Probability Density Function Modeling of Multi-Phase Flow in Porous Media with Density-Driven Gravity Currents

A probability density function (PDF) based approach is employed to model multi-phase flow with interfacial mass transfer (dissolution) in porous media. The joint flow statistics is represented by a mass density function (MDF), which is transported in the physical and probability spaces via Fokker-Planck equation. This MDF-equation requires Lagrangian evolutions of the random flow variables; these evolutions are stochastic processes honoring the micro-scale flow physics. To demonstrate the concept, we consider an example of immiscible two-phase flow with the non-equilibrium dissolution of single component from one phase into the other-a model for solubility trapping during CO2 storage in brine aquifer. Since CO2-rich brine is denser than pure brine, density-driven countercurrent flow is set up in the brine phase. The stochastic models mimicking the physics of countercurrent flow lead to a modeled MDF-equation, which is solved using our recently developed stochastic particle method for multi-phase flow (Tyagi etal. J Comput Phys 227:6696-6714, 2008). In addition, we derive Eulerian equations for stochastic moments (mean, variance, etc.) and show that unlike the MDF-equation the system of moment equations is not closed. In classical Darcy formulation, for example, the mean concentration equation is closed by neglecting variance. However, with several one- and two-dimensional simulations, it is demonstrated that the PDF and Darcy modeling approaches give significantly different results. While the PDF-approach properly accounts for the long correlation length scales and the concentration variance in density-driven countercurrent flow, the same phenomenon cannot be captured accurately with a standard Darcy mode

Application of upscaling methods for fluid flow and mass transport in multi-scale heterogeneous media : A critical review

Physical and biogeochemical heterogeneity dramatically impacts fluid flow and reactive solute transport behaviors in geological formations across scales. From micro pores to regional reservoirs, upscaling has been proven to be a valid approach to estimate large-scale parameters by using data measured at small scales. Upscaling has considerable practical importance in oil and gas production, energy storage, carbon geologic sequestration, contamination remediation, and nuclear waste disposal. This review covers, in a comprehensive manner, the upscaling approaches available in the literature and their applications on various processes, such as advection, dispersion, matrix diffusion, sorption, and chemical reactions. We enclose newly developed approaches and distinguish two main categories of upscaling methodologies, deterministic and stochastic. Volume averaging, one of the deterministic methods, has the advantage of upscaling different kinds of parameters and wide applications by requiring only a few assumptions with improved formulations. Stochastic analytical methods have been extensively developed but have limited impacts in practice due to their requirement for global statistical assumptions. With rapid improvements in computing power, numerical solutions have become more popular for upscaling. In order to tackle complex fluid flow and transport problems, the working principles and limitations of these methods are emphasized. Still, a large gap exists between the approach algorithms and real-world applications. To bridge the gap, an integrated upscaling framework is needed to incorporate in the current upscaling algorithms, uncertainty quantification techniques, data sciences, and artificial intelligence to acquire laboratory and field-scale measurements and validate the upscaled models and parameters with multi-scale observations in future geo-energy research.© 2021 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)This work was jointly supported by the National Key Research and Development Program of China (No. 2018YFC1800900 ), National Natural Science Foundation of China (No: 41972249 , 41772253 , 51774136 ), the Program for Jilin University (JLU) Science and Technology Innovative Research Team (No. 2019TD-35 ), Graduate Innovation Fund of Jilin University (No: 101832020CX240 ), Natural Science Foundation of Hebei Province of China ( D2017508099 ), and the Program of Education Department of Hebei Province ( QN219320 ). Additional funding was provided by the Engineering Research Center of Geothermal Resources Development Technology and Equipment , Ministry of Education, China.fi=vertaisarvioitu|en=peerReviewed

A proof of convergence of a finite volume scheme for modified steady Richards’ equation describing transport processes in the pressing section of a paper machine

A number of water flow problems in porous media are modelled by Richards’ equation [1]. There exist a lot of different applications of this model. We are concerned with the simulation of the pressing section of a paper machine. This part of the industrial process provides the dewatering of the paper layer by the use of clothings, i.e. press felts, which absorb the water during pressing [2]. A system of nips are formed in the simplest case by rolls, which increase sheet dryness by pressing against each other (see Figure 1). A lot of theoretical studies were done for Richards’ equation (see [3], [4] and references therein). Most articles consider the case of x-independent coefficients. This simplifies the system considerably since, after Kirchhoff’s transformation of the problem, the elliptic operator becomes linear. In our case this condition is not satisfied and we have to consider nonlinear operator of second order. Moreover, all these articles are concerned with the nonstationary problem, while we are interested in the stationary case. Due to complexity of the physical process our problem has a specific feature. An additional convective term appears in our model because the porous media moves with the constant velocity through the pressing rolls. This term is zero in immobile porous media. We are not aware of papers, which deal with such kind of modified steady Richards’ problem. The goal of this paper is to obtain the stability results, to show the existence of a solution to the discrete problem, to prove the convergence of the approximate solution to the weak solution of the modified steady Richards’ equation, which describes the transport processes in the pressing section. In Section 2 we present the model which we consider. In Section 3 a numerical scheme obtained by the finite volume method is given. The main part of this paper is theoretical studies, which are given in Section 4. Section 5 presents a numerical experiment. The conclusion of this work is given in Section 6

