Well test analysis for heterogeneous reservoirs

This work presents procedures for analyzing well-test pressure data obtained from a well located in a heterogeneous reservoir where the permeability is an arbitrary function of the distance from the well.; Two procedures to solve the inverse problem of generating permeability distribution directly from well-test pressure data are presented. In the first method, the permeability function is obtained by using a simple, recursive algorithm derived to numerically solve an integral equation of the first kind that represents the wellbore pressure derivative solution. The other procedure is based on considering the "instantaneous permeability", obtained from well-test data, as a volumetrically-weighted harmonic average of the permeability distribution between the wellbore and a radius of investigation. Both procedures can be applied to either transient drawdown data or buildup data corrected for producing-time effects.; A method to obtain the reservoir pressure profile at the instant of shut-in directly from buildup data is presented. It is shown that, under certain conditions, the pressure profile, which is initially computed up to the last radius of investigation, can be reasonably extended up to the reservoir external radius and then used to estimate the average reservoir pressure. In addition, if the pressure profile is established under either a steady-state or pseudosteady state flow condition, one can estimate the productivity index and the equivalent homogeneous permeability.; We present exact solutions for the reservoir pressure response during either steady-state or pseudosteady state flow. Using these solutions, we derived analytical equations for the average reservoir pressure, the stabilized inflow performance and the equivalent homogeneous permeability which characterizes the long-time (boundary-dominated flow) performance. The exact equation for the flow rate during boundary- dominated flow for constant pressure production is also presented.; It is shown that, when our inversion procedures are applied to well-test data obtained from a heterogeneous reservoir, where permeability and/or porosity vary with position, we obtain an equivalent radial permeability distribution with porosity constant. Both reservoir descriptions, when used as input to our simulator, yield essentially identical pressure and pressure derivative responses. This illustrates the nonuniqueness of the inverse problem.