inversion of multi-phase petrophysical properties using...

Copyright 2002, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, 29 September–2 October 2002. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract 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 an abstract of not more than 300 words; illustrations may not be copied. The abstract 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.

Abstract Modern pumpout wireline formation testers (PWFTs) can collect a wide array of data during the pumpout phase of fluid sampling. Both flow rate and pressure are sampled in time during the pumping process and are used to infer apparent permeabilities under the assumption of single-phase fluid flow. Numerical simulation of multi-phase flow has been successfully used to describe filtrate invasion and the resulting pumpout contamination as a function of pumping time and rate. Recent developments in invasion modeling also allow one to simulate the invasion profile for either water-base or oil-base filtrate invasion when the mud properties are coupled to the invasion process. Because of these developments, it is now possible to determine more complex multi-phase petrophysical properties of rock formations. In this paper, we report on a new inversion technique used to estimate petrophysical formation properties from data acquired by a PWFT.

First, a 3D numerical sensitivity study of PWFT data is carried out over a wide range of formation properties including variations of permeability, anisotropy ratio, and porosity. Results from this sensitivity analysis are used as test cases for the inversion algorithm to estimate formation parameters and their uncertainty in the presence of noisy measurements of pressure and flow rate. Inversion is performed making use of a neural network approach. We appraise the robustness and efficiency of the inversion algorithm with actual field data. The estimated formation parameters are further compared to core and wireline data.

Introduction Modern PWFTs are designed primarily to obtain pristine formation fluid samples soon after a well has been drilled. Fluids from the near wellbore region are pumped through a probe for an extended time interval until sensors in the tool determine that a minimal amount of filtrate contamination is present in the sample. Samples taken according to this procedure exhibit less than 5% contamination and hence are considered representative of in-situ PVT conditions. Additional measurements are acquired such as pressure vs. time and flow rates vs. time, while fluid sensors monitor the volume fractions of the fluids present in the flow lines. When all of these parameters are considered as a comprehensive pressure transient data set, it is possible to determine a more complete description of the formation parameters than has been previously attempted. A central objective of this paper is the estimation of formation petrophysical properties from the time record of all measured quantities during the sample pumpout process. Estimation is performed using procedures borrowed from the field of nonlinear inverse theory.

The inversion process typically makes use of a reservoir simulator to reproduce numerically the acquired measurements. One approach is to use a generalized model of the sampled zone that includes the invasion and pumpout boundary conditions. Wireline logs can be used to initialize the inversion. Because of the numerical complexity of the forward problem, the inversion can be rendered very computer intensive and hence slow. To date, inversion algorithms have only been developed in the context of water-base muds (WBM) together with simplifying assumptions made to accelerate the algorithm.1

A more generalized inversion approach is developed in this paper. Forward modeling is used to predict the invasion for both water-base and oil-base muds (OBM). The invasion model was developed previously to include the influence of both filtrate invasion and dynamically coupled mud cake growth.2-3 Subsequently, the pumpout schedule is enforced using a full 3D commercial multiphase fluid-flow simulator (VIP*) that accurately replicates the near wellbore effects associated with both the probe and sealing packer. Pressures

* Mark of Landmark (Halliburton Energy Services).

SPE 77345

Inversion of Multi-Phase Petrophysical Properties Using Pumpout Sampling Data Acquired With a Wireline Formation Tester Jianghui Wu, SPE, Carlos Torres-Verdín, SPE, The University of Texas at Austin Austin; Mark A. Proett, SPE, Halliburton Energy Services; Kamy,Sepehrnoori, SPE, The University of Texas at Austin; David Belanger, SPE, Chevron-Texaco.

2 J. WU, C. TORRES-VERDIN, M. A. PROETT, K. SEPEHRNOORI, D. BELANGER SPE 77345

and filtrate contamination are indicators of how the pumpout process alters the invaded zone. If the formation tester employs a vertical probe located near the source probe, then the additional pressure measurement can provide a more accurate assessment of the cleanup process and can be linked to both sample quality and formation anisotropy. The forward model is also used to perform a general sensitivity analysis and to construct a training data set for a neural network. Up to 720 data sets are used in this paper to construct training sets for neural networks. Three petrophysical properties were used for inversion, namely, absolute permeability, anisotropy, and porosity. These petrophysical parameters are estimated with the aid of the previously constructed neural network using a static feedback protocol. The inversion algorithm based on neural networks becomes much more efficient than alternative gradient-based inversion strategies because of the numerical complexity of the forward problem.

We assess the validity of the neural network inversion algorithm using the reservoir simulator to create a data set for a pumpout sequence acquired in an invaded rock formation with known petrophysical properties. Data input to the inversion include noisy pressures recorded with a dual-probe formation tester, pumping flow rate, and noisy filtrate volume. Examples of inversion are considered for cases of water- and oil-base muds. For a WBM, assumptions are needed regarding the relative permeability of the water and oil phases. To determine how these assumptions can affect the accuracy of the estimated petrophysical parameters, inversions were performed in which errors were made in the input relative permeability curves. Then the changes to the inverted petrophysical parameters were compared to the actual input parameters. In all cases tested, these input variations had only minor effects on the inverted formation properties (i.e., absolute permeability, anisotropy and porosity). A similar technique was performed for OBM invasion models. Finally, several PWFT log data examples were used and the inverted parameters compared with data acquired from independent sources.

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) and more recently the Reservoir Description Tool (RDT). Log examples included in this paper use the RDT with a sampling configuration shown in Fig. 1. The RDT incorporates the latest technological innovations making it ideally suited to the data collection needed for inversion.4

Various modular sections of the RDT are shown in Fig. 1 along with their lengths and weights. The RDT can be arranged in a variety of configurations depending on testing needs; Fig. 1 shows a typical configuration.

PTS - Power Telemetry Section 7 ft, 220 lbs.

DPS - Dual Probe Section 10.6 ft, 385 lbs.

HPS - Hydraulic Power Section 8.8 ft, 296 lbs.

FPS - Flow-control Pumpout Section 12 ft, 325 lbs.

MRILab - NMR Fluid ID Section 14ft, 400 lbs.

