Anomalous dispersion in correlated porous media: A coupled continuous time random walk approach

We study the causes of anomalous dispersion in Darcy-scale porous media characterized by spatially heterogeneous hydraulic properties. Spatial variability in hydraulic conductivity leads to spatial variability in the flow properties through Darcy's law and thus impacts on solute and particle transport. We consider purely advective transport in heterogeneity scenarios characterized by broad distributions of heterogeneity length scales and point values. Particle transport is characterized in terms of the stochastic properties of equidistantly sampled Lagrangian velocities, which are determined by the flow and conductivity statistics. The persistence length scales of flow and transport velocities are imprinted in the spatial disorder and reflect the distribution of heterogeneity length scales. Particle transitions over the velocity length scales are kinematically coupled with the transition time through velocity. We show that the average particle motion follows a coupled continuous time random walk (CTRW), which is fully parameterized by the distribution of flow velocities and the medium geometry in terms of the heterogeneity length scales. The coupled CTRW provides a systematic framework for the investigation of the origins of anomalous dispersion in terms of heterogeneity correlation and the distribution of heterogeneity point values. Broad distributions of heterogeneity point values and lengths scales may lead to very similar dispersion behaviors in terms of the spatial variance. Their mechanisms, however are very different, which manifests in the distributions of particle positions and arrival times, which plays a central role for the prediction of the fate of dissolved substances in heterogeneous natural and engineered porous materials

Debates—Stochastic subsurface hydrology from theory to practice: why stochastic modeling has not yet permeated into practitioners?

This is the peer reviewed version of the following article: [Sanchez-Vila, X., and D. Fernàndez-Garcia (2016), Debates—Stochastic subsurface hydrology from theory to practice: Why stochastic modeling has not yet permeated into practitioners?, Water Resour. Res., 52, 9246–9258, doi:10.1002/2016WR019302], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/2016WR019302/abstract. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-ArchivingWe address modern topics of stochastic hydrogeology from their potential relevance to real modeling efforts at the field scale. While the topics of stochastic hydrogeology and numerical modeling have become routine in hydrogeological studies, nondeterministic models have not yet permeated into practitioners. We point out a number of limitations of stochastic modeling when applied to real applications and comment on the reasons why stochastic models fail to become an attractive alternative for practitioners. We specifically separate issues corresponding to flow, conservative transport, and reactive transport. The different topics addressed are emphasis on process modeling, need for upscaling parameters and governing equations, relevance of properly accounting for detailed geological architecture in hydrogeological modeling, and specific challenges of reactive transport. We end up by concluding that the main responsible for nondeterministic models having not yet permeated in industry can be fully attributed to researchers in stochastic hydrogeology.Peer ReviewedPostprint (author's final draft

Pre-asymptotic corrections to fractional diffusion equations

The motion of contaminant particles through complex environments such as fractured rocks or porous sediments is often characterized by anomalous diffusion: the spread of the transported quantity is found to grow sublinearly in time due to the presence of obstacles which hinder particle migration. The asymptotic behavior of these systems is usually well described by fractional diffusion, which provides an elegant and unified framework for modeling anomalous transport. We show that pre-asymptotic corrections to fractional diffusion might become relevant, depending on the microscopic dynamics of the particles. To incorporate these effects, we derive a modified transport equation and validate its effectiveness by a Monte Carlo simulation.Comment: 6 pages, 3 figure

Effective pore-scale dispersion upscaling with a correlated continuous time random walk approach

International audienceWe investigate the upscaling of dispersion from a pore-scale analysis of Lagrangian velocities. A key challenge in the upscaling procedure is to relate the temporal evolution of spreading to the pore-scale velocity field properties. We test the hypothesis that one can represent Lagrangian velocities at the pore scale as a Markov process in space. The resulting effective transport model is a continuous time random walk (CTRW) characterized by a correlated random time increment, here denoted as correlated CTRW. We consider a simplified sinusoidal wavy channel model as well as a more complex heterogeneous pore space. For both systems, the predictions of the correlated CTRW model, with parameters defined from the velocity field properties (both distribution and correlation), are found to be in good agreement with results from direct pore-scale simulations over preasymptotic and asymptotic times. In this framework, the nontrivial dependence of dispersion on the pore boundary fluctuations is shown to be related to the competition between distribution and correlation effects. In particular, explicit inclusion of spatial velocity correlation in the effective CTRW model is found to be important to represent incomplete mixing in the pore throats

Non-Fickian dispersion in porous media : 2. Model validation from measurements at different scales

