A NEW INVERSION TECHNIQUE DETERMINES IN-SITU RELATIVE PERMEABILITIES AND CAPILLARY PRESSURE PARAMETERS FROM PUMPOUT WIRELINE FORMATION TESTER DATA

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1 A NEW INVERSION TECHNIQUE DETERMINES IN-SITU RELATIVE PERMEABILITIES AND CAPILLARY PRESSURE PARAMETERS FROM PUMPOUT WIRELINE FORMATION TESTER DATA Jianghui Wu, Carlos Torres-Verdín, and Kamy Sepehrnoori, The University of Texas at Austin, Mark A. Proett, Halliburton Energy Services, and Steve C. Van Dalen, ChevronTexaco ABSTRACT Modern pumpout wireline formation testers (PWFT) can monitor and collect a wide array of data including time records of flow rate, pressure, and fluid properties during the pumpout phase of sampling. These transient data are normally only used to help control the sampling process and to estimate sample quality. The inversion algorithm introduced in this paper makes use of the same data to estimate more complex multi-phase formation properties. Such multi-phase formation properties include relative permeability end-points and capillary pressure parameters. Forward models have been developed that can numerically simulate mud-filtrate invasion and the resulting pumpout process as a function of pumping time and rate. Recent developments in invasion modeling make it possible to simulate radial profiles of mud-filtrate invasion for waterbase and oil-base muds where the mud properties are coupled to the invasion process. The same formation parameters can be used to simulate a PWFT pumping sequence thereby providing a forward model for inversion. This paper focuses on the specific case of two-phase immiscible fluid flow in porous media. The main thrust is the application of the two-phase, modified Brooks -Corey parametric model, to describe capillary pressure and relative permeability phenomena. Such a model can be described with a few parameters and hence is amenable to inversion. The free parameters included in the modified Brooks-Corey s model are regarded as unknown variables, while the remaining parameters, e.g., porosity, are treated as a-priori information rendered by ancillary wireline data and/or prior, independent interpretations. Several neural network algorithms are developed to perform the inversion of time domain PWFT pumpout data. The neural network inversion algorithm is also applied to quantify the uncertainty of the estimated modified Brooks -Corey parameters. Inversion examples are described for three representative examples of common field applications. These examples include a synthetic gas -water test and two oil-water field tests with actual PWFT measurements. It is found that reliable estimation of modified Brooks -Corey s parameters is possible in the presence of good quality pumpout data, especially when time-domain sample quality data are available. The stability and reliability of the inversions is also conditioned by the accuracy of a-priori information, such as porosity, absolute permeability, and fluid viscosity. INTRODUCTION One of the primary objectives of modern PWFT has been the collection of pristine formation samples. In the process of collecting a sample, mud filtrate contamination as well as other fluid properties can be monitored using several types of sensors. Additionally, flow rate and pressures are monitored at fluid entry locations. This process is quite similar to rock core measurement techniques designed to determine miscible and immiscible flow properties. While laboratory conditions for core analysis can be controlled precisely, the processes of recovering cores and of preparing them for testing can seriously compromise the measurements. Modern PWFT tools offer the unique opportunity of determining multi-phase formation properties in situ. The evolution of PWFT tools suggests that these devices will eventually become downhole laboratories where fluid samples will be tested and rock properties analyzed with sufficient accuracy and detail to eliminate the need to recover physical samples or core sections. Several recent technical advances are driving such a trend. First, the control of the sampling process is becoming more exacting. Flow control systems and fast-acting motorized valves can make the sampling process a seamless continuous measurement with pressures and flow rates being accurately controlled and monitored. Secondly, nowadays a wider variety of fluid sensors are available for real-time monitoring and control. Future technical developments will include sensors that will drastically improve the physical and chemical characterization of fluid mixtures during the pumpout process. Drilling mud technology and the study of the invasion process have received a great deal of attention in recent years. It is now possible to make predictions of the multiphase filtrate invasion that couples mudcake growth with rock formation properties (Dewan, et al., 2000, Proett, et al., 2001 and Wu, et al., 2001). In this process, the mudcake can dynamically change properties as it becomes compacted against the wellbore. The result is an accurate prediction of the spatial distribution of fluid saturations resulting from the process of mud-filtrate invasion. This spatial distribution of fluid saturations is then used as an initial condition for predicting the time evolution of the pumpout process. Such forward modeling techniques can now be used to build inversion tools where multi-phase formation properties can be determined by matching pumpout measurements (Wu, et al., 2002). 1