MCS - Multi Chamber Section(s) 8.9 ft, 290 lbs. 3 - 1 liter samples each

QGS - Quartz Gauge Section4.2 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

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-cm3 chambers in each section; up to five MCS sections can be configured in the RDT string. The 1,000-cm3 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 (MRILab1) is shown next in Fig. 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 determine when an acceptably low level of contamination

INVERSION OF MULTIPHASE PETROPHYSICAL PROPERTIES USING PUMPOUT SPE 77345 SAMPLING DATA ACQUIRED WITH A WIRELINE FORMATION TESTER 3

exists 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. 5-6

The Flow control Pumpout Section (FPS) shown in Fig. 1 is capable of pumping over 1.0 gal/min at 500 psi with a maximum 4,000-psi pressure differential. A double-acting piston can be easily changed and replaced with a unit that has a 50% larger hydraulic ratio with a 6,000-psi potential and 0.7 gal/min maximum flow rate. It incorporates a precise feedback control system that is capable of controlling the flow rates to within 5% of that specified by the operator. Control of the flow rate is one of the most critical parameters for successful formation evaluation.

The hydraulic power section (HPS) converts electrical power to hydraulic power for the dual probe section (DPS). Whenever required, up to two DPSs can be powered by the HPS. Major components of the DPS include two closely spaced probes (7.25-in. spacing), setting rams, and a 100-cm3-pretest piston pump. Each probe has a high-resolution temperature compensated strain-gauge pressure transducer that can be isolated with a shut-in valve to independently monitor probe pressures. The pretest piston pump also includes a high-resolution, strain-gauge, pressure transducer that can be isolated from the intra-tool flow line and probes. A resistance/capacitance cell is located near the probes to monitor fluid properties immediately after entering either probe. Mud Filtrate Invasion Process Mud filtrate invasion takes place in permeable rock formations drilled with a well that is hydraulically overbalanced by mud circulation. The phenomenon of mud filtrate invasion in permeable rock formations is responsible for both the development of a mud cake on the borehole wall (solids deposition), as well as for the displacement of existing in-situ fluids laterally away from the borehole wall. In general, the mechanical and petrophysical properties associated with mud cake can be characterized by three parameters, namely, reference porosity, reference permeability, and compressibility exponent7.

The following equation relates mud cake permeability with the time pressure change developed across the mud cake itself:

vmc

omc

mc tpk

tk)(

)( = , (1)

where: ko

mc is the permeability (md) of mud cake at a pressure of 1 psi (taken as a reference value),

pmc is the pressure (psi) across the mud cake, and ν is the mud cake compressibility exponent.

In principle, all of the parameters included in Eq. (1) can be determined from an appropriately designed test sequence of mud filtration. At the start of invasion, the filtrate flow rate is at its maximum. As mud cake thickens, the rate is monotonically reduced to an asymptote value. When the mud cake reaches its maximum thickness, the invasion is governed by the fluid saturation front. In the case of mud cake removal or “rub-off”, mud cake thickness is reduced to zero and hence allowed to rebuild. To some extent, mud cake growth is also dynamically coupled to the invasion model and increases at a rate that is controlled by the invasion front that continues to penetrate the formation. A mud cake boundary condition is imposed to predict mud cake growth based on the base mud properties determined in a standard filtration test. 2

In this paper, we make use of a general numerical algorithm to simulate the physics of mud filtrate invasion in vertical and highly deviated boreholes. This algorithm was adapted from an existing multi-phase fluid-flow simulator developed by The University of Texas at Austin, and commercially referred to as UTCHEM. 8

Simulations of mud-filtrate invasion performed with UTCHEM provide an equivalent time-domain flow rate function. Subsequently, this flow rate function can be enforced as a source condition to more comprehensive commercial reservoir simulators such as VIP, for instance. In turn, simulations of invasion with the previously determined equivalent flow rate function provide initial conditions for the simulation of the pumpout process.

Numerical Simulation of Mud-Filtrate Invasion with a Water-Base Mud. Water-base muds (WBM’s) consist of a mixture of solids and liquids in which water becomes the continuous phase. For the simulations considered in this paper, the rock is assumed homogeneous and anisotropic, while mud-filtrate and in-situ oil are assumed immiscible. The simulations also assume that the rock is water wet. Table 1 summarizes the petrophysical and fluid parameters used in the simulation of water-based mud-filtrate invasion.

Parameterization of Relative Permeability Curves. A common quantitative description for relative permeabilities is based on the power functions

wenw

orwrw Skk = , (2)

and oe

noororo Skk = , (3)

where korw is the endpoint relative permeability for residual oil

saturation, Snw is the normalized water saturation and ew is a real number. Similarly, ko

ro is the endpoint relative permeability for irreducible water saturation, Sno is the normalized oil saturation and eo is a real number. Fig. 2 shows the oil-water relative permeability curves assumed in the simulation of mud-filtrate invasion with a WBM.


Fig. 2 – Water-oil relative permeability curves used in the simulation of mud-filtrate invasion with a WBM.

Capillary Pressure Hysteresis. Capillary pressure hysteresis is characterized by bounding imbibition and drainage curves as well as intermediate scanning curves. The capillary pressure will follow the imbibition curve for increasing wetting-phase saturation; and the drainage curve, for decreasing wetting-phase saturation. Assuming water as the wetting phase, capillary pressure becomes governed by the imbibition curve when a WBM invades an oil zone. However, when the pumpout schedule is given its inception, capillary pressure becomes governed by the drainage curve.

In this study, capillary pressure is represented by the power function

penocic SPP = , (4)

where Pci is the maximum capillary pressure and ep is a real number. Fig. 3 shows the imbibition and drainage curves used in this study with Pci=35, ep=3 for imbibition, and ep =1.5 for drainage.

Fig. 3 – Capillary pressure curves used in the simulation of mud-filtrate invasion and pumpout in the presence of a WBM.

Mud-Filtrate Invasion Results. A benchmark example was constructed to simulate the invasion process. Petrophysical properties associated with this benchmark example are summarized in Table 1, including the relative permeability and capillary pressure curves shown in Fig. 2 and Fig. 3. The model consists of a homogeneous bed with thickness equal to 50 ft and limited vertically by no-flow

boundaries. Results from the simulation of mud filtrate invasion are shown in Fig. 4 in the form of radial profiles of water saturation as a function of invasion time. A succession of radial profiles of water saturation is displayed in Fig. 4 representing one-hour increments of invasion time. At the end of 36 hours of mud-filtrate invasion, the mud cake is removed simulating the effect of swabbing or conditioning of the well prior to a logging run. Removal of mud cake results in a sudden increase of water saturation. We remark that hysteresis in capillary pressure is responsible for the abnormally high values of water saturation simulated in the near-wellbore region after the onset of invasion. 9

Fig. 4 – Results from the numerical simulation of mud-filtrate invasion with a WBM. The curves shown correspond to radial profiles of water saturation displayed at time increments of one hour after the onset of invasion. A total of 48 curves are displayed spanning an equal number of hours of invasion after the onset of the invasion process. Numerical Simulation of Mud-Filtrate Invasion with an Oil-Base Mud. Oil-base muds (OBMs) are similar in composition to water-base muds, except that the continuous fluid phase is oil instead of water; existing water droplets become emulsified into the oil phase. We have simulated the invasion process associated with an OBM via Todd and Longstaff’s miscible displacement algorithm10 in which the OBM is treated as a solvent.

