Spectral discretization of Darcy's equations with pressure dependent porosity

International audienceWe consider the flow of a viscous incompressible fluid in a rigid homogeneous porous medium provided with boundary conditions on the pressure around a circular well. When the boundary pressure presents high variations, the permeability of the medium depends on the pressure, so that the model is nonlinear. We propose a spectral discretization of the resulting system of equations which takes into account the axisymmetry of the domain and of the flow. We prove optimal error estimates and present some numerical experiments which confirm the interest of the discretization

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

Solution strategies for nonlinear conservation laws

Nonlinear conservation laws form the basis for models for a wide range of physical phenomena. Finding an optimal strategy for solving these problems can be challenging, and a good strategy for one problem may fail spectacularly for others. As different problems have different challenging features, exploiting knowledge about the problem structure is a key factor in achieving an efficient solution strategy. Most strategies found in literature for solving nonlinear problems involve a linearization step, usually using Newton's method, which replaces the original nonlinear problem by an iteration process consisting of a series of linear problems. A large effort is then spent on finding a good strategy for solving these linear problems. This involves choosing suitable preconditioners and linear solvers. This approach is in many cases a good choice and a multitude of different methods have been developed. However, the linearization step to some degree involves a loss of information about the original problem. This is not necessarily critical, but in many cases the structure of the nonlinear problem can be exploited to a larger extent than what is possible when working solely on the linearized problem. This may involve knowledge about dominating physical processes and specifically on whether a process is near equilibrium. By using nonlinear preconditioning techniques developed in recent years, certain attractive features such as automatic localization of computations to parts of the problem domain with the highest degree of nonlinearities arise. In the present work, these methods are further refined to obtain a framework for nonlinear preconditioning that also takes into account equilibrium information. This framework is developed mainly in the context of porous media, but in a general manner, allowing for application to a wide range of problems. A scalability study shows that the method is scalable for challenging two-phase flow problems. It is also demonstrated for nonlinear elasticity problems. Some models arising from nonlinear conservation laws are best solved using completely different strategies than the approach outlined above. One such example can be found in the field of surface gravity waves. For special types of nonlinear waves, such as solitary waves and undular bores, the well-known Korteweg-de Vries (KdV) equation has been shown to be a suitable model. This equation has many interesting properties not typical of nonlinear equations which may be exploited in the solver, and strategies usually reserved to linear problems may be applied. In this work includes a comparative study of two discretization methods with highly different properties for this equation

Spectral analysis of water content changes in an unsaturated soil layer

Los deslizamientos superficiales localizados en la zona no saturada ocurren en su mayoría por la saturación parcial o total de la zona vadosa y la predicción de los cambios de contenido de agua en la zona vadosa es fundamental para la estimación de la amenaza asociada a este tipo de deslizamientos. Sin embargo, la simulación de los procesos acoplados termo-hidráulicos que tienen lugar en la zona vadosa bajo acciones climáticas requiere cálculos numéricos largos y computacionalmente costosos. Esta tesis busca utilizar un Modelo de Orden Reducido basado en el análisis espectral como una alternativa económica para la predicción de fluctuaciones del contenido de agua. Para ello se determinan dos Funciones de Transferencia del Suelo. La primera es estadística y se basa en correlaciones entre mediciones de contenido de agua en profundidad y en superficie del suelo. La segunda es determinista y se basa en una solución analítica de la ecuación de infiltración unidimensional bajo variaciones periódicas de contenido/presión de agua en la interfase suelo-atmósfera. Ambos enfoques se validan en un conjunto de datos monitoreados en un campo experimental. Los resultados muestran que la estimación del contenido de agua mediante la función de transferencia determinista alcanza una mayor precisión siempre que un procedimiento de back-análisis de los valores característicos de la ecuación (longitud de la columna, coeficiente de difusión) se realiza en un subconjunto de los datos monitorizados.Shallow landslides seated in the unsaturated zone are mostly triggered by soil partial of total saturation and the prediction of the changes of water content in the vadose zone is fundamental for the estimation of landslide hazard. However, the simulation of the full coupled thermo-hydraulic processes taking place in the vadose zone under climatic actions require long and computationally costly numerical computations. This thesis seeks to use a Reduced order Model based on the spectral analysis as a cheap alternative for the prediction water content fluctuations. To this aim, two Soil Transfer Function are determined. The first one bis statistical and based on correlations between water content measurement at depth and at soil surface. The second is deterministic and based on an analytical solution of the one-dimensional infiltration equation under period water content/pressure variations at soil- atmosphere interface. Both approaches are validated on a set of monitored data obtained in an experimental field. Results shown that the estimation of water content using a deterministic soil transfer reached higher accuracy provided that the procedure is trained on a sub-set of measurements in order to determine characteristics values for the equation control parameters (length of the column, diffusion coefficient)