2 A few publications have introduced similar estimation techniques. One recent publication by Zeybek, et al. (2001) approached the immiscible invasion problem where water-base mud (WBM) invaded an oil zone. In that paper, the relative permeabilities of a layered zone were first estimated from induction log data. An array induction log was used to determine the invasion profile and then an analytic invasion model was used to determine relative permeabilities. Subsequently, a forward model was used to simulate the PWFT sampling process neglecting capillary pressure effects. The initial estimates of relative permeability were refined via history matching. This inversion process is quite involved and does not lend itself to a real-time application. For miscible invasion in which an oil-base mud (OBM) invades an oil zone, Wu, et al. (2002) showed that horizontal permeability, anisotropy, and porosity could be estimated using only the time record of pumpout data. The inversion technique described by Wu, et al. (2002) utilized a neural network where the training set was constructed from forward modeling. This paper showed that inversion of pumpout data compared very closely to well-log and core data. Additionally, the inversion technique was computationally efficient and could be used in a real-time setting. A sensitivity analysis was performed to verify the accuracy of the estimation procedure in the presence of both miscible as well as immiscible mud-filtrate invasion. In the present paper, the objective is to obtain a more detailed estimation of petrophysical parameters associated with immiscibly invaded zones. We include the effects of capillary pressure as well as relative permeability. A modified Brooks -Corey model is used to parameterize the relative permeability curves and to link them to capillary pressure curves. The inversion involves determining the endpoint of the relative permeability curves as well as their shape and associated capillary pressures. As in Wu, et al. s (2002) paper, forward modeling is used to build a training set for a neural network while a sensitivity analysis verifies the accuracy of the expected results. Three different types of immiscible sampling results are presented. The first is a gas sample taken in the presence of a WBM. The second is a water sample taken in the presence of an OBM. A third log example considers the problem of an oil sample taken in the presence of WBM filtrate invasion. MODERN PUMPOUT WIRELINE FORMATION TESTER (PWFT) Three tools have been introduced over the past 10 years that can be classified as PWFTs. The earliest is the modular dynamic formation tester (MDT ) followed by the Reservoir Characterization Instrument (RCI u ) and more recently the Reservoir Description Tool (RDT * ). Log examples included in this paper use the RDT with a sampling configuration shown in Figure 1. The RDT Mark of Schlumberger, u Baker Atlas, * Halliburton incorporates technological innovations that make it ideally suited to the data collection strategies required for inversion (Proett, et al., 2001). Various modular sections of the RDT are shown in Figure 1 along with their lengths and weights. The RDT can be arranged in a variety of configurations depending on testing needs. Figure 1 shows but one typical configuration. A Power Telemetry Section (PTS) is placed on top of the RDT. This unit conditions the power for the various tool sections. Two Multi-Chamber Sections (MCS) are shown having three 1,000-cm 3 chambers in each section; up to five MCS sections can be configured in the RDT string. The 1,000-cm 3 chambers can be detached immediately after they pass the rotary table. Both the multi-chamber section (MCS) and the chamber valve section (CVS) contain expulsion ports. The latter are positioned so that the sampled fluid must pass the chamber valves before exiting into the wellbore. This passage eliminates stagnant flow line fluids so that they do not contaminate new samples. Chamber valves are motor driven and can be operated while pumping. The CVS is used in conjunction with two standard 1- to 5-gallon sample chambers available in current formation testers. A recently deployed downhole nuclear magnetic resonance fluid analyzer (MRILab) is shown next in Figure 1 as part of the RDT. It provides fluid properties at in-situ reservoir conditions. Optical, resistivity, and dielectric sensors within formation sampling tools have previously been used to ascertain levels of fluid contamination during the pump -out phase. Viscosity, gas -oil ratio, and hydrogen index can be determined from the nuclear magnetic resonance (NMR) sensor. Fluid capacitance and electrical resistivity measurements are also included in the RDT to provide additional ways to distinguish fluids with overlapping NMR relaxation signatures. With this suite of sensors, it is now possible to determine the volume fraction of contaminating filtrate while pumping (Prammer, et al., 2001, and Masak, et al., 2002). RELATIVE PERMEABILITY AND CAPILLARY PRESSURE MODEL For the sake of simplicity but without sacrifice of generality, the developments presented in this paper assume a modified version of Brooks-Corey (Brooks, et al., 1966, and Erkal, et al., 1997) capillary-pressure and relative-permeability model to describe the behavior of immiscible two-phase fluid mixtures in porous media. One important advantage of the modified Brooks -Corey twophase model is that it provides a simple parametric representation of capillary pressure and relative permeability curves, thereby easily lending itself to inversion. In their seminal work, Brooks and Corey concluded that the capillary pressure curve of a two -phase fluid mixture could be described with the following general equation 2

3 P, (1) * 1/ λ c = Pe ( Swt ) where P c represents capillary pressure, P e is the entry capillary pressure as defined by Brooks and Corey, λ is the pore size distribution index, and S * wt is the effective wetting-phase saturation, given by S S S * w wi wt =, (2) 1 Swi S nwr where S wi is residual wetting phase saturation, and S nwr is residual non-wetting phase saturation. The pore size distribution index, λ, included in equation (1) can be calculated from field capillary pressuresaturation data. A modified set of equations for the Brooks-Corey s relative permeability correlation can be written as and k, (3) * 2/ ) ( ) (3 + λ rw = krwo Swt * 2 * (1+ 2 / λ) k (1 S ) [1 ( S ) ], (4) rnw = wt wt where k rw and k rnw are wetting and non-wetting phase relative permeability, respectively. The endpoint value, k rwo, is wetting phase relative permeability in which wetting phase saturation is given by 1-S nwr. Relative permeabilities in the above equations reflect drainage conditions. Imbibition relative permeabilities are assumed to be the same as drainage in this paper. Therefore, only five free parameters, namely, S wi, S nwr, k rwo, P e, and λ, are required to uniquely and completely describe the modified Brooks -Corey s relative permeability and capillary pressure model. Furthermore, S wi and S nwr could be grouped with porosity given the condition that residual fluids cannot flow during pumping. The main thrust