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

TL;DR

The paper addresses nonlinear flow deviations from Darcy's law in highly heterogeneous porous media by employing an adaptive seepage-law framework that switches between linear and nonlinear constitutive laws based on a flux-threshold. It advances the theory through a regularized, variational formulation to smoothly partition the domain into slow (linear) and fast (nonlinear) regions and demonstrates robustness via extensive numerical validation in two-, one-, and three-dimensional SPE10-inspired test cases. The key contributions include the detailed construction of adaptive and regularized models, explicit flux-threshold definitions with corresponding subdomains, and thorough numerical validation showing accuracy gains over globally nonlinear solutions, with potential for faster, partitioned-domain simulations. The work has practical implications for efficient, accurate simulations of flow in heterogeneous porous media and lays groundwork for future domain-decomposition and data-driven region identification approaches.

Abstract

An adaptive model for the description of flows in highly heterogeneous porous media is developed in~\cite{FP21,FP23}. There, depending on the magnitude of the fluid's velocity, the constitutive law linking velocity and pressure gradient is selected between two possible options, one better adapted to slow motion and the other to fast motion. We propose here to validate further this adaptive approach by means of more extensive numerical experiments, including a three-dimensional case, as well as to use such approach to determine a partition of the domain into slow- and fast-flow regions.

Figures (12)

• Figure 1: On the left, the log-permeability for the example of Section \ref{['subsubsub:two_channels']}; on the right, for Section \ref{['subsubsub:network']}.
• Figure 2: Solutions obtained to the adaptive model for the example in Section \ref{['subsubsub:two_channels']}, in the top row for Scenario a and in the bottom row for Scenario b. The pressure is multiplied by $1000$, and the velocity arrows inside the channels are scaled by $1/6$ and $1/21$ in comparison with those outside, respectively for each scenario.
• Figure 3: Darcy (red) and Darcy--Forchheimer (blue) regions depending on the error tolerance $\delta$ decreasing from left to right according to $\delta=0.05$, $\delta=0.0125$ and $\delta=0.003125$, in the top row for Scenario a and in the bottom row for Scenario b, for the example in Section \ref{['subsubsub:two_channels']}.
• Figure 4: Solutions obtained to the adaptive model for the example in Section \ref{['subsubsub:network']}, in the top row for Scenario a and in the bottom row for Scenario b. The pressure is multiplied by $1000$, and the velocity arrows inside the channels are scaled by $1/11$ and $1/120$ in comparison with those outside, respectively for each scenario.
• Figure 5: Darcy (red) and Darcy--Forchheimer (blue) regions depending on the error tolerance $\delta$ decreasing from left to right according to $\delta=0.05$, $\delta=0.0125$ and $\delta=0.003125$, in the top row for Scenario a and in the bottom row for Scenario b, for the example in Section \ref{['subsubsub:network']}.

Theorems & Definitions (3)

• Example 2.1: equal coefficients and Darcy--Forchheimer
• Remark 2.3: average pressure
• Example 2.5: equal coefficients and full exponential