Transient Behaviour Of Nonhomogeneous Linear Aquifers

Abstract

This paper presents new analytical solutions for the dimensionless pressure drop or cumulative influx for nonhomogeneous aquifers whose thickness, permeability-viscosity ratio. or porosity-compressibility vary linearly with distance. These solutions apply to one-dimensional linear flow conditions with finite aquifer size. The inner boundary condition may be either constant rare or constant pressure. The outer boundary may be either dosed, or at constant pressure. The solutions presented ;n this paper contain homogeneous aquifer solutions as limiting cases. Thus, this study extends the state-of-the-art of analytical aquifer modeling.

Based on this study, a nonhomogeneous aquifer behaves as a homogeneous aquifer at early times. At late times, pseudosteady or steady-state behaviour is observed, depending on the outer boundary condition. But the dimensionless time to late-time state is dependent on the type and severity of the property variation. Limiting solutions at early and late times are presented for each case.

Introduction

Most hydrocarbon reservoirs are at least partly bounded by water bearing rocks called aquifers. The aquifer size may appear infinite or finite compared to the reservoirs they adjoin. In performing material balance calculations as proposed by van Everdingen and Hurst ⁽¹⁾, and Hurst ⁽²⁾, the dimensionless aquifer response to a unit pressure drop or a unit fluid-withdrawal rate is needed. Also, aquifers are frequently represented in reservoir simulators using analytical response functions.

Past studies have considered linear, radial, and spherical flow in the aquifers, Several studies of water influx have been described by Craft and Hawkins ⁽³⁾. Schilthius ⁽⁴⁾ first proposed a steady-state water influx model. A significant advance was made by the van Everdingen and Hurst ^(l) unsteady-state water influx model that used the response of a dimensionless aquifer v..ith time. They Considered the aquifer to be homogeneous and infinite in extent. Later, Hurst ⁽²⁾ presented a simplified solution of a material balance equation which used the response for linear and radial homogeneous, infinite aquifers. Miller ⁽⁵⁾, and Nabor and Barham ⁽⁶⁾ presented analytical solutions for linear, homogeneous aquifers. Nabor and Barham ⁽⁶⁾ considered both constant rate and constant pressure inner boundary conditions. They also considered the outer boundary to be infinite, closed or at a constant pressure.

Mueller ⁽⁷⁾ used finite-difference techniques to compute dimensionless pressure drop or cumulative influx for few cases of a non-homogeneous aquifer. This paper presents analytical solutions for the transient behaviour of nonhomogeneous linear aquifers.

Mathematical Model for Linear Aquifers

A partial differential equation describing the flow of a slightly compressible fluid in porous media is:

Equation 1 (available in full paper)

In writing equation (1), we assume a horizontal formation, no gravity forces, isotropic properties, a single fluid of small but constant compressibility, etc.

For a linear one-dimensional medium, equation (l) can be written as:

Equation 2 (available in full paper)

As indicated by Mueller ⁽⁷⁾, kh/ µ and Φc,h are two groups of variables which can be functions of the space variable. Thickness, appears in both groups. So one may study cases wherein either k/ µ (or k), h or Φc is a function of the space variable. Let us identify these as different cases: