Effect of Oil Production Rate on Performance of Wells Producing from More Than One Horizon

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Abstract
The performance of a two-horizon depletion-type reservoir produced through combination wells is analyzed. By introducing some simplifying approximations it has been possible to obtain formulas which are easy to handle. Only ordinary differential equations are used, and the development of the analysis can he followed without difficulty by the non-specialist. A rigorous analysis has been made of this problem. It has been found that the approximations introduced in the simplified analysis are fully justified and that errors seldom will be more than 2 per cent. The report shows the effect of the rate of withdrawal upon the relative depletion of the two horizons. Numerical examples are given. Introduction In its simplest form an oil reservoir consists of a Continuous, homogeneous body of porous and permeable rock, enclosed at the top and the bottom by impermeable material. The flow of fluid through such a reservoir has been the subject of numerous papers, and the mathematical analysis of well performance, pressure build-up, etc., has reached a high degree of refinement. One of the basic assumptions introduced into these analyses is the homogeneity of the reservoir. In non-homogeneous reservoirs, these computations still can be carried out satisfactorily, provided small irregularities in the physical properties of the rock are distributed in a random fashion. The flow equations are derived as solutions of the continuity equation, and their solutions are dependent upon the assumption of many constant factors in the reservoir: permeability to oil, compressibility of the reservoir fluid, viscosity saturation, etc. When these are assumed to be constant, the continuity equation can be solved rigorously. When these assumptions do not hold and corrections are introduced, it becomes very difficult and often impossible to solve analytically.

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