Academic literature on the topic 'Heterogenous conservation laws'

We extend Brenier’s transport collapse scheme on the Cauchy problem for heterogeneous scalar conservation laws i.e. for the conservation laws with spacetime-dependent coefficients. The method is based on averaging out the solution to the corresponding kinetic equation, and it necessarily converges toward the entropy admissible solution. We also provide numerical examples.

We introduce a notion of stochastic entropy solutions for heterogeneous scalar conservation laws with multiplicative noise on a bounded domain with non-homogeneous boundary condition. Using the concept of measure-valued solutions and Kruzhkov’s semi-entropy formulations, we show the existence and uniqueness of stochastic entropy solutions. Moreover, we establish an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux function and the random source function.

We deal with a single conservation law in one space dimension whose flux function is discontinuous in the space variable and we introduce a proper framework of entropy solutions. We consider a large class of fluxes, namely, fluxes of the convex-convex type and of the concave-convex (mixed) type. The alternative entropy framework that is proposed here is based on a two step approach. In the first step, infinitely many classes of entropy solutions are defined, each associated with an interface connection. We show that each of these class of entropy solutions form a contractive semigroup in L1 and is hence unique. Godunov type schemes based on solutions of the Riemann problem are designed and shown to converge to each class of these entropy solutions. The second step is to choose one of these classes of solutions. This choice depends on the Physics of the problem being considered and we concentrate on the model of two-phase flows in a heterogeneous porous medium. We define an optimization problem on the set of admissible interface connections and show the existence of an unique optimal connection and its corresponding optimal entropy solution. The optimal entropy solution is consistent with the expected solutions for two-phase flows in heterogeneous porous media.

A new mathematical and numerical method of modeling for heterogeneous hydrodynamics processes with take of phase transitions like graphite-diamond will be presented. The method is based on an approximation of conservation laws for masses, momentums, and energies in integral and differential forms. The combination of Harlow's particle-in-cell method and Belotserkovskii's large particles method is used for computing by the modeling method simulation.

A modified method of numerical modeling for heterogeneous fluid dynamics processes with take of phase transitions will be presented. The method is based on the homogenization on cells and approximation of conservation laws for masses, momentums, and energies in integral and differential forms. The combination of Harlow's particle-in-cell method, Belotserkovskii's large particles method and Bakhvalov's homogenization method is used for computing by the modified method simulation for processes with phase transitions.

We have studied a class of (1+1)-dimensional equations that models phenomena with heterogeneous diffusion, advection, and reaction. We have analyzed these fourth-order partial differential equations within the framework of group methods. In this class, the diffusion coefficient is constant, while the coefficients of advection and the reaction term are assumed to depend on the unknown density u(t,x). We have identified the Lie symmetries extending the Principal Algebra along with all the conservation laws corresponding to the different forms of the coefficients, and have derived several brief applications.

