The Solution of Nonlinear Equations

This paper presents procedures for treating problems involving transient flow of gases in porous problems involving transient flow of gases in porous media. The methods involve stepwise calculations using linearized equations derived from the nonlinear, second-order equations that describe transient flow of gas. The distribution of pressure around a gas well at various times can be readily calculated with a desk calculator or a small computer. Equations and procedures are offered for both infinite and limited reservoirs. Solutions by these new technique's are shown to be in good agreement with computer solutions available in the literature. Also discussed is a procedure using relatively few image wells for treating problems in reservoirs with curved, irregular boundaries. Introduction This paper is concerned with solving problems involving transient flow of gas in porous systems. The main contribution of this paper is the development of a technique that permits the effective use of a desk calculator for the computations. The methods presented here will permit individual engineers not having access to one of the larger computers to solve many practical problems that heretofore would have been intractable. problems that heretofore would have been intractable. The equation describing unsteady-state flow of a perfect gas in a horizontal reservoir is reported by perfect gas in a horizontal reservoir is reported by Katz to be (1) This equation is obtained by combining the continuity equation, Darcy's law, and the following density relationship for the gas. =...............................(2) The intractableness of Eq. 1 stems from the fact that it is a nonlinear, second-order differential equation for which no analytical solution is known. In 1953, a pioneering use of computers was presented by Bruce et al., who approximated the presented by Bruce et al., who approximated the differential equation with difference equations and solved these numerically. Their solutions developed for horizontal flow of a perfect gas for circular reservoirs were presented in terms of dimensionless parameters in plots of pressure vs radius for various parameters in plots of pressure vs radius for various times and flow rates. Later, Aronofsky and Porter, using computers, solved radial flow problems for nonideal gases, permitting gas properties to vary as linear functions of pressure. Recently, advances in solving problems involving the flow of nonideal gas have been made by Ramey and his colleagues, who have developed equations and computer solutions for real gases in terms of pseudo-reduced pressures. In the present paper, a method of solving the flow equations for gas is developed for both infinite and limited reservoirs. The methods, which make use of the analytical solutions for the corresponding linear differential equation for radial flow, can be used with a desk calculator or a small computer to solve problems characterized by nonradial as well as by radial reservoir geometry. An example solution is given to illustrate the method for the flow of a perfect gas in a circular, horizontal reservoir; the results have been compared with those of Bruce. The technique can also be applied to the flow of nonideal gases in nonuniform systems, as well as to oil reservoirs above the bubble-point pressure and to aquifers. In all cases, the results of the example problems obtained by the procedures presented in the text are compared with procedures presented in the text are compared with available published and accepted numerical or analytical results. Finally, a discussion is offered with the intent of guiding the reader toward successful application of the technique in solving practical reservoir problems. problems. SOLUTION OF THE NONLINEAR PROBLEM MATHEMATICAL BACKGROUND In early tracts published on the transient flow of fluids in porous media, the following equation is derived for radial flow. SPEJ P. 348