Alexandre M. Tartakovsky

The effect of spatial concentration fluctuations on the reaction of two solutes, A + B ⇀ C, is considered. In the absence of fluctuations, the concentration of solutes decays as A_det = B_det ∼ t⁻¹. Contrary to this, experimental and numerical studies suggest that concentrations decay significantly slower. Existing theory suggests a t^−d/4 scaling in the asymptotic regime (d is the dimensionality of the problem). Here we study the effect of fluctuations using the classical diffusion‐reaction equation with random initial conditions. Initial concentrations of the reactants are treated as correlated random fields. We use the method of moment equations to solve the resulting stochastic diffusion‐reaction equation and obtain a solution for the average concentrations that deviates from ∼t⁻¹ to ∼t^−d/4 behavior at characteristic transition time t*. We also derive analytical expressions for t* as a function of Damköhler number and the coefficient of variation of the initial concentration.

We describe a new, microscopic model for diffusion that captures diffusion induced fluctuations at scales where the concept of concentration gives way to discrete particles. We show that in the limit as the number of particles N$\rightarrow${}$\infty${}, our model is equivalent to the classical stochastic diffusion equation (SDE). We test our new model and the SDE against Langevin dynamics in numerical simulations, and show that our model successfully reproduces the correct ensemble statistics, while the classical model fails.

Existing continuum models for reactive transport in porous media tend to overestimate the extent of solute mixing and mixing-controlled reactions because the continuum models treat both the mechanical and diffusive mixings as an effective Fickian process. Recently, we have proposed a phenomenological Langevin model for flow and transport in porous media [A. M. Tartakovsky, D. M. Tartakovsky, and P. Meakin, Phys. Rev. Lett. 101, 044502 (2008)]. In the Langevin model, the fluid flow in a porous continuum is governed by a combination of a Langevin equation and a continuity equation. Pore-scale velocity fluctuations, the source of mechanical dispersion, are represented by the white noise. The advective velocity (the solution of the Langevin flow equation) causes the mechanical dispersion of a solute. Molecular diffusion and sub-pore-scale Taylor-type dispersion are modeled by an effective stochastic advection-diffusion equation. Here, we propose a method for parameterization of the model for a synthetic porous medium, and we use the model to simulate multicomponent reactive transport in the porous medium. The detailed comparison of the results of the Langevin model with pore-scale and continuum (Darcy) simulations shows that: (1) for a wide range of Peclet numbers the Langevin model predicts the mass of reaction product more accurately than the Darcy model; (2) for small Peclet numbers predictions of both the Langevin and the Darcy models agree well with a prediction of the pore-scale model; and (3) the accuracy of the Langevin and Darcy model deteriorates with the increasing Peclet number but the accuracy of the Langevin model decreases more slowly than the accuracy of the Darcy model. These results show that the separate treatment of advective and diffusive mixing in the stochastic transport model is more accurate than the classical advection-dispersion theory, which uses a single effective diffusion coefficient (the dispersion coefficient) to describe both types of mixing.

Field tracer tests indicate that advective porosity, the quantity relating advective velocity to Darcy flux, may exhibit directional dependence. Theory (S. P. Neuman, 2005) suggests that directional variations in advective porosity could arise from incomplete mixing of an inert tracer between directional flow channels within a sampling volume of soil or rock that may be hydraulically isotropic or anisotropic, rendering advective porosity a function of Peclet number. Whereas Neuman was able to quantify the process for idealized intersecting fracture sets, his theory is too abstract to allow doing so for more complex systems of fractures or pores. We use Smoothed Particle Hydrodynamics (e.g., A. M. Tartakovsky and P. Meakin, 2005, 2006) to simulate advective‐diffusive transport of a passive tracer in the interior of an idealized pore space between impermeable circular or elliptical solid grains of random size, arranged randomly within a square domain. Our numerical experiments reveal a transition from complete mixing of tracer at small Peclet numbers to the gradual evolution of pore‐scale channels of tracer which allow less and less mixing as Peclet numbers grow. Advective porosity, computed as the ratio between total water flux across the domain boundaries and migration rate of the tracer center of mass, is found to diminish asymptotically from a value close to the total porosity at small Peclet numbers to a constant dominated entirely by advection at large Peclet numbers. When total porosity is kept constant, channeling is less pronounced and mixing more prominent when flow takes place parallel to the longer axes of elliptical grains as compared to when the grains are circular. On the other hand, channeling is more pronounced and mixing less so when flow takes place perpendicular to the longer elliptical grain axes. Correspondingly, the rate at which advective porosity diminishes with increasing Peclet number is slower and the asymptote is larger when flow takes place parallel to the longer axes of elliptical grains than when the grains are circular. However, the rate is faster and the asymptote smaller when flow takes place perpendicular to the longer elliptical grain axes.

We present a new model for fluid flow and solute transport in porous media, which employs smoothed particle hydrodynamics to solve a Langevin equation for flow and dispersion in porous media. This allows for effective separation of the advective and diffusive mixing mechanisms, which is absent in the classical dispersion theory that lumps both types of mixing into dispersion coefficient. The classical dispersion theory overestimates both mixing-induced effective reaction rates and the effective fractal dimension of the mixing fronts associated with miscible fluid Rayleigh-Taylor instabilities. We demonstrate that the stochastic (Langevin equation) model overcomes these deficiencies.

Laboratory experiments, pore‐scale simulations, and continuum (Darcy) ‐scale simulations were used to study mixing‐induced precipitation in porous media. In the experimental investigation, solutions containing Na₂CO₃ and CaCl₂ were each injected in different halves of a quasi two‐dimensional flow cell filled with quartz sand. As a result of the in situ mixing between the two solutions, a narrow calcite precipitate layer (less than 5 mm wide) of more or less uniform width was formed between the individual solutions. Pore‐scale simulations were conducted to help understand the mechanism of precipitation layer formation. The effect of the Peclet number, Pe, and the Damköhler number, Da, on mixing induced precipitation was also investigated. Pore‐scale simulations revealed the presence of large pore‐scale concentration gradients. This, and the presence of features, such as the precipitation layer, with characteristic lengths on the order of the average sand grain diameter, indicate the absence of a clear scale separation required for the strict derivation of Darcy‐scale advection‐dispersion equations. Nevertheless, we found that an adaptive high‐resolution model based on advection‐dispersion equations with grid sizes in the mixing zone smaller than the size of the sand grains can qualitatively reproduce the essential features of the experiment. As an alternative to computationally expensive high‐resolution simulations, we proposed new forms for the homogeneous and heterogeneous reaction terms in Darcy‐scale advection dispersion equations. These terms involve transport and mixing indices that account for highly nonuniform pore‐scale concentration distributions and highly localized reactions. The proposed model accurately estimates the changes in solute concentrations due to homogenous and heterogeneous reactions during precipitation of minerals, observed in the pore‐scale simulations, while conventional low‐resolution advective‐dispersion equations produced erroneous results.