Mixed-dimensional geometric multigrid methods for single-phase flow in fractured porous media

This paper deals with the efficient numerical solution of single-phase flow problems in fractured porous media. A monolithic multigrid method is proposed for solving two-dimensional arbitrary fracture networks with vertical and/or horizontal possibly intersecting fractures. The key point is to combine two-dimensional multigrid components (smoother and intergrid transfer operators) in the porous matrix with their one-dimensional counterparts within the fractures, giving rise to a mixed-dimensional geometric multigrid method. This combination seems to be optimal since it provides an algorithm whose convergence matches the multigrid convergence factor for solving the Darcy problem. Several numerical experiments are presented to demonstrate the robustness of the monolithic mixed-dimensional multigrid method with respect to the permeability of the fractures, the grid size, and the number of fractures in the network.The work of the first and fourth authors was supported by Spanish project PGC2018-099536-A-I00 (MCIU/AEI/FEDER, UE). The work of the second author was supported by the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement 705402, POROSOS. The work of the third author was partially supported by the Spanish project FEDER/MCYT MTM2016-75139-R. The work of the fourth author was supported by the DGA (Grupo de referencia APEDIF, ref. E24 17R)

Sedimentation and Flow Through Porous Media: Simulating Dynamically Coupled Discrete and Continuum Phases

We describe a method to address efficiently problems of two-phase flow in the regime of low particle Reynolds number and negligible Brownian motion. One of the phases is an incompressible continuous fluid and the other a discrete particulate phase which we simulate by following the motion of single particles. Interactions between the phases are taken into account using locally defined drag forces. We apply our method to the problem of flow through random media at high porosity where we find good agreement to theoretical expectations for the functional dependence of the pressure drop on the solid volume fraction. We undertake further validations on systems undergoing gravity induced sedimentation.Comment: 22 pages REVTEX, figures separately in uudecoded, compressed postscript format - alternatively e-mail '[email protected]' for hardcopies

Multiphase flow in porous media with phase transitions: from CO₂ sequestration to gas hydrate systems

Ongoing efforts to mitigate climate change include the understanding of natural and engineered processes that can impact the global carbon budget and the fate of greenhouse gases (GHG). Among engineered systems, one promising tool to reduce atmospheric emissions of anthropogenic carbon dioxide (CO₂) is geologic sequestration of CO₂, which entails the injection of CO₂ into deep geologic formations, like saline aquifers, for long-term storage. Among natural contributors, methane hydrates, an ice-like substance commonly found in seafloor sediments and permafrost, hold large amounts of the world's mobile carbon and are subject to an increased risk of dissociation due to rising temperatures. The dissociation of methane hydrates releases methane gas-a more potent GHG than CO₂-and potentially contributes to a positive feedback in terms of climatic change. In this Thesis, we explore fundamental mechanisms controlling the physics of geologic CO₂ sequestration and natural gas hydrate systems, with an emphasis on the interplay between multiphase flow-the simultaneous motion of several fluid phases and phase transitions-the creation or destruction of fluid or solid phases due to thermodynamically driven reactions. We first study the fate of CO₂ in saline aquifers in the presence of CO₂-brine-carbonate geochemical reactions. We use high-resolution simulations to examine the interplay between the density-driven convective mixing and the rock dissolution reactions. We find that dissolution of carbonate rock initiates in regions of locally high mixing, but that the geochemical reaction shuts down significantly earlier than shutdown of convective mixing. This early shutdown reflects the important role that chemical speciation plays in this hydrodynamics-reaction coupled process. We then study hydrodynamic and thermodynamic processes pertaining to a gas hydrate system under changing temperature and pressure conditions. The framework for our analysis is that of phase-field modeling of binary mixtures far from equilibrium, and show that: (1) the interplay between phase separation and hydrodynamic instability can arrest the Ostwald ripening process characteristic of nonflowing mixtures; (2) partial miscibility exerts a powerful control on the degree of viscous fingering in a gas-liquid system, whereby fluid dissolution hinders fingering while fluid exsolution enhances fingering. We employ this theoretical phase-field modeling approach to explain observations of bubble expansion coupled with gas dissolution and hydrate formation in controlled laboratory experiments. Unraveling this coupling informs our understanding of the fate of hydrate-crusted methane bubbles in the ocean water column and the migration of gas pockets in hydrate-bearing sediments