Dans cette thèse, on traite la prise en compte de l'hétérogénéité dans les lois de conservation scalaires, c'est à dire les lois de conservation non invariantes par translation en espace. Ces équations apparaissent notamment dans lesmodèles de trafic. Par exemple, les mécanismes suivants introduisent de l'hétérogénéité : la présence de feux de circulation, des portions de route où la vitesse maximale est limitée, la variabilité de l'état de la route, etc... La prise en compte del'hétérogénéité permet d'enrichir les modèles de trafic. On aborde trois classes de problèmes inhomogènes pour lesquelles on complète et approfondit le cadre mathématique pour l'analyse théorique et l'approximation numérique.Nous explorons en détail le cadre où l'hétérogénéité est matérialisée par l'ajout d'une ou plusieurs interfaces mobiles. Le long des interfaces, on impose une condition de majoration sur le flux de la loi de conservation. Cette classe demodèles permet de tenir compte de la présence d'un petit nombre de véhicules encombrants et lents (ou alors, de véhicules autonomes qui ont pour rôle la régulation du trafic). Dans ce cadre, l'évolution des interfaces et des contraintesest couplée de façon non locale à l'état du trafic et/ou aux paramètres spécifiant l'état du véhicule ou du conducteur. En outre, nous élaborons une description de l'hétérogénéité du trafic résultant des variations du degré d'organisation desconducteurs, dans le cadre des modèles dits "du second ordre". L'aspect numérique est prépondérant pour les modèles de trafic que nous étudions. On construit des schémas numériques robustes et on élabore des techniques decompacité spécifiques. La convergence de ces schémas conduit à des résultats d'existence.Enfin, en lien avec le modèle décrivant l'évolution d'une densité de véhicules sur une route hétérogène, on étudie théoriquement une loi de conservation dans laquelle la dépendance spatiale du flux est explicite. Des résultatsclassiques sur le caractère bien posé ou la correspondance avec l'équation de Hamilton-Jacobi associée sont obtenus sous des hypothèses plus en adéquation avec la modélisation que celles rencontrées dans la littérature. Les applicationsallant au-delà de la description du trafic, on se donne pour objectif l'analyse approfondie des problèmes d'identification de données initiales
This thesis is devoted to the treatment of heterogeneity in scalar conservation laws. We call heterogeneous a conservation which is not invariant by spacetranslation. These equations arise for instance in traffic flow dynamics modeling. The presence of traffic lights or roads that have a variable maximum speed limitare examples of mechanisms which lead to heterogeneous conservation laws. Considering such equations is a way to expand macroscopic traffic flow models. We tacklethree classes of inhomogeneous problems for which we extend the mathematical framework for both the theorical analysis and the numerical approximation.We fully investigate the treatment of heterogeneity when one or several moving interfaces are added in the classic LWR model for traffic flow. Flux constraintsare attached to each interfaces. The resulting class of models can be used to take into account the presence of slow moving vehicles that reduce the road capacityand thus generates a moving bottleneck for the surrounding traffic flow. They can also describe the regulating effect of autonomous vehicles. In this framework,the interfaces and the constraints are linked in a nonlocal way to the traffic density and/or to an orderliness marker describing the state of the drivers. Thedescription of the heterogeneity caused by the variations in the drivers' organization leads to the analysis of a so called second order model. The numericalaspect plays a central role in the analysis of these traffic flow models. We construct robust numerical schemes and establish specific techniques to obtaincompactness of the approximate solutions. Proving convergence of these schemes leads to existence results.Finally, with the space-dependent LWR traffic flow model in mind, we theoretically analyze a class of scalar conservation laws with explicit space dependency.Classical results such as well-posedness or the link to the associated Hamilton-Jacobi equation are obtained under a set of assumptions better fitting themodeling hypothesis. With applications that go beyond traffic modeling in mind, we aim to tackle initial data identification problems

Doktorska disertacija posve¶cena je re·savanju nelinearnih hiperboli·cnih skalarnih zakona odr·zanja u heterogenim sredinama, prou·cavanjem osobina kompaktnosti re·senja familija aproksimativnih jedna·cina. Ta·cnije, u cilju dobijanja re·senja u = u(t; x) problema @ t u + divx f (t; x; u) = 0;uj t=0 = u 0(x); gde su promenljive x 2 R d i t 2 R+, posmatramo familije problema koji na neki na·cin aproksimiraju po·cetni problem, a koje znamo da re·simo, i ispitujemo familije dobijenih re·senja koja zovemo aproksimativna re·senja. Cilj nam je da poka·zemo da je dobijena familija u nekom smislu prekompaktna,tj. da ima konvergentan podniz ·cija granica re·sava po·cetni problem.
Doctoral theses is dedicated to solving nonlinear hyperbolic scalar conservation laws in heterogeneous media, by studying compactness properties of the family of solutions to approximate problems. More precise, in order to obtain solution u = u(t; x) to the problem @ t u + divx f (t; x; u) = 0; uj t=0 = u 0 (x); (4.18) where x 2 R d and t 2 R+, we study the solutions of the families of problems that, in some way, approximate previously mentioned problem, which we know how to solve. We call those solutions approximate solutions. The aim is to show that the obtained family is in some sense precompact, i.e. has convergent subsequence that solves the problem (4.18).

