Upscaling Macroscopic Properties from the Porescale using Surrounding Flow Behavior

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Upscaling Macroscopic Properties from the
Pore-scale using Surrounding Flow Behavior

• Matthew T. Balhoff
• PGE, UT-Austin
• Oct 15, 2009

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Pore-Scale Solution

• Pore-level models used to obtain macroscopic
  properties
• Models used as a stand-alone tool
• Boundary conditions are simple
• No information regarding surrounding media
  included

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Upscalingnot so simple
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Modeling Through the Magnifying Glass
http//www.cpge.utexas.edu/new_generation/
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Mortar Coupling

• 2D FEM mortar spaces are used at interfaces to
  iteratively match pressures at fluxes
• IPARS is designed to allow for coupling of
  different models, physics, and scales
• Improve accuracy by using finer meshes and
  higher-order mortars

• Subdomains are decomposed and solved separately
  using interface pressure bcs
• The subdomains can be different models or even
  scales, which would allow us to couple pore and
  continuum-scale models

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Model Validation Problem
P1.0
P3.0
Mortar Space

• Periodic network model coupled to its replica
• Still want to solve as stand-alone tools
• What pressure field P(x,y) at the interface will
  result in weakly matched fluxes? P 2.0?

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Actual Versus Mortar Approximation
Actual
8x8 Quadratic Mortars
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Upscaling - Single Phase Flow

• Created large (million pore) network models
• Very heterogeneous
• Abrupt changes in pore structure
• Solved for pressures, flows in the network
• Back-calculated permeability using Darcys law
  (KTRUE)

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Traditional Upscaling Approach

• Split the network into several smaller networks
• Solve each network
• Back-calculate each sub-network permeability
• Upscale to get KFD for entire domain using a
  traditional finite difference upscaling
• KFD ? KTRUE

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Upscalinga Mortar Approach

• Split networks at natural boundaries
• Couple using FEM mortars
• Calculate Upscaled K much better than
  traditional approach

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Multiphase Flow (Capillary-dominated Drainage)

• Need relative permeability and capillary pressure
  curves
• Pore-scale models can be a surrogate for
  experiment
• Distribution of fluids may affect macroscopic
  properties
• Traditional boundary conditions may not be
  sufficient

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Relative Permeability Curve
Compare stand-alone network, to one surrounded by
low permeability medium
Relative Permeability
Wetting Phase Saturation
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Capillary Pressure Curve
Capillary Pressure
Wetting Phase Saturation
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Saturation fields are drastically different
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Laminated Networks

• Lamina layering is often encountered in
  sedimentary rock formations
• The layering occurs where high-energy
  environments, carrying large sediment loads,
  deposit their load on undulating, migrating
  surfaces to form a variety of bedforms
  (Ringrose et al.,1993)
• Consists of alternating high and low permeability
  layers

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Qualitative Results
0.300
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Relative Permeability Curves for Laminated
Networks
Relative Permeability
Relative Permeability
Wetting Phase Saturation
Wetting Phase Saturation
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Pc curves for Laminated Networks
Capillary Pressure
Wetting Phase Saturation
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Application Acid Stimulation
Fredd and Fogler (1988)

• Acids injected and consumed near well to improve
  permeability
• Highly conductive wormholes (dependent upon
  injection rate)
• Process occurs inherently at the pore scale which
  makes upscaling very difficult
• How can model this pore-level process in a
  simulator, without losing important information?

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Direct Integration of Pore-scale Models in
Simulators

• Hundreds of pore-level models placed around a
  well
• Models coupled used mortars to ensure continuity
• Parallel and HPC are necessary tools
• Upscaling a posteriori

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Conclusions

• Pore-scale models are typically used as
  stand-alone tools
• Artificial boundary conditions are employed
• Incorrect macroscopic properties are often
  obtained
• Permeability can be underestimated using a
  traditional, 2-scale upscaling
• Mortars allow for coupling of adjacent models
• Permeability estimation improves with higher
  order mortars and finer meshto a point
• Relative Permeability and Pc are also dependent
  on surrounding media
• Saturation fields for capillary dominated flow
  affected by how fluids enter the domain
• Extension to viscous flows will require a mortar
  approach
• Big picture involves direct integration of
  pore-level models in simulators

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Acknowledgements

• Mary Wheeler
• Sunil Thomas
• Jaideep Bhagmane
• Robert Peterson
• Tie Sun