Macroscopic percolation model of immiscible displacement: Effects of buoyancy and spatial structure

Percolation theory, commonly used to study quasi-static immiscible displacement at the microscopic scale, is here extended to simulations of gravity-stable drainage and imbibition in three-dimensional porous media with spatially correlated macroscopic properties. The result of this extension is a macroscopic percolation on a regular lattice of sites, where lattice sites represent regions in a porous medium characterized by different macroscopic properties (e.g., absolute permeability, capillary pressure, and relative permeability curves). These properties are assigned to lattice sites by virtue of parametrizations in terms of local permeability. We present a general formulation of macroscopic percolation that accounts for gravitational effects, which can be important in large-scale immiscible displacements with nonzero density difference. In such cases, we find that the local saturation distribution is markedly different from the distribution of saturation under conditions of negligible buoyancy. Displacements with nonzero density difference proceed with the formation of a transition zone of length inversely proportional to a macroscopic Bond number which characterizes the relative importance of capillary and buoyancy forces at the macroscopic scale. Several important features of percolation at the microscopic scale are also manifested at the macroscopic scale. These include the effects of lattice dimensionality and spatial correlation on the macroscopic percolation threshold and accessibility characteristics. In the absence of buoyancy forces, the large-scale capillary pressure and relative permeability behavior of a heterogeneous system is dictated mainly by the structure of the permeability held and can be explained in terms of macroscopic accessibility. Spatial correlation of permeability is found to have pronounced effects on the large-scale drainage relative permeability curves.