Analytical approximation for nonlinear adsorbing solute transport and first-order degradation

Under some constraints, solutes undergoing nonlinear adsorption migrate according to a traveling wave. Analytical traveling wave solutions were used to obtain an approximation for the solute front shape,c(z, t), for the situation of equilibrium nonlinear adsorption and first-order degradation. This approximation describes numerically obtained fronts and breakthrough curves well. It is shown to describe fronts more accurately than a solution based on linearized adsorption. The latter solution accounts neither for the relatively steep downstream solute front nor for the deceleration in time of the nonlinear front.Notation A parameter - c concentration [mol/m³] - c ₀ ^* depth-dependent local maximum concentration [mol/m³] - c; c ₀;c _i concentration difference, feed, and initial resident concentrations, respectively [mol/m³] - D pore scale diffusion/dispersion coefficient [m²/yr] - f adsorption isotherm - f derivative off toc - f second derivative off toc - G ^* parameter - K nonlinear adsorption coefficient [mol/m³)^1–n ] - l column length [m] - L _d dispersivity [m] - m parameter - n Freundlich sorption parameter - P function ofc ₀ ^* - _q change inq [mol/m³] - q adsorbed amount (volumetric basis) [mol/m³] - q derivative ofq toc - R nonlinear retardation factor - retardation factor for concentrationc - R _l linear retardation factor - R(z ^*) depth-dependent average retardation factor, for front at depthz ^* - s adsorbed amount (mass basis) [mol/kg] - t time [years] - u parameter - v flow velocity [m] - z ^* downstream front depth [m] - z depth [m] - transformed coordinate [m] - ^* reference point value of [m] - first-order decay parameter [y^–1] - dry bulk density [kg/m³] - volumetric water fraction - parameter     

Efficient algorithm for modeling transport in porous media with mass exchange between mobile fluid and reactive stationary media

We compare two approaches to numerically solve the mathematical model of reactive mass transport in porous media with exchange between the mobile fluid and the stationary medium. The first approach, named the “monolithic algorithm,” is the approach in which a standard finite-difference discretization of the governing transport equations yields a single system of equations to be solved at each time step. The second approach, named the “system-splitting algorithm,” is here applied for the first time to the problem of transport with mass exchange. The system-splitting algorithm (SSA) solves two separate systems of equations at each time step: one for transport in the mobile fluid, and one for uptake and reaction in the stationary medium. The two systems are coupled by a boundary condition at the mobile– immobile interface, and are solved iteratively. Because the SSA involves the solution of two smaller systems compared to that of the monolithic algorithm, the computation time may be greatly reduced if the iterative method converges rapidly. Thus, the main objective of this paper is to determine the conditions under which the SSA is superior to the monolithic algorithm (MA) in terms of computation time. We found that the SSA is superior under all the conditions that we tested, typically requiring only 0.3–50% of the computation time required by the MA. The two methods are indistinguishable in terms of accuracy. Further advantages to the SSA are that it employs a modular code that can easily be modified to accommodate different mathematical representations of the physical phenomena (e.g., different models for reaction kinetics within the stationary medium), and that each module of the code can employ a different numerical algorithm to optimize the solution.     

The use of standard deviation and skewness for estimating apparent permeability in a two-dimensional,heterogeneous medium

In this theoretical study, characteristic or effective permeabilities (referred to as ‘apparent permeability’) of a radial/parallel flow system in a heterogeneous medium are calculated by the Monte Carlo method and the finite element method. The permeability distribution in the radial and parallel flow systems are not the same, as going from Cartesian to cylindrical coordinates changes the probability measure. The Bernoulli trials, the normal distribution or the log-normal distribution, is assumed to be the probability density function of permeability. The results are summarized as follows: (1) when the skewness of the distribution function is equal to zero or nearly equal to zero (that is, when the permeability distribution is regarded to be symmetric), the apparent permeability depends on the standard deviation, but not on the kind of distribution function, (2) when the skewness is not equal to zero, the apparent permeability depends not only on the standard deviation, but also on the skewness, (3) the above facts appeared in the radial and the parallel flow systems.