SCHOOL OF TECHNOLOGY
GRADUATE PROGRAM IN MECHANICAL ENGINEERING
Carlos Augusto Soares Ferreira
Analysis of Numerical Schemes in Collocated and Staggered Grids for Problems of Poroelasticity
Florianópolis 2019
Analysis of Numerical Schemes in Collocated and Staggered Grids for Problems of Poroelasticity
Master’s Thesis submitted to the Graduate Program in Mechanical Engineering of the Universidade Fed-eral de Santa Catarina in fulfillment of the require-ments for the degree of Master in Mechanical Engi-neering.
Supervisor: Prof. Clovis Raimundo Maliska, Ph.D. Co-supervisor: Prof. Hermínio Tasinafo Honório, Dr.Eng.
Florianópolis 2019
Soares Ferreira, Carlos Augusto
Analysis of Numerical Schemes in Collocated and Staggered Grids for Problems of Poroelasticity / Carlos Augusto Soares Ferreira ; orientador, Clovis Raimundo Maliska, coorientador, Hermínio Tasinafo Honório, 2019. 174 p.
Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós-Graduação em Engenharia Mecânica, Florianópolis, 2019.
Inclui referências.
1. Engenharia Mecânica. 2. Finite Volume Method. 3. Geomechanics. 4. Poroelasticity. 5. Stabilization Techniques. I. Maliska, Clovis Raimundo. II. Honório, Hermínio Tasinafo. III. Universidade Federal de Santa Catarina. Programa de Pós-Graduação em Engenharia Mecânica. IV. Título.
Analysis of Numerical Schemes in Collocated and Staggered Grids for Problems of Poroelasticity
This master’s thesis was evaluated and approved by examination committee composed of the following members:
Prof. Emilio Ernesto Paladino, Dr.Eng. Universidade Federal de Santa Catarina
Prof. Sérgio Peters, Dr.Eng. Universidade Federal de Santa Catarina
We hereby certify that this is the original and final version of the essay which was
deemed adequate as a thesis for the degree of Master in Mechanical Engineering.
Prof. Jonny Carlos da Silva, Dr.Eng. Graduate Program Coordinator
Prof. Clovis Raimundo Maliska, Ph.D. Supervisor
Florianópolis, December 19, 2019. Documento assinado digitalmente
Clovis Raimundo Maliska Data: 07/04/2020 18:53:26-0300 CPF: 096.177.449-53
Documento assinado digitalmente Jonny Carlos da Silva
Data: 08/04/2020 15:29:33-0300 CPF: 514.515.064-49
do meu crescimento e da minha formação. Sempre ouvi que dar o exemplo é a melhor forma de se influenciar as pessoas. E eu não tenho exemplos, tão somente. Eu tenho os maiores de todos!
To my graduate and lab mates, for their companionship and helpfulness. But mostly for all the amazing discussions. I owe them much of what I learned and what I personally grew up during the development of this thesis.
To my supervisor and professor Maliska, for his support and incredibly inspiring lectures. I hope someday to be able of expressing myself and teaching with such passion.
To my co-supervisor and colleague Hermínio, for his patience and his readiness to help me. If I have seen further it is by standing upon the shoulders of this giant engineer.
To the graduate program in mechanical engineering of the Universidade Federal de Santa Catarina, for providing me the necessary conditions for the completion of this master.
To the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and to Petrobras, for financing this study through scholarships.
To my dear parents and to my sister Mariana, for their unconditional support and love. Because without them, I am nothing.
verdade maior. É o que a vida me ensinou. Isso que me alegra montão.” (João Guimarães Rosa, 1956)
Com o objetivo de desenvolver uma metodologia unificada para a solução numérica do problema acoplado de poroelasticidade, aplicado a reservatórios de petróleo, e a fim de evitar dificuldades que podem surgir devido à negligência dos efeitos da intera-ção sólido-fluido, alguns autores avançaram com o método dos volumes finitos (MVF). Embora vários deles tenham contribuído para o desenvolvimento do método para o pro-blema acoplado de mecânica dos fluidos e geomecânica, apenas alguns abordaram a questão da estabilidade que é inerente à solução numérica do problema consolidação em condição não-drenada. Menos trabalhos ainda foram dedicados ao entendimento do problema de estabilidade e o surgimento das oscilações numéricas. A maioria dos autores concentrou-se na criação de técnicas de estabilização e preocupou-se menos com a origem do problema. Esta dissertação de mestrado contribui nesta direção, ao explorar a natureza física do fenômeno de consolidação e aprofundar nos aspectos numéricos do método. Três técnicas de estabilização são implementadas em um ar-ranjo colocalizado de variáveis e os resultados são comparados aos resultados obtidos com uma formulação não-estabilizada. Além disso, uma formulação não-estabilizada também é implementada num arranjo desencontrado de variáveis e sua estabilidade é testada. Todas as formulações são validadas utilizando os problemas de referência de Terzaghi e de Mandel.
Palavras-chave: Método dos Volumes Finitos. Geomecânica. Poroelasticidade.
INTRODUÇÃO
A teoria da poroelasticidade lida com problemas de interação fluido-estrutura no escoa-mento de fluidos em meios porosos deformáveis. A presença de um fluido que move-se livremente em um meio poroso modifica a resposta mecânica deste meio. Dois meca-nismos desempenham papéis importantes na interação fluido-estrutura: (i) um aumento na pressão de poro induz uma dilatação do meio poroso e (ii) a compressão do meio poroso provoca um aumento instantâneo na pressão de poro. Particularmente, se o fluido é impedido de escapar da rede porosa (condição não-drenada), uma deformação volumétrica permanente induz a uma variação permanente na pressão de poro. Este fenomêno acoplado tem sido o foco de estudo de diversos trabalhos envolvendo intera-ção fluido-estrutura. Exemplos incluem sequestro geológico de CO2 [1], fraturamento
hidráulico [2–5], sismicidade induzida [6–8], deslizamentos de terra [9, 10], subsidência do solo [11, 12] e liquefação do solo [13, 14].
A base da teoria de poroelasticidade são os trabalhos desenvolvidos por Biot [15–19], em que o atuor generalizou a teoria de Terzaghi das tensões efetivas [20, 21]. Outros autores contribuíram para o desenvolvimento desta teoria (da poroelasticidade) e revi-sões de cada contribuição podem ser encontradas na literatura [22, 23]. A descrição matemática dos problemas de poroelasticidade está bem definida mas a solução numé-rica das equações governantes possui algumas condições particulares para garantir a obtenção de soluções estáveis. É largamente reportado na literatura que a teoria fundamental por trás da obtenção de soluções instáveis (e.g. oscilações numéricas) está relacionada à condição não-drenada em que o campo de pressão age como uma restrição incompressível para a deformação. Neste contexto, estratégias para evitar tais instabilidades são necessárias e o desenvolvimento destas tem sido o foco de diversos trabalhos mundialmente [24–35].
É amplamente relatado que as oscilações numéricas podem ocorrer quando os grãos sólidos e a fase fluida são quase incompressíveis e a permeabilidade e/ou o tamanho do passo-de-tempo não são grandes o suficiente para permitir o fluxo relativo entre as fases sólida e fluida. Mesmo para materiais com alta permeabilidade, podem ocorrer oscilações quando o domínio do tempo precisar ser excessivamente refinado para capturar um processo rápido. Alguns exemplos incluem reativação de falhas e fratura-mento hidráulico. Por exemplo, Santillán, Mosquera e Cueto-Felgueroso [36] relataram que o tamanho do passo-de-tempo necessário para resolver o crescimento instável de fraturas geralmente está abaixo de um microssegundo. Prévost e Sukumar [37] investigaram a resposta à falha devido a taxas de carregamento rápidas e lentas, i.e. condições não-drenadas e drenadas, respectivamente. Eles relataram que nenhum excesso significativo de pressão de poro é gerado sobre a falha para taxas de car-regamento lentas, portanto a falha permanece estável. Por outro lado, para taxas de carregamento rápidas, a pressão de poro em excesso gerada é significativa de tal forma que a falha é reativada.
