of permeability and porosity outside the permeable layer.

Figure 2: Water-oil relative permeability and capillary pressure curves assumed in the numerical simulations of both mud-filtrate invasion and formation tester measurements.

14 SPE 95897

Figure 7b: Noise-free simulated pressure transient measurements at the packer (sink) located at 20 ft from the top of the reservoir model.

Figure 6: Time schedule of flow rate assumed for the simulations of formation tester measurements in connection with the base case model.

Figure 8: Comparison of input pressure measurements and pressure measurements simulated with the permeability yielded by the inversion algorithm. Input pressure measurements were simulated for a formation with homogeneous and isotropic permeability of 100 mD, and were contaminated with additive zero-mean random Gaussian noise of standard deviation equal to 1 psi. The permeability yielded by the inversion is equal to 101.6 mD.

Figure 7a: Noise-free simulated pressure transient measurements at the observation probes located at 5 ft (top) and 13 ft (bottom) from the top of the reservoir.

SPE 95897 15

Figure 10: Layered and isotropic formations considered for case examples A and B, respectively. Capillary pressures, relative permeabilities, and rates of mud-filtrate invasion are assumed the same for the two layers. The two observation probes are located within the top layer, whereas the packer flow area is positioned within the bottom layer. Numbers in blue located to the left of the tool schematic indicate the distance in feet from each probe and packer to the top of the reservoir. Reservoir dimensions and remaining formation properties are the same as those of the base case model.

Figure 11a: Simulated pressure differentials (∆P) for the case example A corresponding to a layered and isotropic formation. The plots describe the time sampling used by both the inversion algorithm and the flow simulator. Pressure measurements are contaminated with zero-mean additive random Gaussian noise of standard deviation equal to 0.1 psi. Measurements simulated at the two observation probes (located in the top layer) and the packer (located in the bottom layer) are input to the inversion algorithm.

Figure 9: Simulated pressure transient measurements for three values of permeability anisotropy ratio (=Kh/Kv): 1, 10, and 100 of a rock formation with horizontal permeability equal to 100 mD.

16 SPE 95897

Figure 11b: Simulated pressure differentials (∆P) for the case example B corresponding to a layered and isotropic formation. The plots describe the time sampling used by both the inversion algorithm and the flow simulator. Pressure measurements are contaminated with zero-mean additive random Gaussian noise of standard deviation equal to 0.1 psi. Measurements simulated at the two observation probes (located in the top layer) and the packer (located in the bottom layer) are input to the inversion algorithm.

Figure 13: Simulated pressure differential (∆P) measurements for the case of a multi-layer and isotropic formation. For the case of noise pressure measurements, the latter are contaminated with zero-mean additive random Gaussian noise of standard deviation equal to 0.1 psi. Pressure measurements acquired by the two observation probes and by the packer are input to the inversion algorithm. Flow rate schedule and remaining rock formation properties are the same as those of the base case model.

Figure 12: Geometrical description of a multi-layer formation model. The packer flow area is located in the bottom layer while the two observation probes sample pressure from the upper layers. These layers are separated by low-permeability intermediate layers whose boundaries are known a-priori. Numbers in blue located to the left of the tool schematic indicate the distance in feet from each probe and packer to the top of the reservoir.

Figure 14: Graphical description of the assumed rate of fluid flow in the simulations of formation tester measurements. The objective is to assess the effect of inaccurately measured flow rates on the inversion results.

SPE 95897 17

Figure 16: Pressure transient measurements simulated to assess the effect of presence of a mud-filtrate invaded zone for the case of a formation with isotropic and homogeneous permeability equal to 100 mD. The two plots shown above describe noise-free pressure measurements simulated at one of the observation probes and the packer.

Figure 15: Effect of impermeable bed boundaries on the simulated pressure measurements for a formation of permeability equal to 100 mD. In the figures, ∆h designates the thickness of the permeable rock shouldered by impermeable beds (top and bottom bed boundaries).

18 SPE 95897

Figure 18: Pressure measurements simulated to assess the effect of inaccurate single-phase flow assumptions on the inverted values of permeability for the case of a formation with isotropic and homogeneous permeability equal to 10 mD. The two plots shown above describe noise-free pressure measurements simulated at one observation probe and the packer. Remaining formation properties are the same as those of the base case model.

Figure 17: Pressure transient measurements simulated to assess the effect of the time of mud-filtrate invasion on the inverted values of permeability for the case of a formation with isotropic and homogeneous permeability equal to 100 mD. The two plots shown above describe noise-free pressure measurements simulated at one observation probe and the packer. Remaining formation properties are the same as those of the base case model.

.