Prediction of effective Forchheimer coefficient for one- and two-dimensional flows in heterogeneous geologic media

Non-linear flow in porous media, governed by the Forchheimer equation, interacts with domain heterogeneity when geologic media are involved. In Forchheimer’s law, the pressure gradient is equal to the sum of a linear term in the flow rate (akin to Darcy’s law) and a quadratic term proportional to the second power of the flow rate; the latter coefficient of proportionality is the Forchheimer coefficient. As most experimental values of the Forchheimer coefficient have been derived at the laboratory scale, it is necessary to investigate its upscaling at the field scale in heterogeneous aquifers. Consider a uniform pressure gradient flow in a spatially heterogeneous, perfectly layered permeability random field with a given probability density distribution. The local Forchheimer coefficient β is related to the local permeability k value via an empirical inverse power-law correlation reading β=a/k^c, having an experimental basis and widely adopted in the literature, where a is a constant and c an exponent in the range 0-2. Under the ergodic hypothesis, the effective permeability and Forchheimer coefficient can be derived in two limit cases under one-dimensional flow: (i) a serial arrangement, with flow crossing layers having different permeability, and (ii) a parallel arrangement, with parallel flows within layers of different permeability. Results obtained for the effective permeability recover the lower and upper bounds valid for 1-D Darcy flow in heterogeneous media, i.e., harmonic and arithmetic mean of the permeability distribution for the series and parallel arrangement, respectively. The expressions obtained for the effective Forchheimer coefficient generalize previous formulations derived for a discrete parameter variation. In particular, an expression for the effective Forchheimer coefficient is derived in closed form for the serial arrangement, while numerical values are derived for the parallel arrangement. The impact of spatial variability is assessed adopting a lognormal permeability distribution. The effective Forchheimer coefficient βeff increases with the permeability coefficient of variation for both arrangements; for the serial case, it also increases with the exponent c; the opposite is true for the parallel arrangement. One-dimensional results obtained for serial and parallel arrangements provide lower and upper bounds for evaluating the effective Forchheimer coefficient in 2-D flows. Results for 2-D isotropic media are derived heuristically via geometric averaging of the 1-D expressions.