CAMPINAS
Instituto de Matemática, Estatística e
Computação Científica
PAOLA CUNHA FERRAZ
A Novel Recursive Formulation of Multiscale
Mixed Methods and Relaxation Modeling of
Flow in Porous Media
Uma nova formulação recursiva para métodos
mistos multiescala e modelagem de fluxo em
meios porosos com relaxamento
Campinas
2019
A Novel Recursive Formulation of Multiscale Mixed
Methods and Relaxation Modeling of Flow in Porous
Media
Uma nova formulação recursiva para métodos mistos
multiescala e modelagem de fluxo em meios porosos com
relaxamento
Tese apresentada ao Instituto de Matemática, Estatística e Computação Científica da Uni-versidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do título de Doutora em Matemática Aplicada. Thesis presented to the Institute of Mathe-matics, Statistics and Scientific Computing of the University of Campinas in partial ful-fillment of the requirements for the degree of Doctor in Applied Mathematics.
Supervisor: Eduardo Cardoso de Abreu
Co-supervisor: Luis Felipe Feres Pereira
Este exemplar corresponde à versão
final da Tese defendida pela aluna
Paola Cunha Ferraz e orientada pelo
Prof. Dr. Eduardo Cardoso de Abreu.
Campinas
2019
Biblioteca do Instituto de Matemática, Estatística e Computação Científica Silvania Renata de Jesus Ribeiro - CRB 8/6592
Ferraz, Paola Cunha,
F413n FerA novel recursive formulation of multiscale mixed methods and relaxation modeling of flow in porous media / Paola Cunha Ferraz. – Campinas, SP : [s.n.], 2019.
FerOrientador: Eduardo Cardoso de Abreu. FerCoorientador: Luis Felipe Feres Pereira.
FerTese (doutorado) – Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica.
Fer1. Método dos elementos finitos. 2. Multiescala. 3. Histerese. 4. Métodos de relaxação (Matemática). 5. Programação Paralela. 6. Riemann-Hilbert,
Problemas de. I. Abreu, Eduardo Cardoso de, 1974-. II. Pereira, Luis Felipe Feres. III. Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica. IV. Título.
Informações para Biblioteca Digital
Título em outro idioma: Uma nova formulação recursiva para métodos mistos multiescala
e modelagem de fluxo em meios porosos com relaxamento
Palavras-chave em inglês:
Finite element method Multiscale
Hysteresis
Relaxation methods (Mathematics) Parallel programming
Riemann-Hilbert problem
Área de concentração: Matemática Aplicada Titulação: Doutora em Matemática Aplicada Banca examinadora:
Eduardo Cardoso de Abreu Fabrício Simeoni de Sousa Giuseppe Romanazzi Marcio Arab Murad Márcio Rentes Borges
Data de defesa: 25-07-2019
Programa de Pós-Graduação: Matemática Aplicada
Identificação e informações acadêmicas do(a) aluno(a)
- ORCID do autor: https://orcid.org/0000-0002-1457-705X - Currículo Lattes do autor: http://lattes.cnpq.br/8484916010915279
pela banca examinadora composta pelos Profs. Drs.
Prof(a). Dr(a). EDUARDO CARDOSO DE ABREU
Prof(a). Dr(a). FABRÍCIO SIMEONI DE SOUSA
Prof(a). Dr(a). GIUSEPPE ROMANAZZI
Prof(a). Dr(a). MARCIO ARAB MURAD
Prof(a). Dr(a). MÁRCIO RENTES BORGES
A Ata da Defesa, assinada pelos membros da Comissão Examinadora, consta no SIGA/Sistema de Fluxo de Dissertação/Tese e na Secretaria de Pós-Graduação do Instituto de Matemática, Estatística e Computação Científica.
Agradeço o apoio do Programa de Pós-Graduação em Matemática Aplicada do IMECC/UNICAMP, do Centro de Estudos de Petróleo (CEPETRO) e da Petrobrás via bolsa, no período de 01/2017 até 08/2019, para a realização deste trabalho sob número 2015/00398-0 e 25302/2015.
Agradeço ao meu orientador, o Professor Doutor Eduardo Cardoso de Abreu, e ao meu co-orientador, o Professor Doutor Luis Felipe Feres Pereira, pela oportunidade de desenvolvimento deste trabalho, pela a seriedade e o compromisso com a pesquisa.
Agradeço todos os meus colegas de pesquisa pelo bom ambiente de colaboração do nosso grupo. Em especial, quero agradecer meus queridos Arthur M. E. Santo e Luis G. C. Santos pelas discussões, brigas, risadas e por terem compartilhado de todo suor e lágrimas desse período. Sem eles, não teria chegado ao fim!
Agradeço o CEPID-Centro de Ciências Matemáticas Aplicadas à Industria (CEMEAI/USP) e ao grupo do ICMC/USP pelo acesso ao cluster Euler. Em especial gostaria de agradecer ao Professor Doutor Fabrício Simeoni de Sousa pelo apoio e pronta ajuda no uso do cluster.
Agradeço o Laboratório Nacional de Computação Científica (LNCC) pelo acesso ao cluster Santos Dumont.
Agradeço toda minha família pelo apoio incondicional.
Por fim, agradeço a todos os meus amigos que de alguma forma contribuiram para a conclusão desse trabalho. Em especial agradeço Marcos Alberto, Natália Coutinho, Stefânia Jarosz, Diego Scolfaro, Eduardo Sato, Rafael Guiraldello, Maicon Correa, Camila Lages e Het Mankad por todo o apoio e por terem me aguentado nos momentos de maior insegurança.
That was it. It is over. Did you do it? Have you achieved what you wanted? No? Oh well... – Welcome to NightVale.
O foco deste trabalho é a modelagem matemática e computacional de problemas envolvendo equações diferenciais parciais (PDE) fundamentais em dinâmica de fluidos em meios porosos com coeficientes descontínuos e alto contraste. É proposto uma nova abordagem analítico-numérico para descrever o fluxo bifásico imiscível e incompressível em meios porosos com histerese. Após uma decomposição de operadores baseado na fisíca, conseguimos identificar dois problemas acoplados fundamentais, um problema não-linear de convecção-difusão com convecção dominante e um problema de Poisson vinculado ao sistema de pressão-velocidade. Através de uma decomposição de operadores, voltamos nossa atenção para o estudo e desenvolvimento de estratégias numéricas para resolver ambos os problemas: problema de convecção-difusão e o problema de Poisson. Para o problema de convecção-difusão, nos concentramos em estudos 1D, onde introduzimos um novo método de projeção analítica para a construção da sequência de ondas do problema local de Riemann envolvendo modelagem de relaxamento. Além disso, um novo método computacional 1D é formalmente desenvolvido para validar análise matemática, juntamente com um conjunto representativo de experimentos numéricos, de modo a entender a modelagem da histerese para fluxos bifásicos. Para o Problema de Poisson, nos concentramos em métodos mistos multiescala baseados em abordagens de decomposição de domínios que tem grande potencial no uso em máquinas paralelas multi-core. Problemas de contorno locais (funções da base multi-escala) são calculadas em cada subdomínio para representar soluções discretas, podendo ser eficientemente computadas em clusters CPU-GPU. Esta etapa é usualmente seguida pela solução de um problema de interface global responsável por construir uma solução local que nos dá a solução global procurada. Neste trabalho, apresentamos uma formulação recursiva para substituir o problema de interface global por uma família de pequenos sistemas lineares de interface localizados. A nova técnica constrói esses pequenos sistemas agrupando funções da base multiescala associada a subdomínios adjacentes. Por fim, propomos um novo algoritmo paralelo para implementar a formulação recursiva para métodos mistos multiescala em dispositivos multi-core. Através de vários estudos numéricos, mostramos que o novo algoritmo é muito rápido, e exibe excelente escalabilidade forte e fraca para problemas grandes motivados pela simulação numérica de fluxos. Também fornecemos uma formulação unificada para uma família de métodos mistos multiescala acessíveis para usuários interessados em aplicações.