of this paper consists of the estimation of the modified Brooks -Corey s parameters from RDT time measurements of pressure, flow rate, and fluid contamination. Three representative case studies have been selected to examine and appraise the estimation problem. The first case considers numerically generated RDT data under the assumption of a gas -bearing formation invaded with a WBM. A second test case makes use of field RDT collected in a water-bearing formation that has been previously invaded with an OBM. The final test case examines RDT field data acquired in an oil-bearing formation previously invaded with a WBM. These 3 test cases provide adequate variety to assess the stability and robustness of the inversion procedure on the one hand, and to appraise the ability of noisy RDT data to uniquely estimate modified Brooks -Corey capillary-pressure and relative-permeability parameters, on the other. NUMERICAL SIMULATION OF PUMPOUT DATA Attention is focused on the case of vertical wells and horizontal beds. In all three case studies, the assumption is made of a single-layer rock formation of finite thickness. No hydraulic cross-flow is assumed between the formation layer and its upper and lower shoulders. It is also assumed that the thickness and porosity of the formation are known from ancillary well-log data. Simulation of mud-filtrate invasion is performed assuming an axisymmetric model of fluid saturation and pressure using the procedure described by Wu, et al. (2001). Subsequently, numerical simulations of RDT pumpout data are performed following the method reported by Wu, et al. (2002). All of the simulations of pumpout data are performed with the commercial reservoir simulator VIP * (Landmark, 1998). Figure 2 illustrates the 3D grid model used in the numerical simulation of RDT data. A total of 60x16x27 nodes in the radial, azimuthal, and vertical directions, respectively, were used to construct the finite-difference grid used in the numerical simulations. Pumpout data consist of time samples of (a) pumping probe pressure variations, P 1, (b) monitoring probe pressure variations, P 2, and (c) degree of sample quality measured at the monitoring probe. The degree of sample quality during the pumpout process is represented by the formation fluid fraction. In the examples considered in this paper, simulations of the pumpout process were performed assuming a constant pumping rate. A logarithmic time sampling schedule was adopted for the numerical simulations, as it was better suited to describe the rapid time evolution of pressures and fluid fractions sampled during pumpout. Time simulations were performed at a sampling rate of 10 samples per decade. In the simulations of pumpout, the mud-filtrate invasion process is halted and the flow rate is reversed at the sand face. Therefore, the initial condition for the simulation of pumpout data is obtained from the corresponding simulated state of mud-filtrate invasion. Simulations of pumpout data are performed assuming three-dimensional (3D) flow-rate sources as well as 3D distributions of fluid saturation. INVERSION PROCEDURE Inversion exercises described in this paper are approached using a neural network algorithm. Given the relatively large CPU times involved in the numerical simulations of invasion and pumpout, and given the large volumes of input data, a neural network algorithm represents a good compromise between accuracy and efficiency. The specific configuration of the neural network (i.e. number of nodes and number of layers) used in this paper for inversion depends on the type of unknown parameter (or parameters). However, in all cases data input to the neural network consists of time samples of pumpout data. Each * Mark of Halliburton 3

4 time sample is assigned an input node on the first layer of the network. The neural network algorithms employed in this paper were constructed and trained using MATLAB s neural network toolbox (Demuth, et al., 2000). Training of the neural networks was performed using the Broyden- Fletcher-Goldfarb-Shanno (BFGS) quasi-newton backpropagation algorithm. This required the selection of a multitude of discrete points in parameter space. Numerical simulations of both mud-filtrate invasion and pumpout were carried out for all of the previously selected discrete points in parameter space, thereby creating a data archive for the training of the neural network. This procedure is similar to that described by Wu, et al. (2002). The constructed neural network consisted of three layers. Nodes in the first layer were used to enter time samples of input data, whereas the third layer was constructed with a number of nodes equal to the number of output parameters. A second hidden layer consisting of 3 nodes was used to provide a connection between the first and third layers of the network. In all inversion examples, an assessment of the uncertainty of the estimated parameters is carried out via additive random perturbations to the input data. In turn, the random perturbations are realized with a zero-mean Gaussian random number generator of specified variance. The assessment of uncertainty is relatively fast as usually one inversion can be performed in less than one minute of CPU time. CASE STUDY No. 1: SYNTHETIC GAS-BEARING ZONE INVADED WITH A WBM This is a synthetic case study that makes use of numerically simulated pumpout data. The model consists of a gas -bearing, single-layer, rock formation previously invaded with a WBM. The objective of this case study is to estimate initial water saturation from PWFT data. Initial water saturation of the formation is an important variable when predicting gas reserves. It can be estimated via the equation P c (S wi ) = (ρ w - ρ g ) g Z, (5) where P c is capillary pressure, ρ w and ρ g are density of water and gas, respectively, g is the gravitational acceleration, and Z is the height of formation above the free water level. If there are two unknowns, i.e. water relative permeability endpoint, k rwo, and entry capillary pressure, P e, the best way to estimate S wi is to use a sequential approach. The first step of such an approach is to use PWFT data to estimate k rwo, and the second step is to estimate entry capillary pressure P e, and therefore the unknown value of initial water saturation. Table 1 summarizes the petrophysical and