This paper considers the design of an algorithm for analyzing multi-rate horizontal well pressure transient data using nonlinear regression techniques. Measured data is modeled using the analytical solution for a uniform-flux horizontal well in a rectangular parallelepiped reservoir. Parameter optimization is achieved using the public-domain Levenberg-Marquardt algorithm, LMDER, from Argonne national Laboratory.
We focus on efficient methods for calculation of analytical partial derivatives of the model pressure response which are required by the optimization routine. Complete expressions for all required partial derivatives are provided in Appendices, along with details on a method of evaluating these derivatives which requires minimal computational effort. We demonstrate the utility of the method by analyzing field buildup pressure data.
Pressure Transient data from horizontal wells is (in many cases) difficult to interpret using conventional well test analysis techniques; e.g., semilog analysis, type curve matching and pressure derivative analysis. This is because of the number of reservoir parameters involved (i.e., three directional permeabilities for anisotropic systems) and the complex system geometry. The location of the well in the reservoir and its proximity to the reservoir boundaries can result in pressure and pressure derivative curves that appear practically unanalyzable. Difficulties in successful interpretation are further complicated by rate variations during (or before) the test, and wellbore storage effects.
In this paper, we present details on a nonlinear regression analysis technique to assist the engineer in properly interpreting pressure transient data from a drawdown or buildup test on a horizontal well. Our model is based on the assumption of a "uniform flux" horizontal well in an anisotropic, bounded, rectangular box-shaped reservoirs, with the well axis parallel to one of the principal reservoir axes (see Fig. 1). The boundaries of the reservoir may be sealed or infinite-acting. Production may be at a constant rate or at a sequence of constant (step) rates. Since buildup is a production period where the rate is constant and equal to zero, it is included as a special case of multi-rate production.
Figure 1 - Horizontal Well in a Rectangular Parallelepiped Reservoir.
It must be emphasized that the present model is not a "cure all" for horizontal well pressure transient analysis. The model can lead to erroneous and even non-physical results if the input is not guided by the known physical and geological attributes of the system under study. However, if used intelligently, it can greatly enhance the interpretation engineer's ability to correctly analyze well test data.