SPATIAL SENSITIVITY FUNCTIONS FOR FORMATION-TESTER MEASUREMENTS ACQUIRED IN VERTICAL AND HORIZONTAL WELLS

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1 SPATIAL SENSITIVITY FUNCTIONS FOR FORMATION-TESTER MEASUREMENTS ACQUIRED IN VERTICAL AND HORIZONTAL WELLS Renzo Angeles, Chengwu Yuan, and Carlos Torres-Verdín, The University of Texas at Austin Copyright 2006, held jointly by the Society of Petrophysicists and Well Log Analysts (SPWLA) and the submitting authors. This paper was prepared for presentation at the SPWLA 47 th Annual Logging Symposium held in Veracruz, Mexico, June 4-7, ABSTRACT We develop a conceptual and quantitative methodology to assess the three-dimensional (3D) spatial zone of response of formation-tester measurements acquired in vertical and horizontal wells. Spatial sensitivity functions are calculated numerically from the variation of pressure transient measurements due to perturbations of petrophysical properties at a given point in space and time. Calculations are performed under the assumption of two-phase fluid flow and consider both packer- and point-sources as well as pressure and fractional-flow monitoring probes. We examine perturbations of a range of petrophysical properties to calculate the sensitivity function, including permeability, porosity, and permeability anisotropy. Conventional single-phase spherical- and radial-flow asymptotic solutions often used in the interpretation of formation-tester measurements can lead to significant errors in the construction of the sensitivity maps. Such errors can bias the estimation of permeability and permeability anisotropy because of unaccounted capillary pressure and relative permeability effects. In addition, non-symmetrical flow barriers distort the spatial zone of response, whereas presence of supercharging limits the ability of formation-tester measurements to probe radially deep into the formation. Damaged and stimulated zones near the wellbore can also modify the spatial resolution properties of the acquired measurements and significantly reduce their radial length of investigation. For cases of rock formations penetrated by horizontal wells, the spatial zone of response of formation tester measurements can be highly non-symmetrical. The spatial sensitivity functions described in this paper could be used to design measurement acquisition and interpretation strategies that maximize spatial resolution and depth of investigation in complex geometrical situations that include two-phase flow phenomena. 1 INTRODUCTION Accurate and reliable characterization of heterogeneous permeable media with modular and multi-probe formation testers requires knowledge of the spatial resolution of the measurements. This requirement not only dictates the relative worth of sensor locations in the tool but also reveals the degree of heterogeneity that can be estimated through history-matching methods. At the same time, knowledge of the spatial response of the measurements can assist in the optimal selection of acquisition parameters (e.g. optimum testing time), and facilitate the incorporation of interpretation results into larger-scale reservoir simulators to assess field performance. Several studies have been published concerning interpretation techniques and applications of multiprobe formation testers (Goode et al., 1992; Badaam et al., 1998; Proett et al., 2000) and packer/probe formation testers (Pop et al., 1993; Kuchuk et al, 1998; Hurst et al., 2000; Onur et al., 2004). Similarly, analysis of sensitivity coefficients of pressure transient data has been considered for the purpose of history matching (Bishop et al., 1976; Dogru et al., 1981). Oliver (1993) used a perturbation approach to appraise variations in reservoir transmissivity and storativity on drawdown testing at an observation well. In a recent work, Gok et al. (2005) derived sensitivity coefficient maps for formation tester measurements to study the resolving capabilities of interval pressure transient tests (IPTT) in heterogeneous porous media. The authors also developed and implemented equations for calculating sensitivity coefficients based on 3D gradient and adjoint methods for single-phase fluid flow. In this paper, we introduce sensitivity coefficient maps for two-phase flow in vertical and horizontal wells using point sources (sink probe) and finite line sources (packer). The study also compares the impact of using single-phase and two-phase flow assumptions on sensitivity coefficient maps, including presence of mudfiltrate invasion. We first calculate sensitivity functions for the case of vertical wells in order to gain understanding of the underlying physical process;

