Mathematical pressure distribution expression of a horizontal well in a bounded reservoir at full flow periods

Abstract

In general, pressure distribution in an oil or gas reservoir can be obtained from the solution to the second order, linear, heterogeneous or homogenous partial differential equation called the diffusivity equation. The distribution enables determination of reservoir performance calculations, reserve estimates, and even numerical simulation studies can be validated from the distributions. Information that can be deduced from pressure distributions includes reservoir boundary, fluid and wellbore characterization. Many methods can be deployed to find the eventual distribution based on chosen reservoir and wellbore boundary conditions, depending on the actual nature of the diffusivity equation selected. In this paper, the dimensionless pressure distributions in the unsteady state flow in a completely bounded reservoir system containing a horizontal wellbore, is solved for all the possible flow periods using source and Green’s functions. This choice is founded on the dependence of the selected
diffusivity equation on time. A largely isotropic reservoir is considered. The distribution derived shows there can be as many as five major flow periods without the usual transition periods. A mandatory infinite-acting (radial flow period) initiate the entire flow process. It was found out that transient pressures depend directly on dimensionless pay thickness of the reservoir and inversely on the dimensionless well length during infinite-acting flow in a bounded reservoir. Furthermore, at other flow periods, pressure distributions depend inversely on the reservoir width and reservoir areal extent with time. The final flow period shows complete pseudosteady state behavior since all the no-flow boundaries are felt during this period.