Upscaling Mixing in Highly Heterogeneous Porous Media via a Spatial Markov Model

In this work, we develop a novel Lagrangian model able to predict solute mixing in heterogeneous porous media. The Spatial Markov model has previously been used to predict effective mean conservative transport in flows through heterogeneous porous media. In predicting effective measures of mixing on larger scales, knowledge of only the mean transport is insufficient. Mixing is a small scale process driven by diffusion and the deformation of a plume by a non-uniform flow. In order to capture these small scale processes that are associated with mixing, the upscaled Spatial Markov model must be extended in such a way that it can adequately represent fluctuations in concentration. To address this problem, we develop downscaling procedures within the upscaled model to predict measures of mixing and dilution of a solute moving through an idealized heterogeneous porous medium. The upscaled model results are compared to measurements from a fully resolved simulation and found to be in good agreement.

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Laboratory Experiments on Solute Transport in Non-Homogeneous Porous Media

Hydrology Research ◽

10.2166/nh.1986.0022 ◽

1986 ◽

Vol 17 (4-5) ◽

pp. 305-314 ◽

Cited By ~ 3

Author(s):

Anders Refsgaard

Keyword(s):

Porous Media ◽

Hydraulic Conductivity ◽

Solute Transport ◽

Laboratory Experiments ◽

Heterogeneous Porous Media ◽

Heterogeneous Porous Medium ◽

Dispersion Parameters ◽

Homogeneous Porous Media ◽

Flow Through ◽

Good Agreement

Solute transport in groundwater is a process which has become of major importance during the last decades due to increasing contamination of ground water. This process usually occurs in a medium heterogeneous with respect to hydraulic conductivity and porosity, properties that affect the dispersion of the solutes. The present paper describes an experimental investigation of the solute transport process in heterogeneous porous media, especially the connection between the statistical properties of their hydraulic conductivity distributions and the dispersion parameters governing the spreading of the solutes. The experimental results are compared to theoretical solutions derived for the same case of a solute pulse in an average uniform flow through a heterogeneous porous medium. Generally there is good agreement between the theory and the experiments. In field applications this means that the dispersion parameters can be more readily determined from the soil properties. Furthermore, the deviations between dispersivities determined in laboratory columns and dispersitivies found under field conditions can be explained quantitatively by the differencies in the length scales and in the variances of the hydraulic conductivity distributions.

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Upscaling bimolecular reactive transport in highly heterogeneous porous media with the LAgrangian Transport Eulerian Reaction Spatial (LATERS) Markov model

Stochastic Environmental Research and Risk Assessment ◽

10.1007/s00477-021-02006-z ◽

2021 ◽

Author(s):

Elise E. Wright ◽

Nicole L. Sund ◽

David H. Richter ◽

Giovanni M. Porta ◽

Diogo Bolster

Keyword(s):

Porous Media ◽

Markov Model ◽

Reactive Transport ◽

Heterogeneous Porous Media ◽

Lagrangian Transport

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Multiscale Model Reduction of the Unsaturated Flow Problem in Heterogeneous Porous Media with Rough Surface Topography

Mathematics ◽

10.3390/math8060904 ◽

2020 ◽

Vol 8 (6) ◽

pp. 904

Author(s):

Denis Spiridonov ◽

Maria Vasilyeva ◽

Eric T. Chung ◽

Yalchin Efendiev ◽

Raghavendra Jana

Keyword(s):

Porous Media ◽

Boundary Conditions ◽

Surface Topography ◽

Rough Surface ◽

Heterogeneous Porous Media ◽

Coarse Grid ◽

Basis Functions ◽

Richards Equation ◽

Small Scale ◽

Fine Grid

In this paper, we consider unsaturated filtration in heterogeneous porous media with rough surface topography. The surface topography plays an important role in determining the flow process and includes multiscale features. The mathematical model is based on the Richards’ equation with three different types of boundary conditions on the surface: Dirichlet, Neumann, and Robin boundary conditions. For coarse-grid discretization, the Generalized Multiscale Finite Element Method (GMsFEM) is used. Multiscale basis functions that incorporate small scale heterogeneities into the basis functions are constructed. To treat rough boundaries, we construct additional basis functions to take into account the influence of boundary conditions on rough surfaces. We present numerical results for two-dimensional and three-dimensional model problems. To verify the obtained results, we calculate relative errors between the multiscale and reference (fine-grid) solutions for different numbers of multiscale basis functions. We obtain a good agreement between fine-grid and coarse-grid solutions.