2 subsequently, we calculate sensitivity maps for the case of horizontal wells. Further implementation of this technique is illustrated with the calculation of pressure sensitivity function for porosity and fractional flow sensitivity functions for permeability. METHODOLOGY Reservoir Simulators An efficient in-house two-phase two-dimensional (cylindrical coordinates) implicitpressure explicit saturation finite-difference algorithm (Lee et al., 2004) simulates the pressure measurements and the process of invasion for the case of dual-packer formation tests in vertical wells. In this case, we assume symmetry of petrophysical properties around the axis of the wellbore (axial symmetry). On the other hand, for the case of point-sources and horizontal wells, we use the commercial simulator ECLIPSE in fully implicit, black-oil mode. Base Case Model The base case model describes standard measurement parameters and rock formation properties assumed for most of the synthetic case examples considered in this paper. Fig. 1 shows the tool configuration in a 7.3 m formation penetrated by a vertical well. There is one monitor probe located 2m above the dual-packer module. Table 1 describes the finite-difference grid dimensions along with the physical dimensions and properties of the assumed hydrocarbon-bearing rock formation. Unless otherwise noted, the formation exhibits absolute horizontal (k h ) and vertical permeabilities (k v ) equal to 20 md and 2 md, respectively. The hydraulic test consists of an initial withdrawal of formation fluids from the packer open interval (sink) thereby giving rise to pressure transients at the observation points (at the packer and at the probe). As described in Fig. 2, this stage of the hydraulic test (drawdown) lasts 0.5 hours and is immediately followed by a shut-in phase of 1.5 hours. The packer open interval has a length of 0.98 m. Upper, lower, and external reservoir boundaries are assumed impermeable (flow rate is zero). Table 2 describes the associated rock and fluid properties. Table 3 summarizes the initial conditions prior to formation testing. Figs. 3 and 4 show the assumed relative permeabilities to water and oil, and the capillary pressures, respectively. Mud-filtrate Invasion - An adaptation of INVADE, developed by Wu et al. (2002), is used to simulate the process of mud-filtrate invasion. Simulations include the dynamically coupled effects of mudcake growth and multiphase, immiscible mud-filtrate invasion. In simple terms, the flow rate of mud-filtrate invasion depends on ECLIPSE is a trade mark of Schlumberger. 2 both mud and rock properties. The INVADE algorithm assumes that the rock formation was drilled with a water-based mud (WBM). Table 4 describes the fluid properties, mud properties, and petrophysical parameters assumed to simulate the process of mudfiltrate invasion (pressure, water saturation, and salt concentration) taking place in a single invaded bed. Sensitivity Functions - To calculate the sensitivity functions, we use the so-called time-honored method. This method involves the perturbation of a given petrophysical parameter, π, by an amount Δπ in the i th cell of the discretized spatial medium at a given time t. The corresponding relative perturbation in the measured pressure transient is given by p p( π + Δπ ) p( π ) G = λ, (1) i π π Δπ where p is measured pressure at the sensor and λ is a normalization factor described in the Appendix. Based on the above relation, one can quantify the contribution from each cell in the discretized spatial domain to the pressure response measured at either the packer or at an observation probe. The advantage of using this method is that it can be applied in conjunction with any reservoir simulator. The disadvantage is that it requires m+1 simulation runs for m petrophysical parameters per discretization cell. In equation (1), the size of the petrophysical perturbation (Δπ) was chosen to be 1% of the corresponding parameter, π. Such a parameter selection applies to most of the cases described in this paper. In general, sensitivity functions can be positive or negative depending on the differential pressure response to the petrophysical perturbation in a given discretization cell. However, in this paper, sensitivity functions are plotted with a logarithmic scale to allow the quantitative comparison between the various case examples; hence, values assigned to sensitivity maps will only be positive. Also, note that because pressure is a transient measurement, the corresponding sensitivity functions will vary with measurement time as well. Radius of Investigation This is determined by the ability of the formation tester to sense pressure variations caused by a petrophysical perturbation at a point located a given distance away from the wellbore. The radius of investigation is the maximum radial distance into the reservoir where a nominal perturbation of petrophysical properties can be measured by the tool within a specified measurement noise threshold. In agreement with most commercially available formation testers, we enforced a noise threshold of 0.01 psi to calculate the sensitivity maps. In so doing, we made

