Reservoirs With Spatially Varying Properties: Estimation of Volume From Late Transient Pressure Data
Authors: Gavalas G.R. , Seinfeld J.H. 
Published: December 2007

Abstract
The transient pressure in a reservoir of arbitrary shape and spatially varying properties is expressed in terms of the eigenvalues and eigenfunctions of the region. Although these eigenvalues and eigenfunctions cannot be obtained analytically, except in homogeneous reservoirs of regular shape, the structure of the solution allows the estimation of the pore volume in the general case using data in the pseudosteady or the late-transient period. A least-squares analysis of late transient data is presented which yields, in addition to the pore presented which yields, in addition to the pore volume, the first eigenvalue, which can often be related to reservoir properties. Numerical examples are given using simulated data from a rectangular 4 x 1 reservoir with constant permeability and with spatially varying permeability.

Introduction
The estimation of the pore volume of a petroleum reservoir from well pressure data has heretofore required an analytical expression for the transient pressure. Such analytical expressions can be pressure. Such analytical expressions can be obtained only for reservoirs of regular shape and uniform physical properties. For example, analytical solutions for the pressure in a circular reservoir] with a centrally located producing well were originally derived by Muskat and later by van Everdingen and Hurst. These and similar results for other regularly shaped homogenous reservoirs are presented in the comprehensive monograph of Matthews and Russell. Analytical solutions have also been developed for circular reservoirs composed of two concentric zones each having uniform properties, Carter. In this paper we develop a method for estimating the volume of a reservoir of arbitrary shape, and arbitrary porosity and permeability distributions. Although analytical permeability distributions. Although analytical solutions are now impossible, the structure of the solutions can still be found and this is sufficient for estimating reservoir volume. The pressure/time behavior at a typical well in a reservoir under constant production can be divided into three regimes. The first is the early-transient period, in which the effect of the reservoir boundary is not yet felt; the second is the late-transient period, in which the boundary is felt; and the third is the pseudosteady (semisteady) period, in which the pressure at every point in the period, in which the pressure at every point in the reservoir is decreasing linearly with time. It is well known that the late-transient period is considerably longer in a reservoir of elongated shape than in a circular reservoir of the same volume. Since the early estimation of reservoir volume is of considerable economic interest, we emphasize the estimation of volume using date in the late-transient period, rather than requiring data in the pseudosteady period as in conventional limit tests. We then present a computational example illustrating the estimation using late-transient data.

THEORY
We consider a reservoir of arbitrary cross-section and spatially varying porosity and permeability, as shown schematically in Fig. 1. The thickness of the reservoir will be assumed to vary slowly so that a two-dimensional description can be employed. Thus, the equation for the flow of a fluid of small and constant compressibility can be written as(1)...
where each of the n wells has been considered as a point sink located at (xj, yj). The wells can also be considered as finite, with a similar development as shown in Appendix B. The condition assigned at the external boundary will be that of zero flow,(2)...
We shall consider a period, t greater than or equal to O, of constant production, with an arbitrary initial pressure production, with an arbitrary initial pressure distribution,(3)t=O: p=p (x, y)
SPEJ
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