Trajectory-based modeling of coupled processes with applications to fluid flow and geophysics

Accurate modeling of processes associated with the geophysical monitoring of fluid flow commonly leads to coupled systems of nonlinear equations. Using an asymptotic technique, valid for a medium with smoothly varying heterogeneity, I derive an expression for the velocity of a propagating, coupled front. The asymptotic approach produces an explicit expression for the slowness, the inverse of the velocity, of a propagating front. Due to the nonlinearity of the governing equations, the velocity of the propagating front depends upon the magnitude of the changes across the front in addition to the properties of the medium. Thus, the expression must be evaluated in conjunction with numerical simulation. The semi-analytic expressions for front slowness may be used for characterization and the imaging of flow properties. An application to two-phase flow illustrates this approach and its usefulness. The slowness of the two-phase front is governed by the background saturation distribution, the saturation-dependent component of the fluid mobility, the porosity, the permeability, the capillary pressure function, the medium compressibility, and the ratio of the slopes of the relative permeability curves. Numerical simulation of water injection into a porous layer saturated with a non-aqueous phase liquid indicates that two modes of propagation are important. The fastest mode of propagation is a disturbance that is dominated by the change in fluid pressure. This is followed, much later, by a coupled mode associated with a much larger saturation change. These two modes are also observed in a numerical simulation using a heterogeneous porous layer. A comparison between the propagation times estimated from the results of the numerical simulation and predictions from the asymptotic expression indicates overall agreement.