Well-Test Analysis for Multiphase Flow

Abstract

Summary This paper is a review of well-test analysis under multiphase-flow conditions. The contributions of Perrine and Martin ar noted, and the consequences of using their results are documented. The advantages of incorporating the heterogeneous nature of the porous medium for solution-gas-drive systems with the pseudopressure function are discussed. Methods to predict well-inflow performance are discussed with regard to the works of Vogel and Fetkovich. The long-time-performance prediction of wells wit decline curves that incorporate the heterogeneous prediction of wells wit decline curves that incorporate the heterogeneous nature of the porous medium is addressed. The final part of this paper considers pressure-falloff data. Theoretical developments are reviewed, and problems associated with the identification of fluid banks are listed. Introduction Although several hundred publications have examined the analysis of well-test data, no more than 20 papers deal directly with well-test analysis for multiphase flow. Perhaps most striking is that both monographs devote a total of four pages to this subject. In well-test analysis, the study of single-phase flow in porous media is the norm. The lack of papers pertaining to multiphase flow may be attributed to Perrine and Martin. The Perrine-Martin hypothesis permits wide latitude in analyzing data subject to multiphase flow. For all intents and purposes, these works enable us to extend any technique valid for single-phase flow to multiphase flow. Obviously, there is a limit to which this theory can be extended; thus, these limitations are explored. The past 10 years have seen advances in analysis techniques that use concepts originally used to examine the flow of real gases through porous media. The application of pseudopressure functions to analyze pressure data in solution-gas-drive and gas-condensate systems is pressure data in solution-gas-drive and gas-condensate systems is discussed. Because the objective of every well test ultimately is to predict well deliverability (or injectivity), the seminal works of Vogels and Fetkovich, which discuss the productivity of wells producing by solution-gas drive, are examined. The results of these works arecompared to resolve the apparent differences. Finally, this paper considers the application of the Perrine-Martin theory to the analysis of injection well data. In particular, the utility of methods given in the literature to identify fluid banks and to determine the location of fluid/fluid interfaces is examined. Discussion Basic Equations. The basic equations governing the multiphase flow of fluids were discussed first by Muskat and Meres and Wycoff and Botset in 1936. We consider the flow of oil, water, and gas in a uniform porous medium where the influences of gravity, capillary pressure gradients, and rock compressibility are negligible. Assuming that the -formulation is valid and neglecting capillary effects, the differential equations that govern the flow of oil, gas, and water are given, respectively, by ..........................................(1) where subscript m represents oil or water, and ..........................................(2) Eqs. 1 and 2 may be solved subject to the appropriate constraints given by the initial and boundary conditions to obtain specific solutions of interest. If we restrict our attention to single-phase flow (constant viscosity and slightly compressible liquid), then the partial differential equation that governs the pressure distribution is given by ..........................................(3) Perrine-Martin Theory. As mentioned, virtually all analyses of Perrine-Martin Theory. As mentioned, virtually all analyses of pressure responses subject to multiphase flow are based on the em-pirical pressure responses subject to multiphase flow are based on the em-pirical observations of Perrine He suggested that we can analyze data for multiphase flow conditions if the mobility term in the diffusivity equation for single-phase flow is replaced by the sum of the mobilities of the individual phases and if the single-phase compress-ibility term is replaced by an effective or pseudocompressibiliy that is a function of fluid properties and saturations. Martin (also see Miller et al.) showed that Perrine's suggestions are valid if saturation gradients are negligible. This point is readily seen if we consider flow in an oil/water system. We can expand Eq. 1 to ..........................................(4) If we ignore all second-degree terms, i.e., terms containing (Vp)2 and VpVSm, and add the resulting equations for each phase, we obtain ..........................................(5) similar equation may be derived if we examine the simultaneous flow of oil, gas, and water. The similarity between Eqs- 3 and 4 is readily apparent. The critical assumption made thus far is that second-degree terms are negligible to obtain the left side of Eq- 5. On the basis of the works of Perrine and Martin, we can do the following. 1. We can calculate the total system mobility, (k/u), (assuming three-phase flow), by ..........................................(6) Here, (k/), =(ko/ o, +kg/ g +kw/ w, ), where km (m =o, g, or w,) is the effective permeability of each phase. Although never explicitly stated. Eq. 6 also implicitly forms the basis for obtaining individual phase mobilities, (kill), from the appropriate rates. phase mobilities, (kill), from the appropriate rates. 2. The skin factor is given by ..........................................(7) Here P 1hr is the change in pressure during the test after one hour of test time, and c, is defined by ..........................................(8) Eq. 8 usually is written as ..........................................(9) Here, we assume that Cf is non-negligible. SPEFE P. 585