Modeling Multiphase Flow in Porous Media with Complementary Constraints

Transcription

1 Background Modeling Multiphase Flow in Porous Media with Complementary Constraints Applied Math, Statistics, and Scientific Computation, University of Maryland - College Park October 07, 2014 Advisors Dr. Amir Riaz, Department of Mechanical Engineering, UMCP Dr. David Moulton, Applied Math Group (T-5), Los Alamos National Lab

2 Outline Background 1 Background Applications of Multiphase Flow Continuum Model Project Goals 2 Traditional Approach Multiphase Flow with Nonlinear Complementarity Constraints 3 HPC Framework Unit Tests for Verification Validation and Benchmarking 4 Timeline Deliverables

3 Background Applications of Multiphase Flow Carbon sequestration Groundwater management Contaminant transport Oil and gas recovery Applications of Multiphase Flow Continuum Model Project Goals Source: [Moulton and Coon, 2013] Source: [Hubbard et al., 2011]

4 Mass Balance Background Applications of Multiphase Flow Continuum Model Project Goals We have the general form of the conservation equation: ξ t + ψ = f (1) where ξ is the storage term, ψ the flux term, and f the source term. For mass balance of a component K in an M-phase and N-component system, the above terms read: ξ K = φ ψ K = f = f K M ρ mol,α XαS K α α=1 M α=1 ( ) ρ mol,α Xαq K α + D K pm,αρ mol,α Xα K

5 Energy Balance Background Applications of Multiphase Flow Continuum Model Project Goals The energy conservation equation has the same form as (1). However, the storage, flux, and source terms are ξ K = φ ψ K = f = f h M ρ mass,α u α S α + (1 φ)ρ s c s T α=1 M α=1 (ρ mass,α h K αq α ) + N K=1 α=1 M (D K pm,αρ mol,α h K αm K xα) K λ pm T

6 Constitutive Relations Background Applications of Multiphase Flow Continuum Model Project Goals We assume the flux q obeys an extension to mutiphase of Darcy s law: q α = κ rα µ α K( p α ρ mass,α g) Source: MIT OpenCourseWare The phase pressures are related through capillary pressure law. In two-phase flow with liquid and gas phases, p c (S l ) = p g p l Depending on the structure of the porous medium, various laws for capillary pressures can be used.

7 Algebraic constraints Background Applications of Multiphase Flow Continuum Model Project Goals In addition to mass balance and constitutive equations, we have the constraints for saturation: and for molar fractions: M S α = 1 α=1 0 S α 1 N Xα K = 1 K=1 0 X K α 1

8 Background Applications of Multiphase Flow Continuum Model Project Goals How do we solve all these equations? ( φ ) ( ρ mol,α X K t αs α + ( )) ρ mol,α Xαq K α + D K pm,αρ mol,α Xα K = f K α=l,g α=l,g ( φ ) ρ mass,α u α S α + (1 φ)ρ s c s T + t α=l,g ( M α=1 (ρ mass,α h K αq α ) + q α = κ rα µ α K( p α ρ mass,α g) p c (S l ) = p g p l, S l + S g = 1, 0 S α 1, N K=1 α=l,g ) (D K pm,αρ mol,α h K αm K xα) K λ pm T = f h N Xα K = 1, 0 Xα K 1 K=1

9 Project Goals Background Applications of Multiphase Flow Continuum Model Project Goals Implement the nonlinear complimentarity constraints approach to solve this system of equations in a HPC framework. Provide a capability to model fully coupled 2-phase, 2-component, non-isothermal, miscible flow with phase transitions. Pursue application to realistic problem such as desiccation.

10 Background Primary variable switching - PVS Traditional Approach Discretization Multiphase Flow with Nonlinear Complementarity Constraints Discretize the balance equations to obtain a system of nonlinear equations. For each cell, choose a set of appropriate primary variables [Wu and Forsyth, 2001], e.g., S l and p l when both phases exist. Xl a and p l when only the liquid phase exists. Xg w and p g when only the gas phase exists. Solve the nonlinear system using Newton method Drawbacks Difficult to predict global convergence behavior of the Newton method. Forced to take very small time steps near phase change. Poor handling of phase transitions.