International audienceWe aim at testing and validating a mobile-immobile mass transfer model from a set of single-well injection withdrawal tracer tests in a heterogeneous porous aquifer. By varying the duration of the injection phase, different volumes of aquifer are investigated by the tracer. Hence, we focus the transport model validation not only on reproducing a single breakthrough curve (BTC) but also on the model's capacity to predict the amount of mixing as a function of the volume visited by the tracer. All the BTCs are strongly asymmetric, as expected when dispersion is controlled by diffusive mass transfers between the mobile water and the immobile water part of the porosity. However, the BTC cannot be modeled by a conventional mobile-immobile mass transfer model with a simple power law memory function. To account for that, we implement a continuous time random walk model in which the transition time distribution y (t), which is related to the excursion time probability of the tracer in the immobile domain, is a dual-slope power law distribution. The model best fits the BTC data set with a transitional regime controlled by y(t) t2 and an asymptotic regime characteristic of the conventional double-porosity model with y(t) t1.5 . This work emphasizes that high-resolution concentration measurement and multiple-scale tracer tests are required for assessing solute dispersion models in heterogeneous reservoirs and for subsequently obtaining reliable predictions

Mechanism and stochastic dynamics of transport in Darcy-scale heterogeneous porous media

Solute transport in heterogeneous porous media in general exhibits anomalous behaviors, in the sense that it is characterized by features that cannot be explained in terms of traditional models based on the advection-dispersion equation with constant effective coefficients. Signatures of anomalous transport are the non-linear temporal growth of the variance of solute concentration, non- Gaussian density profiles and heavy-tailed breakthrough curves. Understanding and predicting transport behavior in groundwater systems is crucial for several environmental and industrial applications, including groundwater management and risk assessment for nuclear waste repositories. The complexity of this task lies in the intrinsic multi-scale heterogeneity of geological formations and in the large amount of degrees of freedom. Hence, the predictive description of transport requires a process of upscaling that is based on measurable medium and flow attributes. The time domain random walk (TDRW) and continuous time random walk (CTRW) approaches provide suitable frameworks for transport upscaling. In this thesis, we identify different mechanisms that induce anomalous transport and we quantify their impact on transport attributes. We propose average transport models that can be parameterized in terms of flow and medium properties. Among the mechanisms that induce non-Fickian behaviors, a pivotal role is played by the heterogeneity of the flow field, which is directly linked to medium disorder. Due to its importance, the impact of advective heterogeneity is studied throughout the thesis, alongside with other mechanisms. First, we consider solute trapping due to physical or chemical heterogeneity, which we parameterize in terms of a constant trapping rate and a distribution of return times. We observe three distinct transport regimes that are linked to characteristic trapping time scales. At early times, transport is advection- controlled until particles start to get trapped. Then, the increasing distance between mobile and immobile particles gives rise to a superdiffusive regime which finally evolves towards a trapping-controlled regime. Second, we study transport in correlated porous media. We show that particle motion describes a coupled CTRW that is parameterized in terms of the distribution of flow velocity and length scales. We show that disorder and correlation may lead to similar behaviors in terms of displacement moments, but the difference between these mechanisms is manifest in the distributions of particle positions and arrival times. Next, we study the relationship between flow and transport properties and the impact of different injection conditions on transport. To this end, the relationship between Eulerian and Lagrangian velocities is investigated. Lagrangian statistics evolves to a steady-state that depends on the injection conditions. We study the velocity organization in Darcy flows and we develop a CTRW model for