Abstract
Analysis of fluids in porous media is of great importance in many applications. There are many mathematical models that can be used in the analysis. More realistic models should account for the stochastic variations of the model parameters due to the nature of the porous material and/or the properties of the fluid. In this paper, the standard porous media problem with random permeability is considered. Both the deterministic and stochastic problems are analyzed using the finite volume technique. The solution statistics of the stochastic problem are computed using both Polynomial Chaos Expansion (PCE) and the Karhunen-Loeve (KL) decomposition with an exponential correlation function. The results of both techniques are compared with the Monte Carlo sampling to verify the efficiency. Results have shown that PCE with first order polynomials provides higher accuracy for lower (less than 20%) permeability variance. For higher permeability variance, using higher-order PCE considerably improves the accuracy of the solution. The PCE is also combined with KL decomposition and faster convergence is achieved. The KL-PCE combination should carefully choose the number of KL decomposition terms based on the correlation length of the random permeability. The suggested techniques are successfully applied to the quarter-five spot problem.
Highlights
Fluid flow in porous media is a subject of great interest due to its wide range of applications such as water/oil flow in underground reservoirs, modeling the underground spreading of toxic, and biological applications such as the transport in lungs and arteries [1]
These results illustrate the fast convergence of the Polynomial Chaos Expansion (PCE) solution for partial differential equation (PDE) (8), as assuming a higher-order PCE had a negligible effect on the accuracy of the solution
We considered two cases for the stochastic problem: The first case is solved using both Monte Carlo (MC) and PCE methods and the results are compared
Summary
Fluid flow in porous media is a subject of great interest due to its wide range of applications such as water/oil flow in underground reservoirs, modeling the underground spreading of toxic, and biological applications such as the transport in lungs and arteries [1]. When dealing with porous media, such as geological formations, we do not have complete information about their properties, such as permeability and porosity. This leads to uncertainty about the values of the formation properties and, in turn, in calculating flow in such formations [2]. For evaluating such uncertainties, a probabilistic framework is to be used. Uncertainty quantification (UQ) methods are used for analyzing the system using a model by the parameterization of the uncertain input data (such as permeability) using a set of independent random variables. The model is solved to determine the response of the system to the given random input data [3]
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