In Todd and Longstaff’s algorithm, the effect of partial mixing on effective viscosities is described via the empirical equations

ωω µµµ mooe−= 1 , (5a)

and ωω µµµ msse

−= 1 , (5b)

where µ is fluid viscosity, the subscripts o, and s, stand for oil and solvent, respectively, and ω is a mixing parameter. In the above equations, a null value for the mixing parameter corresponds to the case of a negligible dispersion rate, while a value of 1 for ω corresponds to the case of complete mixing within a simulation grid block. Effective viscosities, symbolized in equations (5a) and (5b) with the secondary subscript e, identify equivalent mixture viscosities, and are governed by the ¼–power fluidity-mixing rule, namely,


44/14/1

⋅+⋅

=

sn

oo

n

s

som

SS

SS

µµ

µµµ , (6)

where So, Ss, and Sn are oil, solvent, and non-wetting phase fluid saturations, respectively.

Mud-Filtrate Invasion Results. A representative benchmark example was chosen to simulate the OBM invasion process. The viscosity of the OBM was assumed to be is 1 cp and the mixing parameter was assigned the value of 0.5. Remaining geometrical, petrophysical, and fluid parameters for this numerical simulation exercise are described in Table 1. Fig. 5 displays radial profiles of solvent (OBM) fraction away from the borehole wall. Radial profiles are shown at one-hour time increments after the onset of the invasion process and spanning a total of 48 hours of mud-filtrate invasion.

Fig. 5 – Results from the numerical simulation of mud-filtrate invasion with an OBM. The curves shown correspond to radial profiles of water saturation displayed at time increments of one hour after the onset of invasion. Forty-eight curves are displayed spanning an equal number of hours of invasion after the onset of the invasion process.

Numerical Simulation of the Pumpout Process The pumpout sampling process is described by the multi-phase/ multi-component flow equations for which the initial pressure and fluid saturation conditions are imposed by the mud-filtrate invasion process. Modern formation testers can monitor the pressure response and, if the flow rate is regulated, can infer information about the formation properties and the invasion front. These results can also predict sample contamination as well as the time required to obtain a pristine formation sample. A commercial reservoir simulator such as VIP can enforce the same 3D grid model both for invasion and pumpout simulation. The pumpout sequence is simulated by stopping the invasion and reversing the flow rate at the sand face.

Three-Dimensional Grid Model. The 3D grid model was constructed to reproduce the geometry of the pressure probes and the surface of the sealing packer elements and probes

against the surrounding cylindrical borehole and rock formations. Probe interfaces were simulated with six vertical grid points per quadrant, amounting to a total of 36 grid points for the complete 3D surface of the probe. Fig. 6 illustrates the 3D grid model. A total of 60x16x27 nodes in the radial, azimuthal, and vertical directions, respectively, were used to construct the finite-difference grid used in the VIP simulations. In similar fashion with the simulations of mud-filtrate invasion, we assumed a homogenous, 50 ft-thick bed, shouldered by impermeable beds. The RDT is centered within this bed. Extensive finite-difference grid studies were conducted to produce internally consistent and accurate numerical simulations.

Fig. 6 – 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.

Numerical Simulation of the Pumpout Process in the Presence of a WBM. We focus our attention to the case of vertical wells and horizontal beds. The degree of sample quality during the pumpout process is represented by the oil fraction. Simulations of the pumpout process were performed assuming a total sampling time of 100 minutes and a constant pumping rate of 20 cc/sec. A logarithmic time sampling schedule was adopted for numerical simulation as it was better suited to describe the rapid time evolution of pressures and fluid fractions. Time simulations were performed at a sampling rate of 10 samples per decade.

Fig. 7 displays a plot of the simulated pumping results in the form of radial profiles of water saturation away from the borehole wall. The red curve describes the initial water saturation condition imposed by the invasion process. A set of 30 curves is used to describe the water saturation profile for the 30 time samples considered in the simulation process between 0.1 and 100 minutes. As time progresses, water saturation decreases and monotonically approaches the value of irreducible water saturation.


Fig. 7 – Plots of water saturation as a function of radial distance away from the borehole wall simulated numerically from a pumpout schedule in the presence of a WBM. Initial pressure and fluid saturation conditions were derived from the simulations of mud-filtrate invasion shown in Fig. 4. Profiles of water saturation are shown for logarithmic sampling time after the onset of the pumpout process. Thirty profiles of water saturation profiles are shown to describe the time evolution of water saturation after the onset of the pumpout process.

Fig. 8 presents the result of a second numerical simulation

and shows the time evolution of sample quality (fractional volume). Sample quality exhibits an approximate linear relationship with respect to logarithmic time.

Fig. 8 – Plot of sample quality vs. time simulated numerically for a pumpout schedule in the presence of a WBM. Initial pressure and fluid saturation conditions were derived from the simulations of mud-filtrate invasion shown in Fig. 4. Numerical Simulation of the Pumpout Process in the Presence of an OBM. Fig. 9 and Fig. 10 summarize the results obtained from the simulation of pumpout in the presence of an OBM. As in the case of a WBM, pumpout was enforced at a constant rate of 20 cc/sec. The behavior of the sample quality curve shown in Fig. 10 is quite different from that simulated for the case of a WBM.

Fig. 9 – Plots of solvent fraction as a function of radial distance away from the borehole wall simulated numerically from a pumpout schedule in the presence of an OBM. Initial pressure and fluid saturation conditions were derived from the simulations of mud-filtrate invasion shown in Fig. 5. Profiles of solvent fraction are shown for logarithmic sampling time after the onset of the pumpout process. Thirty profiles of water saturation profiles are shown to describe the time evolution of water saturation after the onset of the pumpout process.