fluid parameters used in the simulation of WBM invasion and pumpout. After four days of WBM filtrate invasion, the invasion front is located approximately 6 feet into the formation. Pumpout is performed assuming a total sampling time of 360 minutes and a constant pumping rate of 40 cc/sec. The pore size distribution index, λ, is estimated to be 0.45 by fitting equation (1) to experimental capillary pressure data. Step 1: Estimation of the water endpoint of relative permeability curves The water relative permeability endpoint is defined as the water relative permeability value at maximum water saturation. Figure 3 shows modified Brooks-Corey relative permeability curves for different values of k rwo. The degree of sample quality during the pumpout process is here represented by the gas fraction. Simulations of pumpout data are shown in Figures 4 and 5 in the form of time evolution of sample quality and P 1 pumping-probe pressure, respectively. There are a total of 30 sampling times in logarithmic scale from 0.45 to 360 minutes. A neural network algorithm was constructed to take pumpout data as input and to produce estimates of k rwo as output. Two independent neural networks were constructed to perform the inversion. One of the neural networks takes sample quality as input and the other one pumping probe pressure variations, P 1. In these 30 by 1, fully-connected neural networks, 30 input nodes are used to enter an equal number of time-domain samples of the input measurements. The output unit of each neural network is the estimated value of k rwo. We chose a water relative permeability endpoint equal to 0.23 as a test case. In the estimation process, the input sample quality data are perturbed with the addition of zero-mean Gaussian random noise. The objective is to assess the uncertainty of the estimated value of k rwo in the presence of variable amounts of random additive noise. Such a process was performed using zero-mean, additive Gaussian noise of variances equal to 0.2%, 0.4%, 0.6%, 0.8%, and 1.0%, respectively, added to the RDT sample quality data. One thousand realizations of noise were computed for each noise level, thereby resulting in 5,000 noisy data sets input to the neural network. Results obtained from the inversion of these data are shown in Figure 6 in the form of probability curves, one probability curve per value of variance of the additive random noise. For the case of pumping probe pressure data, P 1, were chosen as an input to the inversion. These data were perturbed using zero -mean, Gaussian noise of variance equal to 8, 26, 24, 36, and 40 psi, respectively. One thousand realizations of noise for each noise level resulted in 5,000 noisy input data samples input to the neural network. The corresponding inversion result is shown in Figure 7. Table 2 summarizes the mean value and standard deviation of the estimated value of k rwo. The value of k rwo estimated from the P 1 input data is closer to the true value and the confidence interval is smaller than the corresponding value obtained when using sample quality data as input to the inversion. 4

5 Step 2: Estimation of Initial W ater Saturation Figure 8 shows seven capillary pressure functions corresponding to the same number of values of entry pressure, P e. Figure 9 shows the corresponding values of S wi. The associated simulations of time-domain pumpout data are shown in Figures 10 and 11 in the form of sample quality and pumping probe pressure, P 1, respectively. A neural network was constructed to take time-domain pumpout data as input and to produce a value of S wi as output. Two independent neural networks were considered for this exercise. One of the neural networks takes sample quality data as input and the second one pumping-probe pressure, P 1. Each one of these two neural networks yields independent estimates of S wi. An initial value of water saturation, S wi, equal to was used as a test case. In similar fashion to the preceding inversion of k rwo, the uncertainty of the estimation was appraised by perturbing the input pumpout data with variable amounts of additive zero-mean Gaussian random noise. For the case of time-domain sample quality data, these perturbations comprised 5,000 independent realizations of zero-mean Gaussian noise of variance equal to 0.2%, 0.4%, 0.6%, 0.8%, and 1.0%, respectively. Results obtained from the inversion of these data are described in Figure 12 in the form of probability curves, one probability curve per value of standard deviation of the added noise. For the case of time-domain pumping probe pressure data, P 1 were used as input to the inversion. The uncertainty was appraised by perturbing the input data with 5,000 realizations of zero-mean, additive Gaussian random noise. Variances equal to 8, 26, 24, 36, and 40 psi were used to generate the random Gaussian perturbations. Inversion results are shown in Figure 13 in the form of probability curves, one probability curve per value of variance of additive random noise. Table 3 compares the mean value and standard deviation of the estimated values of S wi using the two neural networks. The two independent estimates of S wi agree extremely well. CASE STUDY No. 2: FIELD DATA ACQUIRED IN A WATER-BEARING ZONE INVADED WITH AN OBM Pumpout data described in this example originate from an offshore hydrocarbon field located in West Africa. The well-log data for this example (refer to Figure 16) were acquired in a well drilled with an OBM through several water-bearing depth intervals. Pumpout data were sampled at a depth of 11,146 ft. A magnetic resonance fluid analyzer (MRILab) was run with the RDT to monitor sample contamination during pumpout. MRILab data were also used to determine in-situ viscosity of mud filtrate and of water in order to better constrain the estimation of absolute permeability from draw-down and build-up pressure data. The first step of the analysis consists of estimating horizontal mobilities and permeability anisotropy. During pumpout, the pumping rate is kept constant at 16 cc/sec. The OBM invasion profile is assumed to be piston-like. At early times, the pumped fluid is OBM filtrate. Early-time records of pumping pressure drop yield a value of horizontal mobility equal to 173 md/cp. Permeability anisotropy was estimated to have a value of 0.04 using the method described