Escoamentos multifásicos em meios porosos são modelados por um sistema de equações diferenciais parciais e o estudo da aproximação das soluções dessas equações desempenha papel crucial na simulação e previsão de problemas de grande interesse prático e impacto econômico e social, tais como a recuperação secundária de petróleo, o armazenamento geológico de CO2 e o transporte de poluentes em aquíferos. O presente trabalho tem como objetivo o desenvolvimento de um simulador numérico tridimensional para avaliar com precisão o transporte de dois fluidos imiscíveis em um meio poroso heterogêneo e que utiliza computação paralela multithread para computadores multiprocessados de memória compartilhada. O sistema de equações diferenciais parciais é decomposto em um subsistema elíptico para a determinação do campo de velocidades dos fluidos e uma equação hiperbólica não-linear para o transporte das fases fluidas. Para esta última, foi utilizado um método numérico de volumes finitos, não-oscilatório de alta ordem baseado em esquemas centrais e que admite uma formulação semi-discreta com coeficientes variáveis no espaço. Experimentos numéricos em modelos tridimensionais foram realizados considerando problemas de escoamentos lineares e não lineares postos em configurações típicas de simulação de reservatórios de petróleo. Os resultados mostraram-se satisfatórios por apresentarem conservação da massa, boa captura das ondas de choque e pequena difusão numérica, independente do passo de tempo.
Multiphase flows in porous media are modeled by a system of partial differential equations and the study of the numerical approximation to the solutions of these plays a crucial role in the simulation and prediction of problems that are of great practical interest and of economic and social impact, such as secondary oil recovery, geological storage of CO2 and transport of pollutants in aquifers. The goal of this work is the development of a three-dimensional numerical simulator that precisely evaluates the transport of two immiscible fluids in a heterogeneous porous media using multithread parallel programming to shared memory multiprocessors computers. The system of partial differential equations is decomposed into a elliptic subsystem used to determine the velocity field and into a hyperbolic equation (nonlinear) to determine the transport of the fluid phases. The approximation to the solution of the latter one is calculated using a high order non-oscillatory finite-differences numerical method based on central schemes that allows a semi-discrete formulation which an extension that enables to work with variable space coefficients. Numerical experiments on three-dimensional models were performed considering linear and nonlinear flow problems in typical settings of oil reservoirs simulations. The results were satisfactory since they presented mass conservation, precise capture of shock waves and small numeric diffusion, regardless of the time step.

