Pressure Behavior of Wells Completed in Multiphase Reservoirs Containing a Constant Pressure Boundary
Authors: Roadifer R.D. 
Published: December 2007

Abstract
This work examines the pressure response for wells producing a multiphase reservoir containing a constant pressure boundary. It is shown, both theoretically and through numerical examples, that the pressure response obtained for such wells is fundamentally different from the pressure response that would be predicted for a single-phase reservoir containing a constant pressure boundary. Based in part on a recent theoretical development by Ref. 1, a new theoretical expression for the wellbore pressure response is derived from Darcy's Law for edge-water drive reservoirs containing a constant pressure outer boundary. The derivative of the wellbore pressure drop with respect to the natural logarithm of time is shown to be a function of the time rate of change in the total volumetric flow rate and the time rate of change in the total relative mobility across the reservoir. In conjunction with fractional flow theory and supported by detailed simulation results, this theoretical expression indicates that the log- derivative of the wellbore pressure drop will not only remain positive, but must in fact increase as the water saturation front progresses towards the well. Following water breakthrough, this theoretical expression, again with supporting numerical results, indicates that the log-derivative of the wellbore pressure drop will become negative (wellbore pressure increasing), monotonically increasing to zero.
In addition to the edge-water drive reservoirs, oil reservoirs underlain by an aquifer containing a constant pressure boundary and oil reservoirs overlain by a gas cap containing a constant pressure boundary are investigated. Theoretical arguments and supporting numerical results indicate that the wellbore pressure response for an oil reservoir underlain by an aquifer containing a constant pressure boundary is similar to that obtained for the edge-water drive case; i.e., the log-derivative of the wellbore pressure drop must remain positive until water breakthrough occurs and following water breakthrough, the log-derivative of the wellbore pressure drop will become negative and monotonically increase to zero. For an oil reservoir overlain by a gas cap containing a constant pressure boundary, numerical results show an almost constant log-derivative of the wellbore pressure drop is obtained (semilog straight line suggestive of pseudoradial flow), indicating that the wellbore pressure response for such systems is dominated by changes in the total relative mobility near the sandface. A complete discussion of these results can be found in Ref. 2.
The theoretical results presented in this work are generally applicable to both single-phase and multiphase reservoirs which are either infinite-acting or contain a constant pressure boundary. The results are particularly useful for analyzing the pressure response of wells producing reservoirs undergoing an immiscible displacement process, either natural or induced.

Introduction
Central to many of the arguments we put forth to explain the pressure response obtained for the multiphase reservoir systems investigated in this study is the classical fractional flow theory, i.e., Buckley-Leverett theory. We, therefore, review the classical theory and examine some of the physical consequences when the assumptions incorporated in the theory approximately hold.
Major assumptions which have typically been incorporated into fractional flow solutions include:one dimensional flow in a homogeneous and isotropic porous reservoir,the fluids are incompressible,gravity and capillarity are negligible,no fingering,Darcy's law applies,local equilibrium exists,the initial distribution of fluids is uniform andat most, two phases are flowing.
Reference 5 notes that even if the fluids are compressible, compressibility effects are negligible if c p&lt;0.01.
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