| Motivated by the problem of finding the minimum threshold path in a lattice of elements with random thresholds, {dollar}tausb{lcub}i{rcub},{dollar} we propose a new class of invasion processes, generally defined as Invasion Percolation with Memory (IPM), in which the front advances by minimizing or maximizing the measure {dollar}Ssb{lcub}n{rcub}=Sigmasb{lcub}i{rcub}tausbsp{lcub}i{rcub}{lcub}n{rcub},{dollar} where n is a real number. Depending on whether the rule minimizes or maximizes {dollar}Ssb{lcub}n{rcub},{dollar} the fronts are either stable and self-affine, connected to Invasion Percolation in a Gradient occurring on a correlated field, or unstable and fractal, where the front is a Self-Avoiding Random Walk (SAW).; We consider IPM applications to two different flow problems. In Bingham plastics, the onset of flow and displacement occurs only after the applied pressure gradient exceeds a minimum value. The effects of yield stress and pore throat size distribution on the flow and immiscible displacement of Bingham plastics in porous media at low capillary numbers are studied. The results compared with visualization experiments of other researchers.; Foam flow in porous media is also controlled by the thresholds of lamellae mobilization from pore throats. Using IPM, the flow simulations of such a fluid in Pore-Network models, for the first time, is studied. The model allows for various parameters or processes, empirically treated in current models, to be quantified and interpreted. In contrary to previous works, here, a dynamic (invasion) in addition to a static process is considered. The onset of foam flow, mobilization, the foam texture, sweep efficiencies and the fraction of the flowing foam, {dollar}chisb{lcub}f{rcub},{dollar} are calculated. Using the snap-off probability as a controlling parameter, high-quality (strong) or low-quality (weak) foams form. In particular, the invasion by foam of low permeability layers during injection in a heterogeneous system is demonstrated.; Finally, for a general quasilinear hyperbolic problem involving reactions of finite rate, to admit as solution a steady-state following the initial equilibrium state, it is found that the generalized flux vector F and the generalized concentration vector C must be colinear. If an additional condition for the overlap of characteristics is valid, the two conditions are sufficient and necessary. |
