Upscaling for porous media flows

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Upscaling for porous media flows

• Andrew Westhead, Advisor Prof. T. Hou
• March 13, 2003

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Introduction and objectives

• The aim of this research is to develop efficient
  numerical schemes for solving the porous media
  transport problem
• The goal of upscaling is to solve for the large
  scale features by correctly incorporating the
  effect of these small scales
• This is important for oil-reservoir simulations
  and groundwater contaminant studies

Cross-section of a heterogeneous permeability
field initially saturated with oil
Water is injected to displace oil
Upscaled flow (our goal)
Oil
Water
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Mathematical model

• Fluid velocity v is modeled using Darcys law
  (pressure equation)
• Buckley-Leverett model for the saturation S
• k is the permeability, ? is the total mobility,
  f(S) is the fractional flow
• 2-phase flow (e.g. oil-water reservoir
  simulations)
• 1-phase flow (e.g. groundwater contaminant flow)
• Pressure equation is elliptic, saturation
  equation is hyperbolic

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Upscaling of the pressure equation

• We are using the multiscale finite element method
  (MSFEM) to solve the pressure equation on a
  coarse grid
• We have found that the 2-phase flow computations
  can be speeded up by only updating the basis
  functions in those regions that the saturation
  front passes through

Coarse grid
x-derivative of MSFEM basis function
MSFEM basis function
For large problems we found that only very few
basis functions need to be updated this leads
to significant savings in computational cost.
S does not change much ahead of the front
S does not change much behind the front
These savings can be achieved without loss of
accuracy.
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Upscaling of the saturation equation

• We are using homogenization techniques for the
  saturation equation
• Working on models of the form
• G models the effect of the small-scale
  large-scale interactions
• Currently, we are working on a homogenization
  model developed by D. Yang (visiting Professor,
  ACM 2002) for the periodic velocity field case
• In this case, we find
• P is a projection into a subspace of fluctuations
• We have investigated various models for the case
  where the velocity field is not period, mostly
  based on heuristic modeling of G
• Here D depends on the fluctuations and flow
  history