Pressure-Transient Analysis for a Slanted Well in a Reservoir With Vertical Pressure Support

Abstract

Summary Analytical solutions are presented for the pressure response of a slanted limited-entry well in an infinite reservoir subject to pressure support from a gas cap, bottomwater drive, or a combination of both. The well's inclination angle ranges from 0 to 90 degrees, covering vertical and horizontal wells as limiting cases. An interpretation method based on nonlinear regression provides a useful analysis procedure for well tests from such reservoirs. The pressure support from the gas cap or bottomwater often masks the late-time reservoir features, making conventional semilog analysis difficult. Field tests are interpreted to illustrate the applicability and limitations of the regression method. Introduction Many reservoirs are subject to gas-cap or strong bottomwater-drive effects. To reduce gas or water coning, the well is completed only in the oil-bearing stratum that is away from the gas cap or bottom-water. As oil is produced, the expanding gas cap or encroaching water maintains a fairly constant pressure at the gas/oil or oil/water interface. The constant pressure boundary flattens the transient pressure response and sometimes completely masks the semilog radial pressure response and sometimes completely masks the semilog radial flow. Thus, conventional semilog interpretation becomes difficult and unreliable for these systems. A substantial amount of numerical and analytical work has focused on the pressure behavior of partially penetrating wells, limited-entry wells, and horizontal drainholes producing from an infinite reservoir with two impermeable boundaries. Various interpretation techniques, including type curves, analyzed well tests from these systems. Analytical solutions for vertical limited-entry wells and horizontal wells in a reservoir with one constant-pressure boundary created by a supporting gas cap or aquifer have been derived. These solutions assume that the flux distribution is uniform across the length of the perforated interval. No comprehensive solutions for inclined wells in perforated interval. No comprehensive solutions for inclined wells in reservoirs with various boundary conditions are available. This paper first presents analytical solutions for the pressure-draw-down behavior of a slanted limited-entry well in an pressure-draw-down behavior of a slanted limited-entry well in an infinite reservoir with different combinations of no-flow and constant-pressure upper and lower boundaries. The solutions are general and degenerate to simpler geometries, such as vertical and horizontal wells. The method of source and Green's functions is used to derive these solutions, and averaging the uniform-flux wellbore solutions over the producing interval is shown to yield good approximations to the more exact infinite-conductivity wellbore solutions. Second, an interpretation procedure based on automatic parameter estimation is presented for well tests from reservoirs parameter estimation is presented for well tests from reservoirs subject to pressure support. Field examples show the applicability of the solutions and the proposed interpretation method. Mathematical Development Fig. 1 shows a schematic of the well/reservoir system considered in this paper. A slanted line-source well is completed at a distance Db from the lower reservoir boundary. The well's angle of inclination is theta w and the perforated interval is of length Lw. The upper and lower reservoir boundaries may be constant-pressure or no-flow boundaries. The reservoir is assumed to be horizontal with thickness h and homogeneous and anisotropic with horizontal and vertical permeabilities kH and kV. Ordinary well testing assumptions also permeabilities kH and kV. Ordinary well testing assumptions also hold.