Técnicas de injeção de traçadores são bastante utilizadas nos estudos de escoamentos em meios porosos heterogêneos, principalmente em problemas relacionados à simulação numérica de escoamentos miscíveis em reservatórios de petróleo e à dispersão de contaminantes em aqüíferos. Neste trabalho apresentamos novos algoritmos para a aproximação numérica do problema de injeção de traçadores. Apresentaremos desenvolvimentos recentes do método Forward Integral-Tube Tracking (FIT) que foi originalmente apresentado em Aquino et al. (2007a). O FIT é um método lagrangeano localmente conservativo utilizado na resolução de problemas de transporte linear. Este método não faz o uso de soluções de problemas de Riemann e baseia-se na construção dos tubos integrais introduzidas em Douglas Jr. et al. (2000b). Além disso, ele possui excelente eficiência computacional e é virtualmente livre de difusão numérica. Resultados numéricos são apresentados com o objetivo de comparar a precisão das soluções fornecidas por novas implementações do método FIT na resolução do problema do traçador em reservatórios de petróleo.
The injection of tracers are used in the investigation of flows in heterogeneous porous media, in studies related to the simulation of miscible dispacements in petroleum reservoirs and the dispersion of contaminants in aquifers. In this work we present new algorithms for the numerical approximation of tracer injection problems. We discuss recent developments of the Forward Integral-Tube Tracking (FIT) scheme which was introduced in Aquino et al. (2007a). The FIT is a locally conservative lagrangian scheme for the approximation of the linear transport problems. This scheme does not use analytic solutions of Riemann problems and is based on the construction of the integral tubes introduced in Douglas Jr. et al. (2000b). The FIT scheme is computationally very eficient and is virtually free of numerical diffusion. Numerical results are presented to compare the accuracy of the solutions provided by new implementation of the FIT scheme for the injection of tracers in petroleum reservoirs.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Neste trabalho é apresentado um novo método acurado com passo de tempo fracionário, baseado em uma técnica de decomposição de operadores, para a solução numérica de um sistema governante de equações diferenciais parciais que modela escoamento trifásico água-gás-óleo imiscível em reservatórios de petróleo heterogêneos no qual os efeitos de compressibilidade do gás não foram levados em conta. A técnica de decomposição de operadores em dois níveis permite o uso de passos de tempo distintos para os três problemas definidos pelo procedimento de decomposição: convecção, difusão e pressão-velocidade. Um sistema hiperbólico de leis de conservação que modela o transporte convectivo das fases fluidas é aproximado por um esquema central de diferenças finitas explícito, conservativo, não oscilatório e de segunda ordem. Este esquema é combinado com elementos finitos mistos, localmente conservativos, para a aproximação numérica dos sistemas de equações parabólico e elíptico associados aos problemas de transporte difusivo e de pressão-velocidade, respectivamente. O operador temporal associado ao sistema parabólico é resolvido fazendo-se uso de uma estratégia implícita de solução (Backward Euler). O modelo matemático para escoamento trifásico considerado neste trabalho leva em conta as forças de capilaridade e expressões gerais para as funções de permeabilidade relativa, campos variáveis de porosidade e de permeabilidade e os efeitos da gravidade. A escolha de expressões gerais para as funções de permeabilidade relativa pode levar à perda de hiperbolicidade escrita e, desta maneira, à existência de uma região elíptica ou de pontos umbílicos para o sistema não linear de leis de conservação hiperbólicas que descreve o transporte convectivo das fases fluidas. Como consequência, a perda de hiperbolicidade pode levar à existência de choques não clássicos (também chamados de choques transicionais ou choques subcompressivos) nas soluções de escoamentos trifásicos. O novo procedimento numérico foi usado para investigar a existência e a estabilidade de choques não clássicos, com respeito ao fenômeno de fingering viscoso, em problemas de escoamentos trifásicos bidimensionais em reservatórios heterogêneos, estendendo deste modo resultados disponíveis na literatura para problemas de escoamentos trifásicos unidimensionais. Experimentos numéricos, incluindo o estudo de estratégias de injeção alternada de água e gás (Water-Alternating-Gas (WAG)), indicam que o novo procedimento numérico proposto conduz com eficiência computacional a resultados numéricos com precisão. Perspectivas para trabalhos de pesquisa futuros são também discutidas, tomando como base os desenvolvimentos reportados nesta tese.
We present a new, accurate fractional time-step method based on an operator splitting technique for the numerical solution of a system of partial differential equations modeling three-phase immiscible water-gas-oil flow problems in heterogeneous petroleum reservoirs in which the compressibility effects of the gas was not take into account. A two-level operator splitting technique allows for the use of distinct time steps for the three problems defined by the splitting procedure: convection, diffusion and pressure-velocity. A system of hyperbolic conservation laws modelling the convective transport of the fluid phases is approximated by a high resolution, nonoscillatory, second-order, conservative central difference scheme in the convection step. This scheme is combined with locally conservative mixed finite elements for the numerical solution of the parabolic and elliptic problems associated with the diffusive transport of fluid phases and the pressure-velocity problem, respectively. The time discretization of the parabolic problem is performed by means of the implicit backward Euler method. The mathematical model for the three-phase flow considered in this work takes into account capillary forces and general expressions for the relative permeability functions, variable porosity and permeability fields, and the effect of gravity. The choice of general expressions for the relative permeability functions may lead to the loss of strict hyperbolicity and, therefore, to the existence of an elliptic region of umbilic points for the systems of nonlinear hyperbolic conservation laws describing the convective transport of the fluid phases. As a consequence, the loss of hyperbolicity may lead to the existence of nonclassical shocks (also called transitional shocks or undercompressive shocks) in three-phase flow solutions. The numerical procedure was used in an investigation of the existence and stability of nonclassical shocks with respect to viscous fingering in heterogeneous two-dimensional flows, thereby extending previous results for one-dimensional three-phase flow available in the literature. Numerical experiments, including the study of Water-Alternating-Gas (WAG) injection strategies, indicate that the proposed new numerical procedure leads to computational efficiency and accurate numerical results. Directions for further research are also discussed, based on the developments reported in this thesis.

Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro
Considera-se neste trabalho um modelo matemático para escoamentos com duas e três fases em reservatórios petrolíferos e a modelagem computacional do sistema de equações governantes para a sua solução numérica. Os fluidos são imiscíveis e incompressíveis e as heterogeneidades da rocha reservatório são modeladas estocasticamente. Além disso, é modelado o fenômeno de histerese para a fase óleo via funções de permeabilidades relativas. No caso de escoamentos trifásicos água-óleo-gás a escolha de expressões gerais para as funções de permeabilidades relativas pode levar à perda de hiperbolicidade estrita e, desta maneira, à existência de uma região elíptica ou de pontos umbílicos para o sistema não linear de leis de conservação hiperbólicas que descreve o transporte convectivo das fases fluidas. Como conseqüência, a perda de hiperbolicidade estrita pode levar à existência de choques não clássicos (também chamados de choques transicionais ou choques subcompressivos) nas soluções de escoamentos trifásicos, de difícil simulação numérica. Indica-se um método numérico com passo de tempo fracionário, baseado em uma técnica de decomposição de operadores, para a solução numérica do sistema governante de equações diferenciais parciais que modela o escoamento bifásico água-óleo imiscível em reservatórios de petróleo heterogêneos. Um simulador numérico bifásico água-óleo eficiente desenvolvido pelo grupo de pesquisa no qual o autor está inserido foi modificado com sucesso para incorporar a histerese sob as hipóteses consideradas. Os resultados numéricos obtidos para este caso indicam fortes evidências que o método proposto pode ser estendido para o caso trifásico água-óleo-gás. A técnica de decomposição de operadores em dois níveis permite o uso de passos de tempo distintos para os quatro problemas definidos pelo procedimento de decomposição: convecção, difusão, pressão-velocidade e relaxação para histerese. O problema de transporte convectivo (hiperbólico) das fases fluidas é aproximado por um esquema central de diferenças finitas explícito, conservativo, não oscilatório e de segunda ordem. Este esquema é combinado com elementos finitos mistos, localmente conservativos, para a aproximação dos problemas de transporte difusivo (parabólico) e de pressão-velocidade (elíptico). O operador temporal associado ao problema parabólico de difusão é resolvido fazendo-se uso de uma estratégia implícita de solução (Backward Euler). Uma equação diferencial ordinária é resolvida (analiticamente) para a relaxação relacionada à histerese. Resultados numéricos para o problema bifásico água-óleo em uma dimensão espacial em concordância com resultados semi-analíticos disponíveis na literatura foram reproduzidos e novos resultados em meios heterogêneos, em duas dimensões espaciais, são apresentados e a extensão desta técnica para o caso de problemas trifásicos água-óleo-gás é proposta.
We consider in this work a mathematical model for two- and three-phase flow problems in petroleum reservoirs and the computational modeling of the governing equations for its numerical solution. We consider two- (water-oil) and three-phase (water-gas-oil) incompressible, immiscible flow problems and the reservoir rock is considered to be heterogeneous. In our model, we also take into account the hysteresis effects in the oil relative permeability functions. In the case of three-phase flow, the choice of general expressions for the relative permeability functions may lead to the loss of strict hyperbolicity and, therefore, to the existence of an elliptic region or umbilic points for the system of nonlinear hyperbolic conservation laws describing the convective transport of the fluid phases. As a consequence, the loss of hyperbolicity may lead to the existence of nonclassical shocks (also called transitional shocks or undercompressive shocks) in three-phase flow solutions. We present a new, accurate fractional time-step method based on an operator splitting technique for the numerical solution of a system of partial differential equations modeling two-phase, immiscible water-oil flow problems in heterogeneous petroleum reservoirs. An efficient two-phase water-oil numerical simulator developed by our research group was sucessfuly extended to take into account hysteresis effects under the hypotesis previously annouced. The numerical results obtained by the procedure proposed indicate numerical evidence the method at hand can be extended for the case of related three-phase water-gas-oil flow problems. A two-level operator splitting technique allows for the use of distinct time steps for the four problems defined by the splitting procedure: convection, diffusion, pressure-velocity and relaxation for hysteresis. The convective transport (hyperbolic) of the fluid phases is approximated by a high resolution, nonoscillatory, second-order, conservative central difference scheme in the convection step. This scheme is combined with locally conservative mixed finite elements for the numerical solution of the diffusive transport (parabolic) and the pressure-velocity (elliptic) problems. The time discretization of the parabolic problem is performed by means of the implicit Backward Euler method. An ordinary diferential equation is solved (analytically) for the relaxation related to hysteresis. Two-phase water-oil numerical results in one space dimensional, in which are in a very good agreement with semi-analitycal results available in the literature, were computationaly reproduced and new numerical results in two dimensional heterogeneous media are also presented and the extension of this technique to the case of three-phase water-oil-gas flows problems is proposed.