Portanto, não é sempre possível utilizar passos-de-tempo grandes o suficiente para evitar o surgimento de oscilações em simulações de reservatórios e alternativas
de-Já na linha dos métodos de volumes finitos, apesar de vários trabalhos terem como foco o desenvolvimento de uma metodologia unificada [33–35, 38–44], apenas algu-nas lidaram com os problemas de estabilidade inerentes ao esquema de discretização [33–35, 41].
OBJETIVOS
Este trabalho tem como foco o estudo das oscilações numéricas empregando dois tipo de arranjos de variáveis na solução numérica de problemas de poroelasticidade e correlacionando os resultados com os erros de aproximação envolvidos em cada abordagem.
Os objetivos específicos deste trabalho são:
1. Correlacionar as equações governantes dos problemas de poroelasticidade com as equações governantes de problemas de fenômenos de transportes para basear a busca por uma interpretação física das constantes de Biot;
2. Prover números adimensionais que expressem de forma concisa o fenômeno da consolidação em termos do mecanismos de transporte envolvidos;
3. Prover e basear novas hipóteses no surgimento de oscilações na solução numé-rica do problema acoplado de geomecânica e escoamento com passos-de-tempo reduzidos;
4. Obter uma formulação numérica de volumes finitos usando malhas bidimensio-nais Cartesianas, em arranjos colocalizado e desencontrado, sem técnicas de estabilização;
5. Obter uma formulação numérica de volumes finitos usando malhas bidimensionais Cartesianas em arranjo colocalizado com técnicas de estabilização;
6. Validar as formulações numéricas através da solução de problemas de benchmark e comparação das soluções numéricas com as soluções analíticas;
7. Testar a estabilidade de cada formulação nas mesmas condições;
8. Comparar os resultados obtidos com as hipóteses previamente formuladas. METODOLOGIA
As simulações numéricas foram realizadas utilizando a linguagem C++ e uma bibli-oteca desenvolvida pelo próprio autor [45]. Esta biblibibli-oteca oferece suporte numérico para implementação do método dos volumes finitos (MVF) na solução de problemas de poroelasticidade. O sistema de equações foi resolvido utilizando a técnica de decom-posição LU, cuja implementação pode ser encontrada na biblioteca PETSc [46–48]. O pós-processamento foi feito através de implementações em Python pelo autor [45].
Foram apresentadas formulações de volumes finitos em malhas Cartesianas com arranjos colocalizado e desencontrado. Para malhas com arranjo colocalizado, três diferentes variações do physical influenche scheme (PIS) baseadas na conservação da quantidade de movimento foram derivadas e implementadas para estabilização da solução. Duas abordagens “não-estabilizadas” foram implementdadas usando apenas diferenças centrais, uma em arranjo colocalizado e outra em arranjo desencontrado. Os problemas de Terzaghi e de Mandel foram resolvidos com o intuito de validação das formulações.
O erro introduzido em cada formulação foi estimado a priori para basear a investi-gação do surgimento das oscilações numéricas. A ordem de aproximação obtida via análise de convergência concordou muito bem com a ordem de aproximação obtida nas estimativas a priori. Foi mostrado que o uso de um PIS aumenta a ordem de aproximação para o deslocamento na conservação da massa. Além disso, o uso de um arranjo desencontrado de variáveis elimina completamente o erro associado à pressão e ao deslocamento na conservação da quantidade de movimento e da massa, respectivamente.
Todos os PIS estabilizaram até certo ponto as oscilações numéricas, quando compa-rados à formulação tradicional que utiliza diferenças centrais. Isto se dá pelo fato de estes esquemas serem derivados utilizando parte das equações diferenciais, gerando melhores aproximações (de mais alta ordem). Os resultados mostraram que a redução dos erros de discretização do deslocamento na conservação da massa minimiza as oscilação pelo enfraquecimento do acoplamento entre os erros.
Por outro lado, foi mostrado que a redução no tamanho do passo-de-tempo provoca um aumento nos erros de aproximação associados ao campo de deslocamento na conservação da massa. Mesmo que todos os PIS aumentem a ordem de aproxima-ção do termo de deslocamento, quando o passo-de-tempo torna-se suficientemente pequeno, o erro associado a este termo domina sobre os demais. O acoplamento dos erros torna-se maior, resultando em soluções instáveis.
Utilizando um arranjo desencontrado de variáveis, os erros associados à pressão e ao deslocamento nas equações de conservação da quantidade de movimento e da massa são eliminados. Os pontos nodais de deslocamento estão localizados exatamente onde eles são necessários para conservação da massa e os pontos nodais de pressão estão localizados onde eles são necessários para conservação da quantidade de movimento. Isto leva a um desacoplamento dos erros e as oscilações numéricas acabam por serem eliminadas.
CONSIDERAÇÕES FINAIS
Nesta dissertação foram resolvidos problemas de poroelasticidade bidimensionais uti-lizando arranjos colocalizado e desencontrado de variáveis. O método dos volumes finitos foi utilizado para discretização das governantes e três formulações estabilizadas foram derivadas e implementadas através do chamado Physical Influence Scheme
caminho para lida com problemas de estabilidade em poroelasticidade, como também em problemas de escoamento incompressível, uma vez que o modelo matemático é bastante similar.
Palavras-chave: Método dos Volumes Finitos. Geomecânica. Poroelasticidade.
On the purpose of developing a unified numerical method for solving the coupled prob-lem of poroelasticity applied to petroleum reservoirs, and avoiding probprob-lems that might occur due to neglection of solid-fluid interaction effects, some authors have advanced with the finite volume method (FVM). Although several of them contributed to the devel-opment of the method to the coupled fluid dynamics and geomechanics problem, only a few addressed the stability issue inherent to the numerical solution of the consolidation problem in undrained condition. And even fewer works were devoted in understanding this stability issue and the emerging of numerical oscillations. Most of the authors fo-cused in devising stabilization techniques and were less concerned with the origin of the problem. This master thesis contributes in this direction by exploring the physical nature of the consolidation phenomenon and deepening into numerical aspects of the method. Three stabilization techniques are implemented in a collocated grid and the results are compared to those obtained with a nonstabilized formulation. Furthermore, a nonstabilized formulation in a staggered grid is implemented and its stability is tested. All formulations are verified against Terzaghi’s and Mandel’s benchmark problems.