Fig. 10 –Plot of sample quality vs. time simulated numerical for a pumpout schedule in the presence of an OBM. Initial pressure and fluid saturation conditions were derived from the simulations of mud-filtrate invasion shown in Fig. 5. Sensitivity Analysis A study was performed to assess the sensitivity of data collected by the RDT during pumpout to petrophysical properties of rock formations. For simplicity, but without sacrifice of generality, we focused our attention to only three petrophysical parameters, namely, absolute formation permeability (k), permeability anisotropy ratio (λ = kv/kh, where kv and kh are vertical and horizontal absolute permeabilities, respectively), and porosity (φ). Remaining petrophysical and geometrical variables were left equal to those described earlier in the context of numerical simulation of mud-filtrate invasion and pumpout. We denote the virtual 3D space of possible petrophysical properties with the triplet (k, λ, φ). The space of possible values spanned by the three parameters is limited by obvious physical bounds (e.g. 0 ≤ φ ≤ 1, and 0 ≤ λ ≤ 1). Moreover, permeability values are assumed distributed along a logarithmic scale. For reference, the benchmark model considered earlier in the numerical


simulation studies of mud-filtrate invasion is located at the point (k = 100, λ = 1.0, φ = 0.25). When describing sensitivity studies in the sections to follow, we assume that variations of a given parameter are made while the remaining two parameters remain equal to those of the benchmark model. Sensitivity of RDT Data to Rock Petrophysical Parameters in the Presence of a WBM. Thirty-six numerical simulations were performed to assess the sensitivity of RDT data to petrophysical variables in the presence of a WBM. Formation permeability values, k , were set at 60, 100, 400, and 1000 md. Values of the permeability anisotropy ratio, λ, were set equal to 0.1, 0.3, and 1.0. Finally, porosity, φ, was set equal to 0.15, 0.25, and 0.35. The combination of 36 sensitivity studies is represented as discrete points in the (k, λ, φ) space shown in Fig. 11. For comparison, graphical results from our sensitivity analysis are displayed together with those yielded by the reference case described by the triplet (k =100, λ =1.0, φ =0.25), and plotted with a continuous curve.

Fig. 11 – Graphical description of the 3D petrophysical parameter space (permeability, permeability anisotropy ratio, porosity) with the location of the discrete points chosen to perform a sensitivity analysis of pumpout data in the presence of a WBM.

Pressure change measured with the pumping probe, P1. As described earlier, the RDT comprises two pumping probes, denoted as P1, and P2. The pumping probe pressure change, p1(t), is defined as the difference between initial formation pressure and pumping probe pressure. Figures 12 through 14 summarize the results of our numerical simulation study to assess the sensitivity of p1(t) to permeability, anisotropy ratio, and porosity, respectively. Simulation results indicate that the change in pumping probe pressure measured by P1, p1(t) , remains more sensitive to formation permeability than to porosity or anisotropy ratio. A larger and more constant (with respect to time) variation in probe pressure is observed with a change in formation permeability than with a change in the two other parameters. (It is also observed that a small value of

permeability anisotropy ratio will produce relatively large values of p2(t)).

Fig. 12 – Plots of the time evolution of P1 probe (pumping probe) and P2 probe (monitoring probe) pressure variation resulting from pumpout in the presence of a WBM. Pressure curves are shown corresponding to four different values of formation permeability.

Fig. 13 – Plots of the time evolution of P1 probe (pumping probe) and P2 probe (monitoring probe) pressure variation resulting from pumpout in the presence of a WBM. Pressure curves are shown corresponding to three different values of anisotropy ratio in formation permeability.

Pressure change measured with the monitoring probe,

P2. The monitoring probe pressure change p2(t) is defined as the difference between initial formation pressure and monitor probe pressure. Figures 12 through 14 describe the simulation of p2(t) for the complete set of 36 discrete points in the parameter space shown in Fig. 11. Similar to p1(t), the time evolution of monitor probe pressure change remains more sensitive to permeability than to either anisotropy ratio or porosity. It is observed that a high value of anisotropy ratio will result in high values of p2(t).


Fig. 14 – Plots of the time evolution of P1 probe (pumping probe) and P2 probe (monitoring probe) pressure variation resulting from pumpout in the presence of a WBM. Pressure curves are shown corresponding to three different values of formation porosity.

Ratio of pressure change. This ratio is defined as p1(t)

over p2(t). Fig. 15 shows results from the simulation of the time evolution of pressure ratio for the complete set of 36 points in parameter space (Fig. 11). The plots are displayed in groups of three families (each family is identified with a distinct color) for an equal number of values of anisotropy ratio. Within each group of curves, plots are shown for the remaining combinations of formation parameters in parameter space. It can be observed from Fig. 15 that the ratio of pressure change remains sensitive to permeability anisotropy. At late sampling times, pressure ratios for all 36 cases will coalesce into three distinct groups with no member variability.

Fig. 15 – Plots of the time evolution of pressure change ratio resulting from pumpout in the presence of a WBM. Families of curves are shown for three different values of anisotropy ratio in formation permeability. Each family of curves consists twelve members (combinations of four different values of permeability with three different values of porosity).

Sample quality. Plots of sample quality vs. time are shown

in Fig. 16 for four values of formation permeability. It is observed that sample quality remains equally sensitive to both permeability and porosity.

Fig. 16 – Plots of the time evolution of sample quality resulting from pumpout in the presence of a WBM. Curves are shown for four different values of formation permeability.

Fig. 17 – Plots of the time evolution of sample quality resulting from pumpout in the presence of a WBM. Curves are shown for three different values of anisotropy ratio in formation permeability.

Fig. 18 – Plots of the time evolution of sample quality resulting from pumpout in the presence of a WBM. Curves are shown for three different values of formation porosity. Sensitivity of RDT Data to Rock Petrophysical Parameters in the Presence of an OBM. The set of 36 simulations considered in the sensitivity study of RDT acquired in the presence of an OBM is graphically described in Fig. 11. Thirty-six simulations were considered in this paper to perform the sensitivity analysis. Formation permeability


values, k , were set at 40, 100, 400, and 1000 md. Values of the permeability anisotropy ratio, λ , were set equal to 0.1, 0.3, and 1.0. Finally, porosity, φ , was set equal to 0.15, 0.25, and 0.35.

Pressure change measured with the pumping probe, P1. Figures 19 through 21 describe the results of our numerical simulation study to assess the sensitivity of p1(t) to permeability, anisotropy ratio, and porosity, respectively. The simulation results indicate that the pumping probe pressure change measured by P1, p1(t) remains more sensitive to formation permeability than to porosity or anisotropy ratio. It is also observed that the influence of porosity on p1(t) is different from that simulated for the case a WBM. Specifically, p1(t) increases with an increase in porosity.

Fig. 19 – Plots of the time evolution of P1 probe (pumping probe) and P2 probe (monitoring probe) pressure variation resulting from pumpout in the presence of an OBM. Pressure curves are shown corresponding to four different values of formation permeability.

Fig. 20 – Plots of the time evolution of P1 probe (pumping probe) and P2 probe (monitoring probe) pressure variation resulting from pumpout in the presence of an OBM. Pressure curves are shown corresponding to three different values of anisotropy ratio in formation permeability.

