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Analysis of the Stochastic Quarter-Five Spot Problem Using Polynomial Chaos
Hesham AbdelFattah1, Amnah Al-Johani2, Mohamed El-Beltagy1
1Engineering Mathematics and Physics Department, Engineering Faculty, Cairo University, Giza 12613, Egypt.
Molecules (Basel, Switzerland)
|July 30, 2020
Summary
This study analyzes fluid flow in porous media using advanced mathematical models. Polynomial Chaos Expansion and Karhunen-Loeve decomposition offer efficient and accurate solutions for stochastic problems, improving predictions in real-world applications.
Area of Science:
- Computational fluid dynamics
- Stochastic modeling in porous media
- Numerical analysis of partial differential equations
Background:
- Fluid flow analysis in porous media is critical for numerous applications.
- Realistic models must incorporate stochastic variations in material and fluid properties.
- Standard porous media models often lack consideration for parameter randomness.
Purpose of the Study:
- To analyze deterministic and stochastic porous media flow problems.
- To evaluate the efficiency of Polynomial Chaos Expansion (PCE) and Karhunen-Loeve (KL) decomposition for stochastic analysis.
- To compare computational techniques against Monte Carlo sampling for validation.
Main Methods:
- Finite volume technique for solving deterministic and stochastic problems.
- Polynomial Chaos Expansion (PCE) for computing solution statistics.
- Karhunen-Loeve (KL) decomposition with exponential correlation function.
- Comparison of PCE and KL-PCE with Monte Carlo sampling.
Main Results:
- PCE with first-order polynomials is accurate for low permeability variance (<20%).
- Higher-order PCE improves accuracy significantly for higher permeability variance.
- Combining KL decomposition with PCE achieves faster convergence.
- KL-PCE performance depends on the number of KL terms and correlation length.
Conclusions:
- PCE and KL-PCE are efficient and accurate methods for stochastic porous media flow.
- The choice of PCE order and KL terms is crucial for optimal performance.
- These techniques are successfully applied to practical problems like the quarter-five spot problem.
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