Palavras-chave: Método Misto Multiescala; Fluxo de Darcy; Fluxo em Meios Porosos;
The focus of this work is the mathematical and computational modeling of mixed differential problems involving partial differential equations (PDE’s) in fundamental problems in fluid dynamics in porous media with discontinuous high contrast coefficients. For concreteness, a new analytical-numerical approach is proposed to describe immiscible and incompressible two-phase flow in porous media with hysteresis. After an operator splitting based on physical processes is applied, we are able to identify two fundamental coupled problems, namely, one is a nonlinear convection-diffusion problem with dominated convection and a second, a Poisson problem linked to pressure-velocity. Taking advantage of the splitting procedure, we turn our attention to study and develop numerical strategies for solving both problems: the convection-diffusion problem and the Poisson problem. For the convection-diffusion problem we concentrate on 1D studies where we introduce a new analytical projection method for construction of the sequence of waves of the solutions of the local Riemann problem involving modeling relaxation. In addition, a new 1D computational method is formally developed to confirm our mathematical analysis along with a representative set of numerical experiments to understand relaxation modeling of hysteresis for biphasic flows. For the Poisson problem we focus on multiscale mixed methods based on a non-overlapping domain decomposition schemes that have great potential to take advantage of multi-core, state of the art parallel computers. Local boundary problems (multiscale basis functions) are calculated in each subdomain to represent the discrete solutions that can be efficiently computed in parallel in CPU-GPU clusters. This step is usually followed by a solution of a global interface problem that assembles the local solutions to give the global solution sought. Here we present a recursive formulation to replace the global interface problem by a family of small, localized, interface linear systems. The new technique constructs small systems by clustering multiscale basis functions associated with nearest neighbor subdomains. Then, we propose a novel parallel algorithm to implement the recursive formulation of multiscale mixed methods in multi-core devices. Through several numerical studies, we show that the new algorithm is very fast, and exhibits excellent strong and weak scaling for large problems motivated by the numerical simulation of subsurface flows. We also provide a unifying presentation for a family of multiscale mixed methods accessible for users interested in applications.
Keywords: Multiscale Mixed Method; Darcy’s Flow; Porous Media Flow; Relaxation;
1 INTRODUCTION . . . 12
1.1 Motivation . . . 12
1.2 Objectives and project proposal . . . 16
1.3 Main results and scientific contribution . . . 17
1.4 Overview of the thesis . . . 18
2 A RELAXATION PROJECTION APPROACH IN HYSTERETIC TWO-PHASE FLOWS . . . 20
2.1 Motivation and Introduction . . . 22
2.2 Equilibrium regions for the two-phase model hysteresis . . . 28
2.3 Computational Modeling with Hysteresis . . . 32
2.4 Numerical procedures for hyperbolic and diffusion models . . . 35
2.4.1 Domain decomposition iteration and mixed finite element approximations for the diffusion equation . . . 35
2.4.1.1 Hybridized mixed finite element approximation . . . 36
2.4.1.2 Algebraic method for solving the discrete parabolic problem . . . 37
2.4.1.3 Discretization in time and a domain-decomposition iterative procedure for the diffusion subproblem . . . 39
2.4.2 Numerical solution of the hyperbolic-convection equation . . . 40
2.5 Numerical Experiments . . . 42
3 RECURSIVE FORMULATION OF MULTISCALE MIXED METH-ODS AND ITS PARALLEL IMPLEMENTATION . . . 50
3.1 Introduction . . . 50
3.2 A review of the Multiscale Robin Coupled Method . . . 52
3.2.1 Mixed multiscale basis functions . . . 54
3.2.2 Interface system . . . 56
3.3 Recursive formulation . . . 56
3.3.1 Two-subdomain decomposition . . . 57
3.3.2 The recursivity . . . 58
3.3.3 Calculation of mixed multiescale basis functions in a level l ` 1 . . . 59
3.4 Parallel Implementation . . . 62
3.4.1 Connection to the Multiscale Mixed Method . . . 63
3.5 Numerical Experiments . . . 66
3.5.2.1 Model Problem 1 - Homogeneous Permeability . . . 74
3.5.2.1.1 Strong Scaling . . . 74
3.5.2.1.2 Weak Scaling . . . 75
3.5.2.2 Model Problem 2 - Heterogeneous Permeability . . . 79
3.5.2.2.1 Strong Scaling . . . 79
3.5.2.2.2 Weak Scaling . . . 79
3.5.3 Qualitative study of the velocity under multi-core configuration . . . 83
4 CONCLUSION REMARKS AND PERSPECTIVES. . . 85
4.1 Perspectives . . . 86
Chapter 1
Introduction
The theme of this work consists of the investigation of fundamental problems in fluid dynamics in porous media. It covers modelling, numerical analysis and computational simulation for numerical resolution of nonlinear PDEs models with highly variable and discontinuous coefficients.
Our objective is to study mathematical models in fluid flow with highly variable, discontinuous and dynamic coefficients in porous media. First, we consider two-phase flow problems where a time dependent coefficient comes from the introduction of hysteresis into our system. Second, we focus our attention in the approximation of the pressure-velocity problem of elliptic nature by the use of a multiscale method motivated by an operator splitting approach and by computational difficulties encountered in the approximation of this problem.
1.1
Motivation
Transport problems in fluid flow in porous media are modeled by nonlinear partial differential equations (PDEs). The mathematical comprehension of the transport phenomena in multi-phase fluid flow dynamics in porous media is very important in many contexts, from pure mathematics (uniqueness, existence and regularity of solutions) to numerical analysis [11,12, 19, 93, 130, 65, 150, 141, 149, 195, 129] and applications [73,
5, 2, 11]. There are many challenges and difficulties in the mathematical modelling and numerical analysis of the PDEs that govern multiphase fluid flow in porous media (see, e.g., [129, 73, 18, 123]). One big difficulty comes from its inherent multiscale nature induced by the geological properties of the rocks that compose the reservoir (e.g., “permeability kpxq” and “porosity φpxq”) and modelled as highly variable and discontinuous coefficients [89]. Also, analytical solutions are very hard (e.g., [73, 89, 103, 130, 65, 93, 163,141]) and sometimes impossible to obtain with traditional techniques due to the nonlinear nature of these PDEs. This means that numerical analysis of algorithms is an alternative in the
mathematical understanding of these PDEs.
Consider the two-phase flow equation for incompressible and immiscible flow, with water and oil phases, in phase formulation [64]
B
Btpφ pxq Swq ` ∇ ¨ pufw` Gwq “ ∇ ¨ ww (1.1)
∇ ¨ u “ 0, u “ ´Kλ∇po` uwo` uG, (1.2)
where Sw, u and po are the water saturation, the Darcy’s flux and the oil pressure. The
term φ is the porosity, and Kpxq is the absolute permeability tensor. The phase relative mobilities λi, total mobility λ and fractional flow fi functions are defined, respectively, as
functions of water saturation given by λipSwq “ kipSwq{µi, λpSwq “
ÿ
i“w,o
λipSwq, fipSwq “ λpSwq{λipSwq, i “ w, o, (1.3)
where kipSwq is the relative permeability and µi the viscosity in each phase. The diffusive
flux ww and the correction terms uwo and uG are also functions of Sw given by,
wwpSwq “ ´K pxq rλwpSwq p1 ´ fwqs ∇pc, uwopSwq “ ´K pxq λwpSwq ∇pc, (1.4)
and Gw “ GwpSwq is the term related to the gravity,
GwpSwq “ rλp1 ´ fwpSwqqρwosg (1.5)
For each phase i “ w, o, pi is the phase pressure and ρi is the phase density where ρwo “ ρw´ ρo. The capillary pressure pc is defined as the pressure difference across the
interface between two immiscible fluids arising from the capillary forces in porous medium. Operator splitting techniques aiming at the approximation of differential op-erators like the systems of partial differential equations (1.1)-(1.2) have been frequently used in problems of simulation of fluid flows in fluid dynamics problems in porous media, such as in natural aquifers and in oil reservoirs, particularly in problems of single-phase and two-phase flows. Physics based operator splitting separates the underlying phenomena and show clearly its mathematical nature, that is, in the form of hyperbolic, parabolic, elliptic or mixed models in the context of (local) classification of PDEs. Essentially, for each time step ∆t we solve sequentially the following subproblems,
1. The elliptic-Poisson subproblem for the oil pressure po and velocity field u,
∇ ¨ u “ q, u “ ´Kpxqλ∇po` uwo` uG. (1.6)
2. The purely convective transport subproblem for water saturation Sw,
B
3. The purely diffusive transport subproblem for Sw,
B
BtpφpxqSwq “ ∇ ¨ pwwpSwqq. (1.8)
In this way, different and appropriate numerical techniques can be employed within the operator splitting formulation to provide efficient numerical procedures, see some relevant work in fluid dynamics [96, 130, 108, 165, 89, 103, 25, 85, 74,147, 76].
