Abstract
Finding a numerical method to model solute transport in porous media with high heterogeneity is crucial, especially when chemical reactions are involved. The phase space formulation termed the multi-advective water mixing approach (MAWMA) was proposed to address this issue. The water parcel method (WP) may be obtained by discretizing MAWMA in space, time, and velocity. WP needs two transition matrices of velocity to reproduce advection (Markovian in space) and mixing (Markovian in time), separately. The matrices express the transition probability of water instead of individual solute concentration. This entails a change in concept, since the entire transport phenomenon is defined by the water phase. Concentration is reduced to a chemical attribute. The water transition matrix is obtained and is demonstrated to be constant in time. Moreover, the WP method is compared with the classic random walk method (RW) in a high heterogeneous domain. Results show that the WP adequately reproduces advection and dispersion, but overestimates mixing because mixing is a sub-velocity phase process. The WP method must, therefore, be extended to take into account incomplete mixing within velocity classes.
Highlights
The advection–dispersion equation (ADE) is the most widely used formulation to model solute transport through porous media
This is why we termed the formulation of Equation (4) multi-advective water mixing approach (MAWMA)
This definition acknowledges that velocity transitions due to heterogeneity do not cause mixing, which helps us to focus on the impact of mixing, which depends exclusively on diffusive processes: dc φ dt
Summary
The advection–dispersion equation (ADE) is the most widely used formulation to model solute transport through porous media It does not adequately characterize transport in heterogeneous media [1,2,3] where dispersion grows with scale [4,5], nonequilibrium occurs [6,7], or breakthrough curves display tailing [8,9]. The most widely extended is the continuous time random walk (CTRW) It consists of random velocity transitions once the solute has travelled a certain space step. The solute dependency of velocity was expressed in a phase space formulation proposed by [41]. Mixing is Markovian in time in contrast to dispersion This observation suggests that solute transport should be localized in space and time, and in velocity. We test the capacity of the WP model and the proposed mass exchange methods to reproduce transport through heterogeneous porous media
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