A Stochastic Model for Hydrodynamic Dispersion

In this chapter we develop one dimensional model without resorting to Fickian assumptions and discuss the methods of estimating the parameters. As of many contracted description of a natural phenomena the model presented in this chapter has its weaknesses. But we model the fluctuation of the solute velocity due to porous structure and incorporate the fluctuation in the mass conservation of solute. Then we need to characterise the fluctuations so that we can relate them to the porous structure

Solute Transport In Heterogeneous Porous Media

Solute mass transport in porous media is strongly correlated with pore fluid flow. The analysis of solute transport is an effective means for studying medium heterogeneities. In this study, we discuss the effects of heterogeneity on the tracer transport. Assuming steady fluid flow, we have simulated tracer transport in various permeability heterogeneities. The results show that the tracer distribution is very closely correlated with the medium heterogeneity, and anisotropy in tracer transport exists when there is permeability lineation and large permeability contrast between low- and high-permeability regions. An important feature by which the tracer transport differs from the fluid flow field is that the tracer transport tends to smear the effects of a thin non-permeable layer (or small permeability barriers) through diffusion into the low-permeability layer, while the fluid flow cannot penetrate the low-permeability layer. In addition, the modeling results also show that the tracer transport strongly depends on the tracer source dimension, as well as the flow source dimension.Massachusetts Institute of Technology. Borehole Acoustics and Logging ConsortiumUnited States. Dept. of Energy (Grant DE-FG02-86ER13636

Resolving Wave Propagation in Anisotropic Poroelastic Media Using Graphical Processing Units (GPUs)

Biot's equations describe the physics of hydromechanically coupled systems establishing the widely recognized theory of poroelasticity. This theory has a broad range of applications in Earth and biological sciences as well as in engineering. The numerical solution of Biot's equations is challenging because wave propagation and fluid pressure diffusion processes occur simultaneously but feature very different characteristic time scales. Analogous to geophysical data acquisition, high resolution and three dimensional numerical experiments lately redefined state of the art. Tackling high spatial and temporal resolution requires a high-performance computing approach. We developed a multi- graphical processing units (GPU) numerical application to resolve the anisotropic elastodynamic Biot's equations that relies on a conservative numerical scheme to simulate, in a few seconds, wave fields for spatial domains involving more than 1.5 billion grid cells. We present a comprehensive dimensional analysis reducing the number of material parameters needed for the numerical experiments from ten to four. Furthermore, the dimensional analysis emphasizes the key material parameters governing the physics of wave propagation in poroelastic media. We perform a dispersion analysis as function of dimensionless parameters leading to simple and transparent dispersion relations. We then benchmark our numerical solution against an analytical plane wave solution. Finally, we present several numerical modeling experiments, including a three-dimensional simulation of fluid injection into a poroelastic medium. We provide the Matlab, symbolic Maple, and GPU CUDA C routines to reproduce the main presented results. The high efficiency of our numerical implementation makes it readily usable to investigate three-dimensional and high-resolution scenarios of practical applications.ISSN:2169-9313ISSN:0148-0227ISSN:2169-935

Evaluating and applying contaminant transport models to groundwater systems

This thesis examines the use of random walk techniques to model the transport of a contaminant in groundwater. These techniques involve the distribution of a plume of contaminant into a discrete number of particles. These particles are then individually subjected to advective and dispersive forces and their progress though the model domain tracked over time. In general, pollution of groundwater is characterised by: 1) being difficult to detect, 2) being complicated and expensive to investigate and monitor, and 3) being expensive to clean up. These factors make the modelling of groundwater contamination an important area of investigation. This thesis presents the principles of groundwater flow and contaminant transport, along with their governing equations (namely, the groundwater flow equation and the advection dispersion equation); methods for the solution of the advection dispersion equation are discussed. These methods include analytic, finite difference and random walk techniques. Three random walk techniques are presented and compared with the analytic solutions for the following cases: 1) one dimensional dispersion 2) one dimensional advection dispersion 3) two dimensional dispersion 4) two dimensional advection dispersion Results of the comparisons have showed that all three random walk schemes presented produce computed results which are consistent with the analytic solution in each of the cases considered. Two finite difference schemes are presented and applied to the case of two dimensional advection dispersion. Through doing so, the problem of numerical diffusion has been highlighted. Random walk techniques have been applied to two physical problems. In the first case, a model has been developed to simulate the movement of a plume of chloride in an aquifer in the province of Saskatchewan, Canada. Results are compared for each of the three random walk schemes, namely time histories and breakthrough curves which plot the concentration of particles at a location in space over the time period modelled. All three random walk techniques have produced results that are very similar, with each modelling the movement of the plume acceptably. The second model uses data from The South Australian Department of Mines and Energy to simulate the movement of salt in the groundwater in a vine growing region near Naracoorte, South Australia. This model produces results which are consistent with available measured data.Thesis (M.Sc.)--University of Adelaide, Dept. of Applied Mathematics, 200

