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Spontaneous imbibition of water into the matrix blocks because of capillary forces is an important recovery mechanism for oil recovery from naturally fractured reservoirs. In modeling this process, it has been assumed classically that local equilibrium is reached and, therefore, capillary pressure and relative permeability functions are only a function of water saturation, resulting in the appearance of the self-similarity condition. In some works published in the last 2 decades, it has, however, been claimed that local equilibrium is not reached in porous media, and therefore, opposite the classical local-equilibrium/self-similar approach, non-equilibrium effects should be taken into... 

Spontaneous imbibition of water into the matrix blocks because of capillary forces is an important recovery mechanism for oil recovery from naturally fractured reservoirs. In modeling this process, it has been assumed classically that local equilibrium is reached and, therefore, capillary pressure and relative permeability functions are only a function of water saturation, resulting in the appearance of the self-similarity condition. In some works published in the last 2 decades, it has, however, been claimed that local equilibrium is not reached in porous media, and therefore, opposite the classical local-equilibrium/self-similar approach, non-equilibrium effects should be taken into... 

An exact analytical solution for one-dimensional fluid flow through rock matrix block is presented. The nonlinearity induced from flow functions makes the governing equations describing this mechanism difficult to be analytically solved. In this paper, an analytical solution to the infiltration problems considering non-linear relative permeability functions is presented for finite depth, despite its profound and fundamental importance. Elimination of the nonlinear terms in the equation, as a complex and tedious task, is done by applying several successive mathematical manipulations including: Hopf-Cole transformation to obtain a diffusive type PDE; an exponential type transformation to get a...