Effective Dispersion of a Solute Cloud in a Chemically Heterogeneous Porous Medium: Comparison of Two Ensemble-Averaging Procedures

1996 ◽  
Vol 32 (11) ◽  
pp. 3311-3319 ◽  
Author(s):  
D. Metzger ◽  
H. Kinzelbach ◽  
W. Kinzelbach
Keyword(s):  
Porous Medium ◽  
Heterogeneous Porous Medium ◽  
Ensemble Averaging ◽  
Effective Dispersion

Temporal behaviour of a solute cloud in a chemically heterogeneous porous medium

1999 ◽  
Vol 386 ◽  
pp. 77-104 ◽  
Author(s):  
S. ATTINGER ◽  
M. DENTZ ◽  
H. KINZELBACH ◽  
W. KINZELBACH
Keyword(s):  
Porous Medium ◽  
Dispersion Coefficient ◽  
Source Size ◽  
Advective Transport ◽  
Centre Of Mass ◽  
Heterogeneous Porous Medium ◽  
Extended Source ◽  
Temporal Behaviour ◽  
Effective Dispersion ◽  
Spatial Dimensions

In this paper we investigate the temporal behaviour of a solute cloud in a heterogeneous porous medium using a stochastic modelling approach. The behaviour of the plume evolving from a point-like instantaneous injection is characterized by the velocity of its centre-of-mass and by its dispersion as a function of time. In a stochastic approach, these quantities are expressed as appropriate averages over the ensemble of all possible realizations of the medium. We develop a general perturbation approach which allows one to calculate the various quantities in a systematic and unified way. We demonstrate this approach on a simplified aquifer model where only the retardation factor R(x) due to linear instantaneous chemical adsorption varies stochastically in space. We analyse the resulting centre-of-mass velocity and two conceptually different definitions for the dispersion coefficient: the ‘effective’ dispersion coefficient which is derived from the average over the centred second moments of the spatial concentration distributions in every realization, and the ‘ensemble’ dispersion coefficient which follows from the second moment of the averaged concentration distribution. The first quantity characterizes the dispersion in a typical realization of the medium as a function of time, whereas the second one describes the (formal) dispersion properties of the ensemble as a whole. We show that for finite times the two quantities are not equivalent whereas they become identical for t→∞ and spatial dimensions d[ges ]2. The ensemble dispersion coefficient which is usually evaluated in the literature considerably overestimates the dispersion typically found in one given realization of the medium. We derive for the first time explicit analytical expressions for both quantities as functions of time. From these, we identify two relevant time scales separating regimes of qualitatively and quantitatively different temporal behaviour: the shorter of the two scales is set by the advective transport of the solute cloud over one disorder correlation length, whereas the second, much larger one, is related to the dispersive spreading over the same distance. Only for times much larger than this second scale, and spatial dimensions d[ges ]2, do the effective and the ensemble dispersion coefficients become equivalent due to mixing caused by the local transversal dispersion. Finally, the formalism is generalized to an extended source. With growing source size the convergence of the effective dispersion coefficient to the ensemble dispersion coefficient happens faster as the extended source already represents an ensemble of point sources. In the limit of a very large source size, convergence occurs on the time scale of advective transport over one disorder length. We derive explicit results for the temporal behaviour in the different time regimes for both point and extended sources.

Analysis of two-dimensional solute transport through heterogeneous porous medium

2018 ◽  
Vol 2 (3) ◽  
Author(s):  
Atul Kumar ◽  
◽  
Lav Kush Kumar ◽  
Shireen Shireen ◽  
◽  
...  
Keyword(s):  
Porous Medium ◽  
Solute Transport ◽  
Two Dimensional ◽  
Heterogeneous Porous Medium

scholarly journals Multidimensional computational models of gas combustion in heterogeneous porous medium

2021 ◽  
Vol 2099 (1) ◽  
pp. 012010
Author(s):  
Yu Laevsky ◽  
T Nosova
Keyword(s):  
Porous Medium ◽  
Parabolic Equations ◽  
Computational Models ◽  
Front Propagation ◽  
Original Model ◽  
Gas Mixture ◽  
Heterogeneous Porous Medium ◽  
Gas Combustion ◽  
Piecewise Constant ◽  
System Of Parabolic Equations

Abstract The processes of filtration gas combustion in heterogeneous porous medium is studying. The presence of two opposite modes of front propagation made it possible to stabilize the combustion front in a composite porous medium with piecewise constant porosity. A feature of this study is the presentation of the original model not in the traditional form of a system of parabolic equations, but in the form of integral conservation laws in terms of the temperature of the porous medium, the total gas enthalpy, and the mass of gas mixture, and the fluxes corresponding to these functions.

MATHEMATICAL SOLUTION OF FINGERING PHENOMENON IN VERTICAL DOWNWARD DIRECTION THROUGH HETEROGENEOUS POROUS MEDIUM

2021 ◽  
Vol 10 (1) ◽  
pp. 483-496
Author(s):  
D.A. Shah ◽  
A.K. Parikh
Keyword(s):  
Porous Medium ◽  
Oil Recovery ◽  
Analytic Solution ◽  
Variational Iteration Method ◽  
Approximate Analytic Solution ◽  
Order Partial Differential Equation ◽  
Heterogeneous Porous Medium ◽  
Fingering Instability ◽  
Oil Flow ◽  
Secondary Oil Recovery

Present study explores the Fingering (Instability) phenomenon's mathematical model that ensues during the process of secondary oil recovery where two not miscible fluids (water and oil) flow within a heterogeneous porous medium as water is injected vertically downwards. Variational iteration method with proper initial and boundary conditions is being used to determine approximate analytic solution for governing nonlinear second order partial differential equation. Whereas MATLAB is applied to acquire the solution's numerical findings and graphical representations.

Comparison of theory and experiment for solute transport in highly heterogeneous porous medium

2007 ◽  
Vol 30 (11) ◽  
pp. 2235-2261 ◽  
Author(s):  
Fabrice Golfier ◽  
Michel Quintard ◽  
Fabien Cherblanc ◽  
Brendan A. Zinn ◽  
Brian D. Wood
Keyword(s):  
Porous Medium ◽  
Solute Transport ◽  
Heterogeneous Porous Medium ◽  
Theory And Experiment
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