In this thesis, the discussion of operator splitting motivates the identification of a transport problem (1.1) (and equations (1.7)-(1.8)) and a coupled Poisson problem (1.2) (and equation (1.6)). Our objective is to discuss appropriate numerical techniques to approximate accurately and/or efficiently each subproblem. For the transport problem we are concerned with accurate numerical approximation of related one-dimensional waterflooding transport flow problems with hysteresis phenomenon in porous media. For the second-order Poisson elliptic problem we are concerned with the development of a computational efficient procedure involving multiscale methods.
In the first half of this thesis we develop a new analytical-numerical approach to deal with relative permeability hysteresis models. The system of equations form a coupled, highly nonlinear system of time-dependent PDEs whose solutions typically exhibit strong gradients which are very difficult to approximate numerically. Mathematical and computational modeling of hysteresis is very challenging and has attracted attention due to the complexity, difficulties and richness as a fundamental problem [4, 42, 43, 67,
72, 82, 102, 128, 202, 174, 184]. In porous media flows, hysteresis is recognized to be present in permeability and in capillarity closure relations [67,135, 155]. The phenomenon of hysteretic permeability during water imbibition is relevant in enhancement of carbon dioxide trapping in deep aquifers [179] and in petroleum engineering [92]. On the other hand, the process of imbibition and drainage behaves differently for water flow in unsaturated soils and capillarity hysteresis becomes relevant [201].
The modeling of hysteresis appears together with a discontinuous flux even in 1D [70,184], where a lack of regularity (non smooth) upon the flux function is observed [4,
17]. In applications, the situation where gravity takes place leads to intricate configurations in the solution [2, 3, 17, 27, 72, 184] and hysteresis modeling might lead to a fundamental comprehension of upscaling of pore-scale forces [178]. However, rigorous mathematical analysis is very limited, in particular, when involving systems of equations as discussed in this thesis. Also, standard mathematical models for flow in porous media such as the two-phase flow system cannot explain the fingering effect alone due to the stability properties of these equations, unless hysteresis or dynamic terms are included [2,125,180]. In situations where we have a flow regime with dominant convection it is possible to ignore the diffusion effects and in this case, the hysteresis effect can be modeled by the relative permeability curves [17, 174]. This is typically the case in Petroleum engineering, and
also the case considered in this thesis. In this context, we propose a two-fold approach to deal with hysteretic two-phase flows in porous media. First, we introduce a new analytical projection method for solution of the Riemann problem for the system of equations for a prototype two-phase flow model via relaxation. Second, a new computational method is formally developed to corroborate our analysis along with a representative set of numerical experiments to improve the understanding of the fundamental relaxation modeling of hysteresis for two-phase flows.
In the second part of this thesis our focus is in multiscale formulations for elliptic problems. Specifically, we are going to construct and implement a parallel recursive general formulation for multiscale mixed methods. The development of multiscale numerical procedures for mixed second order elliptic equations in porous media flow problems has attracted the attention of several research groups and is motivated by the practical needs of the energy and environmental sectors [34,205, 106, 122, 32, 132, 133]. For a review of the many multiscale methods in the literature, see [91, 146, 186]. In general, multiscale methods obtain coarsened models (with much fewer degrees of freedom) that nevertheless incorporate the fine-grid details of the permeability field. See [146] for important multiscale methods in the literature. The choice of a multiscale mixed method to approximate the pressure-velocity problem comes from the necessity of computationally efficient and fast solvers that are able to obtain an accurate solution of high contrast and discontinuous coefficients fluid flow in porous media problems, in a reasonable amount of time. We can cite several techniques for discretization of elliptic problems, such as, for example, Finite Differences [198,188], Finite Volumes [87,203], Mimetic finite differences [158,44,
49, 188], Finite Elements (Classical) [48],Galerkin Discontinuous Finite Elements [79,36,
173], Mixed Hybrid Finite Elements in Hpdivq spaces [181, 182, 50] among others, like generalized finite elements [40]. In general, the idea is to obtain a linear system with good mathematical properties, such as “well conditioned” and a local conservative nature at the discrete level, corresponding to its continuous differential operator.
The resolution of large sparse systems of linear equations is key to many numerical simulations in science and engineering problems and is often the most time consuming part of a calculation since etailed multi-physics simulations, in two- and three-dimensional problems, leads to linear systems that can easily reach the order of hundreds of millions (2D) or even billions (3D) of equations with corresponding unknowns. Even in two-dimensional problems, due to the heterogeneous nature of the porous medium, it is natural to incorporate a notion of uncertainty, which will lead to stochastic PDEs [112,
113, 111, 37, 103, 165, 2] and a Monte Carlo approach is a viable alternative. Essentially, this means calculating statistical moments [112, 113, 111, 103] coming from a set of deterministic experiments in orders of hundreds or thousands (desirable) of computational experiments. This way, fast and robust methods becomes a necessity. For this reason, parallel computing has penetrated the same areas of application. As the computer became
widely available a new incentive was created to develop methods that take advantage of these powerful computational tools. In addition, parallel machines may require different architectures in such a way that the algorithms require different programming paradigms and radically different from the classic and consolidated methods in the literature. As a bibliographical review, it is observed that there are several texts on the subject multigrid and multiscale, as for example [118, 166,51, 172, 91,146]. Although a number of other computational packages are available and ready for immediate use, such as [200, 196, 167,
77, 193, 51, 118,166,172, 193], many of these are not user-friendly. Even so, high quality open source code requires a considerable amount of time for good learning.
Concerning applications, this work covers modelling and numerical solution of nonlinear PDEs with potential application in scientific and technological problems in energy and environment resources, such as
• climate change and global warming containment by carbon capture and sequestration in oil reservoirs and aquifers with high salt concentration [68, 53, 55, 97, 22, 109,
147, 110];
• remediation of subsoil contamination by non liquid phases [53, 55, 97, 22, 109, 147,
110];
• Advanced recovery of hydrocarbons in oil reservoirs, including pre salt layer [68, 97,
22, 109,147,54, 46, 2,5, 16, 18,89];
• Scale transfer problem is a important scientific topic, and appears naturally in the previous areas [165, 97,89, 103, 112, 113, 111].
1.2
Objectives and project proposal
In general, the theme of this work consists of the numerical study of fundamental problems in fluid dynamics in porous media.
The objectives of this work are:
• Discuss a one-dimensional analytical-numerical approach for relaxation hysteresis model for two-phase flow in porous media. .
• Present a recursive formulation for the resolution of interface problem generated by multiscale mixed methods for the numerical resolution of elliptic problems in fluid flow in heterogeneous porous media .
