[Show abstract][Hide abstract] ABSTRACT: We investigate transport on regular fracture networks that are characterized by heterogeneity in hydraulic conductivity. We discuss the impact of conductivity heterogeneity and mixing within fracture intersections on particle spreading. We show the emergence of non-Fickian transport due to the interplay between the network conductivity heterogeneity and the degree of mixing at nodes. Specifically, lack of mixing at fracture intersections leads to subdiffusive scaling of transverse spreading but has negligible impact on longitudinal spreading. An increase in network conductivity heterogeneity enhances both longitudinal and transverse spreading and leads to non-Fickian transport in longitudinal direction. Based on the observed Lagrangian velocity statistics, we develop an effective stochastic model that incorporates the interplay between Lagrangian velocity correlation and velocity distribution. The model is parameterized with a few physical parameters and is able to capture the full particle transition dynamics.
Physical Review E 09/2015; 92(2-1). DOI:10.1103/PhysRevE.92.022148 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper derives and analyzes continuous time random walk (CTRW) models in radial flow geometries for the quantification of non-local solute transport induced by heterogeneous flow distributions and by mobile-immobile mass transfer processes. To this end we derive a general CTRW framework in radial coordinates starting from the random walk equations for radial particle positions and times. The particle density, or solute concentration is governed by a non-local radial advection-dispersion equation (ADE). Unlike in CTRWs for uniform flow scenarios, particle transition times here depend on the radial particle position, which renders the CTRW non-stationary. As a consequence, the memory kernel characterizing the non-local ADE, is radially dependent. Based on this general formulation, we derive radial CTRW implementations that (i) emulate non-local radial transport due to heterogeneous advection, (ii) model multirate mass transfer (MRMT) between mobile and immobile continua, and (iii) quantify both heterogeneous advection in a mobile region and mass transfer between mobile and immobile regions. The expected solute breakthrough behavior is studied using numerical random walk particle tracking simulations. This behavior is analyzed by explicit analytical expressions for the asymptotic solute breakthrough curves. We observe clear power-law tails of the solute breakthrough for broad (power-law) distributions of particle transit times (heterogeneous advection) and particle trapping times (MRMT model). The combined model displays two distinct time regimes. An intermediate regime, in which the solute breakthrough is dominated by the particle transit times in the mobile zones, and a late time regime that is governed by the distribution of particle trapping times in immobile zones.
Advances in Water Resources 04/2015; 82. DOI:10.1016/j.advwatres.2015.04.005 · 3.42 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Flow and transport through fractured geologic media often leads to anomalous (non-Fickian) transport behavior, the origin of which remains a matter of debate: whether it arises from variability in fracture permeability (velocity distribution), connectedness in the flow paths through fractures (velocity correlation), or interaction between fractures and matrix. Here we show that this uncertainty of distribution- vs. correlation-controlled transport can be resolved by combining convergent and push-pull tracer tests because flow reversibility is strongly dependent on velocity correlation, whereas late-time scaling of breakthrough curves is mainly controlled by velocity distribution. We build on this insight, and propose a Lagrangian statistical model that takes the form of a continuous time random walk (CTRW) with correlated particle velocities. In this framework, velocity distribution and velocity correlation are quantified by a Markov process of particle transition times that is characterized by a distribution function and a transition probability. Our transport model accurately captures the anomalous behavior in the breakthrough curves for both push-pull and convergent flow geometries, with the same set of parameters. Thus, the proposed correlated CTRW modeling approach provides a simple yet powerful framework for characterizing the impact of velocity distribution and correlation on transport in fractured media. This article is protected by copyright. All rights reserved.
[Show abstract][Hide abstract] ABSTRACT: We study the nature of non-Fickian particle transport in 3D porous media by simulating fluid flow in the intricate pore space of real rock. We solve the full Navier-Stokes equations at the same resolution as the 3D micro-CT image of the rock sample, and simulate particle transport along the streamlines of the velocity field. We find that transport at the pore scale is markedly anomalous: longitudinal spreading is superdiffusive, while transverse spreading is subdiffusive. We demonstrate that this anomalous behavior originates from the intermittent structure of the velocity field at the pore scale, which in turn emanates from the interplay between velocity heterogeneity and velocity correlation. Finally, we propose a continuous time random walk model that honors this intermittent structure at the pore scale and captures the anomalous 3D transport behavior at the macroscale.