Keywords: Finite Volume Method. Geomechanics. Poroelasticity. Stabilization
1.2.1 Steps of a reservoir study using numerical reservoir simulators. . . 26 1.3.1 One of several fractures on Hogg Island. At the left the subsidence of
the surface has caused the vegetation to be submerged and killed by salt water. . . 27 1.3.2 Contours of equal subsidence for eight-year period shown in light solid
lines; for one-year periods, in heavy broken lines. Dots represents pro-duction wells by the time. . . 28 1.5.1 A checkerboard pressure field. . . 30 1.5.2 Two-dimensional Cartesian grid: (a) collocated grid; (b) staggered grid. 31 2.1.1 Infinitesimal control volume subject to stress and gravitational fields. . 38 2.1.2 Fluid and solid phases velocities and their average value. . . 40 2.1.3 Fluid flow through an infinitesimal deforming control volume. . . 41 2.1.4 Drained and undrained conditions. A drained condition represents a
situation in which fluid is free to move across the boundaries of the element (left element). An undrained condition represents a situation in which the fluid is restrained and cannot move across the boundaries of the element (right element). . . 44 3.1.1 Control volume for a Cartesian grid. . . 54 3.2.1 Variable indexing and control-volumes for a staggered Cartesian grid. 57 3.2.2 Cell-vertex collocated Cartesian grid. . . 59 4.2.1 One-dimensional finite volume in an equally-spaced collocated grid for
momentum conservation equation. . . 69 4.2.2 Error composition for momentum conservation equation in a collocated
grid. . . 70 4.2.3 One-dimensional finite volume in an equally-spaced staggered grid for
momentum conservation equation. . . 71 4.2.4 Error composition for momentum conservation equation in a staggered
grid. . . 71 4.2.5 One-dimensional finite volume in an equally-spaced collocated grid for
mass conservation equation. . . 72 4.2.6 Error composition for mass conservation equation in a collocated grid
with CDS. . . 74 4.2.7 Error composition for mass conservation equation in a collocated grid
with PIS. . . 74 4.2.8 One-dimensional finite volume in an equally-spaced staggered grid for
mass conservation equation. . . 75 4.2.9 Error composition for mass conservation equation in a staggered grid. 75
a collocated grid with Central Differencing Scheme (CDS). . . 76 4.2.11 Error composition for momentum and mass conservation equations in
a collocated grid with Physical Influence Scheme (PIS). . . 76 4.2.12 Error composition for momentum and mass conservation equations in
a staggered grid. . . 76 5.1.1 Extended Terzaghi’s problem. (a) Steady-state problem of the sealed
column loaded by a constant vertical stress at its top. (b) Transient problem of the upper-drained-column loaded by a constant vertical stress at its top. . . 85 5.1.2 Solution of the undrained-step of extended Terzaghi’s problem using a
staggered grid. . . 86 5.1.3 Solution of the undrained-step of extended Terzaghi’s problem using a
collocated grid with CDS. . . 87 5.1.4 Solution of the undrained-step of extended Terzaghi’s problem using
a collocated grid with One-Dimensional Physical Influence Scheme (1D-PIS). . . 87 5.1.5 Solution of the undrained-step of extended Terzaghi’s problem using a
collocated grid with I2D-PIS. . . 88 5.1.6 Solution of the undrained-step of extended Terzaghi’s problem using a
collocated grid with C2D-PIS. . . 88 5.1.7 Solution of the drained-step of extended Terzaghi’s problem using a
staggered grid. . . 89 5.1.8 Solution of the drained-step of extended Terzaghi’s problem using a
collocated grid with CDS. . . 90 5.1.9 Solution of the drained-step of extended Terzaghi’s problem using a
collocated grid with 1D-PIS. . . 90 5.1.10 Solution of the drained-step of extended Terzaghi’s problem using a
collocated grid with I2D-PIS. . . 91 5.1.11 Solution of the drained-step of extended Terzaghi’s problem using a
collocated grid with C2D-PIS. . . 91 5.2.1 Mandel’s problem. (a) Whole domain. (b) Domain reduced to one-quarter. 92 5.2.2 Solution of Mandel’s problem using a staggered grid. . . 93 5.2.3 Solution of Mandel’s problem using a collocated grid with CDS. . . 94 5.2.4 Solution of Mandel’s problem using a collocated grid with 1D-PIS. . . . 95 5.2.5 Solution of Mandel’s problem using a collocated grid with I2D-PIS. . . 96 5.2.6 Solution of Mandel’s problem using a collocated grid with C2D-PIS. . . 97 5.3.1 h-refinement convergence analysis for Terzaghi’s problem solved using
a staggered grid. . . 100 5.3.3 h-refinement convergence analysis for Terzaghi’s problem solved using
a collocated grid with central differencing scheme (CDS). . . 101 5.3.4 t-refinement convergence analysis for Terzaghi’s problem solved using
a collocated grid with central differencing scheme (CDS). . . 101 5.3.5 h-refinement convergence analysis for Terzaghi’s problem solved using
a collocated grid with one-dimensional physical influence scheme (1D-PIS). . . 102 5.3.6 t-refinement convergence analysis for Terzaghi’s problem solved using
a collocated grid with one-dimensional physical influence scheme (1D-PIS). . . 102 5.3.7 h-refinement convergence analysis for Terzaghi’s problem solved using
a collocated grid with incomplete two-dimensional physical influence scheme (I-2D-PIS). . . 103 5.3.8 t-refinement convergence analysis for Terzaghi’s problem solved using
a collocated grid with incomplete two-dimensional physical influence scheme (I-2D-PIS). . . 103 5.3.9 h-refinement convergence analysis for Terzaghi’s problem solved
us-ing a collocated grid with complete two-dimensional physical influence scheme (C-2D-PIS). . . 104 5.3.10 t-refinement convergence analysis for Terzaghi’s problem solved
us-ing a collocated grid with complete two-dimensional physical influence scheme (C-2D-PIS). . . 104 6.1.1 Stability test using a collocated grid with central differencing scheme
(CDS) for Terzaghi’s problem. Pressure profiles shown for the first timestep. . . 106 6.1.2 Stability test using a collocated grid with central differencing scheme
(CDS) for Mandel’s problem. Pressure profiles shown for the first timestep.107 6.1.3 Stability test using a collocated grid with one-dimensional physical
influ-ence scheme (1D-PIS) for Terzaghi’s problem. Pressure profiles shown for the first timestep. . . 108 6.1.4 Stability test using a collocated grid with one-dimensional physical
influ-ence scheme (1D-PIS) for Mandel’s problem. Pressure profiles shown for the first timestep. . . 109 6.1.5 Stability test using a collocated grid with incomplete two-dimensional
physical influence scheme (I-2D-PIS) for Terzaghi’s problem. Pressure profiles shown for the first timestep. . . 110
physical influence scheme (I-2D-PIS) for Mandel’s problem. Pressure
profiles shown for the first timestep. . . 111
6.1.7 Stability test using a collocated grid with complete two-dimensional physical influence scheme (C-2D-PIS) for Terzaghi’s problem. Pressure profiles shown for the first timestep. . . 112
6.1.8 Stability test using a collocated grid with complete two-dimensional physical influence scheme (C-2D-PIS) for Mandel’s problem. Pressure profiles shown for the first timestep. . . 113
6.1.9 Stability test using a staggered grid for Terzaghi’s problem. Pressure profiles shown for the first timestep. . . 114
6.1.10 Stability test using a staggered grid for Mandel’s problem. Pressure profiles shown for the first timestep. . . 115
6.1.11 Comparison of the obtained results using the different approaches for Terzaghi’s problem. Pressure profiles shown for the first timestep. . . . 116
6.1.12 Comparison of the obtained results using the different approaches for Mandel’s problem. Pressure profiles shown for the first timestep. . . . 117
6.2.1 Dimensionless stability test using a collocated grid with central differ-encing scheme (CDS) for Terzaghi’s problem. Pressure profiles shown for the first timestep. . . 121
6.2.2 Dimensionless stability test using a staggered grid for Terzaghi’s prob-lem. Pressure profiles shown for the first timestep. . . 124
B.1.1 Extended Terzaghi’s problem. . . 142
B.2.1 Mandel’s problem. . . 147
B.2.2 Mandel’s problem with domain reduced to one-quarter. . . 147
C.1.1 Finite volume for displacement u and its neighbours. . . 152
C.1.2 Finite volume for displacement v and its neighbours. . . 154
C.1.3 Finite volume for pressure and its neighbours. . . 156
C.1.4 Dirichlet boundary conditions on the left border of a finite volume. . . . 158
C.1.5 Neumann boundary conditions on the left border of a finite volume. . . 160
