Asymptotic dispersion in 2D heterogeneous porous media determined by parallel numerical simulations

Dreuzy, Jean‐Raynald de; Beaudoin, Anthony; Erhel, Jocelyne

doi:10.1029/2006wr005394

Cited by 75 publications

(152 citation statements)

References 41 publications

(73 reference statements)

Supporting

Mentioning

142

Contrasting

Order By: Relevance

“…In the longrange transport most interesting is the case of large fluctuations. To our knowledge, there is only one study [5] where the general case of large fluctuations was considered, and the longrange transport was studied numerically for more than thousands of correlation length scales. In this study, a 2D flow was simulated by an iterative multigrid method where the transport was evaluated up to 1638 correlation length scales, but in this model, the dispersion coefficient was defined by averaging over a discrete cloud of particles moving in one fixed sample of the flow.…”

Section: Introductionmentioning

confidence: 99%

“…This is different from the averaging over an ensemble of Lagrangian trajectories starting from a fixed point. As to the interrelation of these two definitions, see e.g., [21], [5], [23].…”

Section: Introductionmentioning

confidence: 99%

“…The problem of evaluation of the diffusion regime is computationally very intensive since the transport should be simulated for thousands of correlation length scales (e.g., see [21], [5]). Therefore the problem was studied only with the first order approximation ( [21], [4], [23]) which is applicable under strong restrictions of small fluctuations of the log conductivity.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stochastic Flow Simulation and Particle Transport in a 2D Layer of Random Porous Medium

Kurbanmuradov

Sabelfeld

2010

Transp Porous Med

View full text Add to dashboard Cite

A stochastic numerical method is developed for simulation of flows and particle transport in a 2D layer of porous medium. The hydraulic conductivity is assumed to be a random field of a given statistical structure, the flow is modeled in the layer with prescribed boundary conditions. Numerical experiments are carried out by solving the Darcy equation for each sample of the hydraulic conductivity by a direct solver for sparse matrices, and tracking Lagrangian trajectories in the simulated flow. We present and analyze different Eulerian and Lagrangian statistical characteristics of the flow such as transverse and longitudinal velocity correlation functions, longitudinal dispersion coefficient, and the mean displacement of Lagrangian trajectories. We discuss the effect of long-range correlations of the longitudinal velocities which we have found in our numerical simulations. The related anomalous diffusion is also analyzed.

show abstract

Section: Introductionmentioning

confidence: 99%

“…This is different from the averaging over an ensemble of Lagrangian trajectories starting from a fixed point. As to the interrelation of these two definitions, see e.g., [21], [5], [23].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stochastic Flow Simulation and Particle Transport in a 2D Layer of Random Porous Medium

Kurbanmuradov

Sabelfeld

2010

Transp Porous Med

View full text Add to dashboard Cite

show abstract

“…Then, in order to avoid to approximate c at each point in D and to avoid numerical diffusion, a Lagrangian method is preferred to an Eulerian method [12]. This method consists in simulating a cloud of particles in the physical domain: when (2) is considered on R d , c is a law of the process describing the position of the particles.…”

Section: Approximation Of the Transport Problemmentioning

confidence: 99%

“…The dispersion D(ω, t) was estimated by a Finite Difference approximation in [4,11,12,16], but the result is very sensitive to the step taken. Here we use an explicit formula to compute D(ω, t) (see [6,19]).…”

Section: Approximation Of the Transport Problemmentioning

confidence: 99%

A combined collocation and Monte Carlo method for advection-diffusion equation of a solute in random porous media