[Show abstract][Hide abstract] ABSTRACT: Traditionally, seismic interpretation is performed without any account of the flow behavior. Here, we present a methodology to characterize fractured geologic media by integrating flow and seismic data. The key element of the proposed approach is the identification of the intimate relation between acoustic and flow responses of a fractured reservoir through the fracture compliance. By means of synthetic models, we show that: (1) owing to the strong (but highly uncertain) dependence of fracture permeability on fracture compliance, the modeled flow response in a fractured reservoir is highly sensitive to the geophysical interpretation; and (2) by incorporating flow data (well pressures and production curves) into the inversion workflow, we can simultaneously reduce the error in the seismic interpretation and improve predictions of the reservoir flow dynamics.
SEG Technical Program Expanded Abstracts 2013; 09/2013
[Show abstract][Hide abstract] ABSTRACT: Ceramic pot filters (CPFs) are a promising low-cost option for household water treatment, providing a barrier of protection against microbiological contaminants for households with or without reliable piped water supplies. However, as an open-source design, performance of CPFs is not standard across manufacturers and at times can be suboptimal. Furthermore, no scientific study has provided a holistic framework for optimizing filter performance. The goal of this paper is to provide CPF manufacturers with tools to increase their ability to reach performance objectives for flow rate, bacteria removal and strength. This goal is achieved by experimentally determining relationships between performance and three manufacturing parameters: percentage rice husk, rice husk size and wall thickness. These results are translated into design and manufacturing recommendations, which are as follows: 1) tightly control rice husk size to maintain consistent flow rates; 2) maximize wall thickness within the constraints in order to improve bacteria removal; 3) seek alternative methods of increasing bacteria removal if removal levels greater than 2LRV are needed. To go further and provide a more quantitative and universal optimization framework, we then use the identified functional relationships between the manufacturing parameters and filter performance to formulate a single-criterion optimization. This framework enables manufacturers to determine an ideal combination of manufacturing parameters based on the specific situation of each manufacturing site. The systematic approach to CPF design presented in this paper can be further extended to address additional manufacturing parameters and aspects of filter performance to further improve the CPF design. This work has huge potential to better serve the many people around the world who lack safe drinking water.
Global Humanitarian Technology Conference (GHTC), 2013 IEEE; 01/2013
[Show abstract][Hide abstract] ABSTRACT: Anomalous transport, understood as the nonlinear scaling with time of
the mean square displacement of transported particles, is a pervasive
phenomenon during transport through porous and fractured geologic media.
A common signature of anomalous transport in the field is the late-time
power law tailing in breakthrough curves (BTCs). Several conceptual
models, including multirate mass transfer, continuous time random walk
and stream tube models, have been proposed to capture this effective
macroscopic behavior. In general, however, different conceptual models
often produce equally good fits to a single BTC, raising questions about
the predictability power of these effective models. Here we address the
uniqueness of the tracer test interpretation by analyzing BTCs from
various flow configurations, including dipole, convergent and push-pull
tests. We conducted field tracer tests in a saturated fractured granite
formation close to Ploemeur, France. Two boreholes, B1 (83 m deep) and
B2 (100 m deep), which are 6 m apart, were used. A double-packer system
was designed and installed in B1 at 50 m depth to inject tracer into a
single fracture. We measured BTCs under different flow configurations to
demonstrate that the observed tailing is configuration-dependent.
Specifically, the tailing disappears in a push-pull test (Figure 1).
This result indicates that for this fractured granite, the BTC tailing
is controlled by heterogeneous advection and not matrix diffusion. To
explain the change in tailing behavior for different flow
configurations, we developed a lattice network model with heterogeneous
conductivity distribution. The model assigns random conductivities to
the fractures and solves the Darcy equation for an incompressible fluid,
enforcing mass conservation at fracture intersections. We use this model
to investigate whether BTC tailing can be explained by the spatial
distribution of preferential flow paths and stagnation zones, which are
in turn controlled by the conductivity variance and correlation length.
[Show abstract][Hide abstract] ABSTRACT: Flow through lattice networks with quenched disorder exhibits a strong correlation in the velocity field, even if the link transmissivities are uncorrelated. This feature, which is a consequence of the divergence-free constraint, induces anomalous transport of passive particles carried by the flow. We propose a Lagrangian statistical model that takes the form of a continuous time random walk with correlated velocities derived from a genuinely multidimensional Markov process in space. The model captures the anomalous (non-Fickian) longitudinal and transverse spreading, and the tail of the mean first-passage time observed in the Monte Carlo simulations of particle transport. We show that reproducing these fundamental aspects of transport in disordered systems requires honoring the correlation in the Lagrangian velocity.