This chapter discusses electrochemical principles. Electrochemistry is the study of phenomena caused by charge separation. In the context of electrode processes, which are heterogeneous in nature, it deals with the study of charge transfer processes at the electrode/solution interface, either in equilibrium at the interface, or under partial or total kinetic control. Most of the charge transfer processes are transfer of electrons, which can be represented in the simplest case of oxidised species and reduced species, both soluble in solution. In a real situation, both reduction and oxidation must occur to an equal extent, so as not to violate the law of conservation of energy. The chapter then looks at electrochemical cells, the kinetics of electrode reactions, the interfacial region and electrolyte double layer, hydrodynamic electrodes, microelectrodes, and electroanalytical titrations.

Spatial variability and the uncertainty in characterizing the flow domain play an important role in the transport of contaminants in porous media: they affect the pathlines followed by solute particles, the spread of solute bodies, the shape of breakthrough curves, the spatial variability of the concentration, and the ability to quantify any of these accurately. This chapter briefly reviews some basic concepts which we shall later employ for the analysis of solute transport in heterogeneous media, and also points out some issues we shall address in the subsequent chapters. Our exposition in chapters 8-10 on contaminant transport is built around the Lagrangian and the Eulerian approaches for analyzing transport. The Eulerian approach is a statement of mass conservation in control volumes of arbitrary dimensions, in the form of the advection-dispersion equation. As such, it is well suited for numerical modeling in complex flow configurations. Its main difficulties, however, are in the assignment of parameters, both hydrogeological and geochemical, to the numerical grid blocks such that the effects of subgrid-scale heterogeneity are accounted for, and in the numerical dispersion that occurs in advection-dominated flow situations. Another difficulty is in the disparity between the scale of the numerical elements and the scale of the samples collected in the field, which makes the interpretation of field data difficult. The Lagrangian approach focuses on the displacements and travel times of solute bodies of arbitrary dimensions, using the displacements of small solute particles along streamlines as its basic building block. Tracking such displacements requires that the solute particles do not transfer across streamlines. Since such mass transfer may only occur due to pore-scale dispersion, Lagrangian approaches are ideally suited for advection-dominated situations. Let us start by considering the displacement of a small solute body, a particle, as a function of time. “Small” here implies that the solute body is much smaller than the characteristic scale of heterogeneity. At the same time, to qualify for a description of its movement using Darcy’s law, the solute body also needs to be larger than a few pores. The small dimension of the solute body ensures that it moves along a single streamline and that it does not disintegrate due to velocity shear.

Unsteady transonic flows with condensation through steam turbine stator-rotor channels are numerically predicted by using the numerical method developed by our group. Fundamental equations solved here consist of conservation laws of mixed gas, water vapor, water liquid, and the number density of water droplets, coupled with the momentum equations and the energy equation. Also the shear-stress transport (SST) turbulence model is employed to predict the turbulent quantities. The numerical method is based on the high-order high-resolution finite-difference method. The fourth-order monotone upstream-centered schemes for conservation laws (MUSCL) with the total variation diminishing (TVD) scheme, Roe’s approximate Riemann solver, and the lower-upper symmetric Gauss-Seidel (LU-SGS) scheme are employed in the numerical method. As numerical examples, transonic condensate flows of moist air through a turbine and a compressor cascade channel are first calculated. Also wet-steam turbine stator-rotor cascade channels are calculated assuming homogeneous and heterogeneous condensations.

Investigation of the process of a detonation wave (DW) interaction with various obstacles is a fundamental problem. This problem is relevant from the point of view of reducing the destructive e¨ects of heterogeneous explosions in technological disasters and in the studies of the process of de§agration-to-detonation transition and in detonation engines development. In this study, the authors tried to model the heterogeneous detonation of stoichiometric mixture of aluminum particles in oxygen with semi-in¦nite porous insert as a grid of stationary cylinders. The model is based on the system of Euler Euler equations for describing the interaction of continua including the laws of mass, momentum, and energy conservation for each of the phases and components closed by equations of state, momentum exchange (drag forces), and heat transfer between gas, particles, and porous body. Aluminum combustion is described as a reduced reaction initiated after the critical temperature is reached assuming incomplete particle burning. It was assumed that the porous zone is a continuous medium in the form of a grid of stationary cylinders. During the numerical simulation, some §ow regimes were obtained similar to those that were previously obtained in the study of the interaction of detonation waves with inert particles as well as with water sprays. There are regimes with a reduced DW velocity and regime with detonation failure with separation of shock wave and reaction front. Figure compares the results of one- (1D) and two-dimensional (2D) simulations of the propagation regimes of heterogeneous detonation of 1-micrometer aluminum particles in oxygen in a porous zone with a 200-micrometer cylinders. It can be seen that the results are quite similar to each other.

