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Simulating fluid flow in fractured reservoirs with The Matlab Reservoir Toolbox (MRST) Tor Harald Sandve
photo by Chuck DeMets
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Agenda
28. may 2013 2
› Modeling philosophy
› Equations and discretizations
› Modeling concepts for fractures
› Numerical examples
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Mathematical models
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› Darcy’s law (unit viscosity):
𝒗 = − K 𝛻𝛻
› Conservation of mass (single phase, incompressible) → 𝛻 ⋅ 𝒗 = q
› Pressure equation:
−𝛻 ⋅ ( K 𝛻 p ) = q
› Tracer transport:
𝜙𝜕𝜕𝜕𝜕
+ 𝛻 ⋅ 𝜕𝒗 = 𝑞𝑡
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Finite-volume method
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› Integral formulation
� 𝒗 ⋅ 𝒏 d𝑆 = � 𝑞 dΩΩ𝑖𝜕Ω𝑖
› Flux approximation
� 𝒗 ⋅ 𝒏 d𝑆𝑆
≈ � 𝜕𝑘𝛻𝑘
𝜈
𝑘=1
› Linear system 𝑨𝑨 = 𝒒
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Modeling concepts
› Hierarchical modeling: • Large-scale fractures: explicitly • Small-scale fractures: effective permeability (anisotropic)
• Consistent and robust discretization of flow in fractured reservoirs
› incompTPFA_DFM / incompMPFA_DFM 28. may 2013 5
Upscaling
Large- and small-scale fractures
Discrete fracture model + continuum model
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Object:
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› But….. • What if the conditions for upscaling is not valid? (no scale – separation etc. ) • Only coarse transport
› We want a method that is: 1. Comparable with upscaling when the conditions for upscaling apply 2. Otherwise, comparable with standard fine-scale solvers 3. Capable of fine-scale transport
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Modeling concepts
› MsFVM: Multiscale finite volume method (Jenny 2003) – Coarse grids – Post-processing – Direct method / Multiscale method – Inexact solver / Preconditioner in GMRES
› solveMSFV_TPFA_Incomp_DFM
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Upscaling
Fine-scale model Coarse-scale model
Downscaling
Large- and small-scale fractures
Discrete fracture model + continuum model
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Example setup
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› Aperture large-scale fractures: 1mm
› Aperture small-scale fractures: 0.5mm
› Fracture permeability = aperture212
› Fracture porosity: 1
› Matrix permeability: 1mDarcy
› Matrix porosity: 0.01
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Case 1
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› Conforming coarse grid › Non-conforming coarse grid
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Compare as multiscale method
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Compare as multiscale method
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› Multiscale solution › Reference solution
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Compare as preconditioner
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AMG: Y. Notay 2010
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Case 2
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› Conforming coarse grid › Non-conforming coarse grid
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Case 2
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› Direct method › 10 GMRES iterations
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Compare as preconditioners
28. may 2013 15
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Transport upscaling
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› Coarse transport
› Hierarchical models • Large-scale fractures explicit
› Dual models • Small-scale fractures into a fracture continuum
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Transport models
28. may 2013 17
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After thoughts
28. may 2013 18
› Pressure equation: • Linking uncertainty and stopping criteria for the linear solvers
› Transport • Flow based upscaling • aMINC (automatic Multiple INteracting Continua)