transport that is parameterized in terms of flow and medium attributes only. This CTRW accounts for non-stationarity through Markovian velocity models. We study the impact of advective heterogeneity by considering different disorder scenarios. Finally, we quantify the impact of diffusion in layered and fibrous heterogeneous media by considering two disorder scenarios characterized by quenched random velocities and quenched retardation properties, respectively. These mechanisms lead to different, dimension-dependent disorder samplings that give rise to dual transport processes in space and time. Specifically, transport describes correlated Lévy flights in the random velocity model and correlated CTRWs in the random retardation model.El transporte de solutos en medios heterogéneos porosos exhibe comportamientos anómalos, que se caracteriza por rasgos que no pueden ser explicados en términos de modelos tradicionales basados en la ecuación de advección-dispersión con coeficientes efectivos constantes. Las características del transporte anómalo son el crecimiento temporal no lineal de la varianza de la concentración de soluto, los perfiles de densidad no gausianos y la curvas de llegada con colas pronunciadas. Entender y predecir el comportamiento del transporte en hidrología subterránea es crucial para aplicaciones ambientales e industriales, como la gestión de aguas subterráneas o la evaluación de riesgos en repositorios de residuos nucleares. La complejidad de estas tareas se debe a la heterogeneidad intrínseca en múltiples escalas de las formaciones geológicas y del gran número de grados de libertad. Por lo tanto, una descripción predictiva del transporte requiere un proceso de upscaling basado en propiedades medibles del medio y el flujo para el que los modelos time domain random walk (TDRW) y continuous time random walk (CTRW) proporcionan un marco adecuado. En esta tesis, se identifican los mecanismos que inducen transporte anómalo y se cuantifica su impacto en el transporte. Se proponen modelos de transporte parametrizados en términos de las propiedades del medio y el flujo. Entre los mecanismos que inducen comportamientos no fickianos, la heterogeneidad del flujo, relacionada con el desorden del medio, desempeña un papel fundamental. Por lo tanto, su impacto se estudia junto con los de otros mecanismos a lo largo de toda la tesis. Primero, se considera el atrapamiento de soluto debido a heterogeneidades físicas o químicas parametrizadas en términos de un ratio de atrapamiento constante y una distribución de tiempos de retorno. Se observan tres regímenes de transporte relacionados con las escalas temporales características del atrapamiento. A tiempos pequeños, el transporte está controlado por la advección hasta que las partículas comienzan a ser atrapadas. A continuación el incremento de la distancia entre partículas móviles e inmóviles origina un régimen superdifusivo que finalmente evoluciona hacia un régimen controlado por el atrapamiento. Después, se estudia el transporte en medios correlacionados en los que el movimiento de las partículas es descrito por un CTRW acoplado parametrizado según la distribución de velocidades del flujo y de las escalas espaciales. El desorden y la correlación generan comportamientos similares en los momentos del desplazamiento de las partículas, pero diferentes en las distribución de posiciones y de tiempos de llegada. A continuación, se estudia la relación entre flujo y transporte bajo diferentes condiciones de inyección, a través de las velocidades eulerianas y lagrangianas. La estadística lagrangiana evoluciona hacia un estado estacionario que depende de los modos de inyección. Se estudia la organización de las velocidades en flujos de Darcy y se desarrolla un CTRW para el transporte que se parametriza solo en términos de las propiedades del medio y del flujo. Este CTRW considera la no estacionariedad a través de modelos de velocidad markovianos. El impacto de la heterogeneidad advectiva se estudia considerando diferentes escenarios de desorden. Finalmente, se cuantifica el impacto de la difusión en medios heterogéneos estratificados considerando dos escenarios de heterogeneidad que se caracterizan respectivamente por velocidades y propiedades de retraso aleatorias. Estos mecanismos originan diferentes muestreos del desorden que generan procesos de transportes duales en tiempo y espacio. El transporte describe un Lévy flight correlacionado en el modelo de velocidades aleatorias y un CTRW correlacionado en el modelo de retraso

