Nonstationary flow and nonergodic transport in random porous media

Abstract

[1] Transport processes taking place in natural formations are often characterized by the spatial nonstationarity of flow field and the nonergodic condition of solute spreading. The former may originate from statistical inhomogeneity of porous media properties or from the influence of boundary conditions as well as from conditioning procedures on measured data, while the latter is generally due to the finite size of the solute source. In real-world applications these aspects sometimes dominate the transport phenomena, and known literature results based on statistical homogeneity and ergodicity assumptions cannot be properly applied. In this paper a method is proposed to handle different and concurrent causes of the flow field nonstationarity and to describe nonergodic transport of inert solutes by spatial moments in a domain of finite size. The goal is reached by expanding the steady state flow equation in Taylor series limited to the first-order and by the recursive application of finite element (FE) method. The unknowns are the piezometric head mean values and its derivatives in respect to the fluctuating porous media hydraulic conductivity. From the latter the velocity statistics are derived on the basis of Darcy's law according to the first-order approximation. By neglecting the pore-scale dispersion the spatial moments of a solute plume are a posteriori obtained by a consistent Lagrangian analysis starting from the knowledge of the velocity field covariance matrices without any restriction regarding the statistical homogeneity of flow and/or ergodicity conditions. The proposed method is here applied to investigate the influence of boundary conditions on the solute transport developing in a limited domain. The results obtained from different test cases give a velocity field behavior in agreement with literature findings and allow us to define the nonergodic transport in bounded domains. The comparison of the results with some ad hoc developed Monte Carlo simulations ensures the applicability of the proposed approach at least to the small heterogeneity cases here considered.Citation: Darvini, G., and P. Salandin (2006), Nonstationary flow and nonergodic transport in random porous media, Water Resour.

Effective dispersion in conditioned transmissivity fields

“…The last two terms do not exist for stationary Lagrangian flow fields [7] and for any constant expected value of the velocity field [9,11]. In such cases, the particle displacement variance X ij becomes spatially constant, Eq.…”

“…In the Lagrangian approach described in Section 2, the solute line source has been simulated by a discrete set of particles, each of them traveling along a path line of the flow field univocally identified by its initial position. A density of 4 particles for each integral scale is used to properly describe the solute line as discussed in Darvini and Salandin [11].…”

“…We briefly recall the derivation of statistical moments of a inert solute plume by the SFEM approach [11].…”

“…(11) becomes straightforward (e.g. [11]). We focus on the expected value of the second-order spatial moment tensor S ij , that provides the effective dispersion and is computed by ensemble averaging the second moments obtained in each realization.…”

“…[10,11]). Then, the transport moments of a conservative solute plume are computed from the velocity statistics by a Lagrangian approach limited to the first-order.…”