An Adaptive Machine Learning Framework for Fluid Flow in Dual-Network Porous Media

Abstract

Porous materials -- natural or engineered -- often exhibit dual pore-network structures that govern processes such as mineral exploration and hydrocarbon recovery from tight shales. Double porosity/permeability (DPP) mathematical models describe incompressible fluid flow through two interacting pore networks with inter-network mass exchange. Despite significant advances in numerical methods, there remains a need for computational frameworks that enable rapid forecasting, data assimilation, and reliable inverse analysis. To address this, we present a physics-informed neural network (PINN) framework for forward and inverse modeling of DPP systems. The proposed approach encodes the governing equations in mixed form, along with boundary conditions, directly into the loss function, with adaptive weighting strategies to balance their contributions. Key features of the framework include adaptive weight tuning, dynamic collocation point selection, and the use of shared trunk neural architectures to efficiently capture the coupled behavior of the dual pore networks. It is inherently mesh-free, making it well-suited for complex geometries typical of porous media. It accurately captures discontinuities in solution fields across layered domains without introducing spurious oscillations commonly observed in classical finite element formulations. Importantly, the framework is well-suited for inverse analysis, enabling robust parameter identification in scenarios where key physical quantities -- such as the mass transfer coefficient in DPP models -- are difficult to measure directly. In addition, a systematic convergence analysis is provided to rigorously assess the stability, accuracy, and reliability of the method. The effectiveness and computational advantages of the approach are demonstrated through a series of representative numerical experiments.

Figures (22)

• Figure 1: A) The problem is formulated within the framework of the double porosity/permeability (DPP) model, consisting of multiple layers, each characterized by distinct macro- and micro-scale permeabilities, denoted by $k^{(1)}$ and $k^{(2)}$, respectively. The analytical solution within each layer yields constant horizontal components for both the macro- and micro-scale velocities, while the vertical components are zero. B) The macro-scale velocity field, computed using the proposed adaptive Physics-Informed Neural Network (PINN) framework, is visualized across the layers. The framework accurately captures discontinuities in the velocity field, significantly outperforming continuous mixed finite element methods and achieving accuracy levels typically associated with discontinuous Galerkin (DG) methods.
• Figure 1: Fluid flow in a porous medium with dual pore networks, illustrating mass exchange between the networks. The macro-pore network is shown in dark blue and the micro-pore network in light blue. (For interpretation of the references to color in this figure legend, the reader is referred to the online version of this article.)
• Figure 2: Modeling approaches across the spectrum from data-driven to physics-based methods, adapted from tartakovsky2020physics. The position of flow through porous media modeling is highlighted, illustrating its potential to leverage physics-informed machine learning techniques.
• Figure 3: Proposed adaptive modeling framework for the DPP model. The architecture features a shared-trunk, encoded inputs, and adaptive weighting and sampling strategies.
• Figure 4: 1D pressure-driven boundary value problem. The macro/micro-level pressure boundary conditions are prescribed at both ends.

Theorems & Definitions (10)

• Remark 3.1
• Proposition 4.1: Trivial kernel
• proof
• Lemma 4.2: Gårding-type inequality for the DPP model
• proof
• Theorem 4.3: Coercivity
• proof
• Remark 4.1: Weighted least--squares
• Theorem 4.4: Boundedness
• proof