Summary
This paper presents a robust and rigorous method for the numerical solution of the material balance equations of compartmented gas reservoirs. The method is based on the integral form of the material balance equations and employs an implicit, iterative solution procedure. The proposed method enables extension of traditional p/z analysis of single gas reservoirs to complex, compartmented gas reservoirs.
Example calculations of the depletion of a compartmented reservoir show how the p/z is affected by crossflow, reservoir size, and depletion rate. The depletion behavior can be rationalized by the observation that depletion of a compartmented reservoir at a constant rate tends to develop a semisteady state. A field example is presented that illustrates the capabilities of the extended material balance for the analysis of the past performance of compartmented reservoirs.
Introduction
Material balance analysis is a standard reservoir engineering tool for the analysis of the performance of oil and gas reservoirs. Applied to single, tank-type gas reservoirs, the material balance yields a characteristic relationship between the ratio of pressure to z factor (p/z) and cumulative gas production.1 In the ideal case of volumetric depletion, i.e., no changes in the hydrocarbon pore volume during depletion, this relation simplifies to a straight line.
A relatively new development is the application of material balance analysis to more complex, compartmented reservoirs.2–5 A compartmented reservoir is defined here as a reservoir that consists of two or more distinct reservoirs that are in hydraulic communication. A well-known example is a faulted reservoir made up of different fault blocks separated by partially sealing faults.
For the purpose of a material balance analysis, a compartmented reservoir may be modeled as an ensemble of individual tank-type reservoirs, which are connected to one another by thin permeable barriers.2 Each compartment is described by its own material balance, which is coupled to the material balance of neighboring compartments through influx or efflux of gas across the common boundaries. Application of the material balance method to compartmented reservoirs requires a fast, robust, and rigorous method for solving the system of coupled material balance equations. This is the subject of the paper.
Hower and Collins2 presented analytical solutions of the material balance equations for a compartmented reservoir consisting of just two reservoirs. Their solutions hold good under rather restrictive conditions: constant offtake rate from only one reservoir compartment, volumetric depletion, and constant gas properties. Yet the analytical solutions clearly demonstrated the basic features of the depletion of compartmented reservoirs.
Lord and Collins3 generalized the material balance method to multicompartment reservoirs. They solved the material balance equations numerically, without introducing any simplifying assumptions and conditions. They formulated the equations as a system of coupled first-order ordinary differential equations in the pressure. The solution of this system then boils down to numerically solving an initial value problem, for which the authors used the Burlisch-Stoer method. No details were presented on the implementation of this method. Lord et al.4 applied the extended material balance method to the compartmented gas reservoirs in the Frio formation in South Texas.
Payne5 applied the multicompartment reservoir model to single, tight gas reservoirs. He solved the material balance equations by means of an explicit method, ignoring changes in the flow across boundaries and gas properties during a timestep. For the calculation of the crossflow between compartments, Payne used the pressure squared formulation. Payne's calculation method is simple and straightforward, and lends itself very well for implementation in a spreadsheet program. However, the explicit calculation scheme and the use of the pressure-squared approximation might give rise to unacceptable errors.
In this paper, we present a simple but rigorous numerical method for the solution of the material balance equations for compartmented gas reservoirs. It is based on the integral form of the material balance equation for each individual compartment, expressed in cumulative quantities, instead of the differential form as used by Lord and Collins. The solution method employs an implicit calculation scheme that properly accounts for the pressure dependency of gas properties. For reasons of clarity and brevity, we restrict ourselves to gas reservoirs that consist of two compartments. However, the method can be readily generalized to multi-compartment reservoirs.
To illustrate the method we present examples of a compartmented material balance analysis applied in both the prediction mode and in the history-matching mode. The prediction calculations bring out the depletion characteristics of a typical compartmented reservoir. In the history match example, we illustrate the use of the compartmented reservoir model for the analysis of the observed pressure behavior of a real-life compartmented reservoir.
The main advantage of the numerical solution method presented here over previous work is its simplicity. The method can be easily incorporated into existing material balance analysis programs, thereby extending the classic "p over z" analysis to more complex, compartmented reservoir systems. In addition, because of its simplicity the method lends itself very well for automatic history matching of observed reservoir performance. The method is recommended for a first analysis of the performance of compartmented gas reservoirs. Depending on the results a more elaborate analysis may be required by means of a more sophisticated 3D, multigridblock reservoir simulator.
Flowing material balance method with adsorbed phase volumes for unconventional gas reservoirs