Mechanism and stochastic dynamics of transport in Darcy-scale heterogeneous porous media

Solute transport in heterogeneous porous media in general exhibits anomalous behaviors, in the sense that it is characterized by features that cannot be explained in terms of traditional models based on the advection-dispersion equation with constant effective coefficients. Signatures of anomalous transport are the non-linear temporal growth of the variance of solute concentration, non- Gaussian density profiles and heavy-tailed breakthrough curves. Understanding and predicting transport behavior in groundwater systems is crucial for several environmental and industrial applications, including groundwater management and risk assessment for nuclear waste repositories. The complexity of this task lies in the intrinsic multi-scale heterogeneity of geological formations and in the large amount of degrees of freedom. Hence, the predictive description of transport requires a process of upscaling that is based on measurable medium and flow attributes. The time domain random walk (TDRW) and continuous time random walk (CTRW) approaches provide suitable frameworks for transport upscaling. In this thesis, we identify different mechanisms that induce anomalous transport and we quantify their impact on transport attributes. We propose average transport models that can be parameterized in terms of flow and medium properties. Among the mechanisms that induce non-Fickian behaviors, a pivotal role is played by the heterogeneity of the flow field, which is directly linked to medium disorder. Due to its importance, the impact of advective heterogeneity is studied throughout the thesis, alongside with other mechanisms. First, we consider solute trapping due to physical or chemical heterogeneity, which we parameterize in terms of a constant trapping rate and a distribution of return times. We observe three distinct transport regimes that are linked to characteristic trapping time scales. At early times, transport is advection- controlled until particles start to get trapped. Then, the increasing distance between mobile and immobile particles gives rise to a superdiffusive regime which finally evolves towards a trapping-controlled regime. Second, we study transport in correlated porous media. We show that particle motion describes a coupled CTRW that is parameterized in terms of the distribution of flow velocity and length scales. We show that disorder and correlation may lead to similar behaviors in terms of displacement moments, but the difference between these mechanisms is manifest in the distributions of particle positions and arrival times. Next, we study the relationship between flow and transport properties and the impact of different injection conditions on transport. To this end, the relationship between Eulerian and Lagrangian velocities is investigated. Lagrangian statistics evolves to a steady-state that depends on the injection conditions. We study the velocity organization in Darcy flows and we develop a CTRW model for transport that is parameterized in terms of flow and medium attributes only. This CTRW accounts for non-stationarity through Markovian velocity models. We study the impact of advective heterogeneity by considering different disorder scenarios. Finally, we quantify the impact of diffusion in layered and fibrous heterogeneous media by considering two disorder scenarios characterized by quenched random velocities and quenched retardation properties, respectively. These mechanisms lead to different, dimension-dependent disorder samplings that give rise to dual transport processes in space and time. Specifically, transport describes correlated Lévy flights in the random velocity model and correlated CTRWs in the random retardation model.El transporte de solutos en medios heterogéneos porosos exhibe comportamientos anómalos, que se caracteriza por rasgos que no pueden ser explicados en términos de modelos tradicionales basados en la ecuación de advección-dispersión con coeficientes efectivos constantes. Las características del transporte anómalo son el crecimiento temporal no lineal de la varianza de la concentración de soluto, los perfiles de densidad no gausianos y la curvas de llegada con colas pronunciadas. Entender y predecir el comportamiento del transporte en hidrología subterránea es crucial para aplicaciones ambientales e industriales, como la gestión de aguas subterráneas o la evaluación de riesgos en repositorios de residuos nucleares. La complejidad de estas tareas se debe a la heterogeneidad intrínseca en múltiples escalas de las formaciones geológicas y del gran número de grados de libertad. Por lo tanto, una descripción predictiva del transporte requiere un proceso de upscaling basado en propiedades medibles del medio y el flujo para el que los modelos time domain random walk (TDRW) y continuous time random walk (CTRW) proporcionan un marco adecuado. En esta tesis, se identifican los mecanismos que inducen transporte anómalo y se cuantifica su impacto en el transporte. Se proponen modelos de transporte parametrizados en términos de las propiedades del medio y el flujo. Entre los mecanismos que inducen comportamientos no fickianos, la heterogeneidad del flujo, relacionada con el desorden del medio, desempeña un papel fundamental. Por lo tanto, su impacto se estudia junto con los de otros mecanismos a lo largo de toda la tesis. Primero, se considera el atrapamiento de soluto debido a heterogeneidades físicas o químicas parametrizadas en términos de un ratio de atrapamiento constante y una distribución de tiempos de retorno. Se observan tres regímenes de transporte relacionados con las escalas temporales características del atrapamiento. A tiempos pequeños, el transporte está controlado por la advección hasta que las partículas comienzan a ser atrapadas. A continuación el incremento de la distancia entre partículas móviles e inmóviles origina un régimen superdifusivo que finalmente evoluciona hacia un régimen controlado por el atrapamiento. Después, se estudia el transporte en medios correlacionados en los que el movimiento de las partículas es descrito por un CTRW acoplado parametrizado según la distribución de velocidades del flujo y de las escalas espaciales. El desorden y la correlación generan comportamientos similares en los momentos del desplazamiento de las partículas, pero diferentes en las distribución de posiciones y de tiempos de llegada. A continuación, se estudia la relación entre flujo y transporte bajo diferentes condiciones de inyección, a través de las velocidades eulerianas y lagrangianas. La estadística lagrangiana evoluciona hacia un estado estacionario que depende de los modos de inyección. Se estudia la organización de las velocidades en flujos de Darcy y se desarrolla un CTRW para el transporte que se parametriza solo en términos de las propiedades del medio y del flujo. Este CTRW considera la no estacionariedad a través de modelos de velocidad markovianos. El impacto de la heterogeneidad advectiva se estudia considerando diferentes escenarios de desorden. Finalmente, se cuantifica el impacto de la difusión en medios heterogéneos estratificados considerando dos escenarios de heterogeneidad que se caracterizan respectivamente por velocidades y propiedades de retraso aleatorias. Estos mecanismos originan diferentes muestreos del desorden que generan procesos de transportes duales en tiempo y espacio. El transporte describe un Lévy flight correlacionado en el modelo de velocidades aleatorias y un CTRW correlacionado en el modelo de retraso.Postprint (published version

Scale Up Reactive Flow in Heterogeneous Porous Media Using Continuous Time Random Walk Approach