Fig. 21 – Plots of the time evolution of P1 probe (pumping probe) and P2 probe (monitoring probe) pressure variation resulting from pumpout in the presence of an OBM. Pressure curves are shown corresponding to three different values of formation porosity.

Pressure change measured with the monitoring probe, P2. Figures 19 through 21 also describe the simulation of p2(t) for the complete set of 36 discrete points in the petrophysical parameter space (Fig. 11). Similar to p1(t), the time evolution of monitor probe pressure change remains more sensitive to permeability than to either anisotropy ratio or porosity. Rules governing the mixing of fluid viscosities (equations 5a and 5b) are responsible for the markedly irregular behavior of p2(t) with respect to variations in both anisotropy ratio and porosity (Fig. 20 and Fig. 21, respectively).

Ratio of pressure change. Similar to the simulations performed in the presence of a WBM, Fig. 22 shows that the ratio of pressure change remains sensitive to permeability anisotropy. As time progresses, simulations for of all 36 cases will merge into three distinct curves, one curve per value of anisotropy ratio.

Fig. 22 – Plots of the time evolution of pressure change ratio resulting from pumpout in the presence of an OBM. Families of curves are shown for three different values of anisotropy ratio in formation permeability. Each family of curves, plotted with a distinct color, consists of twelve members (combinations of four different values of formation permeability with three different values of formation porosity).


Sample quality. As shown in Fig. 23, the sensitivity of sample quality to a variation in formation permeability becomes negligible. On the other hand, Fig. 24 and Fig. 25 indicate that sample quality exhibits approximately the same sensitivity to both anisotropy ratio and porosity.

Fig. 23 – Plots of the time evolution of sample quality resulting from pumpout in the presence of an OBM. Curves are shown for four different values of formation permeability.

Fig. 24 – Plots of the time evolution of sample quality resulting from pumpout in the presence of an OBM. Curves are shown for three different values of anisotropy ratio in formation permeability.

Fig. 25 – Plots of the time evolution of sample quality resulting from pumpout in the presence of an OBM. Curves are shown for 3 different values of formation porosity.

Inversion of Petrophysical Parameters The inverse problem consists of estimating a set of petrophysical parameters that will best reproduce (in a least-

squares sense) the pumpout sampling data acquired with a wireline formation tester. In other words, the inversion process will locate the point in parameter space (k, λ, φ) that best reproduces the complete time record of p1(t), p2(t), and sample quality. As in any inverse problem, special considerations are necessary to address the deleterious effect of non-uniqueness (multiplicity of solutions) and presence of noise in the data. The sensitivity study presented earlier provides a conceptual basis to guide the inversion process in selecting specific subsets of the available data for the estimation of a given parameter. Using Neural Networks to Solve Inverse Problems. Neural networks themselves embody an estimation problem similar to inversion. The optimal distribution of the weights between the neuron connections of a multi-layer perceptron, for instance, is determined using the same optimization techniques developed in the context of general inversion algorithms. Ideally, one could study a possible relationship between the complexity of training a neural network (i.e., the optimization of its weights) and the complexity of mathematically inverting the network approximates from the training data set.

Training Data Set. A major difficulty in the usage of neural networks to solve inverse problems is the selection of training samples. It is well known that, under certain circumstances, the addition of small amounts of noise to the training data set of a neural network can lead to significant improvements in estimation performance.11-12 For the inversions considered in this paper, training data sets were constructed with simulations for the 36 discrete points of 3D parameter space shown in Fig. 11. For each of the 36 simulations, 20 ancillary data sets were generated by adding independent sets of zero-mean Gaussian noise of standard deviation equal to 2% of the simulated RDT data. The original 36 numerical simulations plus their noisy variants (20 noisy variants per numerical simulation) constituted the final 720 data sets used to train the neural network.

Network Design. A static, feed-forward network was implemented for the inversion. This type of network does not iterate to a final solution but instead directly translates the input signals to an output independent of previous input. The network configuration should allow for an adequate description of the underlying statistical distribution of the spread in the data. For instance, a network that is restricted to too few training sets will not capture sufficient structure in the data, while one that is too complex will replicate the noise on the data (the deleterious phenomenon referred to as over-fitting). In either case, the performance on new data, that is the ability of the network to generalize its parameter space of operation, will be poor. Since the number of input and output neurons is fixed in the application pursued in this paper (30 input units and 1 output unit), our main concern is the number of neurons in the hidden layer. No rules exist for determining the exact number of neurons in a hidden layer. However, Huang and Huang (1991) show that the upper bound of number of neurons needed to reproduce exactly the desired


outputs of the training samples is on the order of the number of training samples.13 Thus, the number of neurons in the hidden layer should never exceed the number of training samples. Moreover, to keep the training problem over-constrained, the number of training samples should always be larger than the number of internal weights. In practice, the best configuration is found by trial and error, starting with a small number of nodes. For the application considered in this paper, extensive testing showed that a neural network with five hidden units was a good compromise between efficiency and accuracy. We made use of Matlab’s neural network toolbox to construct the network.14 Inversion of Petrophysical Parameters in the Presence of a WBM

Step 1: inversion of permeability. From the sensitivity analysis described earlier, it was found that both p1(t) and p2(t) remain sensitive to permeability. In addition, a single-phase spherical well test formula shows that p1(t) is closely related to spherical permeability while p2(t) is closely related to horizontal permeability.4,15 Following these valuable remarks, we opted to make use of p2(t) measurements to train the neural network. The network consists of 30 input units, each unit assigned to a specific time sample of p2(t). A single output unit for the network is designed to take the role of formation permeability.

Step 2: inversion of permeability anisotropy. In principle, the anisotropy ratio can be estimated solely from the time measurements of pressure change ratio, p1/p2. Although the pressure ratio merges into distinct groups of curves at late times (Fig. 15), pressure ratio measurements acquired for different values of anisotropy ratio may overlap at early times. For example, the early time p1/p2 ratios associated with (k = 1000, λ = 0.1, φ = 0.25) and (k = 60, λ = 0.3, φ = 0.25) are close to each other. To reduce the effect of non-uniqueness on anisotropy ratio due to permeability, the estimate of permeability yielded by the previous inversion step should be used as input.

The network constructed for the inversion of permeability anisotropy consists of 31 input units. One unit is assigned to permeability and the remaining units are assigned to time samples of the p1/p2 ratio. A single output unit takes the role played by the anisotropy ratio.