3 use of the dark blue color to identify spatial zones in the probed formation for which the formation tester remains insensitive. As a reference, pressure resolutions of 0.01 psi yield sensitivity values equivalent to in a log 10 scale. SENSITIVITY FUNCTIONS FOR VERTICAL WELLS Pressure Responses Fig. 5 describes the simulated pressure transient responses for the base case model. Fig. 6 shows the corresponding semi-log and log-log plots for transient pressure measurements. During drawdown testing, the pressure wave reaches the radial boundary of the spatial domain in approximately 0.32 hours, after which a radial flow regime ensues. The buildup test, however, only enforces spherical flow during the 1.5 hours of testing. Tool storage due to compressibility of the fluids in the flowline is not considered in this work; however, fluid phases are assumed compressible within the reservoir. As predicted, the magnitude of the pressure measured at the probe is much smaller than that measured at the packer. General Behavior Figures 7 and 9 show the sensitivity function map for packer pressure to loghorizontal and log-vertical permeability, respectively. When evaluated with respect to horizontal permeability, packer pressure sensitivities are high near the packer open interval and extend radially into the formation with time until reaching a radius of 6.8 meters. Then, as the buildup test begins, sensitivities monotonically decrease until the only remaining region of sensitivity is located close to the wellbore, at the same depth as the packer sensor. A similar situation occurs for the case of packer pressure sensitivities with respect to vertical permeability but, their propagation is less conical and also increases in the vertical direction. We also note that the most sensitive spatial region is not only composed of the points located at the same depth as the packer sensor, but also comprises points located in the flow area and their vicinity. Overall, the magnitude of packer pressure sensitivities with respect to horizontal permeability is the largest of all the cases considered in this paper, followed by the packer pressure sensitivities with respect to vertical permeability. Pressure probe sensitivities (Figures 8 and 10) exhibit smaller values than those of the packer but their radius of investigation are approximately the same. In the case of pressure probe sensitivities with respect to horizontal permeability, an arch ensues between the packer flow area and the monitoring probe, with the maximum sensitivity being located at points near the probe and at the same depth as the sensor. As time increases, the arch grows to a maximum of 6.8 meters and monotonically decreases with the buildup test. At the end of the buildup test, the tool only remains sensitive to points located at the same depth as the probe. On the other hand, probe sensitivities with respect to vertical permeability are the highest at locations immediately above and below both the monitoring probe and the packer flow area. In addition, there exists measurable pressure sensitivity to the near-wellbore region not evidenced in the previous sensitivity maps, especially at the beginning of the drawdown test. An alternative way to appraise the behavior of pressure sensitivity functions for packer responses is to plot their radial profile at the packer sensor depth (Fig. 11). It is clear that most of the formation sensitivity during drawdown originates from the near wellbore region (usually overly affected by near-wellbore features such as skin damage). However, during the buildup test, the pressure measurements become more sensitive to spatial zones radially deeper into the formation. Interestingly, the same sensitivity plot emphasizes a clear distinction between the distance traveled by a pressure wave due to fluid withdrawal and the radial distance into the formation where the sensitivity function reaches the noise threshold of the pressure measurement. Effect of Permeability Sensitivity maps are substantially affected by the magnitude of absolute permeability. For low values of permeability, perturbations of pressure will be larger than for high values of permeability (assuming that the production flow rate is kept constant); hence, as shown in Fig. 12, contributions to pressure measurements from regions far from the wellbore will be accentuated with a decrease of formation permeability. The opposite situation occurs for the case of high permeability formations (Fig. 13), where most of the sensitivity recedes to the near wellbore region. In both cases, the permeability anisotropy ratio (k h /k v ) was kept constant at 10. Ironically, the radius of investigation decreases with increasing values of permeability, thereby contradicting the usual tenet of well testing: that the radius of investigation is proportional to the square root of permeability. For instance, in the 200 md formation, the radius of investigation decreased to 3.4 m, whereas for the 2 md formation, the radius of investigation increased to 7 meters. Effect of Permeability Anisotropy We consider two cases of analysis: horizontal permeabilities of 100 md and 20 md with permeability anisotropies of 10 (base case) and 1. In both cases, the characteristic cone in the sensitivity maps at the end of the drawdown test expands vertically for the cases of isotropic 3

4 permeability (Fig. 14). This behavior is predictable because the higher the vertical permeability the stronger the pressure wave will affect the upper and lower spatial regions of the cone. Correspondingly, the radius of investigation decreases from 4.5 to 2.7 meters in the 100 md isotropic formation, and from 6.8 to 4.7 meters in the 20 md isotropic formation. Effect of Single-Phase Flow The base case model considers a pressure test that takes place in a formation with initial water saturation equal to irreducible water saturation. Since only one phase is being displaced (oil), pressure sensitivities for the base case model are equivalent to those of single-phase flow. Effect of Impermeable Bed Boundaries In this example, upper and lower beds of 0.01% porosity and 10-6 md permeability limit above and below the permeable bed (base case model) probed by the formation tester (Fig. 15). The observed effect is an increase in magnitude of all the sensitivity maps as well as an increase of radius of investigation. This effect is not measurable in an anisotropic formation compared to the effect simulated for isotropic formations, where the radius of investigation increased from 4.7 to 7.1 meters. Effect of Skin Factor We consider two types of skin: positive and negative, corresponding to damaged and stimulated regions near the wellbore, respectively. The damaged region is simulated by introducing a radially thin permeability zone with k h =2 md and k v =0.2 md (same porosity) around the wellbore up to a radius of 0.17 m. Using the Furui-modified Hawkins formula (Furui, 2002), the inclusion of such radial zone of low permeability is approximately equivalent to a skin factor of The resulting sensitivity maps (Fig. 16) indicate that pressure measurements become more sensitive to the damaged zone for the case of horizontal permeability; however, for the case of vertical permeability, most of the pressure response originates from the undamaged region. This situation is reversed when the skin factor is negative. To simulate a stimulated zone, we introduce a thin radial layer with the same dimensions as the one used for positive skin; however, in this latter case we use the values k h =200 md and k v =20 md, equivalent to a skin factor of Fig. 17 shows that most of the pressure sensitivity originates from the stimulated zone and the region immediately adjacent to it. The radius of investigation remains approximately the same with and without presence of the damaged zone. Effect of Capillary Transition Zone For measurements acquired within the capillary transition zone, the physics of two-phase fluid flow becomes relevant for the construction of pressure sensitivity maps. In previous cases, the reservoir was located m above the oil-water contact (OWC, in turn located at a depth of 3048 m). For this test case of analysis, we place the bottom of the reservoir at the OWC, thereby forcing the water saturation in the reservoir (assumed to be at capillary equilibrium) to be between 0.39 and 1. Fig. 18 shows the corresponding pressure sensitivity maps, indicating an overall increase of sensitivity magnitude compared to that of the base case model. We note that high values of water saturation decrease the oil relative permeability, hence causing a larger pressure drawdown. As a consequence, pressure measurements become more sensitive to spatial regions that are far from the wellbore. Also, the radius of investigation increases from 6.8 to 7.8 m. Effect of Mud-Filtrate Invasion In this example, as starting point for the sensitivity studies, the simulation algorithm is initialized with the spatial distributions of pressure, saturation, and salt concentration obtained from the simulation of mud-filtrate invasion. The resulting pressure sensitivity map (Fig. 19) consists of abnormally large values of sensitivity extending in the radial direction, thereby indicating that the pressure measurements are uniformly sensitive to perturbations of cell permeabilities. An explanation for this behavior is that the abnormally high values of water saturation near the wellbore increase the resistance of the formation to the flow of oil, acting as though the rock s absolute permeability were much lower than the actual absolute permeability of the formation. Perturbations of pressure due to perturbations of cell permeability are completely masked by the spatial distribution of water saturation resulting from mud-filtrate invasion. Point Sources Probe-type formation tests are analyzed using a similar model to that of the base case. The only difference is that we no longer assume axial symmetry around the wellbore. Thus, the simulation problem becomes three-dimensional, i.e. the formation is now spatially discretized along the azimuthal (θ) direction as well as along the vertical and radial directions in a cylindrical coordinate frame. Table 5 describes the physical dimensions of the finite-difference grid used for the simulation of pressure measurements due to point sources of flow rate. Fig. 20 describes the simulated pressure transient measurements for monitoring and sink probes in a formation with k h =20 md and k v =2 md. As indicated in the figure, during drawdown the sink probe generates a larger pressure differential than a packer source mainly because of its much smaller flow area (the production flow rate is assumed equivalent for the two types of sources). By contrast, the probe pressure decreases during drawdown due to the increased distance between sink and monitoring probes. 4

