Summary
Complex geometry reservoirs can be encountered in the field for a variety of depositional and tectonic processes. For example, fluvial depositional environments may produce interbranching channel reservoirs or reservoirs consisting of relatively high-permeability channels in communication with low-permeability splays.
This paper presents a general methodology for computing pressure responses and flow characteristics in complex geometry reservoirs. The proposed method consists of decomposing the original complex-geometry reservoir into a set of simple-geometry reservoirs, which interact with each other by transfer of fluid and equality of pressure over the regions where they are in hydraulic contact. Analytical solutions are written for each of the simple reservoir components in terms of the unknown pressures and fluxes at their boundaries, and the coupled systems are solved for the desired wellbore pressure responses.
The method of sources and sinks is used to compute the pressure response in the Laplace domain, and the results are inverted numerically with the Stehfest Inversion algorithm.1 We present fast, accurate methods of taking numerical Laplace transforms of the source/sink solutions that make the computations reasonably fast and efficient.
The proposed methodology can be extended to any system (infinite or bounded) in which the Laplace-space solution can be written easily in terms of integrals of real-space source/sink functions, including production at constant bottomhole pressure, wellbore storage effects, or naturally fractured systems. We demonstrate the applicability of the method by modeling branching channels and channel/splay systems.
Introduction
Classical well-test analysis has long been used as a valuable tool in characterizing reservoirs using transient pressure vs. time behavior. In most classical well-test models, the reservoir is idealized as a homogeneous single- or dual-porosity system with a simple reservoir and well geometry. This is done to facilitate generation of analytical solutions to the reservoir problem. In fluvial-deltaic reservoir systems, however, the complex geometry precludes idealizing the reservoir as a simple-shaped homogeneous system and, in general, the transient pressure responses do not resemble those of classical simple-geometry systems. Fluvial-deltaic reservoir systems are one example of general complex-geometry reservoirs for which classical well-test analysis models may not be applicable.
Except for work presented by Larsen2,3 on a network of interconnected linear reservoirs, there appears to have been very little presented in the literature on modeling or describing the pressure transient pressure behavior of these types of reservoirs. Larsen's work was concerned primarily with the long-time productivity of a network of intersecting linear reservoirs, with no special consideration given to the geometry at the regions of intersection. Larsen's work indicated that proper understanding of reservoir type is critical in situations in which extrapolation of short-time test data to possible late-time production characteristics is attempted.
Reservoirs with sealing or partially sealing faults are another type of complex-geometry reservoirs that has received attention in the literature.4–9 Both numerical and analytical solutions have been presented to model the pressure behavior in faulted reservoirs; however, the focus of these studies has been on the effect of the sealing and/or nonsealing faults, with little attention given to the physics of the fluid transfer between the communicating reservoirs. In this work, we attempt to rigorously account for the effects of the hydraulic contact between the connected reservoir components.
Modeling Philosophy and Methodology
In this section, we illustrate our general modeling approach by applying it to a simple reservoir system. The physical model considered in Fig. 1 consists of two semi-infinite reservoirs separated over most of their extent by a hydraulically sealing barrier. One has flow properties of a "channel," which we will refer to as Region 1, while the other has properties of a "splay" (Region 2). The reservoirs are in hydraulic contact only over a small area. A well is producing from one of the reservoirs.
The first step in the modeling procedure is to identify simplegeometry homogeneous reservoir components that make up the complex system. In the preceding example, the complex reservoir is composed of two semi-infinite reservoirs: Region 1 and Region 2. Considering Region 1 only, the reservoir behaves as if it contains two wells: the original producer and a planar injection well at the area of communication between the two reservoirs. Similarly, considering Region 2 only, it behaves as if there were a single producing planar fractured well at the area of communication between the two reservoirs. Because all the fluid leaving Region 2 is entering Region 1, the production and injection rates of the apparent planar "fractured" wells in each of the regions must be the same.
The second step in the modeling process involves separating the various reservoir components from one another at their junction(s) by applying no-flow boundaries. For simple reservoir systems that contain wells, this is achieved by creating a mirror image to the no-flow boundary; that is, the inserted no-flow boundary becomes a plane of symmetry for the extended system. For the problem under consideration, Region 1 and Region 2 are each reflected using the fault as a mirror (see Figs. 2 and 3).
Planar injection or production wells are added along the plane of symmetry to account for fluid transfer from one system to the other. The separate systems are then coupled by equating rates and pressures at the junctions and solving (in Laplace space) for the unknown flux distributions along the introduced planar wells.
Finally, the flux profiles from the preceding step are substituted into the pressure equation for the well of interest to obtain the desired pressure response. We illustrate the procedure in more detail for the simple problem under consideration.
WELL TEST ANALYSIS AND INTERPRETATION: THE USE OF ARTIFICIAL NEURAL NETWORK
