Effective Solvers for Reservoir Simulation Xiaozhe Hu The Pennsylvania State University Numerical Analysis of Multiscale Problems & Stochastic Modelling, RICAM, Linz, Dec. 12-16, 2011 X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 1 / 47
Collaboration with: ◮ ExxonMobil Upstream Research Company ◮ China National Offshore Oil Cooperation ◮ Monix Energy Solutions Main Collaborators: ◮ James Brannick (PSU), Yao Chen (PSU), Chunsheng Feng (XTU), Panayot Vassilevski (LLNL), Jinchao Xu (PSU), Chensong Zhang (CAS), Shiquan Zhang (Fraunhofer ITWM), and Ludmil Zikatanov (PSU) X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 2 / 47
Outline Outline Reservoir Simulation 1 Solvers for Reservoir Simulation 2 Applications & Numerical Tests 3 Ongoing Work and Conclusions 4 X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 3 / 47
Reservoir Simulation Reservoir Simulation 1 Solvers for Reservoir Simulation 2 Applications & Numerical Tests 3 Ongoing Work and Conclusions 4 X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 4 / 47
Reservoir Simulation World without Oil? List of Oil Products: Antiseptics, Aspirin, Auto Parts, Ballpoint pens, Candles, Cosmetics, Crayons, Eye Glasses, Fishing Line, Food Packaging, Glue, Hand Lotion, Insect Repellant, Insecticides, Lip Stick, Global oil discovery peaked in the late Perfume, Shampoo, Shaving 1960s. Now we consume much more Cream, Shoes, Toothpaste, than the amount we discover and this Trash Bags, Vitamin gap is still growing! Capsules, Water Pipes, · · · . ( http://www.aspo-ireland.org/ ) X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 5 / 47
� Reservoir Simulation Oil Recovery � Primary Recovery Secondary Recovery Enhanced Oil Recovery X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 6 / 47
Reservoir Simulation Enhanced Oil Recovery Enhanced Oil Recovery (EOR) is a generic term for techniques for increasing the oil recovery rate from an oil field: Thermal Method Microbial Injection Gas Injection Chemical Flooding ◮ Alkaline Recovery Rate: ◮ Polymer Primary and Secondary ◮ Surfactant ◮ Gel recovery: 20-40% ◮ · · · Enhanced oil recovery: · · · 30-60% X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 7 / 47
Reservoir Simulation Black Oil Model Mass Conservation: � 1 � � S o �� � ∂ + R v S g u o + R v φ = −∇ · u g + ˜ q O ∂ t B o B g B o B g � � S g �� � 1 � ∂ + R s S o u g + R s φ = −∇ · + ˜ u o q G ∂ t B g B o B g B o � 1 � � � ∂ φ S w = −∇ · u w + ˜ q W ∂ t B w B w Darcy’s Law: u α = − kk r α ( ∇ P α − ρ α g ∇ z ) , α = o , g , w µ α Constitutive Laws: S o + S g + S w = 1 , P cow = P o − P w , P cog = P g − P o , X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 8 / 47
Reservoir Simulation Modified Black Oil Model Add polymer and sodium chloride: � φ S ∗ � � C P � ∂ + ∂ w C P ∂ t ( ρ r (1 − φ ) C a ) = −∇ u P + ˜ q W C P ∂ t B r B w B w � φ S w C N � � C N � ∂ = −∇ u N + ˜ q W C N ∂ t B r B w B w Effects on Viscosity: kk rw u w = − ( ∇ P w − ρ w g ∇ z ) µ w , ef f R k Add more components (brine, gel, surfactant, · · · ) X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 9 / 47
Reservoir Simulation Numerical Method Discretization in time: ◮ Fully Implicit Method (FIM) (Douglas, Peaceman & Rachford 1959) ◮ Implicit Pressure Explicit Saturation (IMPES) (Sheldon, Zondek & Cardwell 1959; Stone & Garder 1961) ◮ Sequential Solution Method (SSM) (MacDonald & Coats 1970) ◮ Adaptive Implicit Method (AIM) (Thomas & Thurnau 1983) Discretization in space: finite difference, finite volume, finite element, etc. Linearization: Newton-Raphson Method. Grid: Structured grids are still used by the main-stream; but unstructured grids are catching up. X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 10 / 47
Reservoir Simulation Fully Implicit Method �� �� n � � S g �� n +1 � � S g 1 + R s S o + R s S o φ − φ = ∆ t B g B o B g B o ∇ · ( T n +1 ∇ Φ n +1 + R n +1 T n +1 ∇ Φ n +1 ) g α s o o �� φ S w � n � � n +1 � φ S w 1 = ∇ · ( T n +1 ∇ Φ n +1 − ) w w ∆ t B w B w �� φ S o � n � � n +1 � φ S o 1 = ∇ · ( T n +1 ∇ Φ n +1 − ) o o ∆ t B o B o Potential: Φ α := P α − ρ α g z , α = w , o , g , Transmissibility: T α := k r α k , α = w , o , g . µ α B α X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 11 / 47