by Proett, et al. (2000) and Proett, et al. (2001). Figures 14 and 15 show simulation results performed to assess the sensitivity of monitoring-probe pressure data (P 2 ) to both the modified Brooks-Corey λ parameter and the capillary pressure, respectively. These results clearly suggest that the monitoring-probe pressure P 2 is insensitive to λ, but that it does remain sensitive to capillary pressure, in addition to permeability anisotropy. In this OBM case, after 10 minutes of pumping, fluid around the pumping probe is water while fluid around the monitoring probe is oil. The capillary pressure difference between the two phases adds to the monitoring-probe pressure data, P 2. Observed values of P 2 are close to the numerically simulated results obtained for a pressure entry point, P e, equal to 0.01 psi. This exercise indicates that capillary pressure can be safely neglected in the analysis of RDT pumpout data. The second step consists of estimating k rwo as well as filtrate invasion depth. As suggested by Figure 17, waterbreakthrough time will be sensitive to k rwo. A sensitivity analysis was carried out to study the influence of the depth of k rwo and filtrate invasion on time-domain pumpout measurements. Figure 18 shows how different values of k rwo and of invasion depth will affect the maximum value of both the ratio P 1 /P 2 and water-breakthrough time. A 2x2 neural network was designed for the joint inversion of k rwo and invasion depth. The two input units repres ent the maximum value of P 1 /P 2 and water breakthrough time, respectively. Two output units yield an estimate of both k rwo and invasion depth. In this case, input values of for the maximum measured value of P 1 /P 2, and 1.2 minutes for water-breakthrough time yielded the point (0.12, 0.35 m) in (k rwo, invasion depth) space. The final step consists of estimating the pore-size distribution index, λ. Figure 19 suggests that the P 1 /P 2 ratio is highly sensitive to λ provided that the remaining formation parameters are kept constant. The index λ is estimated to be 1.1. Inversion results were subsequently used to simulate the corresponding transient pumpout measurements. As shown in Figures 20 and 21, the simulated pumpout results are in good agreement with their corresponding field RDT measurements. CASE STUDY No. 3: FIELD DATA ACQUIRED IN AN OIL-BEARING ZONE INVADED WITH A WBM In this example, pumpout data come from a hydrocarbon field located in Australia. The well-log data for this 5

6 example (refer to Figure 27) were acquired in a well drilled with a WBM that intersected several oil-bearing depth intervals. Pumpout data were sampled at a depth of x166 ft. A resistivity fluid analyzer was run with the RDT to monitor sample contamination during pump -out. The pumping rate was kept constant at 20 cc/sec. There are five unknown parameters for the inversion considered in this test case, namely, mud-filtrate invasion volume, horizontal permeability, K h, vertical permeability, K v, endpoint, k rwo, and the pore -size distribution index, λ. Previous sensitivity analyses showed that the RDT measurements were only slightly sensitive to capillary pressure. Therefore, the assumption is made in this case study that capillary pressure effects are negligible. Figure 22 indicates that formation fluid breakthrough occurs at about 9 minutes after the onset of pumpout. Volume of mud-filtrate invasion per formation bed thickness is estimated at 1.65 ft 3 /ft based on the breakthrough time. In this case, sample quality data may be indirectly obtained from the fluid resistivity data shown in Figure 22. The conversion of fluid resistivity to sample quality is performed using a liner-mixing rule of logarithmic resistivity (in this case, BM-filtrate and oil resistivities are equal to and 10 ohm-m, respectively). Before the oil phase breakthrough, the pumpout fluid consists primarily of WBM filtrate and can be considered a single phase. The values of P 2 and P 1 /P 2 during the early pumpout period are then functions of K h, K v, and k rwo. When the pumpout time reaches one minute, the couplet (P 2, P 1 /P 2 ) equal to (35 psi, 14.2) is selected as the reference point. As indicated by Figure 23, given an arbitrary value of k rwo, both K h and K v can be estimated to match the reference couplet (P 2 =35 psi, P 1 /P 2 =14.2). This first step of the inversion yields non-unique estimations: the estimated values of K h and K v depend on the value of k rwo. Sensitivity analysis of pressure responses after breakthrough shows that P 2 remains sensitive to k rwo. As shown in Figure 24, after breakthrough, P 2 may either increase or decrease depending on the specific value of k rwo. A 6x1 neural network was designed for the inversion of k rwo. Six input units are used to enter values of P 2 sampled at 6 pumpout times in the interval from 10 to 31.6 minutes. The output unit yields an estimate of k rwo., in this case equal to By making use of the relationship between k rwo and K h and K v, the values of K h and K v are subsequently estimated to be 260 and 323 md, respectively. The last parameter to be estimated is the pore-size distribution index, λ. All the pressure measurements acquired at late pumpout times will be affected by the value of λ. A 12x1 neural network was designed for the inversion of k rwo. Six input units are used to enter values of P 1 sampled at 6 time locations within the range from 10 to 31.6 minutes. The remaining six input units represent P 1 /P 2 data sampled at the same time locations. The output unit yields an estimate for λ equal to 2.0. Inversion results were subsequently used to simulate the corresponding transient pumpout measurements. As shown in Figures 25 and 26, the simulated transient pumpout data agree well with their corresponding field RDT measurements. CONCLUSIONS PWFT data may provide, at most, five useful pieces of data, namely: two early-time pressure measurements (P 1, P 2 or P 1 /P 2 ); sample quality (breakthrough time); and two late-time pressure measurements (P 1, P 2 or P 1 /P 2 ). This limited information restricts the number of unknown inversion parameters to be equal to or less than five. All other petrophysical parameters must be treated as a-priori information. Numerical studies indicate that wetting-phase fluid may be easily identified from pressure measurements