11 Finite Volume Method Background Traditional Approach Discretization Multiphase Flow with Nonlinear Complementarity Constraints Using cell-centered finite volume for spatial discretization, we have ξ + ψ n = f t C i j η γ ij C i i We introduce some approximation of the volume integrals ξ = 1 ξ, F = 1 f V Ci C i V Ci C i For the surface integrals, we use two-point flux approximation (TPFA) ψ n = γ ij (ρ mol,α X K κ rα α κ) ij+1/2 (ω i ω j ) γ ij µ α ω i = (p α ) i (ρ mass,α ) ij+1/2 g

12 Background Traditional Approach Discretization Multiphase Flow with Nonlinear Complementarity Constraints Discretize in time using backward Euler method, we have a fully discrete system of N nonlinear equations. H(x) = ( ξ(x)) n+1 i ( ξ(x)) n i t t n+1 V ij+1/2 (ωn+1 i ω n+1 j ) F n+1 = 0 Ci j η i where t = γ ij ρ mol,α Xα K κ rα κ is the transmissibility and µ α x = {p 1, X1 1,..., XN 1, S 2,..., S M, T} is the set of N + M + 1 primary variables. The choice of x is not unique, e.g. it is convenient to choose x = {p 1, fg 1,..., fg N, S 2,..., S M, T} [Lauser et al., 2011] in which fg K are the component fugacities defined by Φ K α is the fugacity coefficient. fg K = Φ K αx K αp α, for K {1, 2,..., N}

13 Background Traditional Approach Discretization Multiphase Flow with Nonlinear Complementarity Constraints Nonlinear Complementarity Constraints (NCP) In its simplest form, a nonlinear complementarity constraint problem can be written as follows where x R n, and f : R n R n is a smooth function. A slightly more general form of equation (4) reads x 0 (2) f (x) 0 (3) x T f (x) = 0 (4) g(x) T f (x) = 0 where g : R n R n is another smooth function.

14 Background Multiphase Flow with NCP Traditional Approach Discretization Multiphase Flow with Nonlinear Complementarity Constraints Along with the discretized balance equations, we still need another M equations to complete the system. Recall the constraints for molar fractions N Xα K 1, K=1 N Xα K = 1 if phase α is present. K=1 Combining with the saturation constraints, we get the complementarity conditions 1 N Xα K 0, S α 0, S α (1 K=1 N Xα) K = 0. K=1

15 Background Multiphase Flow with NCP Traditional Approach Discretization Multiphase Flow with Nonlinear Complementarity Constraints The nonlinear complementarity constraints of saturation and molar fractions can be written in compact form as min(s α, 1 N Xα) K = 0 Combining with the discretized mass balance equations, we have the full system for each time step: { H(x) = 0 (from the PDE) R(x) = Ψ(x) = min(f, G) = 0 (from the constraints) with F, and G are discretized functions of S α and 1 N K=1 XK α respectively. K=1

16 Background Traditional Approach Discretization Multiphase Flow with Nonlinear Complementarity Constraints Since Ψ is not differentiable everywhere, we solve the system using a semi-smooth Newton method [Gharbia and Jaffré, 2014]. Algorithm: while k < max iter and res > ɛ do (1) Define the index sets A k and I k : A k := {j : F j (x k ) G j (x k )}, I k := {j : F j (x k ) < G j (x k )} (2) Select an element J k Ψ(x k ) such that its jth line is equal to F j (xk ) if j I k, G j (xk ) if j A k (3) Solve the system H (x k ) x k = H(x k ) J k x k = Ψ(x k ) (4) Update x k+1 x k+1 = x k + x k

17 Background Traditional Approach Discretization Multiphase Flow with Nonlinear Complementarity Constraints Since Ψ is not differentiable everywhere, we solve the system using a semi-smooth Newton method [Gharbia and Jaffré, 2014]. Algorithm: while k < max iter and res > ɛ do (1) Define the index sets A k and I k : A k := {j : F j (x k ) G j (x k )}, I k := {j : F j (x k ) < G j (x k )} (2) Select an element J k Ψ(x k ) such that its jth line is equal to F j (xk ) if j I k, G j (xk ) if j A k (3) Solve the system H (x k ) x k = H(x k ) J k x k = Ψ(x k ) (4) Update x k+1 x k+1 = x k + x k

18 Background Traditional Approach Discretization Multiphase Flow with Nonlinear Complementarity Constraints Since Ψ is not differentiable everywhere, we solve the system using a semi-smooth Newton method [Gharbia and Jaffré, 2014]. Algorithm: while k < max iter and res > ɛ do (1) Define the index sets A k and I k : A k := {j : F j (x k ) G j (x k )}, I k := {j : F j (x k ) < G j (x k )} (2) Select an element J k Ψ(x k ) such that its jth line is equal to F j (xk ) if j I k, G j (xk ) if j A k (3) Solve the system H (x k ) x k = H(x k ) J k x k = Ψ(x k ) (4) Update x k+1 x k+1 = x k + x k

19 Background Traditional Approach Discretization Multiphase Flow with Nonlinear Complementarity Constraints Since Ψ is not differentiable everywhere, we solve the system using a semi-smooth Newton method [Gharbia and Jaffré, 2014]. Algorithm: while k < max iter and res > ɛ do (1) Define the index sets A k and I k : A k := {j : F j (x k ) G j (x k )}, I k := {j : F j (x k ) < G j (x k )} (2) Select an element J k Ψ(x k ) such that its jth line is equal to F j (xk ) if j I k, G j (xk ) if j A k (3) Solve the system H (x k ) x k = H(x k ) J k x k = Ψ(x k ) (4) Update x k+1 x k+1 = x k + x k

20 Amanzi Background HPC Framework Unit Tests for Verification Validation and Benchmarking The simulator of our choice is Amanzi [ASCEM, 2009], which is a part of the Advanced Simulation Capability for Environmental Management (ASCEM) program. Using Amanzi for this course project has many advantages. Open source, still being actively developed. High-level, object-oriented C++ code with extensive documentation. New process kernel can be added easily. Support both structured/unstructured grids. Easy access to advanced linear and nonlinear solvers through flexible interfaces. Suitable for highly parallel, emerging architectures. The code will be used by domain scientists and modelers.