5 Figures 21 and 22 show pressure sensitivity maps calculated for the case of a point source and variations of horizontal permeability. For a point source, pressure sensitivities at the sink probe are higher than those due to packer sources, especially near the wellbore. Also, the radius of investigation increases to 10 m. Pressure sensitivities for the monitoring probe are much lower than those of the packer and exhibit a significant enhancement in the region near the sink probe. This behavior is in contrast with the corresponding packer sensitivities (shown in Fig. 8) where most of the pressure sensitivity originates from the region near the monitoring probe and not from the neighborhood of the sink. The radius of investigation decreases to 1.8 m. Comparison of pressure sensitivity maps for point and packer sources requires a previous normalization for the difference in source flow area. Fig. 23 compares the radial profiles of pressure sensitivity maps calculated along the depth of the sink probe and at the center of the packer. Based on this figure, one can conclude that pressure measurement with point sources are more sensitive to the near-wellbore region than pressure measurements due to packer sources. Also, as expected, pressure sensitivities associated with point sources decrease faster in the radial direction than those of packer sources. SENSITIVITY FUNCTIONS FOR HORIZONTAL WELLS Horizontal Well Case Model In this section, we simulate pressure measurements acquired with a dualpacker formation tester in a horizontal well. Table 6 describes the properties and physical dimensions of the 3D finite-difference grid used to perform the simulations. Rock and fluid properties were assumed equal to those of the base case. Fig. 24 shows the tool configuration in the new measurement acquisition. The rock formation has a thickness of 32 m and is penetrated by a horizontal well at its center. All external reservoir boundaries are assumed impermeable (flow rate is zero). Pressure Responses Fig. 25 shows the corresponding simulated pressure transient responses for this model. We note that it takes a longer time for the monitoring probe pressure to reach its asymptotic steady-state value compared to that of a packer probe in a vertical well. Effect of Permeability Anisotropy In this case, we plot the pressure sensitivity maps in the y-z Cartesian plane, thereby allowing one to calculate sensitivity functions for points above and below the formation tester. Whereas Fig. 26 considers an isotropic formation with k h =20 md and k v =20 md, Fig. 27 shows sensitivity maps for an anisotropic formation with k h =20 md and k v =2 md. We observe that, for the isotropic case, pressure sensitivity maps resemble a semi-circle with a radius of investigation up to 15 m. By contrast, pressure sensitivity maps for the anisotropic case exhibit a semielliptic shape with a distinctive long axis that extends 21 m along the horizontal direction. Effect of Gravity Interestingly, the pressure sensitivity maps shown in Figs. 26 and 27 are symmetric along the center position of the z-axis with and without presence of permeability anisotropy. For the cases studied in this paper we observed no appreciable influence of gravity forces in the spatial distribution of pressure sensitivities. OTHER SENSITIVITY FUNCTIONS Pressure Sensitivity Functions with Respect to Porosity In analogy to previous exercises, we perform perturbations to the porosity of each cell and compare the simulated pressure responses for the perturbed and unperturbed states. Figures 28 and 29 show the corresponding sensitivity maps for packer and probe pressures, respectively. For the packer, pressure sensitivities no longer exhibit the conic shape previously observed for the case of permeability. The sensitivity functions reach their maximum value when the test begins (3.4 m) and monotonically decrease in magnitude with time. We observe the same behavior during the buildup test; however, although not shown here, the sign of the pressure sensitivities changes to negative during this test. It follows that the shape of the sensitivity maps is largely controlled by the packer flow area, especially within the first 0.20 m. Addition of a monitoring pressure probe to the tool configuration does not improve the behavior of the sensitivity maps. Pressure sensitivities still remain high in the vicinity of both packer and monitoring probe. In addition, there exist non-negligible sensitivities below the packer, attributed to negative pressure differentials in those regions at the start of the test and during buildup. Sensitivity Functions with Respect to Fractional Flow Rate Instead of pressures, sensitivities are computed using the fractional flow rate measured at the packer during the drawdown test with respect to variations of horizontal permeability. The assumed model for this case example is the same as that used for the base case model in vertical wells except that we withdraw formation fluid for a longer time (extended drawdown). In addition, to reproduce the effect of mobile water saturation, we arbitrarily inject water as in the process of mud-filtrate invasion, at a constant injection pressure of MPa for 4.8 h. This water injection period 5