Step 3: inversion of porosity. From the sensitivity study presented earlier, it follows that the time record of sample quality is almost exclusively influenced by formation porosity. However, the influence of permeability and anisotropy on sample quality is not negligible. Prior information of both permeability and anisotropy is therefore necessary to reduce non-uniqueness.

The neural network constructed to estimate values of porosity consists of 32 input units. Two units are assigned to values of permeability and anisotropy determined from the previous two sequential inversion steps. The remaining 30 input units are assigned to the individual time samples of sample quality. A single output unit plays the role of porosity.

Iterative Approach. The point in (k, λ, φ) space yielded by sequential inversion steps 1 through 3 can be refined using an iterative approach. With prior information of anisotropy and porosity, the estimation of permeability can be made slightly more accurate. To this end, the neural network of Step 1 is modified to include 32 input units. The two extra input units are used to assign values of anisotropy ratio and porosity. This iterative process continues until there is no improvement on the estimated triplet of petrophysical parameters.

Test case of inversion. The point (150, 0.2, 0.2) in (k, λ, φ) space was chosen as a test case to generate the numerically simulated RDT measurements. Subsequently, random noise was added to the numerically simulated data in order to assess the effect of noisy data on the inversion. The addition of noise was accomplished using 1000 independent realizations of zero-mean, 2.5% time-dependent Gaussian multiplicative noise (the standard deviation of noise was made equal to 2.5% of the absolute value of the time-domain data, one simulation of noise per time sample). A total of 1000 inversions were performed, one per realization of noise added to the input RDT data. Inversion results obtained independently for the 1000 noisy data sets provided valuable statistics to appraise the ability of the data to yield robust estimates of petrophysical parameters.

Fig. 26 illustrates the convergence of the inversion algorithm as a function of iteration count. Only five iterations were necessary to achieve convergence. Petrophysical properties shown in this figure (one panel per petrophysical property) correspond to the mean value obtained from the independent inversions of 1000 noisy data sets. The inversion algorithm yields the point (150.9, 0.195, 0.194) as the optimal estimate of petrophysical properties. Fig. 27 shows a comparison between the input noisy and noise-free data.

Fig. 26 – Plots of the value of inverted petrophysical parameter as a function of iteration count in the neural network inversion algorithm. The inversion was performed from noisy RDT data simulated numerically in the presence of a WBM.


Fig. 27 – (a) Plots of noisy sample quality data input to the inversion of petrophysical parameters. The simulated data (shown in red) were contaminated with Gaussian, zero-mean, 2.5% multiplicative noise when input to the inversion.

(b) Plots of noisy sample monitor-probe pressure data, P2, input to the inversion of petrophysical parameters. The simulated data (shown in red) were contaminated with Gaussian, zero-mean, 2.5% multiplicative noise when input to the inversion. (c) Plots of noisy P1/P2 pressure ratio data input to the inversion of petrophysical parameters. The simulated data (shown in red) were contaminated with Gaussian, zero-mean, 2.5% multiplicative noise when input to the inversion.

Histograms of the inverted petrophysical parameters are

shown in Fig. 28. These histograms indicate that estimates of permeability lie in the interval [130, 163], of anisotropy ratio in the interval [0.182, 0.213], and of porosity in the interval [0.12, 0.33]. The relatively short statistical spread of the inverted petrophysical parameters indicates that the inversion process is adequately robust to produce estimates of both permeability and anisotropy ratio.

Fig. 28 – Histogram of the petrophysical parameters estimated from noisy input RDT data acquired in the presence of a WBM.

Clearly, assumptions are needed regarding the nature and

properties of relative permeability and capillary pressure before attempting the inversion of RDT data into values of permeability, anisotropy ratio, and porosity. To determine how

these assumptions can affect the accuracy of the inversion, we performed inversions of the same data set described above except that erroneous assumptions were made regarding the parameters governing both the relative permeability and capillary pressure curves. Results from this exercise are summarized in Table 2. Case No. 1 in that table describes the inversion results obtained from noisy RDT data in which correct assumptions were made regarding the parameters of both relative permeability and capillary pressure curves. Additional cases reported in the same table consider different types of erroneous assumptions made on the two sets of curves. It is observed that estimates of both permeability and porosity remain relatively unaffected by an error made in the nature of the relative permeability and capillary pressure curves. Inversion of Petrophysical Parameters in the Presence of an OBM

Similar neural networks were designed for the inversion of RDT data in the presence of an OBM. Measurements of p2(t) are used to estimate permeability, of P1/P2 ratio to estimate anisotropy, and of sample quality to estimate porosity.

Test case of inversion. The point (50, 0.2, 0.2) in (k ,λ ,φ) space was chosen as a test case to simulate numerically the input RDT data. As in the case of a WBM, 1000 realizations of zero-mean, Gaussian, 2.5% multiplicative noise, were generated and used to contaminate the numerically simulated data. The inversion of these data yielded the estimate (50.4, 0.195, 0.209) for the unknown petrophysical parameters (k ,λ ,φ).

Fig. 29 shows histograms of inverted petrophysical parameters sampled from 1000 independent inversion results. These histograms indicate that estimates of permeability lie in the interval [50.0, 50.8], of anisotropy ratio in the interval [0.192, 0.198], and of porosity in the interval [0.186, 0.237]. The relatively short statistical spread of the inverted petrophysical parameters lends credence to the robustness of the inversion algorithm in the presence of noisy RDT data.

Fig. 29 – Histogram of the petrophysical parameters estimated from noisy input RDT data acquired in the presence of an OBM.


In the presence of OBMs, assumptions of fluid viscosities and mixing parameter are required to carry out the inversion of permeability, anisotropy ratio, and porosity from RDT data. To determine how these assumptions can affect the accuracy of the inversion, we performed an exercise in which the values of fluid viscosity and mixing parameter were slightly modified from the actual values of the same parameters used in the numerical simulations of RDT data. Results from this exercise are summarized in Table 3. Case No. 1 in that table makes use of the correct values of fluid viscosity and mixing parameters to estimate the three unknown petrophysical parameters from noisy RDT data. It is observed that the estimate of permeability remains relatively stable with a slight change in the assumed fluid-property values. However, the estimated values of anisotropy ratio and porosity are greatly affected by a slight change in the same fluid properties.

