This paper presents a new method of estimating drainage area size and shape from production data (bottom-hole pressures and flowrates). The method is a rigorously derived approximation for variable-rate flow in a closed reservoir. This method requires a graph of Ap/qm vs. the superposition plotting function (which is easily calculated by plotting function (which is easily calculated by hand). The slope and intercept of the graph are used to provide the desired estimates of drainage area size and shape.
The method that we propose is an approximation, however it has been proved to be very accurate for the constant rate, constant pressure, exponential rate, logarithmic rate, hyperbolic rate, sinusoidal rate, and discrete rate cases. The method also gives acceptable results for square wave rate and random rate cases.
The new method is derived for the time after the initial pressure transient has reached the outer boundary. The changes in flowrate cause additional transients, but we assume that this effect is negligible when compared to the influence of the outer boundary. Therefore, if the change in flowrate does not dominate the influence of the outer boundary, the new method should give acceptable results. Also, at present, this method is only derived for single-phase flow of a liquid of small and constant compressibility.
The purpose of this paper is to present a simple, but accurate method of predicting reservoir drainage area size and shape from variable-rate production data. Previous works have dealt with production data. Previous works have dealt with constant or cyclically constant rate and constant bottom-hole pressure production. A summary of these methods is shown graphically in Figure 1. Earlougher defined the cyclically constant or square wave rate case in Figure 2. Rather than focus on a particular rate scheme, we develop a general variable-rate approximation that should give accurate results for typical production situations.
Without a variable-rate solution we would have use the more tedious material balance methods that require average reservoir pressures to estimate reservoir pore volume. This would require the well to be shut-in, which results in lost revenue. However, with the new method, the reservoir pore volume and shape can be estimated directly from the production data, without shutting-in the well. production data, without shutting-in the well. The problem of variable-rate flow in bounded systems isolimited in the literature t work by Earlougher-' and the "stabilized flowing,- U methods (which use average reservoir pressures). Though both approaches give acceptable results for their specific application, Earlougher's case is not realistic and the "stabilized flow" methods again require the well to be shut-in for average reservoir pressure determinations. This suggests the need for a general solution for variable-rate flow in a bounded reservoir.
In the "Description of the New Method" section we will present the general variable-rate solution and the reservoir characteristics which can be derived from it. Also, we will verify the general variable-rate equation (Eq.(2)) using analytical and finite-difference simulation. Then a step-by-step procedure for applying our method and a complete example will be shown in the "Method of Application" section. Finally, we will present the derivation of the exact solution for variable-rate flow in a bounded circular reservoir and the approximate solution for variable-rate flow in any shape reservoir in the Appendix of this report.