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Development of New Design Fatigue Curves in Japan: Discussion of Best-Fit Curves Based on Fatigue Test Data With Small-Scale Test Specimen

Volume 3B: Design and Analysis ◽

10.1115/pvp2018-84052 ◽

2018 ◽

Cited By ~ 1

Author(s):

Yun Wang ◽

Hisamitsu Hatoh ◽

Masato Yamamoto ◽

Motoki Nakane ◽

Akihiko Hirano ◽

...

Keyword(s):

Tensile Strength ◽

Fatigue Tests ◽

Small Scale ◽

Stress Effect ◽

Mean Stress ◽

Mean Stress Effect ◽

Scale Test ◽

The Mean ◽

Best Fit ◽

Good Agreement

Based on the precedent design fatigue curves and recent fatigue data obtained from materials with different mechanical properties, new design fatigue curves with high general versatility in air have been developed by The Japan Welding Engineering Society (JWES). Structural materials with different tensile strength are utilized in fatigue tests to verify the validity of these design fatigue curves and discuss the mean stress effect. The materials employed in this study are austenitic stainless steel (SS) SUS316LTP, carbon steel (CS) STPT370, low-alloy steels (LASs) SQV2A and SCM435H, all of which are used in the structural components of nuclear power plants of Japan. The best-fit curves (BFCs) are formulated by using the parameter of tensile strength to describe the relationship between strain (stress) amplitude and fatigue life [1]. The results of fully reversed axial fatigue tests conducted with small-scale test specimens of those materials in air at ambient temperature show good agreement with the developed BFCs. The results of fatigue tests also indicate that the mean stress effect is remarkable in materials with higher tensile strength. The applicability of Modified Goodman and Smith-Watson-Topper (SWT) approaches to the design fatigue curves is compared and discussed when considering mean stress effect. The correction of mean stress effect with SWT approach shows a good agreement with the developed BFCs.

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Analysis of Numerical Simulation on Near Surface Beneath Clean and Contaminated Small-Scale Wind-Driven Water Waves

Volume 2: Fora, Parts A and B ◽

10.1115/fedsm2007-37558 ◽

2007 ◽

Author(s):

Xiaoxia Hu ◽

Ali Dolatabadi ◽

Kamran Siddiqul

Keyword(s):

Numerical Model ◽

Water Waves ◽

Numerical Study ◽

Surface Region ◽

Surface Flow ◽

Small Scale ◽

Mean Velocity ◽

Near Surface ◽

The Mean ◽

Good Agreement

We report on a numerical study conducted to investigate the near-surface flow beneath clean and contaminated small-scale wind-driven water surfaces. The numerical model is validated in terms of the velocity and surface wave characteristics. A good agreement is observed between the experimental and numerical values. The results from the numerical model show that the mean velocity in the near-surface region is 25–50% higher beneath the contaminated surface as compared to the clear surface. The present trend is also in agreement with the previous experimental observations.

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On the motion induced in a gas confined in a small-scale gap due to instantaneous boundary heating

Journal of Fluid Mechanics ◽

10.1017/s0022112007008658 ◽

2007 ◽

Vol 593 ◽

pp. 453-462 ◽

Cited By ~ 19

Author(s):

A. MANELA ◽

N. G. HADJICONSTANTINOU

Keyword(s):

Early Time ◽

Mean Free Path ◽

Small Scale ◽

Characteristic Time Scale ◽

Relative Temperature ◽

The Mean ◽

Gas Molecules ◽

Hydrodynamic Fields ◽

Gap Width ◽

Good Agreement

We analyse the time response of a gas confined in a small-scale gap (of the order of or smaller than the mean free path) to an instantaneous jump in the temperature of its boundaries. The problem is formulated for a collisionless gas in the case where the relative temperature jump at each wall is small and independent of the other. An analytic solution for the probability density function is obtained and the respective hydrodynamic fields are calculated. It is found that the characteristic time scale for arriving at the new equilibrium state is of the order of several acoustic time scales (the ratio of the gap width to the most probable molecular speed of gas molecules). The results are compared with direct Monte Carlo simulations of the Boltzmann equation and good agreement is found for non-dimensional times (scaled by the acoustic time) not exceeding the system Knudsen number. Thus, the present analysis describes the early-time behaviour of systems of arbitrary size and may be useful for prescribing the initial system behaviour in counterpart continuum-limit analyses.

