Time-dependent Dispersion Coefficient Research Articles

Time-dependent dispersion coefficients for the evolution of displacement fronts in heterogeneous porous media

We present an approach for quantifying displacement fronts in heterogeneous porous media based on the concept of time-dependent apparent dispersion coefficients. The concept of constant asymptotic macrodispersion generally overestimates the area swept by a displacement front and leads to unrealistic upstream dispersion. We show that the large-scale front spreading can be captured by a one-dimensional advection–dispersion equation that is parameterized by a suitably chosen temporally evolving dispersion coefficient. For purely advective front spreading, we derive an analytical expression based on a predictive continuous time random walk approach, which applies to highly heterogeneous porous media. This analysis elucidates the variability of solute travel times as the key longitudinal spreading mechanism. It shows that the evolution of dispersion can be captured as the sum of exponentials that decay on two dominant time scales. In a particle-based picture, these scales mark the short time at which transported particles start exploring the flow variability and the large time at which the slowest particles start decorrelating their transport velocity. Based on these insights, we propose a heuristic formula that accounts for the impact of local-scale dispersion as an additional decorrelation mechanism. The heuristic expression for the longitudinal dispersion coefficient captures solute spreading for a broad range of Péclet numbers and heterogeneity variances. The proposed approach is tested against direct numerical simulations. It provides a robust and fast method for quantifying the evolution of displacement fronts in heterogeneous porous media with possible applications, for example, in groundwater contamination modelling, underground gas storage, and geothermal energy production.

Pre-asymptotic dispersion revisited

We develop a new approach to formulation and solution of mathematical models of one-dimensional advective–dispersive transport that specifies the hydrodynamic dispersion coefficient as a function of time since the solute has entered the flow field, termed ‘age.’ This approach addresses Taylor’s concern about the use of time-dependent dispersion coefficients to model pre-asymptotic dispersion by replacing time-dependence with age-dependence, where age of solute is exposure-time to the flow. We show closed form solutions obtained for transport on the −∞<x<∞ and a numerical solution for transport on the 0<x<∞ domain. We demonstrate how this works by application to an in-silico experiment recently published in a study addressing the same issue in a different manner. Our simple and intuitive approach matches the simulated pre-asymptotic data without additional terms or parameter fitting. The same principle applies to pre-asymptotic dispersion in other important upscaled one-dimensional transports e.g., in river corridor or groundwater flows.

An Analytical Approach to Contaminant Transport with Spatially and Temporally Dependent Dispersion in a Heterogeneous Porous Medium

This study presents an analytical solution to the one-dimensional advection-dispersion equation (ADE) for a semi-infinite heterogeneous aquifer system with space and time-dependent groundwater velocity and dispersion coefficient. The dispersion coefficient is assumed to be proportional to the groundwater flow velocity. In addition, retardation factor, first-order decay and zero-order production terms are also considered. Contaminants and porous media are assumed to be chemically inert. Initially, it is assumed that some uniformly distributed solutes are already present in the aquifer domain. The input point source is considered uniformly continuous and increasing nature in a semi-infinite porous medium. The solutions are obtained analytically using the Laplace Integral Transform Technique (LITT). The nature of the concentration profile of the resulting solution for different parameters in different time domains is illustrated graphically.

Numerical and experimental study on solute transport through physical aquifer model

Abstract Environmental concerns have drawn much research interest in solute transport through porous media. Thus, contaminants of groundwater permeate through pores in the ground, and adsorption attenuates the pollution concentration as the pollutants adhere to the solid surface. Mathematical models based on certain simplifying assumptions have been used to predict solute transport. The transport of solutes in porous media is governed by a partial differential equation known as the advection-dispersion equation. In this study, a two-dimensional numerical model has been developed for solute transport through porous media. Results of spatial moments have been predicted and analysed in the presence of both constant and time-dependent dispersion coefficients. Afterward, a numerical model is used to simulate experimentally observed breakthrough curves for both conservative and non-conservative solutes. Thus, transport parameters are estimated through numerical simulation of observed breakthrough curves. Finally, this model gives the best simulation of observed breakthrough curves, and it can also be used in the field scale.

