Anisotropic effective medium solutions of head and velocity variance to quantify flow connectivity

Most methods for upscaling flow and transport in heterogeneous media from the measurement scale to the simulation scale for field applications generally focus on the prediction of values of effective permeability and apparent dispersion coefficients. Although typically considered as secondary data, measures of head variance σϕ2 and velocity variance σqx2 also contain valuable information on the level of heterogeneity of the hydraulic conductivity (K  ) distribution of the soil. In particular, we investigate the suitability of σϕ2 and σqx2 to yield insight into the potential occurrence of flow barriers and preferential pathways, which significantly affect flow connectivity. Before the application of complex numerical simulators, a proper understanding of the actual link between head and velocity variance, and the spatial distribution of K can be obtained using closed-form solutions. In this paper, semi-analytical expressions of effective permeability, head variance, and velocity variance are derived for saturated flow in two-dimensional anisotropic porous media. The expressions are obtained using the methodology initially proposed by Dagan [Dagan G. Models of groundwater flow in statistically homogeneous porous formations. Water Resour Res 1979;15(1):47–63] for isotropic heterogeneous formations. The solutions are illustrated in the case of binary heterogeneous media and compared to results from numerical simulations of steady-state flow in random K   fields. It is found that the self-consistent solution generally yields relatively poor results when applied to the prediction of head statistics, while both longitudinal and transverse velocity variance are correctly predicted in all cases. The results of the numerical simulations are also used to illustrate the link between σϕ2, σqx2, and flow connectivity. Although head variance is not a stationary property of two-dimensional flow fields, and hence might not exactly represent an intrinsic property of flow such as connectivity, it is found that σϕ2 is negatively correlated to effective permeability Ke,xKe,x for poorly connected K   fields. On the contrary, σqx2 is found to be positively correlated to Ke,xKe,x for all levels of connectivity. Therefore, the results suggest that these statistics of the flow field could be used to quantify flow connectivity when measures of other indicators are not available.