On the validity of effective formulations for transport through heterogeneous porous media

Abstract

Abstract. Geological heterogeneity enhances spreading of solutes and causes transport to be anomalous (i.e., nonFickian), with much less mixing than suggested by dispersion. This implies that modeling transport requires adopting either stochastic approaches that model heterogeneity explicitly or effective transport formulations that acknowledge the effects of heterogeneity. A number of such formulations have been developed and tested as upscaled representations of enhanced spreading. However, their ability to represent mixing has not been formally tested, which is required for proper reproduction of chemical reactions and which motivates our work. We propose that, for an effective transport formulation to be considered a valid representation of transport through heterogeneous porous media (HPM), it should honor mean advection, mixing and spreading. It should also be flexible enough to be applicable to real problems. We test the capacity of the multi-rate mass transfer (MRMT) model to reproduce mixing observed in HPM, as represented by the classical multi-Gaussian log-permeability field with a Gaussian correlation pattern. Non-dispersive mixing comes from heterogeneity structures in the concentration fields that are not captured by macrodispersion. These fine structures limit mixing initially, but eventually enhance it. Numerical results show that, relative to HPM, MRMT models display a much stronger memory of initial conditions on mixing than on dispersion because of the sensitivity of the mixing state to the actual values of concentration. Because MRMT does not restitute the local concentration structures, it induces smaller non-dispersive mixing than HPM. However long-lived trapping in the immobile zones may sustain the deviation from dispersive mixing over much longer times. While spreading can be well captured by MRMT models, in general nondispersive mixing cannot.

Upscaling Mixing in Highly Heterogeneous Porous Media via a Spatial Markov Model

“…Many approaches exist for modeling effective upscaled transport in highly heterogeneous porous media, e.g., multi-rate mass transfer models [41], fractional advection-dispersion equations [10], and continuous time random walks [42]. In this paper, we will focus on one particular model, the Spatial Markov model (SMM).…”

Temporal Mixing Behavior of Conservative Solute Transport through 2D Self-Affine Fractures

“…where c is the solute concentration, t is time, and D m is the molecular diffusion coefficient. The velocity vector in Equation 12is from the flow field based on solution of Equations (10) and (11). It is assumed that the initial concentrations were given by:…”

“…Mixing can be described by the SDR that is a global mixing measure based on the integral of concentration gradients. Recently, several studies have focused on the SDR evolution and scaling properties during solute transport in porous media [7,10,13,21,27,31,32,46]. However, all of those studies on SDR were restricted to porous media.…”

“…Characterizing the spreading and mixing processes of conservative solute through the fractures is very important for the understanding of reaction rates and mass transport rates associated with nuclear waste disposal, enhanced oil recovery, and bioremediation [1][2][3][4][5][6]. Although, in recent decades, many studies have provided new insights into the mechanisms and properties of mixing processes in homogeneous and heterogeneous porous media [7][8][9][10][11][12][13][14][15][16][17], to date, little attention has been focused on mixing behavior in fractures.…”

“…This new decomposition of mixing showed a generic characterization and could offer new ways to establish a transport equation with consideration of both advection, spreading, and mixing. A comparison of different transport models can be found in [10]. A series of analytical solutions for the SDR was derived in non-conservative transport systems by Engdahl, Ginn, and Fogg [32].…”