Modulation instability analysis of a nonautonomous $$(3+1)$$-dimensional coupled nonlinear Schrödinger equation

We have investigated the modulation instability (MI) analysis of a nonautonomous \((3+1)\)-dimensional coupled nonlinear Schrödinger (NLS) equation with time-dependent dispersion and phase modulation coefficients. By employing standard linear stability analysis, we have obtained an explicit expression for the MI gain as a function of dispersion, phase modulation, perturbation wave numbers and an initial incidence power. A nonautonomous coupled NLS equation is found to be modulationally unstable for the same sign of dispersion and phase modulation coefficients. This equation is modulationally stable for zero dispersion and or phase modulation. For nonzero dispersion, the equation is found to be modulationally stable/unstable on distinct bandwidth of wave numbers. The trigonometric, exponential, algebraic functions of time and constant have been chosen as test functions for dispersion and phase modulation to study the effect on the MI analysis. The effect of focusing and defocusing medium on the MI analysis has also been investigated. The MI bandwidth in the focusing medium is found to be larger than defocusing medium. It is found that the MI of the equation can be managed by proper choice of the dispersion and phase modulation parameters.

A fourth-order two-level factored implicit scheme for solving two-dimensional unsteady transport equation with time-dependent dispersion coefficients

This paper analyzes a two-level factored implicit scheme in a numerical solution of two-dimensional unsteady advection-diffusion equation with time dependent dispersion coefficients subjects to initial and boundary conditions. The proposed approach is fast and efficient: unconditionally stable, second order accurate in time, spatial fourth order convergent and it requires less computing time. In fact, the two-level factored technique reduces to solve a tridiagonal system of linear equations at each calculation step. This reduces the computational cost of the algorithm. The analysis of the stability of the numerical scheme considers the -norm while the error estimates and convergence rate use L 2-norm. A wide set of numerical evidences are presented and discussed.

Application of temporal moments to interpret solute transport with time-dependent dispersion

Temporal moments of solute transport through porous media are calculated to analyze the time average spatial distribution of solute plume. Simulation of spatially and temporally distributed breakthrough curves (BTCs) is computationally rigorous and lacking the explanation about overall plume evolution within porous media. However, temporal moment provides an attractive and simple solution to study the plume behavior. In this study, temporal moments are presented to interpret solute plume behavior in heterogeneous porous media such as hydraulically coupled stratified porous media with different time-dependent dispersion models. Governing equations of solute transport have been solved numerically using Crank-Nicolson scheme, and further solute concentration data has been utilized to calculate moments of solute concentration using numerical integration. The effect of various parameters such as mass-transfer coefficient, pore-water velocity, time-dependent dispersion coefficients, and porosity of mobile region on the transport of solute has been studied through sensitivity analyses. Temporal moment analysis revealed that the mass recovery, mean residence time, and variance are sensitive to the estimated parameters. Numerical results suggested that the asymptotic time-dependent dispersion function with mobile–immobile model represents the plume spreading through heterogeneous porous media in a more realistic manner.

Simulating Scale Dependencies on Dispersive Mass Transfer in Porous Media Under Various Boundary Conditions

