An analytical solution for cyclic flow of two immiscible phases

Cyclic injection and production by a single well is a technique used in applications such as natural gas storage, compressed air energy storage and periodic pumping tests. The induced flow and saturation distribution can sometimes be modeled assuming two immiscible and incompressible fluids. An analytical solution is derived for such one-dimensional cyclic flow using the method of characteristics. The solution usually exhibits a front or discontinuity, in which case the contour integral method is applied in the derivation. Saturation (S) is expressed as a function of time, distance from well, mobility ratio and production to injection time ratio t _prod /t _inj . Derivation is carried out for both linear S(x) and radial S(r) flow. Dependence of the solution on these parameters is explored and various cases are analyzed. A simple model for viscosity reduction of the reservoir fluid is presented, accounting for changes in mobility ratio, e.g., following cyclic injection for enhanced oil recovery. It is found that generally, for t _prod /t _inj ⩾1 the solution converges to a periodic steady state, however, for t _prod /t _inj <1 or when accounting for viscosity reduction, such a state is not reached and the injected phase penetrates deeper with each cycle.