Effect of Naturally Fractured Aquifers on Oil Recovery From Stress-Sensitive Naturally Fractured Reservoirs
Authors: Aguilera R. 
Published: March 2010

Abstract
The material balance (MB) equation for stress-sensitive undersaturated and saturated naturally fractured reservoirs (nfr's) has been written taking into account the presence of an unsteady-state naturally fractured aquifer (nfa). The nfa can be of limited or infinite size. The solution is presented in finite difference form to achieve a quick convergence of the iteration process.
Historically, non-fractured aquifers have been attached to material balance calculations of naturally fractured reservoirs. This is not realistic from a geologic point of view as it implies that fractures are present in the oil portion of the reservoir, but disappear the moment the water oil contact is reached. It is shown that the use of an unfractured aquifer in a naturally fractured reservoir can lead to erroneous oil recovery estimates. The effect of water influx on the material balance equation is illustrated with an example.

Introduction
Forecasting the performance of a nfr subject to water entrance from a nfa is a major challenge. The problem is compounded when the nfr and nfa are stress-sensitive. In this paper, the nfr is stress-sensitive, but the nfa is not. The work presented in this paper is not meant to replace a detailed reservoir simulation, which is, in my opinion, the best way to solve the problem, provided that the original oil in place, size of the aquifer, reservoir characterization and quality of the pressure and production data is good. The idea is to have a tool that can provide a quick idea with respect to potential oil recoveries from stress-sensitive nfr's affected by water influx from nfa's.
Various analytical aquifers have been considered in the literature; most notably those developed by Schilthius(1), Hurst(2) and van Everdingen and Hurst(3) for edge water and Allard and Chen(4) for bottom water. All of these aquifers considered matrix porosity but ignored the presence of natural fractures. Aguilera(5) developed equations to account for the presence of natural fractures in unsteady-state edge aquifers. The equations were validated by successfully comparing results against those published by van Everdingen and Hurst(3).
This paper presents MB equations for predicting oil recovery of undersaturated and saturated nfr's affected by water influx from nfa's. The nfa can be of limited or infinite size. The infinite case applies, in practice, to those aquifers that are connected with the external world (for example connected with lakes and rivers). The equations are written taking into account the effective compressibility of matrix and fractures. Stress-sensitive properties such as fracture porosity, fracture permeability, partitioning coefficient and exponent for shape of relative permeabilities are taken into account.

Unsteady-State Naturally Fractured Aquifer
Aguilera(5) presented a solution to the problem of non-stressed nfa's following the work of van Everdingen and Hurst(3). For the case of constant pressure at the inner boundary (constant terminal pressure case) and constant pressure at the outer boundary, the dimensionless water influx is given in Laplace space by:
Equation (1) (Available In Full Paper)
For the case of constant inner pressure and a sealed outer boundary, the dimensionless water influx is given in Laplace space by:
Equation (2) (Available In Full Paper)