Field Data Example Data described in this example was acquired in an offshore operating hydrocarbon field in West Africa. Currently, several mature reservoirs in the field are being subjected to water flood to enhance recovery. Fluid samples acquired with the RDT in these water-flooded reservoirs are used for verifying formation fluid type and water salinity when water

breakthrough is suspected. The log data shown in Fig. 30 comes from a well drilled with an OBM through intervals with free gas, injected sea water, formation water, and oil. The pumpout sample was taken at x104 ft. A magnetic resonance fluid analyzer (MRILab) was run with the RDT to monitor sample contamination during pump-out. The MRI data was also used to determine in-situ viscosity of mud filtrate and of oil to improve estimates of permeability from drawdown and build-up pressure data. Inversion of Petrophysical Properties. The sampling sequence data for this field example is shown in Fig. 31 and Fig. 32 along with the inversion results. Pressure data is shown in Fig. 31 for two probes where one probe was used for pumping the sample; and the second probe, positioned 7.25-in. axial distance from the first, was used to monitor pressure. The filtrate contamination was measured using the MRILab and is plotted in Fig. 32. After the filtrate contamination was reduced below 10%, a 1000 cc sample was taken. Subsequent compositional lab testing on the sample revealed that the actual contamination was less than 4%.

The inversion process consisted of expanding the neural network training set to include the range of conditions presented by this formation. Viscosities for filtrate and crude

GRGAPI0 200

CALIIN6 16

RDT PRESSPSI750 1750

X100

X150

DE

PT

HNPHI

V/V0.45 -0.15

PHIDV/V0.45 -0.15

ILDOHMM0.2 2000

ILMOHMM0.2 2000

T2 DIST

0.3 3000

LOG PERMMD0.1 10000

MRIL PERMMD0.1 10000

RDT PERMMD0.1 10000

COR PERMMD0.1 10000

PHIV/V0.5 0

BVWV/V0.5 0

MTBVITA%50 0

Fig – 30. Field Data Log Example. The RDT sample was taken in an oil zone at x104. These composite log displays show raw log data, core data, and computed results. Caliper, gamma ray, and formation pressure are presented in track 1. Neutron and density porosity are in track 2. Medium and deep resistivity are presented in track 4. The magnetic resonance T2 distribution is presented in track 5. Permeability from various sources is presented in track 6: CORE PERM is air permeability measured on core plugs taken from the whole core; RDT PERM is from pretest drawdown pressure data; MRIL PERM is from the Coates relationship and LOG PERM is from a regression between core air permeability and conventional log data. Track 7 is a bulk volume porosity analysis with total porosity (PHI), bulk volume water (BVW), and bulk volume of bound water from the MRIL.


oil sampled were determined by the MRILab and confirmed later by lab test to be 1.8 cp and 4.0 cp respectively. Inversion of RDT data acquired in this formation yielded the triplet (k = 993, λ = 0.049, φ = 0.22). These estimates were subsequently input to VIP to simulate the corresponding pressure measurements. As shown in Fig. 31 and Fig. 32, the VIP-simulated results are in good agreement with the RDT measurements. Comparing Inversion Results to Core, Log and Lab Tests Table 4 is a summary of the petrophysical parameters yielded by the inversion along with core, log, RDT pretest, and lab sample results. Comparison of all of the values in Table 4 shows a high degree of consistency. The two core air permeabilities were determined close to this depth with values of 550 to 850 md. The integrated “LOG PERM” of 1008 md and the “MRIL PERM” permeability of 668 md were within the range of variance of the core and RDT neural network inversion results.

The RDT pretest horizontal permeability of 750 md was somewhat lower than the inversion results.2 This lower result can be attributed to the fact that the pretests were performed before the pumpout sequence, and the MRILab viscosity for filtrate was used to determine the permeability. It is assumed that filtrate was present in the formation and in the PWFT flow lines during this pretest. Normally the Mobility (md/cp) is reported for pretest because the formation fluid and viscosity is not known. This lower pretest value can be attributed to filtrate fines that partial plug the formation as well as the higher crude viscosity that may be partially mixed with the filtrate near the wellbore. Because these factors are near the wellbore they can be considered a skin effect which can change the anisotropy estimate as well.15

Finally, the contamination results of the sample were confirmed through compositional lab tests that determined the contamination to be less than 4%. Although slightly lower than the MRILab and VIP simulation shown in Fig. 32, all three measurements were in good agreement.

Fig. 31 – Results of inversion using RDT field data. Plots of the time evolution of P1 probe (pumping probe), P2 probe (monitoring probe) pressure variation and P1/P2 calculated with VIP using the inverted petrophysical parameters described in Table 4. For comparison, the measured RDT are shown on the same plots.

Fig. 32 –Results of a VIP simulation using the inversion from the RDT field data. Curves show 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. Summary and Conclusions 1. This paper introduces the first combined miscible and

immiscible multiphase PWFT sampling modeling for OBM and WBM invasion environments. This new model was demonstrated to more accurately predict the complete set of transient data recorded by a PWFT during the sampling process.

2. The invasion modeling includes a coupled mudcake growth model where the properties of the mud cake and the filtrate invasion govern the mud cake thickness. Previous models assume a constant mud cake thickness or a constant invasion rate.

3. The first neural network inversion model was demonstrated where the PWFT pressure data and filtrate contamination sensor data are used to determine the formation properties. Previous methods assume single phase or estimate invasion and contamination using additional wireline log data. In this case, only the PWFT data is required to invert the absolute permeability, anisotropy and porosity.

4. The robustness of this inversion method was demonstrated by using simulations where noise was introduced, and selected input parameters were varied to test the sensitivity to the inversion parameters.

5. An OBM filed example demonstrated that the new inversion technique yields results that compare favorably to core, log, and lab sample analysis data.

6. The advantage of the neural network inversion algorithm over standard nonlinear inversion methods is the efficiency with which estimations can be performed from large transient data sets. The neural network inversion algorithm can efficiently combine dual-probe pressure data with NMR fluid sensor data (MRILab) to provide real-time estimates of petrophysical properties.

7. In principle, a large base of training data sets could be constructed prior to acquiring the PWFT data. The inversion algorithm could then be trained during the preparation of the logging run to activate as soon as transient data have been recorded by the PWFT.


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 Boqin Sun, of ChevronTexaco Exploration Technology Company, for analysis of the MRILab log data and to Cabinda Gulf Oil Company for their permission to publish the field data examples discussed in this paper. Nomenclature

eo = Exponent for oil relative permeability ew = Exponent for water relative permeability ep = Exponent for capillary pressure k = Formation permeability kv = Vertical permeability kh = Horizontal permeability kmc = Mud cake permeability ko

mc = Mud cake permeability reference value kro = Relative permeability for oil ko

ro = Endpoint relative permeability for oil krw = Relative permeability for water ko

mc = Endpoint relative permeability for water Pc = Capillary pressure Pci = Maximum capillary pressure pmc = Pressure across the mud cake p1 = Pumping probe pressure change p2 = Monitor probe pressure change Sno = Normalized oil saturation Snw = Normalized water saturation Sn = Non-wetting phase saturation So = Oil saturation Ss = Solvent saturation t = Time µo = Oil viscosity µo = Mixture viscosity µs = Solvent viscosity µoe = Oil effective viscosity µse = Solvent effective viscosity ν = Mud cake compressibility exponent

Greek Symbols φ = Porosity λ = Permeability anisotropy ω = Mixing parameter

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

References 1. Zybek, M., Ramakrishnan, T.S., Al-Otaibi, S.S., Salamy, S.P.,

Kuchuk, F.J.: “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, prepared for presentation at the 2001 SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, 30 September-3 October 2001.