Studies on mass transfer and transport of dissolved contaminants through heterogeneous hydrogeology rely on realistic expressions of their functional parameters. Hydrodynamic dispersion is one such parameter inheriting significant ambiguities in its estimation when subjected to heterogeneous flow conditions. In this study, a numerical model has been developed to explicitly evaluate the variability of dispersion coefficient on time-based and space-based scales for simulating contaminant transport mechanism in saturated porous media under nonlinear sorption mass transfer. Four types of inlet (solute source) conditions, viz., constant continuous, pulsating, exponentially decaying and sinusoidal varying were considered, and the resulting concentration profiles were compared. The governing partial differential equations were solved using an implicit finite difference technique. The results from sensitivity analysis suggest that the magnitude of the fluid velocity was influenced by the type of dispersion coefficient for different inlet boundary conditions. However, sorption partition coefficients and porosities were generally insensitive to the nature of dispersion under these boundary conditions. When time-dependent dispersion coefficient was adopted, an asymptotic increase in retained solute mass was observed for increased fluid velocity irrespective of the boundary conditions. The behavior of the concentration profile obtained using distance-dependent dispersion coefficient was unique when sinusoidal decaying solute boundary condition was used. The study rightly emphasizes the need for simultaneously updating mass transfer and transport parameters while accounting for the variation in solute inlet conditions.

New insights into tracer propagation in partially saturated porous media

This work deals with the influence of partial saturation on the transport process of a passive tracer. Transport experiments were done in a water-wet glass micromodel combined with specific optical techniques. Full water saturation was achieved by injecting initially the background solution and then the tracer, whereas for the partial saturation conditions, the micromodel was initially saturated with oil, and then sequential the background solution and the tracer were injected at the same flow rate. We have shown that in the investigated range of water saturations it exists a transition in the oil ganglia structure and size. For high water saturations oil ganglia have one or two pores in size, however for lower water saturations they comprise an important number of pores. Transport strongly depends on the size distribution of the oil ganglia as they create large percolating paths and stagnant zones. We also showed the existence of two different types of stagnant zones: zones accessible by diffusion into pores and zones only accessible by spatially limited diffusion in films. The major advantage of using glass micromodels lies in the fact that dispersion coefficients can be computed from concentrations averaged over the pore space or from concentrations at the outlet and simultaneously from spatial concentration profiles. Curves were fitted using the Advection–Dispersion Equation (ADE) with adequate boundary conditions. The fitting quality of the temporal evolution of the average and outlet concentration was very good. However, fitting of the concentration profiles could only be done for the higher water saturations. This is due to the fact that the Representative Elementary Volume (REV) of lower water saturations is larger than the micromodel. The results show that fitting the breakthrough curve in order to determine the dispersion coefficient in a partially saturated porous medium might be misleading. Indeed, when fitting the breakthrough curves we were able to compute a dispersion coefficient even in the case where the REV of the water saturation is larger than the micromodel. Consequently, the knowledge of the local concentration profiles as a function of time is necessary as it provides an additional information on the spatio-temporal behavior of the transport process and therefor a supplementary constraint of the fitting procedure. Finally, we observed a time dependent dispersion coefficient in the regime where oil ganglia comprise several pores. This fact might be attributed to the non-Gaussian nature of the transport.

Analytical Solution for Pollutant Diffusion in Soils with Time-Dependent Dispersion Coefficient

According to the characteristics of the time-dependent dispersion coefficient of pollutant migration in soils, a one-dimensional pollutant migration model considering solute decay is established in the finite thickness soil. Based on the proposed model, the time-dependent boundary condition variable is considered, and different types of time-dependent dispersion coefficients are also assumed. The variable parameter k for pollutant diffusion time is used to control the time-dependent model. The analytical solution is obtained by separating the variable method. The results calculated by the presented analytical solution compared with those in the literature indicate solution accuracy and program reliability. According to the parameter study, variable k in the time-dependent model has a particular influence on the results, and the time-dependent and constant boundary conditions have a significant difference. The proposed method can be a reference for the landfill liner design.