1.3
Main results and scientific contribution
In this work, we developed new computational methods to approximate the solution of fundamental problems in two-phase fluid flow in porous media. Specifically, we focus on two fronts: problems with dynamic coefficients introduced by a hysteresis in the relative permeability and in the application of a multiscale method to solve the elliptic problem.
The study of the Poisson-elliptic problem is motivated by the successful use of an operator splitting approach to separate the numerical resolution of convective, diffusive and elliptic subproblems that compose the two-phase flow problem. In the context of this work, after the application of the operator splitting, the elliptic problem is then fully identified, after a conveniently linearization of the velocity coefficient, or Darcy’s law [181,
182, 50, 48]. Then, we turn our attention in the approximation of the elliptic problem by means of a multiscale method.
The scientific learning of this work embraces the Computational and Applied Mathematics in the fields of Numerical Analysis and Computational Simulation of physical processes. The main scientific works generated by the current thesis are listed as follows:
Accepted papers:
• E. Abreu, A. Bustos, P. Ferraz and W. Lambert, “A Relaxation Projection Analytical-Numerical Approach in Hysteretic Two-Phase Flows in Porous Me-dia”, Journal of Scientific Computing, 79(3) (2019) 1936-1980, [6].
Conference papers:
• Conference proceeding: E. Abreu, A. Bustos, P. Ferraz and W. Lambert, “A computational multiscale approach for incompressible two-phase flow in hetero-geneous porous media including relative permeability hysteresis”, published in Proceedings of the 6th International Conference on Approximation Methods and Numerical Modelling in Environment and Natural Resources (MAMERN), 2015, [4].
Conference participation:
• Conference presentation “A computational multiscale approach for incompress-ible two-phase flow in heterogeneous porous media including relative perme-ability hysteresis”, 6th International Conference on Approximation Methods
and Numerical Modelling in Environment and Natural Resources (MAMERN), 2015.
• Conference presentation “A multiscale approach for pressure-velocity two-phase flow high-contrast porous media”, 2nd IMPA-InterPore Conference on Porous Media Conservation Laws, Numerics and Applications, 2016.
• Poster “A computational approach for incompressible two-phase hysteretic rela-tive permeability flows in porous media”, West Texas Applied Math Graduate Minisymposium 2 at Texas Tech University, April 2018.
• Conference presentation “Recursive parallel implementation of multiscale mixed methods”, InterPore 10th annual meeting and jubilee, May 2018.
• Conference presentation “A Relaxation Projection “A relaxation projection analytical-numerical approach in hysteretic two-phase flows in porous media”, InterPore 10th annual meeting and jubilee, Valencia, 6-10 May, 2019.
• Conference presentation “Multi-core recursive implementation of a multiscale direct solver for flow in high-contrast formations”. InterPore chapter Brazil at LNCC, Petrópolis, 5-8 August, 2019.
In development:
• E. Abreu, P. Ferraz, F. Pereira, A. Santo, L. G. C. Santos and F. S. Sousa, “A Recursive Formulation of Multiscale Mixed Methods and its Parallel Implemen-tation”. Developing manuscript (2019).
Other colaborations:
• One year as a visitor research at the University of Texas at Dallas, US, between September of 2017 and September of 2018.
1.4
Overview of the thesis
An overview summary of scientific achievements about this Doctoral Thesis on computational and analytical methods of a novel recursive formulation of multiscale mixed methods as well as of relaxation modeling of flow in porous media and applications are listed below:
• The design of a new scheme based on conservative discretizations within an operator splitting which is robust and it is able to reproduce effectively the good entropy solutions to the relaxation hysteresis system (Chapter 2).
• Locally conservative mixed finite elements will be used to discretize the spatial operators in the underlying diffusion subproblem resulting from the splitting approach along with a domain decomposition method, and time discretization is performed by the implicit backward Euler method [6] (Chapter 2).
• We make use of the robust Nessyahu and Tadmor [170] scheme along with the first convergence proof for the Lax-Friedrichs given by Karlsen and Towers [145] for the numerical approximation of the hyperbolic problem (Chapter 2).
• The theoretical-computational study along with the provided considerations, give new insights for modeling hysteresis in porous media flow (Chapter 2).
• We develop a novel recursive formulation of multiscale mixed methods and propose a new parallel algorithm to solve the interface problem (Chapter 3).
• In this new formulation the global interface problem is replaced by a family of small interface systems associated with adjacent subdomains, in a hierarchy of nested subdomains (Chapter 3).
• Our novel formulation is general and extends the multiscale direct solver recently introduced in [24] (Chapter 3).
• To show the ready applicability of the novel approach, we implement the proposed new parallel algorithm for the Multiscale Robin Coupled Method [119], that can be seen as a generalization of several multiscale mixed methods. Thus, our novel parallel algorithm can be easily applied to many multiscale mixed procedures (Chapter 3). • We show by means a representative set of numerical studies that the new algorithm is
very fast, and exhibits excellent strong and weak scaling for large problems motivated by the numerical simulation of subsurface flows, in particular for simulation of huge petroleum reservoirs (Chapter 3).
The thesis is organized as follows: In Chapter 2 we present the mathematical model of two dimensional two-phase flow problem with hysteresis in porous media and the computational modeling by operator splitting, the one dimensional hysteresis problem, the numerical approach of each subproblem and numerical results. In Chapter 3 we present a new recursive formulation and parallel implementation to solve interface problems in a family of mixed multiscale methods together with scaling results to show efficiency of the implementation. Some concluding remarks and perspectives are presented in Chapter 4.
Chapter 2
A relaxation projection
analytical-numerical approach in
hysteretic two-phase flows in porous
media
Hysteresis phenomenon plays an important role in fluid flow through porous media and exhibits convoluted behavior that are often poorly understood and that is lacking of rigorous mathematical analysis. We propose a two-fold approach, by analysis and computing to deal with hysteretic, two-phase flows in porous media. First, we introduce a new analytical projection method for construction of the wave sequence in the Riemann problem for the system of equations for a prototype two-phase flow model via relaxation. Second, a new computational method is formally developed to corroborate our analysis along with a representative set of numerical experiments to improve the understanding of the fundamental relaxation modeling of hysteresis for two-phase flows. In [6], the authors introduce a new projection method for Riemann problem analysis for a two-phase transport model via relaxation and a comprehensive study to cover all possible solutions. It provides rigorous and fine analysis for construction of analytical solutions for fundamental two-phase flow problems in porous media with relaxation and a solid basis for the overall computational approach. This approach may lead to an improvement in the comprehension of the nonlinear interactions of fluids in porous media under non-equilibrium conditions. Using the projection method it is shown the existence by analytical construction of the solution. The proposed computational method is based on combining locally conservative hybrid mixed finite method and finite volume discretizations within an operator splitting formulation to address effectively the stiff relaxation hysteretic system modeling two-phase flows in porous media.
analytical and the computational approach for solving hysteresis via relaxation of porous media, and its new contributions are:
• The design of a new scheme based on conservative discretizations within an operator splitting which is robust and it is able to reproduce effectively the good entropy solutions to the relaxation hysteresis system.
• Hybridized mixed finite elements are used for the spatial discretization along with a domain decomposition strategy towards the solution of the resulting algebraic problem. Our choice for the HMFEM as such is related to the use of Robin boundary conditions as natural interfacial transmission conditions in the domain decomposition iterative procedure to the case of numerical simulation of wave propagation in nonlinear three-phase flows in porous media with spatially varying flux functions. • We make use of the robust Nessyahu and Tadmor [170] scheme along with the
first convergence proof for the Lax-Friedrichs given by Karlsen and Towers [145] for the numerical approximation of the hyperbolic problem (2.18)-(2.19) exhibiting discontinuous nonlinear hyperbolic flux functions. The Nessyahu and Tadmor scheme is fully based on the Lax-Friedrichs. This shed light on about our careful choice on the construction of a robust and high-resolution procedure to deal with hyperbolic problem with discontinuous flux.