2. Proett, M.A., Chin, W.C., Manohar, M., Sigal, R., Wu, J.: “Multiple Factors That Influence Wireline Formation Tester Pressure Measurements and Fluid Contacts Estimates,” paper SPE 71566, prepared for presentation at the 2001 SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, 30 September-3 October 2001.

3. Wu, J., Torres-Verdin, C., Sepehrnoori, K., Delshad, M.: “Numerical Simulation of Mud Filtrate Invasion in Deviated Wells,” paper SPE 71739, prepared for presentation at the 2001 SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, 30 September-3 October 2001.

4. Proett, M. A., Gilbert, G. N., Chin, W. C. and Monroe, M. L.: "New Wireline Formation Testing Tool with Advanced Sampling Technology," paper SPE 71317, SPE Reservoir Evaluation & Engineering Journal, April 2001.

5. Prammer, M.G., Bouton, J., Masak, P.: “The Downhole NMR Fluid Analyzer,” paper presented at the 42nd Annual SPWLA meeting, 2001.

6. Masak P.C., Bouton, J., Prammer, M.G., Menger, S., Drack, E., Sun, B., Dunn, K-J., Sullivan, M.: “Field Test Results And Applications of The Downhole Magnetic Resonance Fluid Analyzer ,” paper presented at the 43rd Annual SPWLA meeting, Oiso, Japan June 2–5, 2002.

7. Dewan, J.T., Chenevert, M.E. and Yang X.: “A Model For Filtration of Water-Base Mud During Drilling,” paper presented at the 41st Annual SPWLA meeting, Dallas, Texas, June 4-7,2000.

8. Delshad, M., G.A. Pope, and K. Sepehrnoori: "A Compositional Simulator for Modeling Surfactant Enhanced Aquifer Remediation," Journal of Contaminant Hydrology, 23, 1996, p. 303-327.

9. Killough, J.E.: “Reservoir Simulation With History-Dependent Saturation Functions,” Trans. AIME 261, page 37-48,1976

10. Todd, M.R., Longstaff, W.J.: “The Development, Testing, and Application of a Numerical Simulator for Predicting Miscible Flood Performance,” SPE 3484,1972.

11. Van der Baan, M., Jutten, C.: “Neural networks in geophysical applications,” Geophysics, 65 (4), July-August issue, 2000.

12. Bishop, C. M.: “Training with noise is equivalent to Tikhonov regularization ,”Neural Comp. vol.7, pp.108-116,1995.

13. Huang, S. C., and Huang, Y. F.: “Bounds on the number of hidden neurons in multiplayer perceptrons ,” IEEE Trans. Neur. Networks, vol.2, pp.47-55, 1991.

14. Demuth, H., Beale, M.: Neural network toolbox for use with Matlab, User's Guide, The Math Works, 2000.

15. 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, China, 7–10 November 2000.


Table 1. Summary of Geometrical, Petrophysical, and Fluid Parameters Used in the Construction of the Reference Models of Mud-Filtrate Invasion. Variable Units Base Mudcake Permeability md 0.01 Mudcake porosity Fraction 0.40 Mud solid fraction Fraction 0.50 Mudcake maximum thickness In 1.00 Water viscosity (filtrate) Cp 1.00 Oil viscosity Cp 3.00 Initial water saturation Fraction 0.10 Initial formation pressure Psi 9000.00 Mud hydrostatic pressure Psi 10000.00 Formation permeability Md 100.00 Formation porosity Fraction 0.25 Permeability anisotropy Fraction 1.00 Total invasion time Hours 48.00 Mudcake rub-off time Hours 36.00 Wellbore radius ft 0.40 Formation bed thickness ft 50.00 Top impermeable shoulder location ft 0.00 Bottom impermeable shoulder location ft 50.00 Pumping probe location ft 25.00

Table 2. Summary of input parameters and output results for an inversion exercise performed with correct and erroneous petrophysical assumptions. The inversions were performed from pumpout data simulated in the presence of a WBM. Case No. 1 shows inversion results obtained when making the correct petrophysical assumptions on relative permeability and capillary pressure.

Input Parameters Inversion Output # wiS o

rwk orok ciP k hv kk / φ

1 0.1 0.24 1.0 25.6 150.9 0.195 0.194 2 0.1 0.20 1.0 25.6 137.1 0.149 0.194 3 0.1 0.15 1.0 25.6 124.2 0.110 0.202 4 0.1 0.24 0.9 25.6 142.1 0.211 0.184 5 0.1 0.24 0.8 25.6 130.6 0.230 0.184 6 0.1 0.24 1.0 22.1 148.2 0.173 0.171 7 0.2 0.24 1.0 25.6 154.3 0.212 0.180

Table 3. Summary of input parameters and output results for an inversion exercise performed with correct and erroneous fluid property assumptions. The inversions were performed from pumpout data simulated in the presence of an OBM. Case No. 1 shows inversion results obtained when making the correct assumptions on fluid viscosities and mixing coefficient. Input Parameters Inversion Output # sµ oµ ω k hv kk / φ

1 1 3 0.50 50.4 0.195 0.209 2 1 3 0.25 50.7 0.186 0.262 3 1 3 0.75 50.2 0.204 0.172 4 3 3 0.50 44.1 0.107 0.261 5 5 3 0.50 42.0 0.110 0.290

Table 4. Summary of results for the inversion example of field RDT data along with comparisons to core, Combined Log and MRIL, and RDT pretest permeabilities. Mud-filtrate contamination measurements (solvent fraction, Ss) acquired MRILab and PVT lab testing are compared with those inverted with the neural network algorithm (k, kv/kh , φ). Data Type

sS )(mdk hv kk / φ

Neural network Inversion 7.5% 993 0.049 0.22 Core Data 552 - 852 Log MRIL 668 0.25* Log derived 1008 0.24* RDT Real-time Pretest 750 0.13 RDT MRILab 6.5% Lab Sample Analysis 4% * effective porosity = total porosity – bulk volume of bound water

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