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Upscaling models of solute transport in porous media through genetic programming

Journal of Hydroinformatics ◽

10.2166/hydro.2007.028 ◽

2007 ◽

Vol 9 (4) ◽

pp. 251-266 ◽

Cited By ~ 9

Author(s):

David J. Hill ◽

Barbara S. Minsker ◽

Albert J. Valocchi ◽

Vladan Babovic ◽

Maarten Keijzer

Keyword(s):

Porous Media ◽

Genetic Programming ◽

Solute Transport ◽

Heterogeneous Porous Media ◽

Training Data ◽

Solute Flux ◽

Small Scale ◽

Transport Models ◽

Averaged Equations

Due to the considerable computational demands of modeling solute transport in heterogeneous porous media, there is a need for upscaled models that do not require explicit resolution of the small-scale heterogeneity. This study investigates the development of upscaled solute transport models using genetic programming (GP), a domain-independent modeling tool that searches the space of mathematical equations for one or more equations that describe a set of training data. An upscaling methodology is developed that facilitates both the GP search and the implementation of the resulting models. A case study is performed that demonstrates this methodology by developing vertically averaged equations of solute transport in perfectly stratified aquifers. The solute flux models developed for the case study were analyzed for parsimony and physical meaning, resulting in an upscaled model of the enhanced spreading of the solute plume, due to aquifer heterogeneity, as a process that changes from predominantly advective to Fickian. This case study not only demonstrates the use and efficacy of GP as a tool for developing upscaled solute transport models, but it also provides insight into how to approach more realistic multi-dimensional problems with this methodology.

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Impact of flow conditions on pore-scale solute mixing: experiments in heterogeneous 2D porous media

10.5194/egusphere-egu21-8636 ◽

2021 ◽

Author(s):

Oshri Borgman ◽

Turuban Régis ◽

Baudouin Géraud ◽

Le Borgne Tanguy ◽

Méheust Yves

Keyword(s):

Porous Media ◽

Fluorescence Intensity ◽

Dispersion Coefficient ◽

Reaction Rates ◽

Pore Scale ◽

Flow Conditions ◽

Dispersion Coefficients ◽

Incomplete Mixing ◽

The Mean ◽

Solute Mixing

<p>Solute mixing mediated by flow in porous media plays a significant role in controlling reaction rates in subsurface environments. In many practical cases, incomplete mixing&#8212;inhomogeneous solute concentrations&#8212;occurs at the pore-scale, limiting local and thus upscaled reaction rates, and renders their prediction based on effective dispersion coefficients derived from dispersion models (or by assuming Taylor-Aris dispersion) inaccurate. We perform solute transport experiments in transparent, quasi-two-dimensional, soil analog models to investigate the relationships between pore-scale solute dispersion and mixing under different flow conditions. We use Fluorescein as a conservative tracer and record its fluorescence intensity in monochrome images at fixed time intervals. We convert the fluorescence intensity to solute concentration fields based on a calibration curve obtained with various homogeneous solute concentrations and subsequently compute concentration gradients. Our images provide evidence for incomplete mixing at the pore-scale and show strong gradients transverse to the overall flow direction. We fit the mean longitudinal concentration profile to an analytical solution of the advection-dispersion equation and compute the effective longitudinal dispersion coefficient. Based on the lamellar mixing theory, we also infer an effective diffusion coefficient relevant to the mean concentration gradient&#8217;s dynamics. By comparing these two diffusion/dispersion coefficients in saturated flow conditions, we show that while their values are similar at low P&#233;clet, their scaling behaviors as a function of P&#233;clet are different. Hence, as pointed out by several previous studies, modeling reactive transport processes requires accounting for a mixing behavior driven by a diffusive process that cannot entirely be described by the solute dispersion coefficient. We extend this work by varying the saturation degree in the experiments and our samples' structural heterogeneity to investigate how flow desaturation and porous medium structure impact solute mixing.</p>

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Mathematical Treatment of Saturated Macroscopic Flow in Heterogeneous Porous Medium: Evaluating Darcy’s Law

Hydrology ◽

10.3390/hydrology7010004 ◽

2019 ◽

Vol 7 (1) ◽

pp. 4

Author(s):

R. William Nelson ◽

Gustavious P. Williams

Keyword(s):

Porous Medium ◽

Porous Media ◽

Flow Field ◽

Laboratory Experiments ◽

Heterogeneous Media ◽

Darcy’S Law ◽

Heterogeneous Porous Media ◽

Darcy's Law ◽

Mathematical Treatment ◽

Heterogeneous Porous Medium

We present a rigorous mathematical treatment of water flow in saturated heterogeneous porous media based on the classical Navier-Stokes formulation that includes vorticity in a heterogeneous porous media. We used the mathematical approach proposed in 1855 by James Clark Maxwell. We show that flow in heterogeneous media results in a flow field described by a heterogeneous complex lamellar vector field with rotational flows, compared to the homogeneous lamellar flow field that results from Darcy’s law. This analysis shows that Darcy’s Law does not accurately describe flow in a heterogeneous porous medium and we encourage precise laboratory experiments to determine under what conditions these issues are important. We publish this work to encourage others to perform numerical and laboratory experiments to determine the circumstances in which this derivation is applicable, and in which the complications can be disregarded.

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