Gas Well Testing Analysis Using Finite-Difference Models and Optimization Techniques

In reservoir engineering studies of gas fields, three important parameters are needed: the permeability-thickness product, the skin factor, and the inertial-turbulent factor. Several methods have been proposed to determine these parameters from gas well tests. The most powerful one is the pseudo-steady state analysis, where a linear dependence on a semilogarithmic plot is found between the real gas pseudo-pressure and time. Another one is the type-curve matching. Both have drawbacks because they can be rigorously applied only to specific cases. An automated type-curve match, which provides a general method to analyze well test data, is presented in this paper. It is based on the adjustment of a numerical model to gas well test data, by means of optimization techniques. The numerical model is the solution of the second order strongly non-linear partial differential equation which rules the radial flow of real gases towards a well. The differential equation is solved by finite-differences, taking into account the variation of gas properties with pressure. In order to simulate any gas well test, appropriate initial and boundary conditions are imposed. The least squares method is used to minimize residual differences between actual pressures measured during test and computed pressures. Two optimization techniques are applied to obtain the best estimate of parameters that minimize the sum of the square of the residuals: the univariate method of Fibonacci [1] and the multivariate method of Marquardt [1, 2]. This procedure is applied to a single-rate and a two-rate drawdown test. Numerical results showed excellent agreement with well test data