Non-reactive solute transport modelling with time-dependent dispersion through stratified porous media

We present a numerical solution of the mobile–immobile model (MIM) with time-dependent dispersion coefficient to simulate solute transport through heterogeneous porous media. Observed experimental data of non-reactive solute transport through hydraulically coupled stratified porous media have been simulated using asymptotic and linear time-dependent dispersion functions. Non-Gaussian breakthrough curves comprising long tails are simulated well with the MIM incorporating asymptotic time-dependent dispersion model. The system is under the strong influence of physical nonequilibrium, which is evident by variable mass transfer coefficient estimated at different down-gradient distances. Asymptotic time-dependent functions are capable of capturing the rising limb of the solution phase breakthrough curves with improved accuracy, whereas tailing part simulation capabilities are similar for both asymptotic and linear time-dependent dispersion functions. Further, the temporal moment analysis demonstrated increased spreading, variance for linear dispersion model as compared with asymptotic dispersion model. It is also observed that the first-order mass transfer coefficient varies inversely with travel distance from the input source. It can be concluded from the study that MIM with time-dependent dispersion function is simpler yet sensitive to account for medium’s heterogeneity in a better manner even for small observation distances from the source.

Solution of Gill’s generalized dispersion model: Solute transport in Poiseuille flow with wall absorption

Previous applications of Gill’s generalized dispersion model in the description of solute transport in Poiseuille flow with wall absorption used only the long-time asymptotic values of dispersion coefficients. They are also limited to the simplest second order equation, which resembles Taylor’s classic model with a Gaussian distribution for cross-sectional mean concentration. To obtain the time-dependent dispersion coefficients, in this work the coefficients of dispersion model are concretely related to the moments of concentration distribution. With the readily obtained higher order dispersion coefficients, solution of the Taylor-Gill expansion equation is presented up to the fourth order, accounting for the asymmetry (skewness) and heavy tails (kurtosis) of the cross-sectional mean concentration distribution.

Study of Two-Dimensional, all – Time Dispersion of a Solute in a Fluid – Saturated Porous Medium

The paper presents the mathematical formulation which describes the dispersion of solute in a laminar flow in a sparsely packed porous medium. The effect of interphase mass transfer on dispersion in a unidirectional flow through a horizontally extent of infinite porous channel is examined using the generalized dispersion model of Sankarasubramanian and Gill. The model brings into focus three important coefficients namely the exchange coefficient, the convection coefficient and the dispersion coefficient. The time-dependent dispersion coefficient and mean concentration distribution are computed and results are represented graphically. The problem finds many applications in waste water management, in chromatography and in biomechanical problems.

Influence of Various Dispersion Coefficients on Contaminant Migration in a Fracture-matrix System with Skin Formation

Subsurface contamination due to transient migration of chemicals through weathered fractures is a serious problem in hard rock terrains. In this paper, a numerical model has been used to analyse the effect of various dispersion coefficients on the movement of contaminants in a fracture-matrix syste m with skin formation. A constant continuous source of contaminants is considered at the inlet of a set of parallel fractures and an implicit finite difference technique has been used to solve the model. The transport of contaminants with constant dispersion coefficients were compared with those subjected to distance and time dependent dispersion. Sensitivity analysis has been conducted to study the effect of different fluid velocities, fracture-skin porosities, fracture-skin diffusion coefficients, fracture aperture, retardation factors, and dispersivity-distance ratio on solute transport mechanism within the system. Results suggest that the solute mass retained in the fracture is higher when time-dependent dispersion coefficients are used which may be due to seasonal conditioning of the rocks. When constant dispersion coefficient is considered, the reduction in the solute mass with increment in the fracture-skin porosity is gradual unlike other cases.The solute mass retained in the fracture increases with increase in dispersivity-distance ratio for time-dependent dispersion coefficients.