Erhel

Mghazli²,

Oumouni

2014

ESAIM: ProcS

Self Cite

View full text Add to dashboard Cite

Abstract. In this work, we present a numerical analysis of a method which combines a deterministic and a probabilistic approaches to quantify the migration of a contaminant, under the presence of uncertainty on the permeability of the porous medium. More precisely, we consider the flow equation in a random porous medium coupled with the advection-diffusion equation. Quantities of interest are the mean spread and the mean dispersion of the solute. The means are approximated by a quadrature rule, based on a sparse grid defined by a truncated Karhunen-Loève expansion and a stochastic collocation method. For each grid point, the flow model is solved with a mixed finite element method in the physical space and the advection-diffusion equation is solved with a probabilistic Lagrangian method. The spread and the dispersion are expressed as functions of a stochastic process. A priori error estimates are established on the mean of the spread and the dispersion. Keywords: Uncertainty quantification, elliptic PDE with random coefficients, advection-diffusion equation, collocation techniques, anisotropic sparse grids, Monte Carlo method, Euler scheme for SDE. IntroductionMathematical modeling and numerical simulation are important tools in the prediction of pollutant transport in groundwater and in the evaluation of potential risks of contaminants. The main constraint to the development of these models is the limited knowledge of the geological characteristics and the natural heterogeneity which implies uncertainty in the parameters and data. Stochastic approaches have been developed to deal with this uncertainty, where the permeability field is modeled as a random field a = e G , where G is a correlated Gaussian field [2, 13]. Our objective is to quantify the migration of a contaminant by computing statistics of interest defined by the mean of the spread and of the dispersion of the solute. The permeability a is discretized using a Karhunen-Loève (K-L) truncation up to a suitable and moderately large order.The Monte Carlo method is the most widely used approach to deal with uncertainty. This method is used in [2,4,7,12,13] to approximate the mean spread and dispersion.Recently, stochastic collocation (see [1,9,18,19]), based on sparse tensor product approximation, has gained much attention since it is very effective and accurate for computing statistics from solutions of PDEs with random input data. In this paper, we use a truncated Karhunen-Loève expansion of the random data and an anisotropic sparse grid with Gaussian knots. Then we propose to approximate the mean values by a quadrature rule using this sparse grid. 329The cost is proportional to the number of samples in the Monte Carlo method and to the sparse grid size in the collocation method. Each sample or grid point requires solving a flow PDE and a transport PDE; this can be done in a non intrusive and parallel way. In this paper, we approximate the steady-state flow problem by a mixed finite element method in the physical space to get the velocity field. The tra...

show abstract

Solute transport in low‐heterogeneity sandboxes: The role of correlation length and permeability variance

Heidari

2014

Water Resources Research

View full text Add to dashboard Cite

This work examines how heterogeneity structure, in particular correlation length, controls flow and solute transport. We used two-dimensional (2D) sandboxes (21.9 cm 3 20.6 cm) and four modeling approaches, including 2D Advection-Dispersion Equation (ADE) with explicit heterogeneity structure, 1D ADE with average properties, and nonlocal Continuous Time Random Walk (CTRW) and fractional ADE (fADE). The goal is to answer two questions: (1) how and to what extent does correlation length control effective permeability and breakthrough curves (BTC)? (2) Which model can best reproduce data under what conditions? Sandboxes were packed with the same 20% (v/v) fine and 80% (v/v) coarse sands in three patterns that differ in correlation length. The Mixed cases contain uniformly distributed fine and coarse grains. The Four-zone and One-zone cases have four and one square fine zones, respectively. A total of seven experiments were carried out with permeability variance of 0.10 (LC), 0.22 (MC), and 0.43 (HC). Experimental data show that the BTC curves depend strongly on correlation length, especially in the HC cases. The HC One-zone (HCO) case shows distinct breakthrough steps arising from fast advection in the coarse zone, slow advection in the fine zone, and slow diffusion, while the LCO and MCO BTCs do not exhibit such behavior. With explicit representation of heterogeneity structure, 2D ADE reproduces BTCs well in all cases. CTRW reproduces temporal moments with smaller deviation from data than fADE in all cases except HCO, where fADE has the lowest deviation.

show abstract

Asymptotic dispersion in 2D heterogeneous porous media determined by parallel numerical simulations

Cited by 75 publications

References 41 publications

Stochastic Flow Simulation and Particle Transport in a 2D Layer of Random Porous Medium

Stochastic Flow Simulation and Particle Transport in a 2D Layer of Random Porous Medium

A combined collocation and Monte Carlo method for advection-diffusion equation of a solute in random porous media

Solute transport in low‐heterogeneity sandboxes: The role of correlation length and permeability variance

Contact Info

Product

Resources

About