The use of macroscopic percolation theory to construct large‐scale capillary pressure curves

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Abstract
This paper introduces a macroscopic invasion percolation process suitable for simulating the displacement of one immiscible fluid by another through porous media under conditions of capillary‐dominated flow. The theory is similar to classical percolation theory in that the structure of a real porous medium is represented as an ordered lattice, but differs in that each point of the lattice is assigned a local‐scale porosity, permeability, and capillary pressure‐saturation relationship, rather than microscale pore and pore throat dimensions. Unlike pore‐scale percolation theory, each node of the lattice may be occupied by either one or two phases, thereby allowing bicontinua of fluids in two dimensions. To illustrate an application of the theory, both wetting and non wetting fluids are percolated through a heterogeneous porous medium while accounting for fluid trapping such that a hysteretic, large‐scale capillary pressure‐saturation curve is constructed. The simulations are carried out in spatially correlated, random permeability fields assuming that the local‐scale capillary pressure‐saturation relationships are perfectly correlated with permeability. The resulting large‐scale capillary pressure curves are found to be influenced by the mean and variance of the assigned lognormal distribution of permeabilities. Very little sensitivity to the ratio of correlation lengths was observed. It is found that the threshold saturation giving rise to an initial percolating cluster of nonwetting fluid across the lattice corresponds to between 18.8% and 29.5% nonwetting saturation, depending on the statistics of the permeability field. Comparison of the percolation‐derived capillary pressure curves to those based on a direct arithmetic average demonstrates that an arithmetic average is only valid through the range of fluid saturations where no trapping occurs.