https://www.research.manchester.ac.uk/portal/files/37188690/CONHYD_3228_accepted.pdf

Review of pore network modelling of porous media: Experimental characterisations, network constructions and applications to reactive transport

[1] We consider the transport behavior of a passive solute in a heterogeneous medium, modeled by continuous time random walks (CTRW) and linear multirate mass transfer (MRMT). Within the CTRW framework, we formulate a transport model which is formally equivalent to MRMT. In both approaches the total concentration is divided into mobile and immobile parts. The immobile concentration is given by the convolution in time of the mobile concentration and a memory function. The memory function is a functional of the distribution of transition and trapping times for the CTRW and the MRMT approach, respectively, and determines the transport behavior of the solute. Based on different expressions for the memory function in the two frameworks, we derive conditions for which both approaches describe the same transport behavior. We focus on anomalous transport behavior that can arise if the transition and trapping time distributions behave algebraically in a given time regime. Using an expansion of the Laplace transform of the memory function, we develop explicit expressions for the time behavior of the flux concentration as well as for the center of mass velocity and the (macro) dispersion coefficients of the solute distribution. We observe (anomalous) power law as well as (normal) Fickian transport behavior, depending on the exponents that dominate the trapping and transition time distributions, respectively. The results show that the character of the anomalous transport does not depend on the details of the transport model but only on the exponents dominating the transition or trapping time distributions. The unified transport framework presented here comprising CTRW and MRMT shows new aspects and opens new perspectives for the modeling of transport in heterogeneous media.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2001WR001163

Transport behavior of a passive solute in continuous time random walks and multirate mass transfer

We re-examine-in light of recent theoretical developments-clasaical experiments on dispersion of a passive tracer in fully and partially saturated porous columns. We find that the dispersion breakthrough curves (BTCs) exhibit anomalous (non-Fickian) early arrival times and late time tailing, which can be explained by the Continuous Time Random Walk (CTRW) theory. The CTRW framework includes as a special case the classical advection-dispersion equation (ADE) for Fickian transport. We argue that existing measurements and interpretations of dispersion should be carefully reconsidered in the framework of these advances in conceptual understanding and quantification.

Anomalous Transport in “Classical” Soil and Sand Columns

http://www.mgnet.org/~douglas/Classes/na-sp/2007f-notes/fire_simple.pdf

A wildland fire model with data assimilation

A quantitative methodology to assess the risks to human health from CO2 leakage into groundwater

An efficient multivariate random field generator using the fast Fourier transform

Solute transport studies frequently rely on numerical solutions of the classical advection-diffusion equation. Unfortunately, solutions obtained with traditional finite difference and finite element techniques typically exhibit spurious damping or oscillation when advection dominates. Recently developed variants of these techniques such as the finite analytic method (Chen and Li, 1979; Chen and Chen, 1984) and the optimal test function method (Celia et al., 1989a, b, c) perform well for steady state problems. Extensions of these methods to the transient case have, however, not been successful, primarily because of inadequate approximations of the temporal derivative. The new numerical method proposed in this paper avoids this difficulty by taking the Laplace transform of the transient equation. The transformed expression behaves like a steady state advection-diffusion equation with a first-order decay term. This expression can be solved with either the finite analytic or optimal test function method and the time dependence recovered with an efficient inverse Laplace transform algorithm. The result is an accurate and robust transient solution which performs well over a very wide range of Peclet numbers. We demonstrate this approach by applying the finite analytic method to a Laplace transformed one-dimensional model problem. A comparison with other competing techniques shows that good approximations are required in both space and time in order to obtain accurate solutions to advection-dominated problems. A good space approximation combined with a poor temporal approximation (or vice versa) does not give satisfactory results. The method we propose provides a balanced space-time approximation which works very well for one-dimensional problems. Extensions to multiple dimensions are conceptually straightforward and briefly discussed.

A space-time accurate method for solving solute transport problems

Although groundwater velocities vary over a wide range of spatial scales it is generally only feasible to model the largest variations explicitly. Smaller-scale velocity variability must be accounted for indirectly, usually by increasing the magnitude of the dispersivity tensor (i.e. by introducing a so-called macrodispersivity). Most macrodispersion theories tacitly assume that a macrodispersivity tensor which works well when there is only small-scale velocity variability will also work well when there is larger-scale variability. We analyze this assumption in a high resolution numerical experiment which simulates solute transport through a two-scale velocity field. Our results confirm that a transport model which uses an appropriately adjusted macrodispersivity can reproduce the large-scale features of a solute plume when the velocity varies only over small scales. However, if the velocity field includes both small and large-scale components, the macrodispersivity term does not appear to be able to capture all of the effects of small-scale variability. In this case the predicted plume is more well mixed and consistently underestimates peak solute concentrations at all times. We believe that this result can be best explained by scale interactions resulting from the nonlinear transformation from velocity to concentration. However, additional analysis will be required to test this hypothesis.

/pdf/macrodispersivity-and-large-scale-hydrogeologic-variability-4igh06apru.pdf

Macrodispersivity and Large-scale Hydrogeologic Variability

Numerical simulation of solute transport in heterogeneous porous media is greatly complicated by the large velocity and concentration gradients induced by spatial variations in hydraulic conductivity. Eulerian-Lagrangian methods for solving the transport equation can give accurate solutions to heterogeneous problems if their interpolation algorithms are properly selected. This paper compares the performance of four Eulerian-Lagrangian solvers that rely on linear, quadratic, cubic spline, and taut spline interpolators. In each case a tensor product decomposition is used to reduce the general n -dimensional interpolation problem to a sequence of n one-dimensional problems. Comparisons of a set of test problems indicate that the linear and taut spline interpolators are dispersive while the quadratic and cubic spline interpolators are oscillatory. The cubic and taut spline interpolators give consistently better accuracy than the more conventional linear and quadratic alternatives. Simulation experiments in two- and three-dimensional heterogeneous media indicate that the taut spline interpolator, which is applied here for the first time to a solute transport problem, is able to yield accurate essentially nonoscillatory solutions for high grid Peclet numbers. The cubic spline interpolator requires significantly less computational effort to achieve performance comparable to the other methods.

/pdf/an-investigation-of-eulerian-lagrangian-methods-for-solving-2idjhinxca.pdf

An investigation of Eulerian-Lagrangian methods for solving heterogeneous advection-dominated transport problems

solve the advection-diffusion equation with spatially variable coefficients. The Laplace transform is used to evaluate the temporal derivative in the advection-diffusion equation ana- lytically, thereby eliminating the effects of the time deriva- tive on accuracy and stability. Because the temporal deriv- ative is evaluated using the Laplace transform, there is no need for the time-stepping that is associated with more traditional techniques for the solution of the advection- diffusion equation. Therefore it is a potentially more efficient technique. The authors do not provide numerical simulations involv- ing the spatially variable coefficient advection-diffusion equation. A reason for this might be that there are very few analytical solutions to the advection-diffusion equation with spatially variable coefficients. The purpose of this discussion is to present what we believe to be a new analytical solution to a particular form of the advection-diffusion equation with spatially variable co- efficients and to highlight a potential mass conservation

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/94WR01492

Feng Ruan

Papers

An efficient multivariate random field generator using the fast Fourier transform

A space-time accurate method for solving solute transport problems

Macrodispersivity and Large-scale Hydrogeologic Variability

An investigation of Eulerian-Lagrangian methods for solving heterogeneous advection-dominated transport problems

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