21 Background HPC Framework Unit Tests for Verification Validation and Benchmarking Hardware Code development on Personal laptop, Macbook Pro 2GHz i7 quad-core processor with 8GB RAM. Poblano - a 64-processor, 256GB shared memory server at the New Mexico Consortium. For realistic problem, we might get some time on Hopper, a petaflop supercomputer at NERSC. Source: NERSC

22 Background Unit tests for verification HPC Framework Unit Tests for Verification Validation and Benchmarking Unit test for pressure equation. Unit test for 1D advection. (λ P) = f u t + f (u) x = 0 Buckley-Leverett problem for one-dimensional 2-phase, 2-component, immiscible, incompressible flow. S t = U(S)S x U(S) = Q df φa ds Method of manufactured solution (MMS) for more complex flow.

23 Background Validation and Benchmarking HPC Framework Unit Tests for Verification Validation and Benchmarking Validation Physical data very difficult to obtain. Data not publicly available. Might not be able to validate. Benchmarking Benchmark against existing codes and test problems [Bourgeat et al., 2013]. Open Porous Media (OPM) - ewoms [Lauser, 2013].

24 Timeline Background Timeline Deliverables Phase I October Gain confidence working with Amanzi coding standard, build tools, interfacing with other libraries, etc. Get this new PK to interface correctly with both the multi-process coordinator (MPC) and input files Start with building a simple solver for the pressure equation. November Implement the code for 2-phase, single-component, isothermal flow. Implement the code for 2-phase, 2-component, isothermal flow. December Start adding nonlinear complementarity constraints for phase transitions. Prepare mid-year report and presentation.

25 Timeline (cont.) Background Timeline Deliverables Phase II January: Develop active sets and semi-smooth Newton method. February: Add miscibility effect. March: Incorporate energy equation for thermal effect. April Collect the unit tests and make a test suite. Benchmark with existing codes or pursue realistic problems, such as desiccation if time permits. May: Prepare final report and presentation.

26 Deliverables Background Timeline Deliverables An implementation of the NCP approach for a two-phase, two-component, non-isothermal, miscible flow as a PK in Amanzi. A semi-smooth Newton solver for the resulting system. Unit tests for verification. Results of validation and testing. Documentation for all the codes. User guide for multiphase flow PK. Weekly reports. Project proposal and presentation. Mid-year presentation and report. Final presentation and report.

27 References ASCEM. http: //esd.lbl.gov/research/projects/ascem/thrusts/hpc/, Alain Bourgeat, Farid Smaï, and S. Granet. Compositional Two-Phase Flow in Saturated-Unsaturated Porous Media: Benchmarks for Phase Appearance/Disappearance. In Simulation of Flow in Porous Media, pages De Gruyter, URL Ibtihel Ben Gharbia and Jérôme Jaffré. Gas phase appearance and disappearance as a problem with complementarity constraints. Mathematics and Computers in Simulation, 99(0):28 36, S. Hubbard, C. Steefel, H. Beller, and K. W. Williams. http: //esd.lbl.gov/files/research/projects/sustainable_ systems/sustainsystemsbrochure2011small.pdf, A. Lauser. ewoms module A. Lauser, C. Hager, R. Helmig, and B. Wohlmuth. A new approach for phase transitions in miscible multi-phase flow in porous media. Advances in Water Resources, 38: , 2011.

28 References J. D. Moulton and E. Coon. From multi-physics to many-physics: A paradigm shift in environmental and climate modeling. Technical report, Los Alamos National Laboratory, Y-S. Wu and P.A. Forsyth. On the selection of primary variables in numerical formulation for modeling multiphase flow in porous media. Journal of Contaminant Hydrology, 48: , 2001.

29 References List of Unknowns and Physical Quantities S α saturation of phase α P α pressure of phase α Xα K molar fraction of component K in phase α fg K fugacity of component K T temperature φ porosity µ α viscosity of phase α h α specific enthalpy of phase α ρ s density of the solid phase c s specific heat capacity of the solid phase D K pm,α diffusion coeff. of comp. K in phase α u α specific internal energy of phase α λ pm heat conduction coefficient κ rα relative permeability of phase α