Approximating Well-to-Fault Distance from Pressure Build-Up Tests
Authors: Gray K.E. 
Published: December 2007

Abstract
The method used by Horner to calculate distance from a well to a linear barrier fault in an otherwise infinite reservoir is an approximation, the accuracy of which depends upon transient duration relative to fault distance and reservoir rock and fluid properties. Using calculated build-up curves for an undamaged well, Horner's method and several alternative methods not requiring a computer are illustrated and discussed.
Introduction
The unsteady-state pressure behavior of a given well indicates the net effect of all transients operating to produce pressure changes at that well. Only, one or two transients would be involved, for example, during a constant-rate drawdown or build-up test, respectively, of a single well in a new reservoir of large extent. On the other -hand, a multitude of individual transients might be contributing to the pressure behavior at the well. For example, rate changes at the well in question, interference from an offset well or wells, a permeability, or phase change within the reservoir, faults, pinchouts, etc., influence a well's pressure-time history. It is not always possible to isolate each pressure transient from the composite pressure-time history at a test well, since both pressure-increasing and -decreasing transients might be involved, and since some "interference" transients might arrive at the well simultaneously. Depending upon the situation, however, prior pressure changes may, have stabilized at the well before the arrival of another transient, in which case we can usefully analyze that transient. The presence of a fault in a reservoir is obviously of great importance, and this paper is concerned with the pressure behavior of a well near a sealing fault, and how, under idealized conditions, distance to the fault might be calculated from measured well-pressure data. No implication that other interference phenomena are unimportant is intended. An ideal well is used to provide the best situation we could hope for. If problems in analyses arise here, then we might suspect that unambiguous analyses of actual well data might be at times impossible; this implication is intentional. During shut-in time delta t, the pressure behavior at the sand face of a single oil well which has been produced at a constant rate q, for a time t, in a horizontal formation of constant thickness h, uniform permeability k, and porosity phi, at a distance d, from a linear barrier fault in an otherwise infinite reservoir, is given in dimensionless form (in cgs units) by
(1)
where and the dimensionless quantities pressure, rate and time are defined in cgs unity by
where pw is the wellbore pressure and pw is the stabilized formation shut-in pressure. The remaining quantities are defined in the nomenclature. Using the "constant terminal rate" solution to the radial diffusivity equation and the Lord Kelvin point source given by Hurst and van Everdingen, as well as the principle of superposition, Eq.1 can be derived by the method of images. The effect of the fault is duplicated by an image well identical to the real well and located at a mirror image of the real well from the fault plane. The physical significance of the image well is that there is no fluid flow across the fault. i.e., it is a sealing one. While such an assumption is necessary to the mathematical development of Eq. 1, it also limits its strict application, because obviously not all faults are sealing. The real and image wells flow at constant rate for a time t and are then shut in for time delta t. There are, then, four individual constant-rate transients, the superposed effects of which are to be measured at the real well.
JPT