• The theoretical-computational study along with the provided considerations, give new insights for modeling hysteresis in porous media flow and helps towards a deeper comprehension on the design of new analytical and computational tools on more complex mathematical models as such for two-phase as well as for three-phase flow problems in discontinuous heterogeneous porous media with high-contrast.
This Chapter is structured as follows. Introduction and motivation are given in Section 2.1. The key ideas of mathematical modeling of two-phase flow with hysteresis relaxation model in porous media are discussed. In Section 2.2, we explain waves (shock, rarefactions and contact discontinuities) and some bifurcations structures used to obtain the Riemann solution. We also indicate the projection method and the admissibility criterion for shocks and the behavior of interaction waves with the source terms [6] . The computational modeling by operator splitting technique is introduced in Section 2.3. Then, numerical procedures used for solving the diffusion and hyperbolic problems, are shown in Section 2.4. Finally, numerical experiments for the one-dimensional two-phase flow with hysteresis model are presented in Section2.5 along with semi-analytical Riemann solutions.
2.1
Motivation and Introduction
We consider a combination of Riemann problem analysis and a computational approach for solving a two-phase hysteresis flow model via relaxation.
Mathematical and computational modeling of hysteresis is very challenging under theoretical and applied point of view, in a wide range of problems and has attracted attention due to the complexity, difficulties and richness as a fundamental problem [4, 42,
43, 67,72, 82, 102, 128, 202, 174,184].
In porous media flows, modeling hysteresis is also recognized to be present in both permeability and capillarity closure relations [67, 135, 155]. The phenomenon of hysteretic permeability during water imbibition is very relevant in enhancement of carbon dioxide trapping in deep aquifers [179]. See [92] for a good recent survey for relative permeability hysteresis on related to petroleum engineering. On the other hand, for water flow in unsaturated soils the process of imbibition and drainage behaves differently and capillarity hysteresis is also relevant [201]. In addition, capillarity hysteresis is relevant for the Richards equation incorporating non-equilibrium effects in the capillarity pressure [88].
Modeling hysteresis appears with discontinuous flux even in 1D [70, 184], where a lack of regularity (non smooth) upon the flux function is observed [4, 17]. In applications, the situation where gravity takes place leads to intricate configurations in the solution [2, 3, 17, 27, 72, 184] and altogether modeling hysteresis might lead to fundamental comprehension of upscaling of pore-scale forces [178]. However, a lack of rigorous mathematical analysis in hysteresis modeling is very limited and in particular when involving systems of equations as discussed in this work. In [174] the authors introduced a hysteresis modeling for a scalar two-phase flow in porous media via relaxation and in [8] a formalism to deal with relaxation system for flow in porous media was introduced and the analysis includes the latter equations.
Generically, the models used to study the dynamics of hysteresis in porous media are hyperbolic or hyperbolic-parabolic system of equations with hyperbolic dominance, using scalar modeling [4, 27,42,43, 67, 82,88,59, 83, 84, 108, 136,141, 75,144, 202,148,
174, 184] or system of equations [2, 3, 8, 9, 14, 15, 38, 62, 64, 63, 73, 86, 101, 100, 131,
39, 130, 137,138, 139, 153,161, 163, 190]. From the mathematical viewpoint, for systems with hyperbolic dominance there is a challenge in finding a notion of unique solution [17,
70, 202, 184]. The non-uniqueness is very common for the weak solutions in hyperbolic systems and this situation gives rise to several admissibility criteria, or entropy conditions, that select the unique solution; see also for distinct and more general concepts of entropy solutions linked to hyperbolic-parabolic flux functions and the issue of Riemann problems for classical and non-classical solutions [9,10, 26, 27, 30, 31, 38, 61, 63, 71,73, 78, 101,
an alternative modeling process to account for the “exact” physical problem along with uniqueness of solution for a given initial datum? For problems in multiphase flow, in [8], we introduce a novel modeling of phase transitions in thermal flow in porous media by using hyperbolic system of balance laws, instead of a system of conservation laws. Roughly speaking, relaxation encompass entropy conditions in a natural and general form for scalar and systems (see [8, 7,136,161, 174] and references therein). It is worth mentioning that our relaxation interpretation of entropy conditions discussed in this Chapter (see [8]) agree with a previous work of Natalini & Tesei [169], which was proposed by G. I. Barenblatt [41] to describe non-equilibrium two phase fluid flow in permeable porous media; see also [1, 137]. In addition, we will discuss this approach related to issues in non-equilibrium effects that is very relevant and recognized by many authors in distinct models [20,41, 42,
43,63,80,82,88,117, 124, 137,169,185]. We mention the recent works [60,175,176,177,
187] for a novel techniques to deal with equilibrium-type models.
For concreteness, in this work we employ an numerical approach to study relaxation modeling hysteresis loops, imbibition and drainage in the relative permeability, first introduced in [174] for a two-phase porous media flow model without gravity. We extended the modeling considering gravity and physical parameters involved to control all possible flows. For this model, we study the interaction between waves (shock and rarefactions) in equilibrium and far from equilibrium. The new numerical scheme results are corroborated by the analytical approach obtained in [6]. Since the fluids are assumed to occupy the whole pore space, we have Sw` Sq “ 1, i.e., Sq “ 1 ´ Sw (q “ oil or q “ gas).
Thus, we can write all functions only on water saturation. The model problem is, BSw Bt ` B BxpufwpSw, πq ` GpSw, πqq “ B BxwpSw, πq, (2.1) Bπ Bt “ g pSw, πq “ $ ’ & ’ % pπIpSwq ´ πq{τ, if π ă πIpSwq , pπDpSwq ´ πq{τ, if π ą πDpSwq , 0, otherwise. (2.2)
The 2 unknowns for the system (2.1)-(2.2) are the water saturation Sw “ Swpx, tq and the
parameter π “ πpx, tq modeling hysteresis, defined in R ˆ R`. The gravity GpS
w, πq and
the diffusive wpSw, πq terms are given by,
GpSw, πq “ rθp1 ´ fwqρwos g, w pSw, πq “ DpSw, πq BSw Bx , (2.3) where DpSw, πq “ ´ rθwpSwqp1 ´ fwqs Bpc BSw
, along with the closure of the equations (2.1 )-(2.2), the relative mobilities θi “ θipSwq, the total mobility θ “ θpSwq, the fractional flow
functions fi “ fipSwq, and the total velocity u (i “ w, o): θipSwq “ kipSwq µi , θpSwq “ kw µw ` ko µo , fipSwq “ θi θ, u “ uo` uw. (2.4)
The two-phase water-gas case is similar. For each phase i, ki is the relative permeability, µi
the viscosity, pi denotes the pressure and ρi is the density (where ρwq “ ρw´ ρq, q “ oil or g “ gas); g is the gravity number. In 1D the incompressibility condition implies Bu{Bx “ 0, i.e. the total fluid velocity u is independent of position, and thus we take it to be constant in space and time. In (2.2), πI and πD relate to the imbibition and drainage processes, respectively, via the hysteresis mechanism and the scanning curves modeling the hysteresis loops (see Figure1), and τ is the relaxation time. Following [174], we take the water relative permeability function kwpSwq, the oil drainage kDo and imbibition k
I
opSwq permeability
functions given by,
kwpSwq “ Sw2, k D
o pSwq “ p1 ´ Swq3 and kIopSwq “ p1 ´ Swq2. (2.5)
Each scanning permeability curve is given by [174],
koSpSw, πq “ p1 ´ πq2p1 ´ α πq´1p1 ´ αSwq . (2.6)
Using relations kIopSwq “ koSpSw, πq and koDpSwq “ kSo pSw, πq, one can derive the following
explicit formulas for πIand πD, where 0 ď Sw ď 1 and 0 ď πI, πD ď 1. The water fractional
flow function f pSw, πq is expressed by (2.5)-(2.6), and the capillary pressure function pc“ pcpSwq are given by:
f “ kwpSwq {µw
kwpSwq {µw` kSo pSw, πq {µo
and pc“ 5Pcp2 ´ Swq p1 ´ Swq , (2.7)
where Pc is a characteristic pressure. The capillary pressure difference pcpSwq “ pw´ po is
measured empirically to be a decreasing function of Sw [42,43,82,202]. For the theoretical
analysis we use Riemann conditions for the two unknowns (the water saturation Sw and
the hysteresis parameter π)
pSwpx, 0q, πpx, 0qq “ $ & % L “ pSL, πLq if x ă 0; R “ pSR, πRq if x ą 0. (2.8)
Remark 1. For numerical purposes, we assume Spx, tq and πpx, tq are defined in Ω “
ra, bs ˆ R`, thus we need to supply the system of equations with initial and boundary conditions in the form (of course, fully compatible with (2.8)):
Spx, 0q “ η1pxq, πpx, 0q “ η2pxq, x P Ω, (2.9)
Spa, tq “ SL, Spb, tq “ SR, πpa, tq “ πL, πpb, tq “ πR, t ą 0. (2.10)
The approximation of these conditions is discussed in details in Section2.3 through step 1) until step 5) along with the operator operator splitting.