Abstract Physics-informed neural networks (PINNs) have shown success in solving physical problems in various fields. However, PINNs face major limitations when addressing fluid flow in heterogeneous porous media, related to discontinuities in rock properties. This is because automatic differentiation is inadequate for evaluating the spatial derivatives of hydraulic conductivity where it is discontinuous. This study aims to devise PINN implementations that overcome this limitation. This work proposes decoupling the mass conservation equation from Darcy's law and utilizing the residuals of these decoupled equations to train the loss function of the PINN, rather than using a single residual from the combined equation. As a result, we circumvent the need to find the spatial derivative of the discontinuous hydraulic conductivity, and instead, we impose the continuity of fluxes. This decoupling necessitates that each primary unknown (pressure and velocity components) be computed by the neural networks (NNs) rather than deriving the velocity (or fluxes) from the pressure. We examined three NN configurations and compared their performance by analyzing their accuracy and training time for various 2D scenarios. These scenarios explored various boundary conditions, different hydraulic conductivity fields, as well as different orientations of the heterogeneous media within the domain of interest. In these problems, the pressure and velocity field are the primary unknowns. The three configurations include: (a) one NN with the three unknowns as its outputs, (b) two NNs, one outputting pressure and the other outputting the velocity, and (c) three NNs, each having one primary unknown as an output. Utilizing these NN architectures, we were able to solve the heterogeneous problems with varying levels of accuracy when compared to results from numerical simulators. While maintaining a similar number of training parameters for a fair assessment, the configuration with three NNs yielded the most accurate results, with a comparable training time to the other configurations. Using this optimal configuration, we performed a sensitivity analysis to demonstrate the effect of modifying the NN(s) hyperparameters, such as the number of layers, the number of nodes per layer, and the learning rate, on the accuracy of the results. We introduce a novel PINN approach for modeling fluid flow in heterogeneous media. This proposed method not only preserves the inherent discontinuity of rock petrophysical properties but also leverages the benefits of automatic differentiation. By incorporating this PINN architecture, we have opened up new possibilities for extending the application of PINN to realistic reservoir simulations that capture the complexities of the subsurface.

Abstract Physics-aware machine learning (ML) techniques have been used to endow data-driven proxy models with features closely related to the ones encountered in nature. Examples span from material balance and conservation laws. Physics-based and data-driven reduced-order models or a combination thereof (hybrid-based models) can lead to fast, reliable, and interpretable simulations used in many reservoir management workflows. We built on a recently developed deep-learning-based reduced-order modeling framework by adding a new step related to information of the input-output behavior (e.g., well rates) of the reservoir and not only the states (e.g., pressure and saturation) matching. A Combination of data-driven model reduction strategies and machine learning (deep- neural networks – NN) will be used here to achieve state and input-output matching simultaneously. In Jin, Liu and Durlofsky (2020), the authors use a NN architecture where it is possible to predict the state variables evolution after training an autoencoder coupled with a control system approach (Embed to Control - E2C) and adding some physical components (Loss functions) to the neural network training procedure. In this paper, we extend this idea by adding the simulation model output, e.g., well bottom-hole pressure and well flowrates, as data to be used in the training procedure. Additionally, we added a new neural network to the E2C transition model to handle the connections between state variables and model outputs. By doing this, it is possible to estimate the evolution in time of both the state variables as well as the output variables simultaneously. The method proposed provides a fast and reliable proxy for the simulation output, which can be applied to a well-control optimization problem. Such a non-intrusive method, like data-driven models, does not need to have access to reservoir simulation internal structure. So it can be easily applied to commercial reservoir simulations. We view this as an analogous step to system identification whereby mappings related to state dynamics, inputs (controls), and measurements (output) are obtained. The proposed method is applied to an oil-water model with heterogeneous permeability, 4 injectors, and 5 producer wells. We used 300 sampled well control sets to train the autoencoder and another set to validate the obtained autoencoder parameters.