Abstract

We analyze the problem of uncertainty propagation for nonlinear two-phase transport in heterogeneous porous media. Specifically, we study the evolution of the saturation field associated with nonlinear immiscible two-phase transport (i.e., Buckley--Leverett problem) in the presence of a stochastic velocity field. The uncertainty in the velocity field is due to the limited information that is usually available about the heterogeneous porosity and permeability fields of the particular subsurface formation of interest. The uncertainty in the total-velocity field leads to uncertainty in the saturation of the injected fluid phase, both in space and time. Given information about the spatial statistics of the correlated heterogeneity, we derive the multipoint cumulative distribution functions (CDF) of saturation. The methodology takes full account of the nonlinear hyperbolic nature of the species conservation law. To obtain the multipoint CDF, we first derive the partial differential equation (PDE) of the “raw” CDF of saturation at a given point. Then, we describe the development of the PDE that governs the evolution of the multipoint raw CDF of saturation. The resulting equation is linear in the space-time variables and is solved semianalytically for problems in one spatial dimension and numerically for higher spatial dimensions. Ensemble averaging of the raw CDF leads to the multipoint CDF of saturation. We then compute the two-point saturation CDF profiles in one spatial dimension. We also use the two-point CDF to compute the saturation autocovariance function. We demonstrate the accuracy of our new “distribution method” by comparing the predictions with exhaustive high-resolution Monte Carlo simulations.

Keywords

  1. multiphase flow
  2. heterogeneous porous media
  3. stochastic methods
  4. stochastic partial differential equations
  5. Buckley--Leverett equation
  6. multipoint statistics
  7. CDF method
  8. saturation

MSC codes

  1. 35L60
  2. 60G60
  3. 65M75
  4. 76D06
  5. 86A05

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Information & Authors

Information

Published In

cover image SIAM/ASA Journal on Uncertainty Quantification
SIAM/ASA Journal on Uncertainty Quantification
Pages: 353 - 377
ISSN (online): 2166-2525

History

Submitted: 28 September 2016
Accepted: 28 January 2017
Published online: 13 April 2017

Keywords

  1. multiphase flow
  2. heterogeneous porous media
  3. stochastic methods
  4. stochastic partial differential equations
  5. Buckley--Leverett equation
  6. multipoint statistics
  7. CDF method
  8. saturation

MSC codes

  1. 35L60
  2. 60G60
  3. 65M75
  4. 76D06
  5. 86A05

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