6 takes place prior to the formation test. Sampling times are different for this case too: 0.3, 0.45, 0.75, and 3.0 hours. We applied a measurement threshold of 4.5e-8 s/md/m 6 in the fractional-flow sensitivity maps shown in Fig. 30, which corresponds to a measurement accuracy (for water cut) of 0.01%. Fig. 31 shows the behavior of fractional flow rate with time for different values of formation permeability. This figure helps one to understand the behavior of the calculated sensitivity maps. The spatial region of the formation contributing to the fractional flow rate measured at the packer first increases dramatically to a maximum and then monotonically decreases as the fractional flow stabilizes toward the value of 0 (water that remains in the reservoir is irreducible). We note that most of the sensitivity to fractional flow rate originates in the packer flow area and at points located immediately above and below this region. DISCUSSION There is an important distinction between the radius of investigation obtained from sensitivity function maps and the radius of investigation derived from conventional analytical solutions. The latter is determined by the speed at which the pressure disturbance propagates into the reservoir which, in theory, is proportional to the square root of permeability. Whether the pressure measurements are sensitive to this perturbation is a completely different issue. Our work shows that the radius of investigation decreases with increasing values of permeability because the contribution to pressure measurements due to locations far from the wellbore decreases with an increase of permeability. As expected, the addition of a pressure monitoring probe in the dual-packer tool configuration enhances the sensitivity of pressure measurements to spatial zones located deep into the probed formation. The simulations described in this paper indicate that the radius of investigation remains the same for the two modalities of pressure measurement acquisition. However, if only the packer is used (without the monitoring probe), most of the pressure measurement will originate from the near-wellbore region, and hence could be biased by skin and other near-wellbore damage conditions. Capillary pressure and relative permeabilities could also play a dominant role in the shape and magnitude of pressure sensitivity maps. High values of water saturation decrease the effective permeability of oil during formation testing, whereupon the formation behaves in the same manner as a formation with lower values of permeability. In turn, this causes enhanced pressure sensitivity to radially deeper spatial regions into the probed formation. Such an observation stresses the importance of calculating pressure sensitivity maps with simulations of two-phase flow instead of singlephase flow approximations. Whenever formation testing is performed in the vicinity of no-flow barriers, transient pressures measured at the monitoring probe increase in magnitude and distort the sensitivity function maps, especially when the formation is isotropic. As a consequence, pressure measurements become more sensitive to radially deeper spatial regions in the probed formation. Super-charging effects caused by mud-filtrate invasion limit the ability of formation-tester measurements to probe radially deep into the formation. It was observed that presence of high values of water saturation and pressure near the wellbore decreased the effective permeability to oil, thereby enhancing the pressure sensitivities in a manner similar to the case of positive skin (damaged area). This behavior of the pressure sensitivity functions indicates that pressure measurements could be significantly biased by radial and vertical variations of permeability in the nearwellbore region, hence substantially diminishing the ability of the measurements to probe deep into the formation. Despite their longer radius of investigation, compared to that of packer-sources at the same production flow rate, pressure sensitivity maps associated with point sources exhibit significant enhancement in the spatial region surrounding the sink probe. Even with the addition of a monitoring probe, pressure sensitivity values remain high in the vicinity of the sink probe. Compared to pressure sensitivity maps associated with packer sources at equal fluid velocities, sensitivity maps for point sources exhibit a large enhancement near the wellbore and a shorter radius of investigation. Formation tests acquired in horizontal wells entail additional geometrical complexity for the calculation and interpretation of pressure sensitivity maps. The simulations described in this paper indicate that permeability anisotropy can have a significant effect in the shape of the pressure sensitivity functions around the perimeter of the wellbore. Sensitivity maps for measurements of fractional flow rate offer an interesting alternative to the estimation of petrophysical parameters. Unlike their analogous pressure-derived maps, fractional-flow derived sensitivity maps are evenly influenced by the nearwellbore region and the virgin rock at the same depth. This property could help to reduce the non-uniqueness 6