Numerical assessment of 3D macrodispersion in heterogeneous porous media

Hydrodynamic dispersion is a key controlling factor of solute transport in heterogeneous porous media that critically depends on dimensionality. It has been shown that the transverse macrodispersion (asymptotic dispersion transverse to the mean velocity direction) vanishes only in 2D and not in 3D. Using classical Gaussian correlated permeability fields with a lognormal distribution of variance σ²y, we determine numerically the longitudinal and transverse dispersivities as functions of heterogeneity and dimensionality. We show that the transverse macrodispersion steeply increases with σ²y underlying the essential role of flow lines braiding, a mechanism specific to 3D systems. The transverse macrodispersion remains however at least two orders of magnitude smaller than the longitudinal macrodispersion, which increases even more steeply with σ²y. At moderate to high levels of heterogeneity, the transverse dispersion also converges much faster to its asymptotic regime than do the longitudinal dispersion. Braiding cannot be thus taken as the sole mechanism responsible for the high longitudinal macrodispersions. It could be either supplemented or superseded by stronger velocity correlations in 3D than in 2D. This assumption is supported by the much larger longitudinal macrodispersions obtained in 3D than in 2D, up to a factor of 7 for σ²y = 7.56

Applications of coarse-graining approaches : volume averaging, macrotransport theory and scaling concepts

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1999.Includes bibliographical references.by Venkatraghavan Ganesan.Ph.D

Enhanced NAPL removal and mixing with engineered injection and extraction

Aquifer remediation with in-situ soil washing techniques and enhanced oil removal typically involves the injection of liquid solutions into the geological formation to displace and mobilize non-aqueous phase liquids (NAPLs). The efficiency of these systems is oftentimes low because the displacing fluid bypasses large quantities of NAPL due to the inherent complexity of a heterogeneous natural system. Here, engineered injection and extraction (EIE) generated by rotating periodic injection is proposed as a method to enhance NAPL removal and mixing. To evaluate the method, we perform two-phase flow simulations in multiple realizations of random permeability fields with different correlation structures and connectivity between injection and extraction wells embedded in a five-spot pattern. Results show that EIE can significantly improve removal efficiency and mixing depending on several controlling factors. The effects of EIE are more significant under unfavorable conditions, that is, when injection and extraction wells are well-connected through preferential channels, permeabilities are highly heterogeneous, and/or the mobility ratio between the wetting and the non-wetting fluids is larger than one. Removal efficiency reaches its maximum value when the Kubo number is close to one, that is, when the saturation front travels one range of the permeability field in an injection pulse. These effects can develop in just a few cycles. However, removal efficiency should undergo first an early stage with detrimental effects in order to maximize removal in the long term. EIE not only enhances NAPL removal and mixing but also reduces the uncertainty, making the system more reliable and less dependent on heterogeneity.This work was partially supported by the European Commission, through project MARSOLUT (grant H2020-MSCAITN-2018); by the Spanish Ministry of Economy and Competitiveness, through project MONOPOLIOS (RTI 2018-101990-B-100, MINECO/FEDER); and by the Catalan Agency for Management of University and Research Grants through FI 2017 (EMC/2199/2017).Peer ReviewedPostprint (published version

A multiscale analysis of flow and transport in the human placenta

The human placenta is characterised by a unique circulatory arrangement, with numerous villous trees containing fetal vessels immersed in maternal blood. Placental tissue therefore manifests a multiscale structure balancing microscopic delivery of nutrients and macroscopic flow. The aims of this study are to examine the interaction between these scales and to understand the influence of placental organisation on the effectiveness of nutrient uptake, which can be compromised in pathologies like pre-eclampsia and diabetes. We first systematically analyse solute transport by a unidirectional flow past an array of microscopic sinks, taking up a dissolved nutrient or gas, for both regular and random sink distributions. We classify distinct asymptotic transport regimes, each characterised by the dominance of advective, diffusive or uptake effects at the macroscale, and analyse a set of simplified model problems to assess the accuracy of homogenization approximations as a function of governing parameters (Peclet and Damkohler numbers) and the statistical properties of the sink distribution. The difference between the leading-order homogenization approximation and the exact solute distribution is determined by large spatial gradients at the scale of individual villi (depending on transport parameter values) and substantial fluctuations that can be correlated over lcngthscales comparable to the whole domain. In addition, we consider the nonlinear advective effects of solute-carriers, such as red blood cells carrying oxygen. Homogenization of the solute-carrier-facilitated transport introduces an effective Peclet number that depends on the slowly varying leading-order concentration, so that an asymptotic transport regime can be changed within the domain. At large Peclet and Damkohler numbers (typical for oxygen transport in the human placenta), nonlinear advection due to solute-carriers leads to a more uniform solute distribution than for a linear carrier-free transport, suggesting a "homogenizing" effect of red blood cells on placental oxygen transport. We then use image analysis and homogenization concepts to extract the effective transport properties (diffusivity and hydraulic resistance) from the microscopic images of histological sections of the normal human placenta. The resulting two-dimensional tensor quantities allow us to assess the anisotropy of placental tissue for solute transport. We also show how the pattern of villous centres of mass can be characterised using an integral correlation measure, and identify the minimum spatial scale over which the distribution of villous branches appears statistically homogeneous. Finally, we propose a mathematical model for maternal blood flow in a placental functional unit (a placentone), describing flow of maternal blood via Darcy's law and steady advective transport of a dissolved nutrient. An analytical method of images and computational integration along streamlines are employed to find flow and solute concentration distributions, which are illustrated for a range of governing system parameters. Predictions of the model agree with experimental radioangiographic studies of tracer dynamics in the intervillous space. The model supports the hypothesis that basal veins are located on the periphery of the placentone in order to optimise delivery of nutrients. We also explain the importance of dilatation of maternal spiral arteries and suggest the existence of an optimal volume fraction of villous tissue, which can both be involved in the placental dysfunction. Theoretical studies of this thesis thus constitute a step towards modelling-based diagnostics and treatment of placental disorders