Fluid penetration through a crack in a pressure gradient

We investigate invasion percolation fingers in'a quenched medium in which the randomness has a gradient following a power law both in the direction of the flow and perpendicular to it. The first gradient corresponds to a pressure gradient, the second one to the density of microcracks that arises in a self-organized way around a large crack. We, give an argument for the value of the fractal dimension as found in previous simulations. We calculate the roughness exponent of the fingers and find a value consistent with 4 as predicted by the argument. The penetration of fluids in porous media [l] is usually acompanied by the creation and propagation of maior cracks through which this fluid will flow. This coupling between fluid transport and cracking has hardly ever been attacked theoretically, although it seems to be the main mechanism in technologically very important applications like hydraulic fracturing. It is the purpose'of this letter to present a very simple model for this problem and its numerical analysis. We want to consider a non-wetting fluid penetrating into a two-dimensional quenched disordered medium. The disorder is modelled by placing random mumbers zi on the sites of a square lattice distributed according to a distribution P(z,) . Fluid penetration into a random solid has often been described by invasion percolation. In particular, in [2], a model has been proposed in which a gradient is introduced by assuming the random numbers to be distributed as P(zJ ~ 9 . with r the distance from the injection point. This graded distribution of random permeabilities models, to some degree, a pressure gradient. A numerical simulation of that model showed that, for positive a the clusters are fractal with a fractal dimension of df=1.44f0.02, independent of a. This result has not been understood yet. Here we will give an argument for this value. The penetration of a fluid is, however, usually accompanied by the opening of a crack in the medium. It has been shown [3,4] that brittle crack formation is accompanied by the formation of a cloud of microcracks around the main crack and that the density of crack surface generated by the microcracks decays like a power law as a distance p of the position of the main crack. We want to include this effect into the model proposed in [2] by assuming that P(zJ a p B . This will be implemented into our model by the following prescription: We consider a square lattice of horizontal length L1 and vertical height L, and with periodic boundaries in the vertical direction. At line jo on the left boundary we inject the fluid, i.e. we occupy this site. Then on each site of the first two columns a random 0305-4470/93/221145 +04 $07.50