7 of permeability inversion when simultaneously honoring pressure and flow rate measurements. The synthetic examples considered in this paper suggest that locations of point sinks and pressure probes could be adjusted to improve the resolving properties of formation tester measurements. One possibility is to consider a small array of pressure probes to dynamically control the spatial zone of response of formation testers thereby improving the estimation of petrophysical properties. We could also consider specific flow-rate pulsing sequences that could improve the resolving capabilities of early- and late-time pressure measurements. A proper source-sensor measurement acquisition design should provide selective radial deepening and azimuthal orientation of the 3D spatial zone of pressure response. CONCLUSIONS The following conclusions stem from the simulation cases described in this paper: 1. Sensitivity maps are a useful tool to understand the rock-fluid interactions occurring during a formation test. Whether used for planning or for measurement interpretation, information contained in pressure measurements can be more adequately interpreted with knowledge of the spatial regions in the formation affecting the measurements. 2. Single-phase flow assumptions, although practical, disregard important fluid-flow effects due to capillary pressure and relative permeability. These effects are significant when analyzing the influence of skin, supercharging, and other fluid displacement mechanisms that involve more than one fluid phase. 3. We have redefined the concept of radius of investigation for formation tester applications. Instead of using the speed at which the pressure disturbance travels into the formation, we suggest that the radial length of investigation be determined by the pressure sensitivity functions calculated at a given time during the pressure test. 4. Complex geometries and fluid-flow conditions such as mud-filtrate invasion in horizontal wells can have a significant effect in the estimation of petrophysical properties, including permeability and permeability anisotropy. Sensitivity function maps shed considerable light to the resolving properties of formation-tester measurements. Moreover, pressure and fractional-flow sensitivity functions could be used to design optimal measurement acquisitions in horizontal and highlydeviated wells, including the use of additional sources and sensor for active steering of the spatial zone of response. ACKNOWLEDGEMENTS We are thankful to Dr. Kamy Sepehrnoori and Hee-Jae Lee from the University of Texas at Austin for valuable discussions throughout the course of this project. Funding for the work reported in this paper was provided by UT Austin's Research Consortium on Formation Evaluation, jointly sponsored by Aramco, Baker Atlas, BP, British Gas, ConocoPhillips, Chevron, ENI E&P, ExxonMobil, Halliburton Energy Services, Marathon, Mexican Institute for Petroleum, Norsk- Hydro, Occidental Petroleum Corporation, Petrobras, Schlumberger, Shell International E&P, Statoil, Total, and Weatherford. REFERENCES Badaam, H., Al-Matroushi, S., Young, N., Ayan, C., Mihcakan, M., and Kuchuk, F.J., 1998: Estimation of Formation Properties Using Multiprobe Formation Tester in Layered Reservoirs, paper SPE presented at the 1998 SPE Annual Technical Conference and Exhibition, New Orleans, September Bishop, K.A., Brett, V.S., Green, D.W., and McElhiney, J., 1976: The Application of Sensitivity Analysis to Reservoir Simulation, paper SPE 6102 presented at the 51 st Annual Fall Technical Conference and Exhibition of the Society of Petroleum Engineers of AIME, New Orleans, October 3-6. Dogru, A.H. and Seinfeld, J.H., 1981: Comparison of Sensitivity Coefficient Calculation Methods in Automatic History Matching, Society of Petroleum Engineers Journal, October issue, Furui, K., Zhu, D., and A.D. Hill, 2002: A Rigorous Formation Damage Skin Factor and Reservoir Inflow Model for a Horizontal Well, paper SPE presented at the SPE International Symposium and Exhibition on Formation Damage Control, Lafayette, Louisiana, February Gok, I.M., Onur, M., and Kuchuk, F.J., 2005: Estimating formation properties in heterogeneous reservoirs using 3D Interval Pressure Transient Test and Geostatistical data, paper SPE presented at the 14 th SPE Middle East Oil and Gas Show and Conference, Bahrain, March. Goode, P.A., and Thambynayagam, R.K.M., 1992: Permeability Determination With a Multiprobe Formation Tester, paper SPE 20737, SPE Formation Evaluation, December issue, Hurst, S.M., McCoy, T.F., and Hows, M.P., 2000: Using the Cased-Hole Formation Tester Tool for Pressure Transient Analysis, paper SPE presented at the 2000 SPE Annual Technical Conference and Exhibition, Dallas, October

