Two-Phase Pressure Test Analysis

Abstract

Summary A theoretical basis is given for well test analysis of solution-gas andgas-condensate reservoirs in the infinite-acting period. The study is limitedto radial flow with a fully penetrating well in the center of the drainagearea. Porosity and absolute permeability are constant, and gravitational andcapillary effects are neglected. The tests are conducted at constant surfacerate, and the skin factor is zero. An analytical expression for thepressure/saturation relationship is derived from the time-dependent gas and oilflow equations. This relationship can be used to generate pseudopressurefunctions that allow test interpretation with liquid analogy. Theinfinite-acting drawdown case is treated in detail, while the buildup case isbriefly discussed. Theoretical developments are exemplified by simulateddrawdown and buildup tests in a solution-gas-rive reservoir. Introduction Many press-transient tests can be interpreted by using solutions of thediffusivity equation that are based on the liquid analogy of reservoir fluids. In the liquid model, the reservoir fluids are represented by a single-phaseliquid with small and constant compressibility and constant viscosity. Thecorresponding diffusivity equation is linear, and solutions for a variety ofboundary conditions have been presented in the literature. Single-phase gastests can be interpreted within the liquid analogy by introducing an integraltransform, the pseudopressure function, as suggested by Al-Hussainy etal.1 This pseudopressure function is uniquely dependent onpressure and can therefore be used for both drawdown and buildup analyses. To a certain extent, multiphase flow effects also can be adapted to theliquid model solutions if total mobility and compressibility areused.2,3 The interpretation of the test will yield the effectivepermeability. For solution-gas reservoirs, this method becomes less reliablewith increasing gas saturation.4 A better adaptation to the liquidanalogy can be achieved by introducing a pseudopressure function into themultiphase-flow equations, as suggested by Fetkovich,5 in analogywith the single-phase gas case. This suggestion was pursued byRaghavan,6 who gave practical methods for calculating thepseudopressure function for oil in solution-gas-drive reservoirs and showedthat the standard liquid-analogy semilog plots could be used to calculate theabsolute formation permeability from computer-generated test data. To evaluatethe oil pseudopressure function, the relation between oil saturation andpressure must be known. Raghavan, however, did not present a theoreticallybased relation but demonstrated, through examples, that the instantaneousproducing GOR could be used for drawdown interpretation. He also suggested thatthe producing GOR at shut-in could be used for build-up analysis. The objective of this paper is to present theoretical relationships betweenpressure and saturation that can be used to evaluate pseudopressure functionsin the infinite-acting period. The suggested methods are valid for anymultiphase system provided that the fluid flow can be described by diffusivityequations based on "beta" formulation - i.e., FVF's. The examples, obtainedfront computer-generated test data, are limited to solution-gas-drivereservoirs. Theory For simplicity, consider only the oil and gas phases to be mobile and theirreducible water to be incompressible. The flow equations for gas and oil, respectively, are thenEquation 1Equation 2 where So+Sg+Siw=1,Rs =gas dissolved in the oil phase, and rs=oil dissolved in the gas phase. This last term is includedto make the system of equations applicable to gas-condensate reservoirs. Inthis case, the FVF's, Bo, rs,Rs, and Bg, can be derived from aconstant-volume-depletion experiment, as proposed by Whitson and Torp.7 The following simplifying notation is introduced.Equations 3a-3e In these expressions, Rs, rs,Bo, Bg, µo, andµg depend only on p, while krg andkro depend only on S. Consistent units areassumed. We also introduce (?x/?S)p and(?x/?p)S for the partial derivative withrespect to saturation at constant pressure and the partial derivative withrespect to pressure with saturation held constant, where x (a, a,b, ß). Eqs. 1 and 2 may then be written asEquations 4 and 5 The nonlinearities of Eqs. 4 and 5, given by coefficients a and a, can beeliminated by introducing an integral transform of the pressure - thepseudopressure function, pp - defined by Fetkovich.5Actually, several pseudopressure functions can be used:Equations 6–8 Any linear combination of a and a can be used to defineppt - e.g., normalizing Eqs. 4 and 5 before adding. Inpractical use, the choice of definition depends on reservoir characteristicsand test boundary conditions. Eq. 6 is used for an oil reservoir with constantsurface oil rate; Eq. 7 is used with constant surface gas rate during the test;and Eq. 8 requires constant total rate of gas plus oil at surfaceconditions. To evaluate the integrals in Eqs. 6 through 8, the relationship betweenS and p must be known. This relationship must be consistent withthe pressure and saturation profiles developed around the well during the test. In contrast to the single-pseudopressure function, the multiphasepseudopressure functions generally are not uniquely dependent on pressure butdepend on the history of the test; e.g ppo(p) for adrawdown test is different from ppo(p) for buildupfollowing the drawdown.6