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

The XDEM Multi-physics and Multi-scale Simulation Technology: Review on DEM-CFD Coupling, Methodology and Engineering Applications

The XDEM multi-physics and multi-scale simulation platform roots in the Ex- tended Discrete Element Method (XDEM) and is being developed at the In- stitute of Computational Engineering at the University of Luxembourg. The platform is an advanced multi- physics simulation technology that combines flexibility and versatility to establish the next generation of multi-physics and multi-scale simulation tools. For this purpose the simulation framework relies on coupling various predictive tools based on both an Eulerian and Lagrangian approach. Eulerian approaches represent the wide field of continuum models while the Lagrange approach is perfectly suited to characterise discrete phases. Thus, continuum models include classical simulation tools such as Computa- tional Fluid Dynamics (CFD) or Finite Element Analysis (FEA) while an ex- tended configuration of the classical Discrete Element Method (DEM) addresses the discrete e.g. particulate phase. Apart from predicting the trajectories of individual particles, XDEM extends the application to estimating the thermo- dynamic state of each particle by advanced and optimised algorithms. The thermodynamic state may include temperature and species distributions due to chemical reaction and external heat sources. Hence, coupling these extended features with either CFD or FEA opens up a wide range of applications as diverse as pharmaceutical industry e.g. drug production, agriculture food and processing industry, mining, construction and agricultural machinery, metals manufacturing, energy production and systems biology

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

Development and implementation of the generalized continuum model for transport in porous media

Fluid flow phenomena in porous media have always attracted a lot of attention of scientists and engineers. Attempts to quantify the average transport in homogeneous media with a simple partial differential equation with constant coefficients disclosed significant inconsistencies comparing to experiments. Modern numerical simulations of porous networks confirmed that those inconsistencies are systematic and not caused by the observation error. The error appeared as a result of the, so called, anomalous or non-Fickian transport, which was in contrast to the normal regime, described by the Fick’s laws. The problem has been addressed through the introduction of more complex and substantial models to describe the phenomena. Although, these new approaches have resolved the problem of quantification, they have raised another question for researchers and engineers, how to choose the most suitable approach and, if it is possible, to parametrize the modeling choice at all. The models general lack of physical consistency makes it difficult to distinguish the model parameters. This leaves judging of suitability to the general accuracy of quantification only, which is often not the most important criterion. In other words, the model parameters are typically estimated by fitting the model to the experimental data, and are often not related to the real properties of the medium. Therefore, a model is often chosen a priory, based only on the experience of the researcher. In this work, we address the problem of model selection by introducing a new model: the Generalized Continuum Transport model. This model transforms into existing models at certain limits and, therefore, constrains the modeling choice through the introduction of the parameter space. It is shown that the Generalized Continuum Transport model limits to the advection-dispersion equation, the Continuous Time Random Walk, the Multi-Rate Mass Transfer and the Multiple-Porosity models, when corresponding configurations of the parameter space are applied. The model’s accuracy is studied by quantifying the breakthrough curves obtained from a fine scale porous network modeldemonstrating significant appearance of anomalous transport phenomena. The results show that the error of quantification is smaller than the error of the existing models. It is discussed that the parameters of the Generalized Continuum Transport model are related to the physical properties of porous media. Finally, it is presented that the parameter space of GCT can be constrained and related to the transport phenomena studied. Hence, the limits of GCT are controlled by the transport complexity and the desired accuracy and the modeling choice can be parametrized

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Using the Sharp Operator for edge detection and nonlinear diffusion

In this paper we investigate the use of the sharp function known from functional analysis in image processing. The sharp function gives a measure of the variations of a function and can be used as an edge detector. We extend the classical notion of the sharp function for measuring anisotropic behaviour and give a fast anisotropic edge detection variant inspired by the sharp function. We show that these edge detection results are useful to steer isotropic and anisotropic nonlinear diffusion filters for image enhancement

Use of groundwater lifetime expectancy for the performance assessment of a deep geologic waste repository: 1. Theory, illustrations, and implications

Long-term solutions for the disposal of toxic wastes usually involve isolation of the wastes in a deep subsurface geologic environment. In the case of spent nuclear fuel, if radionuclide leakage occurs from the engineered barrier, the geological medium represents the ultimate barrier that is relied upon to ensure safety. Consequently, an evaluation of radionuclide travel times from a repository to the biosphere is critically important in a performance assessment analysis. In this study, we develop a travel time framework based on the concept of groundwater lifetime expectancy as a safety indicator. Lifetime expectancy characterizes the time that radionuclides will spend in the subsurface after their release from the repository and prior to discharging into the biosphere. The probability density function of lifetime expectancy is computed throughout the host rock by solving the backward-in-time solute transport adjoint equation subject to a properly posed set of boundary conditions. It can then be used to define optimal repository locations. The risk associated with selected sites can be evaluated by simulating an appropriate contaminant release history. The utility of the method is illustrated by means of analytical and numerical examples, which focus on the effect of fracture networks on the uncertainty of evaluated lifetime expectancy.Comment: 11 pages, 8 figures; Water Resources Research, Vol. 44, 200