Mathematical modeling of groundwater contamination with varying velocity field

Abstract In this study, analytical models for predicting groundwater contamination in isotropic and homogeneous porous formations are derived. The impact of dispersion and diffusion coefficients is included in the solution of the advection-dispersion equation (ADE), subjected to transient (time-dependent) boundary conditions at the origin. A retardation factor and zero-order production terms are included in the ADE. Analytical solutions are obtained using the Laplace Integral Transform Technique (LITT) and the concept of linear isotherm. For illustration, analytical solutions for linearly space- and time-dependent hydrodynamic dispersion coefficients along with molecular diffusion coefficients are presented. Analytical solutions are explored for the Peclet number. Numerical solutions are obtained by explicit finite difference methods and are compared with analytical solutions. Numerical results are analysed for different types of geological porous formations i.e., aquifer and aquitard. The accuracy of results is evaluated by the root mean square error (RMSE).

Analytical Model for Contaminant Migration with Time-Dependent Transport Parameters

AbstractA general analytical model for one-dimensional (1D) contaminant transport in infinite domain with time-dependent transport parameters is presented in this paper. The model is based on the advection-dispersion equation with a time-dependent flow velocity, a time-dependent dispersion coefficient, and a time-dependent distribution coefficient due to sorption. It takes into account a first-order irreversible reaction (decay), an arbitrary initial distribution of the contaminant, and an arbitrary space- and time-dependent sink/source term. The model can handle any time-dependent transport parameter. Analytical solutions are provided for both the advection dominant and the advection-dispersion transport equations. The proposed analytical solutions are general and they can be used for problems where one or more parameters are time-dependent. It is shown that the presented solutions can be reduced to other existent solutions where one transport parameter is assumed to be time-dependent. The general analyt...

Analysis of a splitting scheme for a class of random nonlinear partial differential equations

In this paper, we consider a Lie splitting scheme for a nonlinear partial differential equation driven by a random time-dependent dispersion coefficient. Our main result is a uniform estimate of the error of the scheme when the time step goes to 0. Moreover, we prove that the scheme satisfies an asymptotic-preserving property. As an application, we study the order of convergence of the scheme when the dispersion coefficient approximates a (multi)fractional process.

Effect of distance-dependent and time-dependent dispersion on non-linearly sorbed multispecies contaminants in porous media

In this study, the effect of distance-dependent and time-dependent dispersion coefficient on the transport of multispecies contaminants subjected to Freundlich non-linear adsorption in porous media has been analyzed. An implicit finite difference has been used to develop the numerical model. A constant continuous source of first species contaminant has been assumed at the inlet of the porous domain. The results obtained from distance-dependent and time-dependent dispersion coefficient has been compared with that obtained using constant dispersion coefficient for different fluid velocities and dispersivity–distance ratios. The first species of the multispecies transport is subjected to Freundlich non-linear sorption. Sensitivity analysis has been conducted to study the effect of different simulation time, fluid velocities, sorption exponent, and dispersivity–distance ratio on solute transport mechanism within the porous media. Results suggest that the effect of distance-dependent dispersion coefficient on multispecies contaminant movement is negligible when the fluid velocity is very low. The fluid velocity has a significant impact on the multispecies solute movement when non-linear sorption and distance-dependent dispersion coefficient are considered. The effect of variation of the Freundlich sorption exponent on contaminant concentration is negligible when distance-dependent and time-dependent dispersion coefficients are considered.

On shallow water waves in a medium with time-dependent dispersion and nonlinearity coefficients

In this paper, we studied the progression of shallow water waves relevant to the variable coefficient Korteweg–de Vries (vcKdV) equation. We investigated two kinds of cases: when the dispersion and nonlinearity coefficients are proportional, and when they are not linearly dependent. In the first case, it was shown that the progressive waves have some geometric structures as in the case of KdV equation with constant coefficients but the waves travel with time dependent speed. In the second case, the wave structure is maintained when the nonlinearity balances the dispersion. Otherwise, water waves collapse. The objectives of the study are to find a wide class of exact solutions by using the extended unified method and to present a new algorithm for treating the coupled nonlinear PDE’s.

Special Issue on Toxics and Pathogens in the Environment

Special Issue on Toxics and Pathogens in the Environment