Eqs. (2.1)-(2.2) are of parabolic-type with non-smooth functions (non-differentiable hyperbolic/parabolic flux functions modeling the loops; see also Figure 1b with respect
to quantities Sw and π. However, the proposed numerical method for the full problem
(2.1)-(2.8) is designed to handle discontinuous flux functions induced by the dynamics of the hysteresis model as in [70] (even in a 1D model; see also [145,199]) or by the influence of the discontinuous permeability in a multi-D inhomogeneous porous medium problem as in [2].
(a)Oil imbibition koIpSw, πq, drainage kDo pSw, πq,
water kwpSwq permeabilities.
(b)Scanning curves for oil permeability curves of constant π.
Figure 1 – Oil permeability depends not only on the fluid saturation but also on the hysteresis parameter and on the saturation tendency (imbibition and drainage processes).
Re-laxation (SHMR) and was proposed in [174]; see also [8,4]. It describes the phenomenology of relative permeability hysteresis by using a hysteresis parameter π to parameterize the scanning curves and allowing non-equilibrium states relax quickly to equilibrium. The equation contains functional relationships which depends on the flow mode (drainage or imbibition) and the history parameters. The solution consists of continuous waves (expansion waves and constant states), shock waves (possibly connecting different modes)
and stationary discontinuities (connecting different saturation histories).
In [117], a non-equilibrium model linked to hyperbolic discontinuous flux functions was discussed. This brings to attention the relevance of non-standard models where the flux function is written in the form F pS, x, tq; see also e.g., [2,21,30, 31,58, 70,
144, 142, 143, 199].
Here we discuss a nonstandard model with hysteresis that can also be modeled by discontinuous flux functions [2,3, 17, 27,70,72, 184]. During the recent years the case where the flux function is discontinuous has been an active research area [2, 56,57,61, 78,
81, 107, 116, 121, 145, 148, 160, 164, 197, 199]. The survey paper [26] and other recent developments [9,21, 30,31,58, 63,71,143,145,189,190,199] allows to better understand several relevant issues and new conceptual notions of weak entropy solutions with specific modeling assumptions for scalar and multi-D problems with discontinuous nonlinearities in space-time.
Problems of the type (2.1), linked to nonlinear degenerate parabolic system of convection-diffusion equations (see [11]) or for a non-classical three-phase flow problem with discontinuous coefficients in case of gravity in 2D (see [2]), or for a nonlinear system with discontinuous coefficients (see also [10,28,29,144] and references cited therein), when related to a formulation similar the one treated in this work, occur in several applications. However it is always closely linked to the purely hyperbolic counterpart. These problems are also challenging because, upon discretization, the differential-algebraic result in very high-dimensional nonlinear constraints to be solved [60,175, 176, 177, 187]. Essentially, in these works a global approach were considered based on combining in a clever way Newton’s method (e.g., like inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion) with a slower converging technique like Picard iteration or the L-scheme from [176] to improve robustness for initial approximation while recapturing quadratic convergence on the overall iterates approach the true solution (see also [159]).
Thus, exploiting the structure of these models in their basic nature is essential to make the (approximation) solution of the differential model problem tractable. Here we describe an operator splitting technique based on physics rather than on dimension for the numerical solution of a nonlinear system of partial differential equations which models two-phase and three-phase flow through heterogeneous porous media.
formulations [2, 14, 15] (see [145] and also Section 2.4.2) to deal with two-phase and three-phase problems in porous media flows by means of an operator splitting formulation. An important feature of this approach is a simple straightforward generalization for non-classical systems and for multidimensional heterogeneous flow problems. In [2, 3], the case of non-classical three-phase flow with gravity were considered where the hyperbolic flux functions and the static capillary pressure difference model are both discontinuous in space (induced by the permeability and the porosity properties of the porous medium). Many others successfully numerical methods based upon an operator splitting formulation have also been discussed in the literature [130, 141,75]. The study of operator splitting techniques has a long history and has been pursued with various methods (see the recent survey on this subject [115]).
Distinct operator splitting techniques for two- and three-phase transport prob-lems in multiscale heterogeneous porous media flow are established upon an operator splitting formulation, where convective-diffusive forces and pressure velocity mechanisms are accounted for in separate substeps [2, 3, 4, 11, 14,15, 20, 83, 84,130, 141, 75].
In those works, different techniques are used to deal with nonlinear parabolic convection-diffusion equations (without focus on the issue of discontinuous flux functions). However, classical operator splitting might fail in solving nonlinear standard non-equilibrium models in porous media [20]. In this regard, modeling problems with relaxation (under non-equilibrium conditions) might help to shed additional light on this subject to better understand the fundamental nonlinearities interplay in porous media systems [8].
The following works proposed new directions and trends for the discretization of porous media models based on the mixed finite element method and multiscale mortar mixed finite element discretizations [34, 47], adaptive enriched Galerkin methods [95,151], multipoint flux mixed finite element method for Darcy flow on non-matching hexahedral grids [105], novel adaptive algorithm based on conjugacy on a posteriori error estimates with stopping criteria [94, 204], multiscale mixed methods [Valentin13, 119], generalized multiscale finite element methods [90] and also high-order conservative method based on finite volume/element discretization [13]. We also mention a mixed finite element discretization of some degenerate parabolic equations including the class of Richards problems [176, 177]. In this present work, the concretate in improving comprehension on the analytical and computational approach for solving a relaxation system modeling hysteretisis π in which the first order flux functions is non-smooth and thus challeging. However, in case of the parabolic degeneracy, we might use the approach developed in [11] to deal with a nonlinear non-classical three-phase flow system in 1D and 2D heterogeneous porous media flows. In this latter work, it uses an operator splitting combined with mixed finite element, and a decomposition of the domain into different flow regions. Compatibility conditions are obtained to bypass the degeneracy convection dominated parabolic system
of equations and be numerically tractable without any mathematical trick to remove the singularity and without use of a parabolic regularization.
Indeed, we mention some domain decomposition methods and related procedures on the rich literature, but more closely linked to porous media transport flow problems as treated in this work, such as space-time domain decomposition in mixed formulations for advection-diffusion problems [127, 126] and application to partially saturated flow [104,
187].
