A perturbation method for mixed boundary-value problems in pressure transient testing

Data recorded during a well test is interpreted for formation parameters, such as permeability, by comparing the measured pressure transient with that predicted by a mathematical model of the system. In single-phase homogeneous situations, this model is based on the linear diffusion equation. Despite the simplicity of this equation, it is often difficult or laborious to construct exact solutions, in some cases because the relevant problem is of mixed boundary-value character. Here we describe an approximate calculational method which, with little effort, gives good results in a variety of well test problems of this type.We first prove a rather general perturbation theorem, and then apply it in three illustrative cases. The first example provides a test of our basic result for the case of a fully penetrating vertical fracture, for which an exact (though nonelementary) solution is known. The second example is that of partially penetrating or horizontal wells; it is shown that our general result can provide a mathematical basis for the so-called pressure-averaging technique in which the pressure-flux relationship is approximated by assuming that the flux is uniform along the well and then computing the spatial average of the wellbore pressure. Our third example is a new result for the steady-state pressure drop due to flow into a small circular hole on the surface of an impermeable cylinder. This example has relevance for the Schlumberger RFT tool, which withdraws fluid from the formation via a circular probe which penetrates the mudcake surrounding the borehole wall. Numerical results for the shape factor which represents the effect of the borehole curvature are provided.