An Approximate Method for Computing Nonsteady-State Flow of Gases in Porous Media

An approximate method of calculation is developed in this paper which allows duplication of radial unsteady-state gas flow computer results where Darcy's law applies, such as those reported by Aronofsky and Jenkins and Bruce, Peaceman, Rachford, and Rice. Moreover, the new calculation method can be used to obtain results for radial unsteady-state gas flow obeying the quadratic flow law proposed by Duwez and Green. Means are discussed for predicting well behavior at single or superimposed flow rates in finite or infinite reservoirs, determining reservoir rock properties from well-test data, reproducing and interpreting back-pressure test data, and determining the radial extent and reserves of gas reservoirs from well-test data. Example calculations are presented for gas flow following both Darcy's law and the quadratic flow law. Introduction Since the publishing of U. S. Bureau of Mines Monograph 7, most gas-well testing methods have been based on the equation where q= production rate, pf and Pw are formation pressure and sandface pressure, respectively, and y and a are constants to be obtained from test data. These methods, used for predicting both deliverability and "open-flow" capacity of gas wells have been useful and accurate in many cases but unsatisfactory in others. Even at best, however, they do not supply information about the formation or lead to an understanding of nonsteady-state gas flow in porous media. Many theoretically based studies of gas flow obeying Darcy's law have been made. Since the partial differential equations which result from combining Darcy's flow law with the continuity equation are nonlinear, most of the published research consists of either numerical solutions or analytical solutions for linear approximating equations. Such solutions have been of limited value in field work due to their unhandy form and their failure to correlate most field data. There is evidence which indicates that Darcy's law is inadequate to describe gas flow at some flow rates of practical interest. A quadratic flow law, which reduces to Darcy's law at low rates, is more successful in accounting for experimentally observed behavior. This flow law has been applied successfully to a few hypothetical reservoir cases in work which has not yet been published. However, these numerical solutions of the equations involved have been successful only on a special analog computer. Routine applications to field cases would be awkward and have not been attempted. The present paper describes an approximate method for computing nonsteady-state gas flow solutions which has been completely successful in predicting the results for both Darcy flow and quadratic flow obtained by elaborate numerical methods. The new calculation method allows determination of the observable variables in gas-well testing at constant rates. It is similar to the scheme of using a succession of steady states suggested by Muskat in that it makes use of steady-state and material-balance equations. It also is similar to the work of Aronofsky and Jenkins in that the new method includes Aronofsky and Jenkins stabilized flow equation as a special case. It improves upon both of these calculation schemes in that it accurately describes all portions of reservoir history and suggests means of determining reservoir rock properties from well-test data. This paper deals only with production from a reservoir, in which case the rate is defined as being negative. The reservoir studied here is a homogeneous, disk-shaped porous body of uniform thickness, with all boundaries sealed except the inner radial boundary of the well. SPEJ P. 264^