8 Kuchuk, F.J., 1998: Interval Pressure Transient Testing With MDT Packer-Probe Module in Horizontal Wells, SPE Reservoir Evaluation & Engineering, December issue, 509. Lee, H., Wu, Z., and Torres-Verdín, C., 2004: Development of a Two-Dimensional, Axi- Symmetric Single Well Code for Two-Phase Immiscible Fluid Flow, Salt Mixing, and Temperature Equilibration in the Near Wellbore Region with applications to the Simulation of Mud- Filtrate Invasion And Formation Tester Data, Fourth Annual Report, Formation Evaluation Program at The University of Texas at Austin, Appendix II, August 19, 3 Oliver, D.S., 1993: The Influence of Nonuniform Transmissivity and Storativity on Drawdown, Water Resources Research, 29, No. 1, 169. Onur, M., Hegeman, P.S., and Kuchuk, F.J., 2004: Pressure-Transient Analysis of Dual Packer-Probe Wireline Formation Testers in Slanted Wells, paper SPE presented at the SPE Annual Technical Conference and Exhibition, Houston, September Pop, J.J., Badry, R.A., Morris, C.W., Wilkinson, D.J., Tottrup, P., and Jonas, J.K., 1993: Vertical Interference Testing With a Wireline-Conveyed Straddle-Packer Tool, paper SPE presented at the 1993 SPE Annual Technical Conference and Exhibition, Houston, October 3-6. 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 presented at the SPE International Oil and Gas Conference and Exhibition in China, Beijing, November Wu, J., Torres-Verdín, C., Proett, M., Sepehrnoori, K., and Belanger, D., 2002: Inversion of Multi-phase Petrophysical Properties Using Pumpout Sampling Data Acquired with a Wireline Formation Tester, paper SPE presented at the Annual Technical Conference and Exhibition, San Antonio, September 29-October 2. APPENDIX Equation (1) includes a normalization factor, λ, defined as the ratio of production flow rate and cell volume. For instance, to calculate the pressure sensitivity G ik of the i th cell with respect to permeability, k, we make use of the formula p 1 p 1 G = ik = k, (A1) log k qvi k qvi where G ik is the sensitivity in MPa-s/mD/m 6, p is pressure measured at the sensor in Pa, k is permeability 8 in md, q is the production flow rate in m 3 /s, and V i is the numerical cell volume in m 3. The purpose of λ is to normalize the calculated cell sensitivities for (a) the non-uniform nature of the numerical grid used in the simulations (logarithmic steps in the radial direction) and (b) different values of production flow rate or cross-sectional area of source flow. An analogous formula is used for the calculation of pressure sensitivities with respect to porosity, and for fractional flow-rate sensitivity with respect to permeability. In the latter cases, the formula for the calculation of sensitivities does not include logarithmic values of the petrophysical property. For the calculation of the sensitivities in Fig. 23, we modified the value of λ, defined above, to compare point-source and packer-source pressure sensitivities with equivalent fluid velocities at the source flow area. In this case, the calculation of pressure sensitivities is performed with the formula G ik p 1 Ap =, (A2) logk qvi Al where the additional multiplicative term included on the right-hand side of the equation compensates for the difference in flow area between a probe (A L ) and a packer (A p ). Thus, the comparison of sensitivity functions is performed for hydraulic tests that produce fluid at the same velocity and not at the same flow rate. Measurement units in equation (A2) are the same as those in equation (A1). ABOUT THE AUTHORS Renzo Angeles is a Graduate Research Assistant and a PhD candidate at the University of Texas at Austin. From , he worked as a field engineer for Schlumberger, receiving training in Peru, Colombia, Ecuador, USA, and Canada. Recently, he held a summer internship with Chevron at the Energy Technology Company. He is a recipient of the 2003 Presidential Endowed Osmar Abib Scholarship and the Chevron Scholarship. His interests involve pressure transient testing, numerical simulation of reservoirs, and formation characterization. His PhD research focuses on the quantitative analysis and inversion of formation tester measurements acquired in highly deviated wells using two- and three-phase flow analysis including the effects of mud-filtrate invasion. Chengwu Yuan is a Graduate Research Assistant and PhD student at the University of Texas at Austin. In 2005, he received an MS degree in Petroleum Engineering from Texas A&M University. Previously,

9 he worked as a field engineer in Sheng Li (China) and conducted research at the China University of Petroleum. From February to August, 2005, he interned with Schlumberger Data and Consulting Services. His experience includes automatic history matching, finitedifference and streamline simulation, stochastic modeling, and uncertainty assessment. Mr. Yuan s professional interests include data inversion and integration, and reservoir simulation. Carlos Torres-Verdín received a PhD 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 has been with the Department of Petroleum and Geosystems Engineering of The University of Texas at Austin, where he currently holds the position of Associate Professor. He conducts research on borehole geophysics, formation evaluation, and integrated reservoir characterization. Torres-Verdín 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. Variable Units Value Wellbore radius (r w ) m 0.11 External radius (r e ) m Reservoir thickness m 7.32 Datum depth (bottom of reservoir) m Water/oil contact depth m Number of nodes radial axis Number of nodes vertical axis Grid cell size radial axis m variable Grid cell size vertical axis m 0.12 Table 1: Finite-difference grid and physical dimensions of the assumed hydrocarbon-bearing rock formation. Variable Units Value Porosity Water STP Kg/m Water viscosity mpa.s 0.50 Water compressibility 1/Pa 3.63e-10 Water formation volume factor Oil STP Oil viscosity mpa.s 0.50 Oil compressibility 1/Pa 3.92e-9 Oil formation volume factor Production flow rate m 3 /s 5.52e-5 Table 2: Summary of input rock and fluid properties for the base case formation model. Variable Units Value Initial datum Pa 3.88e+7 Initial water saturation Connate water salinity ppm Table 3: Summary of the assumed initial conditions for the base case formation model. Variable Units Value Mudcake Permeability md 0.3 Mudcake Porosity Mud solid fraction Mudcake maximum thickness m 7.62e-3 Mudcake compressib. exponent Mudcake exponent multiplier Mud hydrostatic pressure Pa 4.34e+7 Mudcake rub-off time Mud overbalance pressure Pa 0.46e+7 Table 4: Summary of mud properties assumed for the sensitivity studies of mud-filtrate invasion.. Variable Units Value Number of nodes radial axis Number of nodes vertical axis Number of nodes azimuth. axis Grid cell size radial axis m variable Grid cell size azimuthal axis degree 15 Grid cell size vertical axis m 0.12 Table 5: Finite-difference grid and physical dimensions of the simulation domain for the 3D point-source case. Variable Units Value Wellbore radius (r w ) m 0.11 Reservoir thickness m 32.0 Water/oil contact depth m No. of nodes x axis No. of nodes y axis No. of nodes z axis m 21 Grid cell size x axis m 1.52 Grid cell size y axis m 1.52 Grid cell size z axis m 1.52 Table 6: Finite-difference grid and physical dimensions for the horizontal well case. (refer to Fig. 24 for the orientation of the-axes). 9

