Adaptive Model for Determining Nonlinear-Flow Regions in Highly Heterogeneous Porous Media: Numerical Validation

Core Concepts

An adaptive model is developed and numerically validated to determine the regions of nonlinear flow in highly heterogeneous porous media.

Abstract

The paper presents an adaptive model for describing flows in highly heterogeneous porous media. The model selects between a linear (Darcy) and a nonlinear (Darcy-Forchheimer) constitutive law depending on the magnitude of the fluid's velocity.
The key highlights are:

The adaptive model switches between the linear and nonlinear laws based on a flux threshold, which is determined by an error tolerance parameter δ. This allows identifying the slow (Darcy) and fast (Darcy-Forchheimer) flow regions in the porous medium.

Numerical experiments are conducted on four test cases, including 2D and 3D scenarios. The first two cases are inspired by an application in landfill management, while the remaining two are based on the SPE10 benchmark reservoir scenario.

The adaptive approach is shown to be more accurate than using a global Darcy model, with errors reduced by several orders of magnitude compared to the reference Darcy-Forchheimer solution.

By decreasing the error tolerance δ, the Darcy-Forchheimer region expands, capturing more of the heterogeneous high-velocity flow paths in the porous medium.

The adaptive model provides a computationally efficient way to determine the partition of the domain into slow and fast flow regions, which can be leveraged in future work for domain decomposition algorithms.

Stats

The seepage flux magnitude ∥u∥ is used to determine the Darcy and Darcy-Forchheimer regions.
The Forchheimer number Fo = c^(1/m)_F √(k/μ) ∥u∥ quantifies the weight of the nonlinear term relative to the linear term.

Quotes

"By decreasing the error tolerance δ, the Darcy-Forchheimer region expands, capturing more of the heterogeneous high-velocity flow paths in the porous medium."
"The adaptive approach is shown to be more accurate than using a global Darcy model, with errors reduced by several orders of magnitude compared to the reference Darcy-Forchheimer solution."

Key Insights Distilled From

Numerical validation of an adaptive model for the determination of nonlinear-flow regions in highly heterogeneous porous media

by Alessio Fuma... at arxiv.org 05-06-2024

https://arxiv.org/pdf/2405.02094.pdf

Deeper Inquiries

How could machine learning techniques be leveraged to efficiently determine the Darcy and Darcy-Forchheimer regions without solving the regularized model

Machine learning techniques can be leveraged to efficiently determine the Darcy and Darcy-Forchheimer regions without solving the regularized model by training a machine learning model on a dataset of simulated or observed flow data. The model can learn the patterns and characteristics of flow behavior in different regions of the porous medium and classify them as either Darcy or Darcy-Forchheimer regions.
One approach could be to use supervised learning algorithms, such as classification algorithms like Support Vector Machines (SVM), Random Forest, or Neural Networks. The input features for the model could include parameters like permeability, velocity, pressure gradient, and other relevant properties of the porous medium. The model would be trained on labeled data where the regions are already classified as Darcy or Darcy-Forchheimer based on the adaptive model results.
Another approach could be to use unsupervised learning techniques like clustering algorithms to identify patterns in the flow data and automatically group similar regions together. This could help in identifying regions with similar flow characteristics without the need for predefined labels.
By leveraging machine learning techniques, the process of determining the Darcy and Darcy-Forchheimer regions can be automated, efficient, and potentially more accurate than manual classification based on predefined thresholds or rules.

What are the potential limitations of the adaptive model in capturing anisotropic effects in the porous medium beyond the permeability tensor

The potential limitations of the adaptive model in capturing anisotropic effects in the porous medium beyond the permeability tensor include:

Limited Representation: The adaptive model primarily focuses on the magnitude of the fluid velocity to determine the flow regime, but it may not fully capture the directional dependencies or anisotropies present in the porous medium. Anisotropic effects related to the orientation of the permeability tensor or other physical properties may not be adequately addressed by the model.

Complexity of Anisotropy: Anisotropic effects in porous media can be complex and multifaceted, involving variations in permeability, porosity, and other parameters in different directions. The adaptive model's simplistic approach based on velocity thresholds may not be sufficient to capture the nuanced anisotropic behavior of the flow.

Homogenization Assumptions: The model's reliance on homogenization methods and averaging techniques to derive constitutive laws may oversimplify the anisotropic behavior of the porous medium. Anisotropy in real-world porous media can be highly variable and may not conform to the assumptions made in the model.

Tensor Permeabilities: While the adaptive model can handle tensor permeabilities to some extent, the representation of anisotropic effects solely through permeability tensors may not fully capture the intricate flow behaviors associated with anisotropy in porous media.

To address these limitations, more advanced models incorporating detailed anisotropic properties, such as tensorial representations of permeability, directional dependencies in constitutive laws, and sophisticated numerical techniques, may be required.

How could the adaptive model be extended to handle time-dependent flows in heterogeneous porous media

To extend the adaptive model to handle time-dependent flows in heterogeneous porous media, several modifications and considerations can be made:

Transient Formulation: The adaptive model can be reformulated to include time derivatives in the governing equations to account for transient flow behavior. This would involve introducing terms for time-dependent pressure and velocity changes in the constitutive laws.

Temporal Adaptation: The adaptive approach can be extended to dynamically adjust the classification of Darcy and Darcy-Forchheimer regions based on temporal variations in flow conditions. Machine learning algorithms can be trained to recognize temporal patterns and adapt the region classification accordingly.

Incorporating Time Scales: Different time scales of flow phenomena, such as fast transient events or long-term trends, can be considered in the adaptive model to capture the temporal dynamics of flow in heterogeneous porous media.

Validation with Time-Resolved Data: The adaptive model can be validated using time-resolved flow data from simulations or experiments to ensure its accuracy in capturing time-dependent behaviors. This validation process can help refine the model and improve its predictive capabilities over different time intervals.

By incorporating time-dependent considerations into the adaptive model, it can provide a more comprehensive understanding of flow dynamics in heterogeneous porous media and offer insights into how flow behavior evolves over time.