Fractional-order integral generalization of the Rapoport–Leas equation

Abstract

A one-dimensional time-fractional integral model for a two-phase flow through a porous medium with power-law memory in the presence of capillary forces is considered. Based on the time-fractional integral generalization of Darcy’s law and the classical mass balance equations for phases, neglecting gravity forces, a time-fractional integral generalization of the Rapoport–Leas equation is derived. It is shown that in the limiting case of constant capillary pressure, the obtained equation coincides with the classical Buckley–Leverett equation. An example of initial boundary value problem for the derived fractional integral equation is given for the case of a finite reservoir. Unlike the classical Rapoport–Leas equation, its time-fractional integral analogue does not have a traveling wave solution in general case. It is due to the fact that the fractional integral with a finite lower limit is not invariant with respect to translations in time. However, for the model problem of two-phase displacement in an unbounded reservoir, considered as a discontinuity breakup problem, the full memory approximation is valid. In this case the obtained equation has a traveling wave solution. In this approximation, relations for the main characteristics of the saturation jump on the shock are obtained. In particular, it is shown that the speed of the shock can be found from an equation that includes the time-fractional integral characteristic of the Leverett function. The behavior of the shock near its left and right boundaries is also asymptotically studied for different saturation levels at the boundaries. It has been proven that if the initial saturation exceeds the residual saturation of the displacing phase, then on the right boundary the saturation at the jump exponentially tends to the initial one. On the left boundary, the asymptotic behavior of the jump is also exponential, with the exception of one special case in which it becomes a fractional power law. For the equation in question, an approximate solution has been analytically constructed for the case of «weak» full memory, when the order of fractional integration is close to zero and can be considered as a small parameter.