C.1.6 Prescribed normal stress on the right border of a finite volume. . . 161 C.2.1 Finite volume for pressure P , displacement u and v and its neighbours. 163
5.0.1 Input data for the solution of the extended Terzaghi’s problem and
Man-del’s problem for Gulf Mexico shale and water. . . 84
5.3.1 Size and characteristic length for the grids used on the h-refinement convergence analysis. . . 98
5.3.2 Timestep sizes used on the t-refinement convergence analysis. . . 99
6.1.1 Input data for stability tests, for soft sediment and water. . . 105
6.1.2 Compiled results for stability tests. . . 116
6.2.1 Input data for dimensionless stability tests. . . 119
6.2.2 Stiffness contrast between drained and undrained response, Fourier and Reynolds numbers for poroelasticity for dimensionless stability tests.121 A.1.1 Relationship among bulk continuum and micromechanical coefficients. 139 A.2.1 Relationship among bulk continuum and micromechanical coefficients for incompressible solid phase. Source: Detournay and Cheng [76]. . . 140
A.2.2 Relationship among bulk continuum and micromechanical coefficients for highly compressible fluid phase. Source: Detournay and Cheng [76]. 141 C.1.1 Coefficients for momentum balance in x-direction in a staggered grid (Equation C.1.3). . . 154
C.1.2 Coefficients for momentum balance in y-direction in a staggered grid (Equation C.1.6). . . 155
C.1.3 Coefficients for continuity equation in a staggered grid (Equation C.1.9). 157 C.1.4 Coefficients for momentum balance in x-direction in a staggered grid, reduced for a prescribed normal stress on the right border (Equation C.1.18). . . 162
C.1.5 Coefficients for momentum balance in y-direction in a staggered grid, reduced for a prescribed normal stress on the upper border (Equation C.1.19). . . 162
C.2.1 Coefficients for momentum balance in x-direction in a collocated grid (Equation C.2.1). . . 163
C.2.2 Coefficients for momentum balance in y-direction in a collocated grid (Equation C.2.2). . . 164
C.2.3 Coefficients for continuity equation in a collocated grid (Equation C.2.3). 164 C.2.4 Coefficients for momentum balance in x-direction in a collocated grid, for a prescribed shear stress on the lower border (Equation C.2.4). . . 165
C.2.5 Coefficients for momentum balance in y-direction in a collocated grid, for a prescribed normal stress on the upper border (Equation C.2.5). . 166
C.2.6 Coefficients for continuity equation in a collocated grid, for a prescribed fluid flow on the right border (Equation C.2.6). . . 166
with CDS (Equation C.2.7). . . 167 C.2.8 Coefficients for momentum balance in y-direction in a collocated grid
with CDS (Equation C.2.8). . . 167 C.2.9 Coefficients for continuity equation in a collocated grid with CDS
(Equa-tion C.2.9). . . 168 C.2.10Coefficients for continuity equation in a collocated grid with 1D-PIS
(Equation C.2.10). . . 169 C.2.11Coefficients for continuity equation in a collocated grid with I-2D-PIS
(Equation C.2.11). . . 171 C.2.12Coefficients for continuity equation in a collocated grid with C-2D-PIS
1D-PIS One-Dimensional Physical Influence Scheme BES First-order backward Euler scheme
C-2D-PIS Complete Two-Dimensional Physical Influence Scheme CDS Central Differencing Scheme
CVFEM Control Volume Finite Element Method EbFVM Element-based Finite Volume Method FDM Finite Difference Method
FEM Finite Element Method
FVM Finite Volume Method
I-2D-PIS Incomplete Two-Dimensional Physical Influence Scheme PIS Physical Influence Scheme
h Characteristic length of the grid σ Total stress (tensor) field
g Gravitational (vector) field ρ Solid-fluid continuum density ρf Fluid-phase density
ρs Solid-phase density
φ Medium porosity
σ1 Effective stress (tensor) field
α Biot-Willi’ coefficient
p Pore-pressure (scalar) field I Second-order identity tensor
G Shear modulus
λ Lamé’s first parameter (drained) Strain (tensor) field
u Displacement (vector) field
∇u Displacement gradient (tensor) field E Young’s modulus (drained)
K Bulk modulus (drained) ν Poisson’s ratio (drained) ε Volumetric strain
q Rate of fluid volume injection/withdrawal
vf Absolute fluid-phase velocity
vs Absolute solid-phase velocity
ζ Variation in fluid content
Q Biot’s modulus
v Fluid-phase velocity relative to the solid-phase (Darcy’s velocity) κ Medium absolute permeability
µ Fluid dynamic viscosity
u Component in x of the displacement (vector) field v Component in y of the displacement (vector) field w Component in z of the displacement (vector) field σ Total stress (scalar) value
Ku Undrained bulk modulus
M Longitudinal (or p-wave) modulus (drained) cb Compressibility of the bulk volume
cs Compressibility of the solid-phase
cp Compressibility of the pore volume
K Stiffness matrix
M Fluid storage capacity matrix F Fluid flow matrix
A Coupling matrix of the geomechanics problem B Coupling matrix of the fluid dynamics problem C Coupling matrix of the geomechanics problem D Coupling matrix of the fluid dynamics problem H1 Stabilization matrix related to the 1D-PIS
c Consolidation coefficient
Reφ Reynolds number for poroelasticity
Foφ Fourier number for poroelasticity
1 INTRODUCTION . . . . 24 1.1 BIOT’S CONSOLIDATION . . . 24 1.1.1 Applications . . . . 24 1.2 RESERVOIR SIMULATION . . . 25 1.3 SUBSIDENCE OF OILFIELDS . . . 27 1.4 HYDRAULIC FRACTURING . . . 29 1.5 NUMERICAL STABILITY ISSUES . . . 30 1.6 OBJECTIVES AND CONTRIBUTIONS . . . 32
1.6.1 Specific Objectives . . . . 33
1.7 OUTLINE OF THE THESIS . . . 34
2 POROELASTICITY THEORY . . . . 36
2.1 BIOT’S SOIL CONSOLIDATION . . . 36
2.1.1 Momentum Conservation . . . . 37 2.1.2 Mass Conservation . . . . 40 2.1.3 Improvements on Biot’s Theory . . . . 43
2.1.3.1 Limiting Behaviours - Drained and Undrained Conditions . . . 44 2.1.3.2 Micromechanical Approach . . . 46
2.1.4 Porosity Variation Estimate . . . . 49
2.2 MATHEMATICAL MODEL . . . 50
3 NUMERICAL FORMULATION . . . . 53
3.1 THE FINITE VOLUME METHOD . . . 54
3.1.1 Momentum Conservation . . . . 55 3.1.2 Mass Conservation . . . . 55
3.2 GRID TYPES AND INTERPOLATION SCHEMES . . . 56
3.2.1 Staggered Grid . . . . 56
3.2.1.1 Linear System . . . 58
3.2.2 Collocated Grid . . . . 59
3.2.2.1 Central Differencing Scheme (CDS) . . . 59 3.2.2.2 One-Dimensional Physical Influence Scheme (1D-PIS) . . . 61 3.2.2.3 Incomplete Two-Dimensional Physical Influence Scheme (I-2D-PIS) . 62 3.2.2.4 Complete Two-Dimensional Physical Influence Scheme (C-2D-PIS) . 63
4 STABILITY ISSUES . . . . 65
4.1 MINIMUM TIMESTEP SIZE . . . 65 4.2 A PRIORI ERROR ESTIMATES . . . 66
4.2.1 Momentum Conservation Error Estimates . . . . 69 4.2.2 Mass Conservation Error Estimates . . . 72
4.5 DIMENSIONLESS POROELASTICITY AND ERROR ESTIMATES . . 81
5 NUMERICAL VERIFICATION . . . . 83
5.1 EXTENDED TERZAGHI’S PROBLEM . . . 84
5.1.1 Undrained Stage . . . . 85 5.1.2 Drained Stage . . . 89 5.2 MANDEL’S PROBLEM . . . 92 5.3 CONVERGENCE ANALYSIS . . . 97 6 STABILITY TESTS . . . 105 6.1 STABILIZATION EFFECTIVENESS . . . 105 6.2 DIMENSIONLESS NUMBERS AND STABILITY . . . 118
7 CONCLUSION . . . 125
7.1 SUGGESTIONS FOR FUTURE WORKS . . . 127
REFERENCES . . . 129 APPENDIX A – MICROMECHANICAL APPROACH . . . 139
A.1 EXPRESSIONS FOR POROELASTICITY PROPERTIES . . . 139 A.2 SIMPLIFIED EXPRESSIONS FOR LIMITING CASES . . . 139
APPENDIX B – BENCHMARK PROBLEMS . . . 142
B.1 EXTENDED TERZAGHI’S PROBLEM . . . 142
B.1.1 Undrained one-dimensional consolidation . . . 143 B.1.2 Drained one-dimensional consolidation . . . 144
B.2 MANDEL’S PROBLEM . . . 147
APPENDIX C – FINITE VOLUME METHOD IN CARTESIAN GRIDS 152
C.1 STAGGERED GRID . . . 152
C.1.1 Dirichlet Boundary Conditions . . . 158 C.1.2 Neumann Boundary Conditions . . . 160
C.2 COLLOCATED GRID . . . 163
C.2.1 Dirichlet Boundary Conditions . . . 164 C.2.2 Neumann Boundary Conditions . . . 165 C.2.3 Central Differencing Scheme (CDS) . . . 166 C.2.4 One-Dimensional Physical Influence Scheme (1D-PIS) . . . 167 C.2.5 Incomplete Two-Dimensional Physical Influence Scheme (I-2D-PIS)169 C.2.6 Complete Two-Dimensional Physical Influence Scheme (C-2D-PIS)171
1 INTRODUCTION
The main purpose of this master’s thesis is to deal with the numerical aspects of poroelasticity problems, focused on the origin of numerical oscillations on the pore pressure field. Many of the earlier works point out that this issue arises from an unstable approximation of the incompressibility constraint in the initial condition1 (known as
undrained consolidation). The current master’s thesis aims to explore the numerical oscillations’ origin, based on the approximations of the different terms in the partial differential equations.
