Abstract

Reservoir deposition occurs over geologic periods of time. Although reservoirs are assumed homogeneous for simplicity of analysis, most reservoirs are heterogeneous in nature. Some common forms of heterogeneity are the presence of layers and the presence of different zones of fluids and/or rocks in the formation.

In this study, a new analytical solution for multi-layered composite reservoirs with pseudosteady state interlayer crossflow has been developed. Fluid flow in the reservoir has been treated as a generalized eigenvalue problem. The developed analytical solution for an n-layered composite reservoir is applicable for any irregularly-shaped discontinuity boundary, and for closed, constant-pressure and infinite outer boundary conditions. This new solution is computationally very efficient. Using eigenvalues and eigenvectors of the system, this method requires solution of an order of magnitude fewer simultaneous equations as compared to other methods proposed in the literature. This method is also very versatile and can handle multiple composite regions (more than two), and partially-penetrating wells subject to bottom-water and/or gas-cap drives for well testing purposes. This analytical model has been validated by comparing the results with those of some simple, well-known models in the well testing literature. Solution methodology and future possibilities of the new solution have also been discussed.

Introduction

Most reservoirs are heterogeneous in nature. The presence of layers and zones of different fluid and/or rocks is a common cause for reservoir heterogeneity. Figures 1a and 1b show a layered reservoir and a layered composite reservoir, respectively. The horizontal lines show the layering and the arrows show the presence of crossflow. The layers may be communicating or noncommunicating. Formation crossflow is present when the layers are communicating. When the layers do not communicate with each other, except through the wellbore, then the reservoir is termed a "commingled reservoir". A layered, composite reservoir may result because of artificial as well as natural processes. Enhanced oil recovery processes such as steam flooding, CO2 flooding, in-situ combustion, tectonic movements, phase changes, acidizing, and temperature differences may cause a reservoir to behave as a composite reservoir. The tilted line in Figure 1b shows the discontinuity boundary or the fluid front. A layered, composite reservoir situation occurs when all or some of the layers have two or more regions of different rock and/or fluid properties.

Numerous studies have been reported in the literature on layered reservoirs and Table 1 lists relevant papers on layered reservoirs. One aspect in which the studies differ is the way they model crossflow between the layers. Formation crossflow has been modelled mainly by two methods: pseudosteady state crossflow and transient crossflow. Pseudosteady state crossflow assumes that the resistance to crossflow is confined to the interlayer boundary and the flow is horizontal within each layer. This assumption reduces a two-dimensional problem to a one-dimensional problem. Transient crossflow utilizes the two-dimensional diffusivity equation for each layer. Table 1 also shows that, although numerous studies have appeared on layered reservoirs with formation crossflow, very little work has been reported for layered, composite reservoirs with formation crossflow. Responses of layered reservoirs may be summarized as follows: for commingled reservoirs, the time needed to reach pseudosteady state is an order of magnitude higher than that for homogeneous reservoirs; semi-log analysis can be used to estimate the average permeability-thickness product and the skin effect; initially, a crossflow system and a commingled system have the same responses; then there is a transition period and, finally, the crossflow system behaves like an equivalent homogeneous system.

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