Numerical modeling of toxic nonaqueous phase liquid removal from contaminated groundwater systems: mesh effect and discretization error estimation

Self Cite

Summary A theoretical study of reactive infiltration instability is conducted on the dissolution timescale. In the present theoretical study, the transient behavior of a dissolution‐timescale reactive infiltration system needs to be considered, so that the upstream region of the chemical dissolution front should be finite. In addition, the chemical dissolution front of finite thickness should be considered on the dissolution timescale. Owing to these different considerations, it is very difficult, even in some special cases, to derive the first‐order perturbation solutions of the reactive infiltration system on the dissolution timescale. To overcome this difficulty, an interface‐condition substitution strategy is proposed in this paper. The basic idea behind the proposed strategy is that although the first‐order perturbation equations in the downstream region cannot be directly solved in a purely mathematical manner, they should hold at the planar reference front, which is the interface between the upstream region and the downstream region. This can lead to two new equations at the interface. The main advantage of using the proposed interface‐condition substitution strategy is that through using the original interface conditions as a bridge, the perturbation solutions for the dimensionless acid concentration, dimensionless Darcy velocity, and their derivatives involved in the two new equations at the interface can be evaluated just by using the obtained analytical solutions in the upstream region. The proposed strategy has been successfully used to derive the dimensionless growth rate, which is the key issue associated with the theoretical study of dissolution‐timescale reactive infiltration instability in fluid‐saturated porous rocks.

Section: Introductionmentioning

confidence: 99%

An interface‐condition substitution strategy for theoretical study of dissolution‐timescale reactive infiltration instability in fluid‐saturated porous rocks

Zhao

Hobbs

Ord

2019

Self Cite

“…It is also noted that in recent years, a significant advance in dealing with anomalous solute transport in porous media is to consider the chemical dissolution reactions in a solute advection‐dispersion system and to investigate the anomalous solute transport phenomena that are caused by the instability of the system . For instance, the anomalous diffusion and flow channels of fingering patterns can result from the chemical dissolution‐front instability in a solute advection‐dispersion system including chemical dissolution reactions .…”

Section: Introductionmentioning

confidence: 99%

Using a shifted Grünwald‐Letnikov scheme for the Caputo derivative to study anomalous solute transport in porous medium

Mascarenhas

Moraes

Cavalcante

2019

Summary We propose an extension of the shifted Grünwald‐Letnikov method to solve fractional partial differential equations in the Caputo sense with arbitrary fractional order derivative α and with an advective term. The method uses the relation between Caputo and Riemann‐Liouville definitions, the shifted Grünwald‐Letnikov, and the traditional backward and forward finite difference method. The stability of the method is investigated for the implicit and explicit scheme with homogeneous boundary conditions, and a stability criterion is found for the advective‐dispersive equation. An application of the method is used to solve contaminant diffusion and advective‐dispersive problems. The numerical solution for the fractional diffusion and fractional advection‐dispersion is compared with their respective analytical solutions for different time and space grid refinements. The diffusion simulation exhibited a good fit between the analytical and numerical solutions, with the explicit scheme going from stable to unstable as the time and space refinement changes. The fractional advection‐dispersion application produced small deviations from the analytical solution. These deviations, however, are analogous to the numerical dispersions encountered in conventional finite difference solutions of the advection‐dispersion equation. The new method is also compared with the traditional L2 method. Notably, an example that involves asymmetrical fractional conditions, a fractional diffusivity that depends on time, and a source term show how the methods compare. Overall, this study assesses the quality and easiness of use of the numerical method.

“…In particular, analytical solutions can also play a unique role in validating many new numerical methods . For these reasons, analytical solutions have been derived in recent years for many scientific problems such as (i) the convective pore‐fluid flow problem that can be treated as a coupled pore‐fluid flow and heat transfer problem in a half‐space consisting of fluid‐saturated porous media ; (ii) the chemical dissolution–front instability problem that can be treated as a coupled pore‐fluid flow, heat transfer, mass transport, and chemical reaction problem in a full space consisting of fluid‐saturated porous media ; (iii) the physical dissolution–front instability problem that can be encountered in the effective treatment of non‐aqueous phase liquid in the geo‐environmental engineering field ; and (iv) the problem of pore‐fluid focusing within a crack in a full space consisting of fluid‐saturated porous media . Because of the aforementioned important roles of analytical solutions, we will derive the analytical solutions for the problem considered in this study.…”

Section: Introductionmentioning

confidence: 99%

Propagation of Rayleigh waves in fluid‐saturated non‐homogeneous soils with the graded solid skeleton distribution

Zhou

2016

Summary The propagation characteristic of Rayleigh waves in a fluid‐saturated non‐homogeneous poroelastic half‐plane is addressed. Based on Biot's theory for fluid‐saturated media, which takes the inertia, fluid viscosity, mechanical coupling, compressibility of solid grains, and fluid into account, the dispersion equations of Rayleigh waves in fluid‐saturated non‐homogeneous soils/rocks are established. By considering the shear modulus of solid skeleton variation with depth exponentially, a small parameter, which reflects the relative change of shear modulus, is introduced. The asymptotic solution of the dispersion equation expressing the relationship between the phase velocity and wave number is obtained by using the perturbation method. In order to analyze the effects of non‐homogeneity on the propagation characteristic of Rayleigh waves, the variation of the phase velocity with the wave number is presented graphically and discussed through numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.