This chapter begins by briefly reviewing Biot’s consolidation theory, which is the basis for poroelasticity and its applications. Further on, we explore the subsidence in oilfields and the hydraulic fracturing problems, some applications of Biot’s consolidation theory. Following, we present a discussion about numerical stability issues and some recent works towards this problem. This chapter ends with the presentation of the goals of the work developed and the general structure of the current master’s thesis.
1.1 BIOT’S CONSOLIDATION
A soil undergoing an external load does not assume an instantaneous deformed state but settles gradually at variable rate. Such settlement is caused by a gradual adaptation of the soil to the load variation. This phenomenon was first explained by Terzaghi [20, 21], by looking at the soil as a porous medium with elastic properties and voids filled with water.
The presence of a freely moving fluid in a porous medium modifies its mechanical response. Two mechanisms play key roles in the fluid-rock interaction: (i) an increase of pore pressure induces a dilation of the rock and (ii) a compression of the rock causes an increase of pore pressure, if the fluid is prevented from escaping the pore network (undrained condition).
Terzaghi’s theory was first generalized by Rendulic [49], however it was Biot [15, 16] that first developed a linear poroelasticity theory that is consistent with both mechanisms outlined above. These works became the main foundation on representing different porous media in studies involving fluid-structure coupling and the fundamental basis of poroelasticity theory.
1.1.1 APPLICATIONS
The poroelasticity theory is applied in reservoir engineering, since an oil reservoir rock is composed of a solid matrix of sedimentary nature and fluid saturated pores. Any load applied to this structure is partly transferred to the solid matrix (effective stress) 1Refers to the condition at an initial time. The consolidation problem with an incompressibility
and partly transferred to the fluid within (fluid pressure). During oil production, the fluid pressure inside the reservoir tends to deplete, and the load supported by the fluid phase is transferred to the solid matrix, increasing the effective stress and causing the rock to deform.
As the fluid flows through the porous medium, the pressure field (and furthermore the stress field) changes and a new equilibrium state is reached by deforming the porous medium, which directly affects the fluid flow. Thus the consolidation of a porous medium is deeply coupled with the fluid flow. When the consolidation takes place in a very short period, such that the fluid does not have enough time to flow, the consolidation is defined as undrained.
Numerical simulation is a powerful tool for addressing problems that involve fluid-structure interaction in porous media. Examples include geologic CO2 sequestration [1],
hydraulic fracturing [2–5], induced seismicity [6–8], landslides [9, 10], land subsidence [11, 12] and soil liquefaction [13, 14]. Further on, we discuss some of these problems.
1.2 RESERVOIR SIMULATION
The reservoir engineering is multidisciplinary and powered by theories from the most diverse areas of science, among them geology, fluid mechanics, thermodynam-ics and geomechanthermodynam-ics2. Given the importance of the petroleum on the global energy market and the challenges involved on searching for new oilfields and technology de-velopment for enhancing the oil production, research on geology, fluid mechanics and thermodynamics were put forward while geomechanics stood in the shade for decades.
With the oil production development, several serious problems began to arise, like well losses by collapsing and casing shear, subsidence (as discussed in section 1.3) in the seabed causing serious damages to the platforms and lifting equipment, flooding in coastal regions, sand production, fractures and rock permeability changes, among others. Thus, the oil industry realized the real necessity of expanding research in the geomechanics to predict these unwanted effects.
Besides the importance of geomechanics on the above mentioned undesired effects, it plays an important role on the fluid flow physics, since it affects the permeabil-ity of the medium, and therefore cannot be neglected on a reservoir study, especially when this is performed by using a numerical simulator. Using a numerical simulator enables the attainment of information about the performance of an oilfield under several production scenarios, thus allowing to find the best production strategy.
More specifically, the reservoir behaviour can be analysed when subjected to the injection of different fluid types (water, gas, vapour and others), when under the influence of different production flow rates and/or injection, and it can figure the effect 2Geomechanics is the theoretical and applied science of the mechanical behaviour of geological
of the wells place and the distance between these on the final oil recovery. According to Rosa, Carvalho, and Xavier [50], the steps generally followed on the execution of a reservoir study, using numerical reservoir simulators, can be summarized as shown on Figure 1.2.1.
Figure 1.2.1 – Steps of a reservoir study using numerical reservoir simulators.
Rock and fluid data
Geology data Production and
completion data Grid generation
Data digitalization Data collection and preparation Numerical model History matching Extrapolation
Source: Adapted from Rosa, Carvalho, and Xavier [50].
The development of the numerical reservoir simulator must comprehend the physics involved on the production process, which includes the fluid flow through porous media and may include the stress/strain field on the reservoir solid matrix. In other words, the fluid mechanics and possibly the geomechanics. In the field of petroleum engineering, the solid deformation impact on fluid flow is often neglected.
Still, as it will be discussed in sections 1.3 and 1.4, there are quantitative ev-idences that suggest a demand for a numerical simulation tool which considers the solid-fluid interaction. In addition, the oil production prediction will not be accurate due to the shrinkage of the pore space if the solid-fluid coupling is not considered.
This coupled problem is usually solved using two different numerical methods on each problem: Finite Volume Method (FVM) or Finite Difference Method (FDM) for the fluid flow in porous media and Finite Element Method (FEM) for stress field in the solid matrix. The disadvantages of using two different numerical methods are the requirement of interpolation of the coupling parameters from one problem to another and the lack
of warranty regarding the conservative property of the approximate equations of both problems.
Furthermore, under certain circumstances numerical oscillations may occur when solving the geomechanics and fluid flow coupled problem. The oscillations arising is credited to an unstable approximation of the pore pressure profile in nearly undrained conditions. For that reason, strategies for minimizing these instabilities are needed and this has been the focus of several works worldwide [24–28, 30, 33, 34, 51–54].
1.3 SUBSIDENCE OF OILFIELDS
“In 1917 a prolific oil field was developed near the mouth of Goose Creek, and during 1918 and subsequent years, millions of barrels of oil were removed from beneath its surface. Beginning in 1918 it became apparent that Gaillard Peninsula, near the center of the field, and other nearby low land was becoming submerged. Elevated plank roadways or walks were built from the mainland to the derricks. Derrick floors had to be raised. Vegetation was flooded and killed, and finally all of the peninsula disappeared beneath water.” From Pratt and Johnson [55, p. 578]
Figure 1.3.1 shows the subsidence3 of the surface on Hogg Island due to oil
exploitation on the Goose Creek oilfield.
Figure 1.3.1 – One of several fractures on Hogg Island. At the left the subsidence of the surface has caused the vegetation to be submerged and killed by salt water.
Source: Pratt and Johnson [55].
Ranking high among the many causes of land subsidence are those related to man’s exploitation of the earth’s natural resources. The withdrawal of fluids and the consequent reduction of subsurface pressures in reservoirs causes compression in the oil zones. The land subsidence is explained by the effective stress concept introduced by Terzaghi [20, 21].
A reservoir rock is made of a solid matrix of sedimentary nature, such as sand-stones, carbonates mud, dolomites, among others, and fluid-filling pore channels. Any load applied to a reservoir rock is partially transferred to its solid matrix (effective stress) and partially transferred to its fluid within (fluid pressure). During the oil production, the fluid pressure inside the reservoir tends to deplete, and the load share supported by the fluid phase is transferred to the reservoir’s rock, increasing the effective stress and causing the rock compression.
The subsidence in oilfields can, therefore, be related to the oil production, as shown in Figure 1.3.2, a subsidence map of Goose Creek oilfield in Baytown and the production wells position. Compression is the geomechanical response of the reservoir rock to the fluid pressure depletion. Surface subsidence is one of the possible effects caused by this response.
Figure 1.3.2 – Contours of equal subsidence for eight-year period shown in light solid lines; for one-year periods, in heavy broken lines. Dots represents pro-duction wells by the time.
Source: Pratt and Johnson [55].
The same phenomenon was observed in several other oilfields, for instance, we can cite the Wilmington oilfield in California/USA [56], the Bolivar Coastal fields on
the eastern margin of Lake Maracaibo, Venezuela [57], the Groningen gas field in the north-eastern part of the Netherlands [58] and the Ekofisk field in the north sea [59, 60].