Capillary modeling under non-equilibrium conditions (e.g., hysteresis, dynamic capillarity, interfacial areas, percolation processes in general) has been recognized to be relevant in the modeling porous media flow processes; a non-exhaustive list of some relevant works can be found in [4, 42, 43, 67, 82,102, 128, 202, 174, 184]; we highlight paper [88] that considers an extension of the Richards equation, where non-equilibrium effects like hysteresis and dynamic capillarity are incorporated in the relationship between water pressure and saturation. Our approach here is based on a relaxation modeling hysteresis as discussed in [174], but with a more general analysis with an addition in the design of a effective numerical scheme that can be extended to the case of three-phase flow as discussed in [2, 14, 64] that also consider a close connection to the purely hyperbolic counterpart to the case of systems lacking a general theory (see [2, 3, 62] and references cited therein). See papers [39,138,139] for a relevant discussion on modeling of thee-phase flow relative permeabilities. Notice that our computational modeling choice is due to the generality of the so-called phase formulation [64] (see also [2]), in which we do not care about the form of the relative permeabilities [69,86,163,194] and allows the use of general capillary pressure models employed on practical applications [52, 154, 155]; see also [20]. The correct modeling of capillarity effects plays an important role in three-phase models [38, 101,100, 131,163] including the notions of entropic solutions as well as existence and uniqueness results linked to the the definition of the capillary pressures in the interior of the triangle of saturations. The three-phase flow problem is very challenging and the technique introduced in this work can be extended for such problems.
2.2
Equilibrium regions for the two-phase model hysteresis
We define three important mathematical structures: iq´ scanning region pSw, πq
where πIpSwq ă π ă πDpSwq; iiq´ the imbibition curve pSw, πIpSwqq; and the iiiq´
drainage curve pSw, πDpSwqq. The union of these structures forms the equilibrium region,
EQR“ tpSw, πq such that πIpSwq ď π ď πDpSwqu. (2.11)
The expression for πI, is obtained by setting koIpSwq “ koSpSw, πq; after algebraic
manip-ulations we obtain a functional form for πI, such that 0 ď πI ď 1 and 0 ď Sw ď 1. A
Figure 2 – The phase space pSw, πq and the curves πI and πD and the equilibrium region
EQR.
of relative permeabilities with hysteresis koDpSwq, koIpSwq, kSopSw, πq and kwpSwq used in
this paper are presented in the numerical experiments in Section 2.5. In Figure 2, we present the phase plane pSw, πq, where we draw EQR and the curves πI and πD.
Remark 2. In [174] and [184], the authors give a physical interpretation for the construc-tion of drainage and imbibiconstruc-tion curves. These physical condiconstruc-tions are used to select, in some situations, the correct wave in the Riemann solution. In our work, we obtain the correct wave using our projection method and the study of existence of traveling waves and interaction waves, see [6].
Equation (2.1) can be nondimensionalized by identifying characteristic numbers for each variable and two dimensionless groups appear. The first one, designated by σg, quantifies the ratio between the gravity effects and convective effects. The second dimensionless group, designated by σp, represents the ratio between convective and diffusive effects, respectively by σg “ kcρwig vcµi , σp“ pckc vcµiL , i “ o, g, (2.12)
where for two-phase flow systems, the dimensionless group σp “ 1{Péclet is of order 10´4 to 10´2 [42]. After nondimensionalizing, the system (2.1) reads,
BSw Bt ` B Bxrufw` σgGs “ B Bxpσpwq . (2.13)
Since the total velocity u is considered constant, we can, after nondimensionalizing time and space variables, remove its explicit dependence from the governing system. We will
specify the values σp and σg in our numerical experiments. To be closer to more realistic real-life transport problems, the characteristic pressure Pc in equation (2.7) is taken to be 1.18215ˆ105 Pa, so that the corresponding dimensionless value 1{Péclet, is of order 10´3
for two-phase flow [42]. Indeed, more general capillary pressure curves based on J-Leverett function can be handled by our approach (see [2] for an immiscible three-phase flow in porous media with discontinuous capillary pressure in 2D). We take µo “ 1.0 cP and µw “ 0.5 cP. By taking advantage of the dimensionless system (2.13) we performed a general
analysis to cover the two flow situations, namely, water-oil and water-gas flows. Indeed, we were able to give a simple geometric interpretation for further use in numerical experiments and elsewhere. After a simple analysis one can show mathematically that there are 8 possible cases (see Figure 3covering water-gas and water-oil systems) depending on the relationship between σg and the speed u that are pσg ă 0, u ă 0q (Fig. 3a), pσg ą 0, u ă 0q (Fig. 3b), pσg ă 0, u ą 0q (Fig. 3c), pσg “ 0, u ă 0q (Fig. 3d), pσg ą 0, u “ 0q (Fig.
3e), pσg ą 0, u “ 0q (Fig. 3f), pσg “ 0, u ą 0q (Fig. 3g) and pσg “ 0, u ă 0q (Fig. 3h). Moreover, we also identify three different regimes which are more representative and we show some Riemann solutions for these three cases which are: pσg ą 0, u “ 0q (Fig.3f), pσg ă 0, u ă 0q (Fig. 3a) and pσg ą 0, u “ 0q (Fig. 3e) (see [6] for all the details). The Riemann solution for the other 5 remain relationships between σg and u are very similar (in some cases, particular solution) to the ones obtained for the three representative cases.
We note that for each phase i “ w, g the dimensionless groups σp and σg (in 1D as well as in Multi-D; see [2]) are embedded in the pressure-velocity, and the transport diffusive and gravity-convection terms. Indeed, notice that for the case of water-gas two-phase problem, the resulting system of equations for water-gas transport flow problem is similar, replacing the subscript o by g.
We stress here that the phase plane described in Figure 2 does not depend on the parameters σp or u, for all 8 regimes the phase plane is given by Figure 2.
Riemann solution is a powerful tool to analyze the behavior of solution of hyperbolic systems. Moreover, it is very useful to validate the quality and accuracy of numerical method. The calculation of the rarefaction and shock waves is fundamental to obtain the Riemann solution and is the main ingredient to construct the Riemann solution [73]. We introduce a projection method for construction of the sequence of waves of the solution of the local Riemann problem involving relaxation modeling. To justify our method, we studied the viscous profile for admissibility of shock waves and the rarefaction behavior in the interaction waves, for details on the projection method see [6]. The projection algorithm introduced in [6] is general in the sense this approach might be used to deal with other relaxation systems [8, 7, 136, 161] and extended for three-phase flow classical [137,138,139] and non-classical problems [2, 39, 62].
(a) pσgă 0, u ă 0q. (b) pσgą 0, u ă 0q.
(c) pσgą 0, u ą 0q. (d) pσgă 0, u ą 0q.
(e) pσgą 0, u “ 0q. (f) pσgă 0, u “ 0q.
(g) pσg“ 0, u ą 0q. (h) pσg“ 0, u ă 0q.
2.3
Computational Modeling with Hysteresis
In (2.2), πI and πD relate to imbibition and drainage processes, respectively, via hysteresis, and τ is the relaxation time. For the numerical viewpoint, using the scanning hysteresis model with relaxation (SHMR) has an advantage in that the non-equilibrium states (extending beyond the drainage and imbibition curves) induced by inherently numerical instabilities relax quickly to equilibrium (treating these curves merely as attractors of states outside the scanning region) [174]. The numerical solution of the non-linear system (2.1)-(2.2) is achieved by an operator splitting technique (see, e.g., [84,
108, 141, 75]) supported by our analysis of the projection method introduced previously (according to [2] we can apply this computational method for multi-D problems).