if the fluid viscosities are known from external information. Invasion depth or invasion volume becomes important for the ensuing inversion process. In cases of shallow invasion, such as Case No. 2, a joint inversion of both invasion depth and endpoint k rwo is advised. For the case of relatively deep invasion, such as Case No. 3, invasion volume could be estimated easily from breakthrough time. Pressure measurements usually remain a strong function of relative permeability endpoint and a weak function of the pore-size distribution index. Because of these functional relationships, the inversion algorithm should first attempt to estimate an endpoint before using the pore-size distribution index to fine-tune the results. Values of entry capillary pressure may affect all of the measurements slightly, and therefore this parameter is difficult to estimate. Two exceptions are (a) all the other parameters are known, such as in Case No. 1, (b) the monitoring probe is surrounded by non-wetting-phase fluid while the pumping probe is surrounded by wetting-phase fluid at late pumpout times, such as in Case No. 2. Conclusions from this inversion study are summarized as follows: 1. This paper demonstrates that transient PWFT data exhibit the necessary degrees of freedom to estimate immiscible two -phase flow parameters contained in a modified Brooks-Corey s model of relative permeability and capillary pressure. 2. While the accuracy of the inversion depends on the reliability of a-priori information about ancillary variables, such as porosity, fluid viscosity, etc., adding Gaussian noise perturbations to the transient data shows that the neural network inversion method yields a reasonable inversion error tolerance. 3. Up to five unknown parameters may be inverted from transient PWFT data. For example, mud-filtrate invasion volume, horizontal permeability, K h, vertical permeability, K v, endpoint, k rwo, and the pore-size distribution index, λ were determined using two field case examples. 6

7 4. It is possible to construct an inversion algorithm in which the initial condition (i.e., the invasion profile), the relative permeability endpoint, and the pore-size distribution index, λ, can be estimated in a sequential manner. 5. Entry capillary pressure can be estimated under very special conditions. The best case is when fluid surrounding the pumping probe fluid changes from non-wetting to the wetting phase while the monitoring probe remains surrounded with the non-wetting fluid during the pumpout process. NOMENCLATURE g = Gravitational acceleration K v = Vertical permeability K h = Horizontal permeability k rw = Relative permeability for wetting phase k rwo = Wetting phase relative permeability endpoint k rnw = Relative permeability for non-wetting phase P c = Capillary pressure P e = Entry capillary pressure P 1 = Pumping probe pressure change P 2 = Monitor probe pressure change S nwr = Residual non-wetting phase saturation S w = Wetting phase saturation S wi = Residual wetting phase saturation * S wt = Effective wetting phase saturation t = Time Z = Formation height above the free water level Greek Symbols φ λ ρ g ρ w = Porosity = Pore-size distribution index = Density of gas = Density of water Acronyms CVS = Chamber valve section DPS = Dual probe section FPS = Flow control pumpout section HPS = Hydraulic power section MCS = Multi-chamber section MDT = Modular dynamic formation tester NMR = Nuclear magnetic resonance OBM = Oil-base mud PWFT = Pumpout wireline formation testers PTS = Power telemetry section RCI = Reservoir characterization instrument RDT = Reservoir description tool WBM = Water-base mud ACKNOWLEDGEMENTS We would like to express our gratitude to Halliburton, Baker Atlas, Schlumberger, and Anadarko for funding of this work through UT Austin s Center of Excellence in Formation Evaluation. Special thanks are also extended to David Belanger, of ChevronTexaco Exploration Technology Company, for providing log data for case No. 2 and to Cabinda Gulf Oil Company for their permission to publish one of the field data examples discussed in this paper. REFERENCES Brooks, R.H. and Corey A.T., 1966, Properties of Porous Media Affecting Fluid Flow, Journal of the Irrigation and Drainage Division, Proc. Of ASCE 92, NO. IR2, p Demuth, H. and Beale, M., 2000, Neural Network Toolbox for Use with MatLab, User's Guide, The Math Works. Dewan, J.T., Chenevert, M.E. and Yang X., 2000, A Model For Filtration of Water-Base Mud During Drilling, proceedings of the 41 st Annual SPWLA meeting, Dallas, TX, June 4 7. Erkal, A. and Numbere, D.T., 1997, Relative Permeability Effects on the Migration of Steamflood Saturation Fronts, Paper SPE 38299, proceedings of SPE Western Regional Meeting, Long Beach, CA, June Landmark, 1998, VIP-EXECUTIVE Reference Manual. Masak, P.C., Bouton, J., Prammer, M.G., Menger, S., Drack, E., Sun, B., Dunn, K-J. and Sullivan, M., 2002, Field Test Results and Applications of The Downhole Magnetic Resonance Fluid Analyzer, Proceedings of the 43 rd Annual SPWLA meeting, Oisa, Japan, June 2 5. Prammer, M.G., Bouton, J. and Masak, P., 2001, The Downhole NMR Fluid Analyzer, proceedings of the 42 nd Annual SPWLA meeting. Proett, M. A., Chin, W. C. and Mandal, B., 2000, Advanced Dual Probe Formation Tester with Transient, Harmonic, and Pulsed Time Delay Testing Methods Determines Permeability, Skin, and Anisotropy, Paper SPE 64650, proceedings of the SPE International Oil and Gas Conference and Exhibition in China held in Beijing, China, Nov Proett, M. A., Gilbert, G. N., Chin, W. C. and Monroe, M. L., 2001, New Wireline Formation Testing Tool with Advanced Sampling Technology, Paper SPE 71317, SPE Reservoir Evaluation and Engineering Journal, April. Proett, M.A., Chin, W.C., Manohar, M., Sigal, R. and Wu, J., 2001, Multiple Factors That Influence Wireline Formation Tester Pressure Measurements and Fluid Contacts Estimates, Paper SPE 71566, proceedings of the SPE Annual Technical Conference and Exhibition held in New Orleans, LA, Sept. 30 Oct. 3. Wu, J., Torres-Verdín, C., Sepehrnoori, K. and Delshad, M., 2001, Numerical Simulation of Mud Filtrate Invasion in Deviated Wells, Paper SPE 71739, proceedings of the SPE Annual Technical Conference and Exhibition held in New Orleans, LA, Sept. 30 Oct. 3. Wu, J., Torres-Verdín, C., Proett, M. A., Sepehrnoori, K. and Belanger, D., 2002, Inversion of Multi-Phase Petrophysical Properties Using Pumpout Sampling Data Acquired With a Wireline Formation Tester, Paper SPE 77345, proceedings of the SPE Annual Technical Conference and Exhibition held in San Antonio, TX, Sept. 29 Oct. 2. 7

