A Generalized 3D Analytical Model for Transient Flow in Compartmentalized Reservoirs
Authors: Rahman N.M. Anisur , Ambastha A.K. 
Published: December 2007

Abstract
A new generalized transient flow model for a compartmentalized system in three-dimensional coordinate system with N compartments has been developed analytically. Each compartment may have distinct rock and fluid properties and may produce through a number of partially-penetrating wells. A partially-communicating fault or barrier causing poor hydraulic communication between a pair of adjoining compartments is modeled as a thin skin at the interface. Production rates and conditions at the extreme boundaries are considered time-variant. The solution has also been validated. An example problem with stacked channel realization has been studied with the new solution. A simple method for detecting the poorly-drained compartments from the extended drawdown data has been developed. This has also lead to an estimation of hydrocarbon volume in each compartment.

Introduction
Compartmentalized reservoirs are made up of a number of hydraulically-communicating compartments or regions. The communication between these compartments may be poor due to the presence of faults or low-permeability barriers. Evidence of reservoir compartmentalization both in oil and gas reservoirs has been presented in the literature. The idea of compartmentalization has been developed initially from the observations of discontinuities in producing fields. Reservoir compartmentalization has been observed in both areal and vertical extent. The horizontal barriers may be due to the presence of shales, micaceous streaks or stylolytes while the vertical barriers may be due to the presence of faults or of stratigraphic changes. In the mathematical models for transient flow in areally-compartmentalized reservoirs, the effects of gravity are neglected. However, in the presence of massive vertical compartmentalization, such effects are important. In this study a generalized model for transient flow in compartmentalized reservoirs in three-dimensional Cartesian coordinate system is presented. This new solution takes care of the effects of gravity by formulating the entire problem in terms of potential, rather than in terms of pressure. The new solution has been further generalized by considering time-dependent production rates and Cauchy-type conditions at the extreme boundaries. Since various types of boundary conditions are encountered in the field, the time-dependent Cauchy-type of boundary conditions are capable of dealing with wide ranging situations. For instance, the Cauchy-type conditions can yield the Dirichlet- and Neumann-types as special cases. The Dirichlet-type condition arises if an extreme boundary is maintained at a constant potential and the Neumann-type condition arises if the extreme boundary is closed to communication of any fluid. Here, we consider that there are N compartments that might be arranged vertically and areally for the purpose of developing the solution. This kind of formulation has been considered particularly for the purpose of evaluating reservoir properties and of detecting complex geological structures of compartmentalized systems including cellular system, stacked channel realization, and aeolian dune sand realization. In these kinds of geological structures, the areal models, as developed in Refs. 3 and 4, will not be adequate to describe the mechanics of flow because of high degree of compartmentalization in vertical extent. Ref. 6 has used the solution from Ref. 4 for evaluating the communication resistance between a system of a small compartment (producing) in communication with a big compartment (supporting) from extended drawdown data.
A number of models based on the material balance technique have been proposed in the literature for studying compartmentalized reservoirs. However, these models work reasonably well for the gas reservoirs having permeability in the range of moderate to high (greater than 5 md). Ref. 9 also concludes that these material balance models are not appropriate for formation permeability being less than 5 md. Thus, it is obvious that the criterion for using such models will be much more restrictive in oil reservoirs.
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