In this context, reservoir simulation stands as an important tool in prevention and reduction of the land subsidence, as scientists can predict the subsidence trend and devise strategies for mitigation such as pressure maintenance practices.
1.4 HYDRAULIC FRACTURING
“Since Stanolind Oil introduced hydraulic fracturing in 1949, close to 2.5 million fracture treatments have been performed worldwide. Some believe that approximately 60% of all wells drilled today are fractured. Fracture stimulation not only increases the production rate, but it is credited with adding to reserves - 9 billion bbl of oil and more than 700 Tsef of gas added since 1949 to US reserves alone - which otherwise would have been uneconomical to develop.” From Montgomery and Smith [61, p. 27]
Hydraulic fracturing (or fracking) is a common practice in oil industry that aims at opening up fractures in rocks by a pressurized fluid, thus stimulating the fluid flow and increasing the recovered volume. It basically consists of producing fractures and keeping them open using small particles/grains. Hydraulic fracturing is an effective well stimulation technique and largely used worldwide.
Many fields might not exist today without hydraulic fracturing. And as the global balance of supply and demand forces the hydrocarbon industry toward more uncon-ventional resources such as shale gas, shale oil, tight sands, coal bed methane, un-derground coal gasification, among others, hydraulic fracturing will continue to play a substantive role in unlocking otherwise unobtainable reserves [61].
“Directional drilling and hydraulic-fracturing technologies are dramatically increasing natural-gas extraction. In aquifers overlying the Marcellus and Utica shale formations of northeastern Penn-sylvania and upstate New York, we document systematic evidence for methane contamination of drinking water associated with shale-gas extraction. In active gas-extraction areas (one or more gas wells within 1 km), average and maximum methane concentrations in drinking-water wells increased with proximity to the nearest gas well and were 19.2 and 64 mg CH4 L´1 (n=26), a
potential explosion hazard [...].” From Osborn et al. [62, p. 8172]
There are risks associated with this practice, for example contamination of groundwater, blowout due to gas explosion, fracking-induced earthquakes, among oth-ers. Fracking process might generate new fractures or enlarge existing ones above the target formation, increasing the connectivity of the fracture system. Osborn et al. [62] have reported that the reduced pressure following the fracking activities could release
natural gas in solution, leading to gas exsolving4 rapidly from solution, allowing it to po-tentially migrate upward through the fracture system and contaminate the groundwater. The problems mentioned above highlight the importance of including geome-chanical effects into reservoir simulators, which must be used to predict these unwanted effects and devise strategies for avoiding them.
1.5 NUMERICAL STABILITY ISSUES
As discussed by Prakash and Patankar [63], stability issues (e.g. oscillations) may appear when using equal-order methods for both pressure and velocity to solve incompressible fluid flow problems, such as those represented by
µ∇2v ´ ∇p ` f “ 0, (1.5.1)
∇ ¨ v “ 0, (1.5.2)
where µ is the fluid viscosity, v is the fluid flow velocity, p is the pressure and f is the body force. The numerical oscillations are consequence of the fact that only gradients of pressure appear in the momentum conservation equation (Eq. 1.5.1) and pressure does not appear explicitly in the mass conservation equation (Eq. 1.5.2) although this last one is used as constraint on the determination of the pressure field.
Because of this, if pressure is interpolated using linear shape functions, only pressure differences between alternate grid points are involved in the overall system of equations. Hence, the equations reveal no difference between a uniform pressure field and a checkerboard pressure field [63]. An example of a checkerboard field is shown in Figure 1.5.1.
Figure 1.5.1 – A checkerboard pressure field.
Source: Prakash and Patankar [63].
4Exsolve is used in geology to describe the separation or precipitation of a substance from a solid
With the adoption of staggered grids by Harlow and Welch [64], where pressure and velocity are calculated at different positions over the grid, the velocity field satisfies both momentum and mass conservation equations. With the pressure in the center of the control volume and velocity at its facets, the velocity field is calculated where it is required for mass conservation therefore avoiding the need of an approximation and truncation errors.
According to Maliska [65], the staggered grid does not present the stability prob-lems faced by collocate arrangements of variables. On the first case, pressure field is stored in a way that its gradient is the driving force of the velocity stored in between two pressure points. Figure 1.5.2(a) presents an example of a 2D collocated Cartesian grid and Figure 1.5.2(b) presents an example of a 2D staggered Cartesian grid.
Figure 1.5.2 – Two-dimensional Cartesian grid: (a) collocated grid; (b) staggered grid.
x x x x (a) x x x x (b) x u-node v-node p-node Integration point
Source: Own authorship.
However, for unstructured grids it becomes more complicated using a staggered formulation and collocated formulations are often used, in spite of techniques using staggered grids that were developed recently [66]. Thus, since pressure and velocity usually end up locating at the same point, the velocity field is not calculated where it is required for mass conservation, demanding an interpolation between nodal points. Strategies for obtaining proper approximations for avoiding stability issues were devel-oped by Rhie and Chow [67], Schneider and Raw [68], among others.
The same stability issues might also occur at the beginning of a simulation near drained boundaries or at the interface between materials of different permeability. As it will be discussed on the next sections, the governing equations of the consolidation phenomenon might reduce to
G∇2u ´ α∇p ` ρg “ 0, (1.5.3)
∇ ¨ u “ 0, (1.5.4)
where G is the shear modulus, u is the displacement, α is Biot-Willis’ coefficient [19], p is the fluid pore-pressure, ρ is the porous medium density and g is the gravity.
As in the fluid mechanics problem described by Eq. 1.5.1 and 1.5.2, only gra-dients of pressure appear in Eq. 1.5.3 (also known as geomechanics equation) and pressure does not appear explicitly in Eq. 1.5.4. Because of this, the same stability issues observed when solving incompressible fluid flow problems (coupled pressure-velocity fields) might also be observed when solving poroelasticity problems (coupled pressure-displacement fields). In this situation, using interpolation functions of equal order for both pressure and displacement will likely result in instabilities when the mini-mum timestep requirement is not satisfied, as reported by Vermeer and Verruijt [24].
It is widely reported that the numerical oscillations may occur when the solid and fluid are nearly incompressible and the permeability and/or the timestep are not large enough to allow relative flow between the solid and fluid phases. And even for materials with high permeability, oscillations may arise when the time domain is finely refined to capture a fast process.
Some examples include fault activation and fracking. Santillán, Mosquera, and Cueto-Felgueroso [36] have reported that the timestep required for solving unstable fracture growth is typically below a microsecond. Prévost and Sukumar [37] investigated the fault response due to rapid and slow loading, i.e. undrained and drained loading. They reported that no significant excess pore-pressure are built on the fault for slow loading, and the fault remains stable. On the other hand, for rapid loading, significant excess pore-pressure are built on the fault, reactivating it.
Therefore, it is not always possible to use a larger timestep for avoiding the numerical oscillations in reservoir simulation and other strategies must be developed to mitigate this problem. In order to overcome this problem, several stabilization techniques have been devised in the finite element framework [27, 28, 30, 51, 52]. Conversely, very few alternatives are available for finite volume formulations. On the purpose of developing a unified numerical method in the finite volume framework, several works were developed [33, 34, 38, 39, 53, 54, 69–74], but only a few approached the problem of stability inherent to the numerical method [33, 34, 53, 54, 72].
In some of these works [33, 53, 54], the physical influence scheme (abbrevi-ated to PIS), originally developed for stabilizing the Navier-Stokes equations [68], has been successfully applied to geomechanics. This method consists of obtaining the dis-placements values at the interface of a control volume by using a special interpolation scheme derived from the momentum conservation equation. This way, the PIS incorpo-rates part of the physical phenomena into the approximation, specially the influence of the pressure field.
1.6 OBJECTIVES AND CONTRIBUTIONS
The main purpose of this master thesis is to contribute with the development of the FVM for the solution of poroelasticity problems, by studying the numerical
os-cillations that might occur when solving the approximated equations under certain conditions, devising and evaluating strategies for stabilization of the solution.
Different finite volume formulations applied to Cartesian grids are presented for solving poroelasticity problems. This work aims to investigate different versions of the physical influence scheme for addressing stability issues during undrained consolida-tion. Furthermore, staggered grids are adopted for studying the approximation errors impact on the arising of numerical oscillations.