8 Zeybek, M., Ramakrishnan, T.S., Al-Otaibi, S.S., Salamy, S.P. and Kuchuk, F.J., 2001, Estimating Multiphase Flow Properties Using Pressure and Flowline Water-Cut Data from Dual Packer Formation Tester Interval Tests and Openhole Array Resistivity Measurements, Paper SPE 71568, proceedings of the SPE Annual Technical Conference and Exhibition held in New Orleans, LA, Sept. 30 Oct. 3. ABOUT THE AUTHORS Jianghui Wu is a research assistant and PhD. Candidate in the Department of Petroleum and Geosystems Engineering at The University of Texas at Austin. He received both B.E. and M.S. degrees from the University of Petroleum in China. Formerly, he was a reservoir engineer working for CNPC. Carlos Torres -Verdín received a Ph.D. in Engineering Geoscience from the University of California, Berkeley, in During he held the position of Research Scientist with Schlumberger-Doll Research. From , he was Reservoir Specialist and Technology Champion with YPF (Buenos Aires, Argentina). Since 1999, he is an Assistant Professor with the Department of Petroleum and Geosystems Engineering of The University of Texas at Austin, where he conducts research in formation evaluation and integrated reservoir characterization. He has served as Guest Editor for Radio Science, and is currently a member of the Editorial Board of the Journal of Electromagnetic Waves and Applications, and an associate editor for Petrophysics (SPWLA) and the SPE Journal. Mark A. Proett received a BSME degree from the University of Maryland and a MS degree from Johns Hopkins. He has been involved with the development of formation testing systems since the early 1980s, and has published extensively. Proett holds 16 patents, 14 of which deal with well testing and fluid flow analysis methods. He has served on the SPWLA and SPE technical committees and served as the Chairman for the SPE Pressure Transient Testing Committee. He is currently a Senior Scientific Advisor for Halliburton Energy Services in the Strategic Research group. Kamy Sepehrnoori is the Bank of America Centennial Professor in the Department of Petroleum and Geosystems Engineering of The University of Texas at Austin. His teaching and research interests include computational methods, reservoir simulation, parallel computations, applied mathematics, and enhanced oil recovery. Sepehrnoori holds a PhD degree in petroleum engineering from The University of Texas at Austin. Steven C. Van Dalen is a Senior Formation Evaluation Specialist based in Luanda, Angola. He has a B.S. in Geology from the University of Houston and a M.A. in Geology from the University of Texas at Austin. He joined Chevron in 1981 and has worked as a geologist in development and exploration assignments in Denver, Colorado, Hobbs, New Mexico, and Midland, Texas. Table 1. Summary of Geometrical, Petrophysical, and Fluid Parameters Used in the Construction of Case Study No. 1 (Synthetic Gas-Water Case). Variable Units Base Mudcake Permeability md 0.03 Mudcake porosity fraction 0.60 Mud solid fraction fraction 0.20 Mudcake maximum thickness in 0.10 Water viscosity (filtrate) cp 1.00 Gas viscosity cp Rock compressibility 1/psi 1.0E-6 Water compressibility 1/psi 1.0E-6 Initial formation pressure psi Mud hydrostatic pressure psi Formation permeability md Formation porosity fraction 0.14 Permeability anisotropy fraction 0.14 Total invasion time hours Mudcake rub-off time hours 0.25 Wellbore radius ft 0.35 Formation bed thickness ft Top impermeable shoulder location ft 0.00 Bottom impermeable shoulder location ft Pumping probe location ft 8.00 Table 2. Inversion Case Study No. 1: Summary of input and inversion results for k rwo. # Neural Network Input Mean of k rwo Standard Deviation 1 Sample quality 0.2% noise Sample quality 0.4% noise Sample quality 0.6% noise Sample quality 0.8% noise Sample quality 1.0% noise P 1 with 8 psi noise P 1 with 16 psi noise P 1 with 24 psi noise P 1 with 36 psi noise P 1 with 40 psi noise Table 3. Inversion Case Study No. 1: Summary of input and inversion results for S wi. # Neural Network Input Mean of S wi Standard Deviation 1 Sample quality 0.2% noise Sample quality 0.4% noise Sample quality 0.6% noise Sample quality 0.8% noise Sample quality 1.0% noise P 1 with 8 psi noise P 1 with 16 psi noise P 1 with 24 psi noise P 1 with 36 psi noise P 1 with 40 psi noise

9 Table 4. Summary of Geometrical, Petrophysical, and Fluid Parameters Used in the Construction of Inversion Case Study No. 2 (OBM-Water Case). Variable Units Base Water relative permeability end point fraction 0.12 OBM filtrate invasion depth ft 0.35 Pore size distribution index number 1.10 Capillary entry pressure psi 0.00 Water viscosity (filtrate) cp 5.00 Oil viscosity cp 7.50 Rock compressibility 1/psi 2.5E-5 Water compressibility 1/psi 2.5E-5 Initial formation pressure psi Formation horizontal mobility md/cp Formation porosity fraction 0.20 Permeability anisotropy fraction 0.04 Wellbore radius ft 0.35 Formation bed thickness ft 8.00 Top impermeable shoulder location ft 0.00 Bottom impermeable shoulder location ft 8.00 Pumping probe location ft 4.00 Table 5. Summary of Geometrical, Petrophysical, and Fluid Parameters Used in the Construction of Inversion Case Study No. 3 (WBM-Oil Case). Variable Units Base Water relative permeability end point fraction 0.45 WBM filtrate invasion volume/thickness ft 3 /ft 1.65 Pore size distribution index number 2.00 Capillary entry pressure psi 0.00 Water viscosity (filtrate) cp 1.60 Oil viscosity cp 2.00 Rock compressibility 1/psi 2.5E-5 Water compressibility 1/psi 2.5E-5 Irreducible water saturation fraction 0.10 Residual oil saturation fraction 0.10 Initial formation pressure psi Formation horizontal permeability md Formation porosity fraction 0.31 Permeability anisotropy fraction 1.24 Wellbore radius ft 0.35 Formation bed thickness ft 5.00 Top impermeable shoulder location ft 0.00 Bottom impermeable shoulder location ft 5.00 Pumping probe location ft 2.50 PTS - Power Telemetry Section 7 ft, 220 lbs. MCS - Multi Chamber Section(s) 8.9 ft, 290 lbs. 3-1 liter samples each MRILab - NMR Fluid ID Section 14ft, 400 lbs. FPS - Flow-control Pumpout Section 12 ft, 325 lbs. HPS - Hydraulic Power Section 8.8 ft, 296 lbs. DPS - Dual Probe Section 10.6 ft, 385 lbs. QGS - Quartz Gauge Section 4.2 ft, 102 lbs. CVS - Control Valve Section 4.2 ft, 102 lbs. 2-1 to 5 gal. samples Fig. 1 Diagram showing the various modular components of the RDT implemented with a bottom-hole configuration. Fig. 2 Graphical rendering of the 3D finite -difference grid constructed to simulate the tool s geometry, the surface of the probes, and the surrounding borehole and rock formations. The RDT is centered within a 16 ft-thick bed, shouldered by impermeable beds. 9