Reservoir Simulation Newton-Raphson Method In each Newton Iteration, we need to solve the following Jacobian system: J gP J gS w J gS o δ P R g = δ S w J wP J wS w J wS o R w δ S o J oP J oS w J oS o R o Different blocks have different properties, for example J gP = 1 ∆ t c gP − ∇ · D gP ∇ − C gP · ∇ − R gP , usually the diffusion term in J gP is dominating, which makes J gP like an elliptic problem. However, J oS o = 1 ∆ t c oS o − C oS o · ∇ − R oS o , here convection term is dominating, which makes J oS o like a transport problem. X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 12 / 47
� � Reservoir Simulation Including Wells Peaceman Model One-dimentional radial flow Single-phase flow in the wellbore N perf 2 π h perf Kk r α X q i ,α = − r w + s ) µ α ( P j − P bhp − H wj ) r e ln( j Jacobian system with wells: � J RR � � δ R � � R R � J RW = J WR J WW δ W R W X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 13 / 47
Reservoir Simulation Numerical Challenges Strongly coupled and complicated PDE system High nonlinearity Complicated geometry: irregular domain with faults, pinch-out and erosion Heterogeneous porous media; large permeability jumps Complex wells Unstructured grids: corner point grid, anisotropic mesh Highly non-symmetric and indefinite large-scale Jacobian system Different properties of the physical quantities (the equation that describes the pressure is mainly elliptic, the equation that describes saturation is mainly hyperbolic) Commercial Simulators set a high bar: ECL2009 (Schlumberger), STARS (CMG), VIP (Halliburton), · · · Goal: develop fast solvers, deliver a solver package X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 14 / 47
Solvers for Reservoir Simulation Reservoir Simulation 1 Solvers for Reservoir Simulation 2 Applications & Numerical Tests 3 Ongoing Work and Conclusions 4 X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 15 / 47
Solvers for Reservoir Simulation Solvers “For a reservoir simulation with a number of gridblocks of order 100,000, about 80% − 90% of the total simulation time is spent on the solution of linear systems with the Jacobian.” —- Chen, Huan, & Ma, Computational Methods for Multiphase Flows in Porous Media , 2005 X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 16 / 47
Solvers for Reservoir Simulation Solvers “For a reservoir simulation with a number of gridblocks of order 100,000, about 80% − 90% of the total simulation time is spent on the solution of linear systems with the Jacobian.” —- Chen, Huan, & Ma, Computational Methods for Multiphase Flows in Porous Media , 2005 Direct solvers: (Price & Coats 1974) Incomplete factorization preconditioner: (Watts 1981, Behie & Vinsome 1982; Meyerink 1983; Appleyard & Cheshire 1983) Constrained Pressure Residual (CPR) preconditioner: (Wallis 1983; Wallis,Kendall, Little & Nolen 1985; Aksoylu & Klie 2009) Algebraic Multigrid (AMG) Method : ◮ Used in CPR preconditioner (pressure equation): (Klie 1997; Lacroix, Vassilevski & Wheeler 2001, 2003; Scheichl, Masson & Wendebourg 2003; Cao, Tchelepi, Wallis & Yardumian 2005; Hammersley & Ponting 2008; Dubois, Mishev & Zikatanov 2009; Al-Shaalan, Klie, Dogru & Wheeler 2009; Jiang & Tchelepi 2009) ◮ For whole Jacobian system: (Papadopoulos & Tchelepi 2003; St¨ uben 2004; Clees & Ganzer 2007) X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 16 / 47
Solvers for Reservoir Simulation Auxiliary Space Preconditioning 1 construct preconditioner via auxiliary (simpler) problems 2 solve the auxiliary problems by efficient solvers (such as multigrid) 3 apply preconditioned Krylov subspace methods Examples: Fictitious Domain Methods (Nepomnyaschikh 1992) Method of Subspace Correction (Xu 1992) Auxiliary Space Method (Xu 1996) Nodal Auxiliary Space Preconditioning in H ( curl ) and H ( div ) spaces (Hiptmair & Xu 2007) · · · · · · X. Hu (Penn State) Effective Solvers for Reservoir Simulation Linz,Dec.14, 2011 17 / 47
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