In addition, the approximation error introduced in each approach is estimated a priori in order to substantiate the investigation of numerical oscillations emerging and scale analysis of the partial differential equations are performed to support the discussion of consolidation’s physical phenomenon.
Firstly, the physical influence scheme is obtained by considering a one-dimensional momentum equation. Next, the two-one-dimensional momentum equation is considered but specific terms of the equation are neglected. Finally, the PIS is derived from the full two-dimensional momentum equation. These three techniques are compared against the non-stabilized formulation, which uses CDS for approximating the displacements. Terzaghi’s and Mandel’s problems [20, 21, 75] are solved for verification purposes.
1.6.1 SPECIFIC OBJECTIVES
The specific objectives of this thesis are:
1. Correlate the governing equations of poroelasticity problems with the governing equations of general transport phenomena in order to back up the searching for physical interpretation of Biot’s constants;
2. Provide dimensionless numbers that express in a shorter form the consolidation phenomenon in terms of transport phenomena involved;
3. Provide and support new hypotheses on the oscillations arising when numerically solving the geomechanics and fluid flow coupled problem with reduced timestep sizes;
4. Obtain numerical finite volume formulations using two-dimensional Cartesian col-located and staggered grids without stabilization techniques;
5. Obtain numerical finite volume formulations using two-dimensional Cartesian col-located grids with stabilization techniques;
6. Verify the numerical formulations by solving benchmark problems and comparing with their analytical solutions;
8. Compare the obtained results with the formulated hypotheses.
1.7 OUTLINE OF THE THESIS
Biot’s soil consolidation theory, the basis of poroelasticity theory, is presented in chapter 2. The chapter begins by deriving the governing equations of the consolidation problem under Biot’s assumptions and discuss the meaning of the constants introduced by Biot. Next, a brief review of undrained and drained condition and micromechanical approach on the contributions of the constituents of the porous medium. A short dis-cussion regarding the invariance of porosity assumption is presented and the chapter ends with the mathematical model used on the development of the present work.
Chapter 3 focuses on discussing and deriving the numerical formulations used on the development of the present work. A concise review of the method of weighted residuals is brought and the governing equations are discretized with each one of the proposed formulations. First, the governing equations are discretized for a staggered grid. Next, the governing equations are discretized for a collocated grid using central differencing schemes in all terms. Finally, three different physical influence schemes are derived and used on the approximation of the displacement field in the mass con-servation equation for a collocated grid.
Chapter 4 is centered on addressing the stability issues arising from the solution of the geomechanics and fluid flow coupled problem with reduced timestep sizes. First, the theory regarding the minimum timestep size is reviewed. Next, a numerical anal-ysis of the one-dimensional consolidation problem is provided and the approximation errors resulting from the discretization process are estimated a priori. Furthermore, a discussion regarding undrained and drained conditions in light of a scale analysis is pre-sented. Finally, a dimensionless set of governing equations is obtained to deepen the discussion and explore the nature of the numerical oscillations. Some dimensionless numbers summarizing the consolidation phenomenon are provided.
In chapter 5, results obtained by using the finite volume framework on the solution of benchmark problem are compared to analytical solutions, for verification purposes. First, results are presented for extended Terzaghi’s problem, which considers the body force effect, neglected on the classical version of the problem. Next, results are pre-sented for Mandel’s problem, which presents a non-monotonic behaviour thus being interesting for verification purposes. Chapter 5 ends with the results of convergence analysis which indicate the error decay rate and should be consistent with the a priori error estimates provided in chapter 4.
Stability test results are presented and discussed in chapter 6. These tests are carried using different timestep sizes and fixed grids, for all the numerical formulations presented in chapter 3 and verified in chapter 5. Terzaghi’s and Mandel’s problems are used for the tests whose purpose is to check the effectiveness of each numerical
formulation in obtaining stable solutions and to explore the nature of the numerical oscillations. First, results of some tests are presented with the purpose of checking the stability of each formulation. Next, results of dimensionless tests with different materials are presented with the purpose of exploring the origin of the numerical oscillations.
Chapter 7 ends this work with a succinct review of the results obtained and the final conclusions. Suggestion for future works are presented.
2 POROELASTICITY THEORY
The mechanical strength resisting compression of a fluid-filled porous medium is composed of contributions from different sources. It is somewhat obvious that com-pressibility of solid and fluid-phases both contribute to the strength of the material. Furthermore, the mobility1of the fluid affects the mechanical response of the material.
These coupled mechanisms bestow an apparent time-dependent character to the mechanical properties of the rock. Indeed, if excess pore pressure induced by compression of the rock is allowed to dissipate through diffusive fluid mass transport, further deformation of the rock progressively takes place. It also appears that the rock is more compliant under drained conditions than undrained conditions [76].
The earliest theory to account for the influence of pore fluid on the quasi-static de-formation of soils was developed by Terzaghi [20, 21]. This theory was first generalized by Rendulic [49], however it was Biot [15, 16] that first developed a linear poroelasticity theory that is consistent with both mechanisms outlined by Detournay and Cheng [76]. These works became the main foundation on representing different porous media in studies involving fluid-structure coupling.
Over the years, other researchers contributed to the expansion of the consoli-dation theory. These contributions formed the basis of the poroelasticity theory which is nowadays used in poromechanics problems in the petroleum industry. Among these works, can be cited Biot [17, 18], Biot and Willis [19], Detournay and Cheng [76], Cryer [77], Verruijt [78], Rice and Cleary [79], Green and Wang [80], and Cheng [81].
Rice and Cleary [79] linked the poroelasticity parameters to concepts that are well understood in rock and soil mechanics. In particular, their work emphasizes two limiting behaviours of a fluid-filled porous medium: drained and undrained conditions. Detournay and Cheng [76] formulated the micromechanical approach in which individ-ual contributions of solid and fluid are explicitly considered. Further discussion on this topic is presented in the following sections.
This chapter reviews Biot’s soil consolidation theory, the basis of poroelasticity theory, and discuss the meaning of the constants introduced by Biot. Next are presented the relationships used in the present work for the description of the porous medium and the hypothesis of porosity invariance is supported by a scale analysis of the presented relationships. Finally, the mathematical model for two-dimensional poroelasticity, under the assumption of plane-strain state, is addressed.
2.1 BIOT’S SOIL CONSOLIDATION
It is well known to engineering practice that a soil under load does not assume an instantaneous deflection but settles gradually at a variable rate. Such settlement
is very apparent in clays and sands saturated with water. The settlement is caused by a gradual adaptation of the soil to the load variation. This process is known as soil consolidation [16].
This phenomenon was first explained by Terzaghi [20, 21]. By looking at the soil as a porous medium with elastic properties and voids filled with water, Terzaghi [20, 21] was able to analyse the settlement of a column of soil under a constant load and oedometric conditions2. The remarkable success of this theory in predicting the
settlement for many types of soils has been one of the strongest incentives in the creation of a science of soil mechanics [16].
Following laboratory experiments, Terzaghi [20, 21] first explained the soil con-solidation by analysing the sedimentation of a soil column subject to a constant load and restricted to any lateral displacement. Then, Biot [16] expanded the theory to a three-dimensional case, considering a time-dependent load. Biot [16] modelled the soil as a porous medium filled with water and assumed the medium to behave as a continuous mass rather than as discrete particles.
Biot [16] considered the following assumptions in his model: i Isotropic and homogeneous material;
ii Linear-elastic stress-strain3relationship;
iii Small strains; iv Isochoric fluid flow;
v Fluid flow in accordance with Darcy’s law.
2.1.1 MOMENTUM CONSERVATION
Consider an infinitesimal control volume as shown in Figure 2.1.1 and subject to a stress field σ and a gravitational field g. A differential momentum balance yields to the equilibrium criteria given by
∇ ¨ σ ` ρg “ 0, (2.1.1)
where ρ is the solid-fluid continuum density given by
ρ “ φρf ` p1 ´ φqρs. (2.1.2)
2Oedometer tests simulate one-dimensional deformation. The sample is confined such that lateral
displacement is prevented, but allowed to swell or compress vertically in response to changes in applied load.