An adaptive stabilized finite element method for the Darcy's equations with pressure dependent viscosities

This work aims to introduce and analyze an adaptive stabilized finite element method to solve a nonlinear Darcy equation with a pressure-dependent viscosity and mixed boundary conditions. We stated the discrete problem's well-posedness and optimal error estimates, in natural norms, under standard assumptions. Next, we introduce and analyze a residual-based a posteriori error estimator for the stabilized scheme. Finally, we present some two- and three-dimensional numerical examples which confirm our theoretical results

A Comprehensive Review of the Simulation Methods for Analysis at the Pore-scale

Fluid flow through porous material is relevant in different fields of engineering, such as in engine and vehicle development, and can be supported through CFD simulation. Numerical simulations at the pore-scale can be used to replace or reduce expensive laboratory measurements. These methods offer a valuable opportunity to connect the pore-scale properties of the porous material with displacement processes on the continuum-scale. Furthermore, they allow researchers to specify crucial flow properties, e.g., capillary pressure, which are crucial for REV-scale research. Three main methods, direct numerical, pore network modeling, and hybrid approaches, are widely used in order to analyze the pore-scale mechanics of fluid flow behavior through porous materials with CFD simulations. The present comprehensive review demonstrates and highlights the significant advantages, disadvantages, and critical challenges in the pore-scale fluid flow simulations. The main challenges include the characterization of material properties, and up-scaling process from pore to continuum or field-scale

Thermal degradation of solid porous materials exposed to fire

Thermal degradation processes in solid materials are crucial in provoked or hazardous fire events, since pyrolysis is the main source of gaseous combustible matter which feeds the fire. The simulation of fire events on the global scale requires a good description of these processes as a function of the ambient parameters such as temperature and oxygen concentration. But characterizations of thermal decomposition by small or large scale measurements often yield different responses, due to the coupling in the latter case between the chemical reactions and the various heat and mass transport mechanisms. The purpose of this work is to make the connection between these two points of view, and to predict the macroscopic behavior as a function of the constituent properties, of the micro-structural characteristics and of the ambient conditions. A chemical model, including a reaction scheme and associated thermo-kinetic parameters, can be obtained by thermogravimetric measurements on samples small enough to prevent any limiting influence of heat and mass transports. Numerical simulations can then be conducted on a larger scale, to determine the material response in a given geometrical configuration, undergoing any scenario of ambient conditions. These simulations are performed on the Darcy scale in prescribed scenarios which correspond to standardized physical tests. This will allow establishing a typology of behaviors, identifying the key processes and their governing parameters, and validating the numerical predictions. In a later stage, microscopic investigations could be conducted for a better characterization of some processes and for the determination of relevant effective coefficients. Ultimately, this description of the material degradation could be directly coupled with a fire simulation tool on the global scale, which would provide the time dependent ambient conditions, and would account for the influence of the emitted species on the fire development