[Show abstract][Hide abstract] ABSTRACT: We study stochastic transport through a lattice network with quenched disorder and evaluate the limits of predictability of the transport behavior across realizations of spatial heterogeneity. Within a Lagrangian framework, we perform coarse graining, noise averaging, and ensemble averaging, to obtain an effective transport model for the average particle density and its fluctuations between realizations. We show that the average particle density is described exactly by a continuous time random walk (CTRW), and the particle density variance is quantified by a novel two-particle CTRW.
[Show abstract][Hide abstract] ABSTRACT: We study stochastic transport through a lattice fracture network with quenched disorder, and evaluate the limits of predictability of the transport behavior across realizations of spatial heterogeneity. As shown in the figure, We consider a two-dimensional regular fracture network model characterized by a constant fracture length, fracture aperture and fracture angle. We assign independent and identically distributed random particle velocities to each link. This implies a random uncorrelated particle transport velocity field. Different values of particle velocity are assumed to be the result of microscale processes, such as different conductance or adsorption rate. Within a Lagrangian framework, we perform coarse graining, noise averaging, and ensemble averaging, to obtain an effective transport model for the average particle density and its fluctuations between realizations. We show that the average particle density is described exactly by a continuous time random walk (CTRW), and the particle density variance is quantified by a novel two-particle CTRW. An important question regarding predictability of transport is how the variance --- especially, the variance where the particle density is maximum --- evolves in time. Simulations using the two-particle CTRW can answer to this question. We generalize our model for the correlated velocity field. This second model assigns random transmissivities to the fractures and solves the Darcy equation for an incompressible fluid, enforcing mass conservation at fracture intersections. The latter yields a correlated random flow through the fracture system. To incorporate the impact of heterogeneity and the velocity correlation on effective transport, we study Lagrangian velocity transitions in space and time. We demonstrate that Lagrangian velocities are Markov process in space but not in time. Using the property of spatial Markov process, we derive a CTRW in phase space characterized by a correlated velocity increment. We introduce the concept of 4D transition matrix which has information of both directionality and velocity transitions to capture full 2D particle density distribution. We obtain good representations for both transverse and longitudinal particle density. (a) Schematic of the lattice fracture network considered here, with two sets of links with orientation {-alpha, +alpha} with respect to the x-axis, and lattice spacing l = 1. (b) Representation of transport through the network, from particles released at the origin at t = 0. Shown is the particle density (represented by circle size) at t = 30 for a single realization with beta = 1.5.
[Show abstract][Hide abstract] ABSTRACT: We study transport in a random fracture network using a stochastic modeling approach. We consider a two-dimensional regular fracture network model characterized by a constant fracture length and fracture angle. The transport velocity in the fractures is a random variable. We consider two models. The first one is characterized by a constant fracture aperture and thus constant flow velocity and a retardation factor that is assigned randomly for each fracture. The latter implies a random uncorrelated particle velocity. In each realization, the spatial distribution of retardation values is fixed. The second model assigns random transmissivities to the fractures and solves the Darcy equation for an incompressible fluid, enforcing mass conservation at fracture intersections. The latter yields a correlated random flow through the fracture system. Within a Lagrangian transport framework, we derive effective equations for particle transport by stochastic averaging and compare the obtained mean behavior with direct numerical simulations of particle transport in single medium realizations and the corresponding ensemble average. We determine analytically and numerically the concentration variances in both fracture network models and thus probe the self-averaging behavior of concentration. The first model (uncorrelated transport velocities) describes effectively an uncoupled continuous time random walk, which is obtained by coarse graining and ensemble averaging of the local scale Langevin equations. The second model (correlated flow velocity) describes a continuous time random walk characterized by a transition matrix for the (correlated) random time increment.
[Show abstract][Hide abstract] ABSTRACT: We study transport in a lattice fracture network with uncorrelated velocity fields using a stochastic modeling approach. We consider a two-dimensional regular fracture network model characterized by a constant fracture length and fracture angle. The transport velocity in the fractures is a random variable. Here, we present an exact derivation of effective equations for the average particle density and concentration variance from the microscopic disorder model. Within a Lagrangian transport framework, we derive effective equations for particle transport by coarse graining, noise averaging and ensemble averaging of the local scale Langevin equations. We rigorously show that average particle density describes effectively an uncoupled continu-ous time random walk (CTRW) and the concentration variance is quantified by a two particle CTRW. The obtained mean behavior and concentration variance are compared to direct nu-merical simulations of particle transport in single medium realizations and the corresponding ensemble averages.