10 monitor probe 3.66 m 2.01 m packer 0.98 m Fig. 1: Tool configuration of the packer-probe formation tester used in the base case model. Pressure measurements are obtained at the two vertical locations in response to time-dependent flow rates imposed through the packer open interval. Fig. 4: Water-oil capillary relative permeability curves assumed in the numerical simulations of formation tester measurements. Fig. 2: Time schedule of flow rate assumed for the simulations of formation tester measurements in connection with the base case model. Fig. 5: Simulated pressure transient measurements at the observation probe and packer interval. Fig. 3: Water-oil capillary pressure curve assumed in the base case model. 10 Fig. 6: Log-log plot of buildup pressure change and pressure derivative for probe and packer (base case).

11 Fig. 7: Sensitivity maps for packer pressure with respect to log-horizontal permeability (base case). Fig. 9: Sensitivity maps for packer pressure with respect to log-vertical permeability (base case). Fig. 8: Sensitivity maps for probe pressure with respect to log-horizontal permeability (base case). Fig. 10: Sensitivity maps for probe pressure with respect to log-vertical permeability (base case). 11

12 packer log k h probe log k h packer log k v probe log k v Fig. 11: Sensitivity radial profile at the packer sensor with respect to log-horizontal permeability (base case). Fig. 13: Sensitivity maps for packer and probe pressures for a formation with k h =200mD and k h /k v =10 at 0.5 h of drawdown. packer log k h probe log k h k h =20 md, k v =2 md k h =k v =20 md (isotropic) packer log k v probe log k v k h =100 md, k v =10 md k h =k v =100 md (isotropic) Fig. 12: Sensitivity maps for packer and probe pressures for a formation with k h =2mD and k h /k v =10 at 0.5 h of drawdown. Fig. 14: Sensitivity maps for packer pressure with respect to log-horizontal permeability at 0.5 h of drawdown (effect of permeability anisotropy). 12

13 packer log k h probe log k h packer log k h probe log k h packer log k h (isotropic) probe log k h (isotropic) packer log k v probe log k v Fig. 15: Sensitivity maps for packer and probe pressures at 0.5 h of drawdown (impermeable beds). Fig. 17: Sensitivity maps for packer and probe pressures at 0.5 h of drawdown (skin=-0.4). packer log k h probe log k h packer log k h probe log k h packer log k v probe log k v packer log k v probe log k v Fig. 16: Sensitivity maps for packer and probe pressures at 0.5 h of drawdown (skin=+4.1). Fig. 18: Sensitivity maps for packer and probe pressures at 0.5 h of drawdown (capillary transition zone). 13

14 packer log k h probe log k h packer log k v probe log k v Fig. 19: Sensitivity maps for packer and probe pressures at 0.5 h of drawdown (mud-filtrate invasion). Fig. 21: Sensitivity maps with respect to log horizontal permeability for a sink probe in a vertical well. Fig. 20: Simulated pressure transient measurements using a point source (sink probe) in a vertical well. Fig. 22: Sensitivity maps with respect to log horizontal permeability for a sink probe in a vertical well. 14

15 Fig. 23: Comparison of sensitivity radial profiles for point and line (packer) sources with respect to loghorizontal permeability (base case). Source fluid velocity is the same in both cases. 32 m monitor probe z x 2.01 m 0.98 m packer Fig. 26: Sensitivity maps for packer pressures to log horizontal permeability in a horizontal well (isotropic case). Fig. 24: Tool configuration of the packer-probe formation tester in a horizontal well (base case). Fig. 25: Simulated pressure transient measurements enforced by 2.76e-5 m 3 /s fluid withdrawal from the dual-packer open interval in a horizontal well. Fig. 27: Sensitivity maps for packer pressures to log horizontal permeability in a horizontal well (anisotropic case) 15

16 0.3 h (drawdown) 0.45 h (drawdown) Fig. 28: Sensitivity maps for packer pressures with respect to porosity (base case) h (drawdown) 3.0 h (drawdown) Fig. 30: Sensitivity maps for packer pressures with respect to fractional flow rate (base case). Fig. 31: Simulated fractional flow rate measurements at the packer during an extended drawdown test. Fig. 29: Sensitivity maps for probe pressures with respect to porosity (base case). 16