A theoretical study of gas flow in porous media with a spherical source

Abstract

Gas flow behavior from a spherical source is explored by using linear and nonlinear models, not only in terms of pressure but also in terms of flux. The approach considers dimensionless parameters scaling both radius and time. Specific observations are made for large, moderate, and small time conditions. At large time, the nonlinear model becomes a linear ordinary differential equation with pressure solution independent of the material. However, for moderate and small scaled times this is not the case. The nonlinear model must be solved by using either linear approximations, semi-analytical, or numerical procedures. This model is nonlinear in the primary variable (pressure). However, appropriate mathematical manipulations allow one to change the nonlinearity into a single coefficient, depending on pressure. Focusing on the effects of this coefficient, the nonlinear solution can be confined between two linear solutions obtained by using atmospheric and boundary pressures. Appendix A is an exploration of the errors arising between the nonlinear solution and these two solutions. In Appendix B, a nonlinear model is used to find solutions for large, moderate, and small times. For large time, the case corresponds to the steady state case, and coincides with the solution presented in Appendix A. For moderate and small times the quasi-analytical approximation and the asymptotic solutions of linear and quadratic normalizations of pressure are presented. In Appendix C, simulations of gas flows in linear and nonlinear situations are made. The problem is to determine the change of air pressure in a tank when it is connected to a spherical cavity embedded in a porous medium. These changes in pressure occur when the air moves through the porous media, either for gas extraction or air injection. Both linear and nonlinear analyses require calculations of the pressure and the mass in the tank when the initial and boundary conditions change with time. For each case, gas extraction or air injection, the differences between the linear and the nonlinear models are examined to determine the suitability of the linear model.