10 Fig. 3 Case No. 1. Plots of six water-gas modified Brooks-Corey relative permeability curves for different values of endpoint, k rwo. Fig. 6 Case No. 1. Probability of the K rw endpoint estimated from noisy input sample quality data. Curves are shown for estimated probability from five sets of noisy input pumpout data (0.2%, 0.4%, 0.6%, 0.8%, 1.0% zeromean Gaussian additive noise). Fig. 4 Case No. 1. Plots of the time evolution of sample quality resulting from pumpout in the presence of a WBM. Curves are shown for six different values of water relative permeability endpoint. Fig. 7 Case No. 1. Probability of the K rw endpoint estimated from noisy input P 1 (pumping probe pressure variation) data. Curves are shown for estimated probability from five sets of noisy input data (8 psi, 16 psi, 24 psi, 36 psi, 40 psi additive noise). Fig. 5 Case No. 1. Plots of the time evolution of P 1 (pumping probe pressure variation) resulting from pumpout in the presence of a WBM. Curves are shown for six different values of water relative permeability endpoint. Fig. 8 Case No. 1. Plots of seven capillary pressure functions corresponding to an equal number of entry capillary pressures, P e. 10

11 Fig. 9 Case No. 1. Plot of S wi vs. entry capillary pressure, P e. Fig. 12 Case No. 1. Probability of the S wi estimated from noisy input sample quality data. Curves are shown for estimated probability from five sets of noisy input data (0.2%, 0.4%, 0.6%, 0.8%, 1.0% zero-mean Gaussian additive noise). Fig. 10 Case No. 1. Plots of the time evolution of sample quality resulting from pumpout in the presence of a WBM. Curves are shown for seven different values of S wi. Fig. 13 Case No. 1. Probability of the S wi estimated from noisy input P 1 (pumping probe pressure variation) data. Curves are shown for estimated probability from five sets of noisy input data (8 psi, 16 psi, 24 psi, 36 psi, 40 psi additive noise). Fig. 11 Case No. 1. Plots of the time evolution of P 1 (pumping probe pressure) variation resulting from pumpout in the presence of a WBM. Curves are shown for seven different values of Swi. Fig. 14 Case No. 2. Plots of the time evolution of P 2 (monitoring probe pressure variation) resulting from pumpout in the presence of an OBM. Curves are shown for three different values of l (P e=0.01 psi). 11

12 PRESSURE 4000 PSI 5000 CALI 6 IN 16 GR 0 GAPI 200 DEPTH FEET RHOB 1.95 G/C NPHI 0.45 V/V CPORE T2 Spectrum 50 0 HMRS 50 MTBVIA V/V 0 OHMM Bulk Vol Oil 0.5 V/V 0 HDRS BVW 0.2 OHMM V/V 0 COREPERM MRIL PERM 0.1 MD K 0.1 MD Fig. 16 Case No. 2. The water sample was taken at a depth of 11,146 ft. Core data displayed in the last two tracks were acquired in the original vertical wellbore near this sidetrack. Formation pressure is displayed in track no. 1 (red points). Fig. 15 Case No. 2. Plots of the time evolution of P 2 (monitoring probe pressure variation) resulting from pumpout in the presence of an OBM. Curves are shown for three different values of capillary entry pressure, P e (l=1.0) together with the measured P 2 data. Fig. 18 Case No. 2. Plots of water breakthrough time vs. maximum value of P 1/P 2. Curves are shown for four different value of invasion depth. Each curve has been constructed using five different values of k rwo. Fig. 17 Case No. 2. Plots of the time evolution of sample quality resulting from pumpout in the presence of an OBM. Curves are shown for a combination of three different values of K h and two different values of k rwo. 12

13 Fig. 19 Case No. 2. Plots of the time evolution of P 1/P 2 resulting from pumpout in the presence of an OBM. Curves are shown for three different values of l together with the measured field data. Fig. 22 Case No. 3. Plot of the measured time evolution of pumpout fluid resistivity. Fig. 20 Case No. 2. Numerical simulation (VIP) of RDT data performed with the inverted relative permeability parameters. Plots of the time evolution of P 1 probe (pumping probe), P 2 probe (monitoring probe) pressure variation and P 1/P 2 calculated with VIP using the inverted petrophysical parameters described in Table 4. For comparison, the measured RDT data are shown on the same plots. Fig. 23 Case No. 3. Plot of horizontal and vertical permeability, K h and K v, respectively, as functions of k rwo. These plots were used to match measured pumpout data (P 2, P 1/P 2) with the values (35 psi, 14.2). Fig. 21 Case No. 2. Inversion results obtained from RDT field measurements. Curves are shown of the time evolution of sample quality resulting from pumpout in the presence of an OBM. For comparison, sample quality values estimated from MRILab data are shown on the same plot. Fig. 24 Case No. 3. Plots of the time evolution of P 2 (monitoring probe pressure variation) resulting from pumpout in the presence of a WBM. Curves are shown for seven different values of k rwo together with the measured P 2 data. 13

14 Fig. 25 Case No. 3. Numerical simulation (VIP) of RDT data performed with the inverted relative permeability parameters Plots of the time evolution of P 1 probe (pumping probe), P 2 probe (monitoring probe) pressure variation and P 1/P 2 calculated with VIP using the inverted petrophysical parameters described in Table 5. For comparison, the measured RDT data are shown on the same plots. Fig. 26 Case No. 3. Numerical simulation (VIP) of RDT data performed with the inverted relative permeability parameters. Curves are shown of the time evolution of sample quality resulting from pumpout in the presence of a WBM. For comparison, sample quality values estimated from resistivity fluid analyzer data are shown on the same plot. Fig. 27 Field LWD Data Log for an oil sample with WBM (Case No. 3). 14