Reservoir heterogeneities strongly affect the fluid flow in porous media. The behavior of transport and reaction of fluid varies at different scales, leading to the discrepancies between laboratory experiments and field observations. The reactive processes in porous media may alter reservoir properties with different spatial and temporal scale, varying future transport and reaction behaviors. This thesis provides an efficient probabilistic approach to scale up coupled transport and reaction processes in heterogeneous porous media to field scale based on laboratory-scale information. The continuous time random walk (CTRW) is a probabilistic framework which is always incorporating with particle tracking (PT) approach to model solute transport in heterogeneous porous medium. In CTRW-PT approach, the motion of solute particles is described as combination of random independent spatial and temporal increments in each walk step. The spatial and temporal increments, or normally called as transition distance and time, are chosen from a joint space-time probability density function by a stochastic process. The CTRW-PT approach simulates reactive fluid transport as non-reactive fluid. The modelling of reaction and dissolution is followed by in each time step, updating the change of porous medium and its effect on following transport and reaction. The characteristic probability density function (pdf) is used to simulate the transport of fluid. Adjusting the ensemble parameter β and t_2 accounts for the effects of heterogeneity which leads to anomalous flow behavior: the fluid propagates along the “preferential” pathways with short transition times and “trapped” in some zones with long transition times. It mimics the macroscopic behavior that fluid has the tendency to propagate in high-permeability zones and bypass the low-permeability zones. Simulations of non-reactive tracer flow and nanofluid flow under various conditions are performed at core-scale to obtain the key parameters in characteristic pdf by matching the experimental results. The effect of reactive process, heterogeneity and flow rate on flow behavior is analyzed. The CTRW-PT simulation captures the characters of anomalous behavior of delayed breakthrough. The model is run at larger scale as reservoir properties are scaled up properly. The core-scale simulation based on the characteristic pdf agrees with the experimental results. The large-scale simulation is implemented by using the characteristic pdf to describe flow behaviors in a large-scale domain. It is shown that CTRW-PT approach is more effective in large-scale modeling than solving advection-diffusion-reaction equation (ADRE) by finite difference method (FDM). The simulation results at large scale show that the flow response is spatial-dependent. Compared to solving traditional ADRE, the utilization of CTRW-PT approach to model reactive fluid flow captures the characters of anomalous flow behavior, especially in highly heterogeneous porous media. By the probabilistic framework and stochastic process, this approach is more computational-efficient for scaling up lab-scale results to larger scale. It can consolidate the lab-scale understanding with field prediction to optimize the field treatment design

Anomalous transport in disordered fracture networks: Spatial Markov model for dispersion with variable injection modes

We investigate tracer transport on random discrete fracture networks that are characterized by the statistics of the fracture geometry and hydraulic conductivity. While it is well known that tracer transport through fractured media can be anomalous and particle injection modes can have major impact on dispersion, the incorporation of injection modes into effective transport modeling has remained an open issue. The fundamental reason behind this challenge is that-even if the Eulerian fluid velocity is steady-the Lagrangian velocity distribution experienced by tracer particles evolves with time from its initial distribution, which is dictated by the injection mode, to a stationary velocity distribution. We quantify this evolution by a Markov model for particle velocities that are equidistantly sampled along trajectories. This stochastic approach allows for the systematic incorporation of the initial velocity distribution and quantifies the interplay between velocity distribution and spatial and temporal correlation. The proposed spatial Markov model is characterized by the initial velocity distribution, which is determined by the particle injection mode, the stationary Lagrangian velocity distribution, which is derived from the Eulerian velocity distribution, and the spatial velocity correlation length, which is related to the characteristic fracture length. This effective model leads to a time-domain random walk for the evolution of particle positions and velocities, whose joint distribution follows a Boltzmann equation. Finally, we demonstrate that the proposed model can successfully predict anomalous transport through discrete fracture networks with different levels of heterogeneity and arbitrary tracer injection modes. © 2017 Elsevier Ltd.PKK and SL acknowledge a grant (16AWMP- B066761-04) from the AWMP Program funded by the Ministry of Land, Infrastructure and Transport of the Korean government and the support from Future Research Program (2E27030) funded by the Korea Institute of Science and Technology (KIST). PKK and RJ acknowledge a MISTI Global Seed Funds award. MD acknowledges the support of the European Research Council (ERC) through the project MHetScale (617511). TLB acknowledges the support of European Research Council (ERC) through the project Re- activeFronts (648377). RJ acknowledges the support of the US Department of Energy through a DOE Early Career Award (grant DE-SC0009286). The data to reproduce the work can be obtained from the corresponding author.N