3The sign convention for the entire present work shall be the same as it follows. Compression stress
shall be considered negative while tension stress shall be considered positive. Volumetric strain shall be positive for expansion and negative for compression.
Figure 2.1.1 – Infinitesimal control volume subject to stress and gravitational fields.
σ
xxσ
xyσ
xzσ
yxσ
yyσ
yzσ
zxσ
zyσ
zzx
y
z
g
Source: Own authorship.
In Equation 2.1.2, ρf and ρs are the density of the fluid-phase and the solid-phase,
respectively, and φ is the medium porosity4, defined as the ratio of the pore volume and the bulk volume.
Terzaghi [20, 21] defined the effective stress σ1 as an internal stress field
devel-oped in the solid structure which must balance the external loads. The effective stress is given by
σ1
“ σ ` αpI, (2.1.3)
where σ is the total stress, α is Biot-Willis’ coefficient [19], p is the pore-pressure and I is the second-order identity tensor. Substituting Equation 2.1.3 into Equation 2.1.1, we obtain
∇ ¨ σ1
´ ∇ pαpq ` ρg “ 0. (2.1.4)
Following the linear-elastic stress-strain relationship assumption, we can write σ1
“ 2G ` λtr pq I, (2.1.5)
where G is the shear modulus, λ is Lamé’s first parameter5, and is the strain tensor field, which under small strains assumption is given by
“ 1 2
”
∇u ` p∇uqTı, (2.1.6)
4In the present work, we shall use no subscript for bulk properties, and subscripts "f" and "s", for
fluid-phase and solid-phase properties, respectively.
5The Lamé parameters are two material-dependent quantities denoted by λ and µ that arise in
strain-stress relationships. In general, λ and µ are individually referred to as Lamé’s first parameter and Lamé’s second parameter, respectively. In the context of elasticity, µ is called shear modulus and is sometimes denoted by G instead of µ, as in the present work.
where u is the displacement vector field and ∇u is the displacement gradient tensor given by ∇u “ » — — — — — — — – Bu Bx Bu By Bu Bz Bv Bx Bv By Bv Bz Bw Bx Bw By Bw Bz fi ffi ffi ffi ffi ffi ffi ffi fl . (2.1.7)
For isotropic elastic porous materials, only two independent parameters are needed to describe the relationship between effective stress σ1 and strain . These
parameters are related by
νE “ 2νp1 ` νqG “ 3νp1 ´ 2νqK “ λp1 ` νqp1 ´ 2νq, (2.1.8) where E is the Young’s modulus, K is the bulk modulus6and ν is the Poisson’s ratio.
Substituting Equation 2.1.5 into Equation 2.1.4, we obtain
∇ ¨ p2Gq ` ∇ pλε ´ αpq ` ρg “ 0, (2.1.9) where ε is the volumetric strain given by
ε “ tr pq “ xx` yy` zz “ ∇ ¨ u. (2.1.10)
Substituting Equation 2.1.6 in Equation 2.1.9, we obtain ∇ ¨”G∇u ` G p∇uqT
ı
` ∇ pλε ´ αpq ` ρg “ 0, (2.1.11) which is the conservative form of the momentum conservation equation. The conser-vative form is important in the FVM, as it will be discussed on Chapter 3. Following the assumption of homogeneous material, we can take G, λ and α off the differential operators, obtaining
G∇2u ` pG ` λq∇ p∇ ¨ uq ´ α∇p ` ρg “ 0. (2.1.12) Equation 2.1.12 is very similar to the well-known Navier-Stokes equation for a compressible fluid. This vectorial equation can be split into three scalar equations. In Cartesian coordinates, these equations are
G∇2u ` pG ` λq Bε Bx ´ α Bp Bx ` ρgx“ 0, G∇2v ` pG ` λq Bε By ´ α Bp By ` ρgy “ 0, G∇2w ` pG ` λqBε Bz ´ α Bp Bz ` ρgz “ 0. (2.1.13)
6Not be confused! Bulk modulus describes the material’s response to a volumetric isotropic stress. In
the present work, the bulk modulus K is actually the bulk modulus of the bulk material (or bulk volume), since it describes the solid-fluid continuum’s response to volumetric isotropic stresses. To avoid sounding redundant, "bulk modulus of the bulk volume" will be shorted to "bulk modulus". The present work will be explicit when describing any other situation.
Equations 2.1.13 are the governing equations of the geomechanics problem for the displacement field components and represent the momentum conservation, relating displacement, fluid pore pressure and body forces.
2.1.2 MASS CONSERVATION
Fig. 2.1.2 shows a cross section area of a porous medium with fluid-phase and solid-phase velocities and their average value. We denote v1
f, v1sand v1 as real profiles of
the absolute fluid-phase velocity, absolute solid-phase velocity and fluid-phase velocity relative to the solid-phase. In order to obtain these profiles, the conservation equations should be solved for the pore channels, which is not the objective of the present work. Following the assumption of a solid-fluid continuum, we consider average values of these velocities, denoted by φvf, φvs and φvf {s.
Figure 2.1.2 – Fluid and solid phases velocities and their average value.
v
1v
1s
v
1 f
v
1φv
f {sφv
sφv
fSource: Adapted from Dal Pizzol [69].
Consider a fluid flow through a deforming and fully-saturated porous medium, as depicted in Fig. 2.1.3. A mass balance for the fluid-phase yields
B Btpρfφq∆x∆y ´ ρfq∆x∆y “ ` 9 mx|x´ 9mx|x`∆x ˘ ` ´ 9 my|y ´ 9my|y`∆y ¯ , (2.1.14) where q is the rate of fluid volume injection/withdrawal at reservoir conditions, per unit of control volume. Considering the average values of the fluid-phase velocities, as depicted in Fig. 2.1.2, the mass fluxes 9mx and 9my are given by
9 mx “ ży`∆y y ρfv1f xdy “ ρfφvf x∆y, (2.1.15) 9 my “ żx`∆x x ρfv1f ydx “ ρfφvf y∆x. (2.1.16)
Figure 2.1.3 – Fluid flow through an infinitesimal deforming control volume. 9 mx|x`∆x 9 my|y`∆y 9 mx|x 9 my|y x y
Source: Own authorship.
Substituting Eq. 2.1.15 and 2.1.16 into Eq. 2.1.14, and rearranging, yields B Btpρfφq ´ ρfφvf x|x´ ρfφvf x|x`∆x ∆x ´ ρfφvf y|y ´ ρfφvf y|y`∆y ∆y “ ρfq,
and in the limit, when ∆x, ∆y Ñ 0, B
Btpρfφq ` ∇ ¨ pρfφvfq “ ρfq,
wherevf is the absolute fluid-phase velocity. This equation may be rewritten as
B
Btpρfφq ` ∇ ¨ rρfφpvf ´ vsqs ` ∇ ¨ pρfφvsq “ ρfq, (2.1.17) wherevs is the solid-phase velocity. We can note that ρfφ is the partial density of the
fluid-phase [82], defined as mass of fluid per unit of the total volume. As stated by McTigue [82], in linearised elasticity theory, the partial density and the volumetric strain of the fluid εf are related by
ρfφ “ ρf 0φ0p1 ´ εfq, (2.1.18)
where ρf 0 and φ0 are reference values (unstrained) of the fluid density and the medium
porosity. In fact, ρfφVf “ ρf 0φ0Vf 0, where Vf is the volume occupied by the fluid, and
εf “ pVf ´ Vf 0q{Vf 0. Combining these relationships, we can write
ρfφp1 ` εfq “ ρf 0φ0 6 ρfφ “ ρf 0φ0 1 ` εf “ ρf 0φ0p1 ´ εfq 1 ´ ε2 f « ρf 0φ0p1 ´ εfq,
since εf ! 1 and 1 ´ ε2f « 1. Substituting Eq. 2.1.17 into Eq. 2.1.18, we obtain
B Btrρf 0φ0p1 ´ εfqs ` ∇ ¨ rρf 0φ0p1 ´ εfqpvf ´ vsqs ` ∇ ¨ rρf 0φ0p1 ´ εfqvss “ ρfq, or ´B Btpφ0εfq ` ∇ ¨ rφ0p1 ´ εfqpvf ´ vsqs ` ∇ ¨ rφ0p1 ´ εfqvss “ ρf ρf 0 q.