In our splitting technique we take into account the convection and diffusion effects separately and sequentially. It separates the numerical resolution of the hyperbolic and parabolic operators we can identify in equations (2.1)-(2.2) solving each subproblem sequentially until it reaches the total simulation time. To apply the operator splitting technique we introduce two time steps: ∆tc for the solution of the convective subproblem
and ∆td for the solution of the diffusive subproblem. We take ∆td “ ic∆tc, where ic is a
positive integer. Define tn “ n∆td and tn,κ“ tn` κ∆tc, the diffusive and convective times
steps respectively, where the convective microsteps are determined dynamically by a CFL stability condition. Also, let T “ ns∆td be the total simulation time. For simplicity of
notation, we drop the subscript w in the next sections. All variables are related to the water saturation, otherwise stated. Water is injected at a constant rate q along the left boundary of the domain, x “ 0. The initial data of water saturation and hysteresis are given by right state pS, πqR and left state pS, πqL, at the initial simulation time.
In the algorithm, the water saturation Snpxq is calculated as final values of the calculation in rtn´1, tns for n ą 0 or initial saturations in n “ 0. For tn ă t ă tn`1 solve,
sequentially, the system (2.1)-(2.2) with initial condition,
Spx, tnq “ Snpxq, (2.14)
and the equation (2.2) for the relaxation parameter with initial condition,
πpx, tnq “ πnpxq, (2.15)
with πnpxq in equation (2.15) defined by solution of equation (2.16). This is calculated by first solving the hyperbolic equation for the convective transport in the time step ∆tc, and
second, by solving the parabolic equation for the diffusive transport in time step ∆td. Due
to the stiff time scale, we use the following analytical solution for Eq. (2.2) to account for the hysteresis equilibrium state from the initial data So0pxq “ 1 ´ S0pxq considering ∆tc; in particular for initial data outside the non-equilibrium region (i.e., scanning region,
drainage and imbibition curves), πn“ $ ’ & ’ % πI`So0˘``π0´ πI`So0˘˘ e´∆tc{τ, if π0 ă πI`So0˘ , πD`So0˘``π0´ πD`So0˘˘ e´∆tc{τ, if π 0 ą πD`So0˘ , π0, otherwise. (2.16)
Remark 3. The first step solution of hysteresis equation (2.2) is quite important to avoid non-physical oscillations on the boundary conditions. Notice from equation (2.2) that data outside the non-equilibrium region must achieve equilibrium quickly due to stiff time scale nature of hysteresis equation (see numerical experiments reported in Figures 6, 7 and 8).
To incorporate hysteresis for ∆td we also use for (2.2),
πn`1“ $ ’ & ’ % πIpSonq ``π n ´ πIpSonq˘ e´∆td{τ, if π n ă πIpSonq , πDpSonq ``π n ´ πDpSonq˘ e´∆td{τ, if π n ą πDpSonq , πn, otherwise. (2.17)
The initial condition for the analytical calculation of the hysteresis step is defined by Sonpxq “ 1 ´ Snpxq.
1) Set S0, the initial saturation. Lets calculate the water saturation Sn`1. For n “ 0, 1, . . . , pns´ 1q, do:
2) For κ “ 0, 1, . . . , pic´ 1q, calculate:
a) For t P rtn,κ, tn,κ`1s solve, sequentially, both the following hyperbolic
subsys-tem and the ODE (2.2) of relaxation. Bsκ Bt ` B Bxrf ps κ , πκq ` G psκ, x, πκqs “ 0, (2.18) with initial condition,
sκ`x, tn,κ˘“ #
Snpx, tnq , κ “ 0,
sκ´1`x, tn,κ´1˘ , κ “ 1, ..., ic´ 1.
(2.19)
The quantity sκ represents the approximation for the water saturation obtained by the hyperbolic solver for equation (2.18)-(2.19) for time t P rtn, κ, tn, κ`1s. The convective
prob-lem (2.18)-(2.19) will be approximated by a finite volume method to handle discontinuous flux (see also [2, 30, 31, 145]). In each time step κ “ 0, . . . , pic´ 1q linked to convective
transport, the equation for π is locally solved
Bπ Bt “ g ps κ o, πq :“ $ ’ & ’ % pπIpsκoq ´ πq{τ if π ă π I psκoq , pπDpsκoq ´ πq{τ if π ą π D psκoq , 0, otherwise, (2.20)
to deal with hysteresis effects of the convective step ∆tc, by the formula, πκ`1 “ $ ’ & ’ % πIpsκoq ``πκ´ πIpsoκq˘ e´∆tc{τ if πκ ă πIpsκoq , πIpsκoq ``π κ ´ πDpsκoq˘ e´∆tc{τ if π κ ą πDpsκoq , πκ, otherwise. (2.21)
3) Set ¯spx, tnq “ sic´1px, tn, ic´1q and ¯πpx, tnq “ πic´1px, tn, ic´1q, as initial conditions to
solve the parabolic problem.
4) Calculate ¯spx, tn`1q in rtn, tn`1s by the solving the parabolic subsystem,
B¯s Bt ´
Bwp¯s, x, ¯πq
Bx “ 0, (2.22)
with boundary conditions
wp¯s, x, ¯πq “ 0, x P BΩ. (2.23)
Locally conservative mixed finite elements will be used to discretize the spatial operators in problem (2.22)-(2.23) along with a domain decomposition method, and time discretization is performed by the implicit backward Euler method [2]. Again, we use the formula for hysteresis (2.2) into the diffusion step ∆td,
¯ πpx, tn`1q“ $ ’ & ’ % πIp¯soq`` ¯πpx, tnq´πIp¯soq˘ e´∆td{τ, if ¯πpx, tnq ă πIp¯soq , πDp¯soq`` ¯πpx, tnq´πDp¯soq˘ e´∆td{τ, if ¯πpx, tnq ą πDp¯soq , ¯ πpx, tnq, otherwise. (2.24)
The initial condition for the analytical calculation of the hysteresis step is defined by ¯
sopx, tnq “ 1 ´ ¯spx, tnq.
5) Set Sn`1 “ ¯spx, tn`1q and πn`1 “ ¯πpx, tn`1q.
2.4
Numerical procedures for hyperbolic and diffusion
models
In this section, we overview the numerical procedure that we employ for the solution of the diffusion equation (2.22)-(2.23). Hybridized mixed finite elements are used for the spatial discretization along with a domain decomposition strategy towards the solution of the resulting algebraic problem. Indeed, hybridized mixed finite elements is locally conservative by construction and is quite adequate for accurate velocity field computation in the case of porous media transport problems [66, 64, 45, 83, 84]. It is completely feasible and can be computationally advantageous to define the finite element method for the saturation equation over different partitions of the domain. Moreover, our choice for the HMFEM as such is related to the use of Robin boundary conditions as natural interfacial transmission conditions in the domain decomposition iterative procedure as discussed by [45, 83,157] and its application [2] to the case of numerical simulation of wave propagation in nonlinear three-phase flows in porous media with spatially varying flux functions. Moreover, the implementation of the code is simple because the domain decomposition of elements of the finite element method localizes the computations and the scheme is also naturally parallelizable [64, 45, 83, 157]. We also point out the issue of discontinuous flux for convection-diffusion models with convection-dominated problems is not well understood, see, e.g., [27,31, 30, 59].
2.4.1
Domain decomposition iteration and mixed finite element approximations
for the diffusion equation
Consider the diffusion equation to be solved as seen in Section 2.3, BS Bt ´ BwpS, x, πq Bx “ 0, wpS, x, πq “ DpS, x, πq BS Bx, (2.25)
along with @x P Ω “ r0, Xs Ă R, t ą 0, and proper initial and boundary conditions. We shall take the computational domain to be a uniformly partitioned interval and employ the simplest RT space. We notice that the same partition will be used to the discretization of the convective transport system. Domain decomposition methods have been studied for the most part as algebraic tools for solving problems on parallel machines (see, e.g., [84] and references therein). This is not what we do here. Instead, we employ a spatial decomposition of the diffusion problem (2.25) to construct a simple and efficient iterative method for the numerical solution.
To discretize equation (2.25) we consider the spaces V “ tv P H1pΩqu and W “ L2pΩq. The global weak form for the diffusion equation (2.25) for a time tnă t ă tn`1,
