Source-linked AI summary
Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training
Ehsan Haghighat, Danial Amini, Ruben Juanes
TL;DR
Forward PINN solutions for coupled single- and multiphase poromechanics are difficult because their multi-objective losses can optimize unstably. The paper uses dimensionless governing equations and stress-split sequential training, which provides stable and robust convergence across benchmark problems, although training remains slow and accuracy is difficult to control.
Problem
PINN training for coupled poromechanics is challenging because multiple differential relations create an unstable, complex multi-objective loss optimization.
Method
The paper approximates poromechanics variables with neural networks, uses dimensionless coupled equations, and trains them sequentially with a fixed-stress split.
Results
Stress-split sequential training is reported as the most stable and accurate approach across Mandel’s, Barry-Mercer’s, and two-phase drainage problems.
Takeaways & Limitations
The results support stress-split sequential training as a robust PINN strategy for fully coupled single-phase and two-phase poromechanics.
Takeaways & Limitations
Training remains very slow, and controlling PINN accuracy is challenging; inverse coupled multiphase problems remain a proposed next step.
Abstract
from arXiv · showhide
Physics-informed neural networks (PINNs) have received significant attention as a unified framework for forward, inverse, and surrogate modeling of problems governed by partial differential equations (PDEs). Training PINNs for forward problems, however, pose significant challenges, mainly because of the complex non-convex and multi-objective loss function. In this work, we present a PINN approach to solving the equations of coupled flow and deformation in porous media for both single-phase and multiphase flow. To this end, we construct the solution space using multi-layer neural networks. Due to the dynamics of the problem, we find that incorporating multiple differential relations into the loss function results in an unstable optimization problem, meaning that sometimes it converges to the trivial null solution, other times it moves very far from the expected solution. We report a dimensionless form of the coupled governing equations that we find most favourable to the optimizer. Additionally, we propose a sequential training approach based on the stress-split algorithms of poromechanics. Notably, we find that sequential training based on stress-split performs well for different problems, while the classical strain-split algorithm shows an unstable behaviour similar to what is reported in the context of finite element solvers. We use the approach to solve benchmark problems of poroelasticity, including Mandel's consolidation problem, Barry-Mercer's injection-production problem, and a reference two-phase drainage problem. The Python-SciANN codes reproducing the results reported in this manuscript will be made publicly available at https://github.com/sciann/sciann-applications.
1. Introduction
The paper applies PINNs to coupled flow and mechanics in porous media, addressing challenges from complex multiphysics optimization. It reports dimensionless equations and stress-split sequential training as stable approaches for benchmark problems.
- Motivation: Coupled flow and mechanics matters in geotechnical, biomechanical, and reservoir engineering, including liquefaction and production-induced subsidence.These applications motivate computational models for porous-media problems.
- PINNs: PINNs integrate governing equations, initial and boundary conditions, and data within a single neural-network loss function.Unknown solution variables are approximated by feed-forward neural networks and optimized at sampling and data points.
- Challenges: Training PINNs for complex coupled problems is slow and difficult because multiple loss terms create a challenging multi-objective optimization problem.Network design and hyper-parameter selection often require trial and error.
- Contribution: The study finds dimensionless equations and stress-split sequential training suitable for coupled single-phase and two-phase poromechanics.The approach is applied to Mandel’s, Barry-Mercer’s, and two-phase drainage problems.
2. Governing Equations
The governing formulation combines conservation laws, poroelastic constitutive relations, and dimensionless single- and two-phase equations. Sequential iterations use either fixed-stress or fixed-strain splitting to couple flow and mechanics.
- Balance Laws: Porous-media response is governed by fluid mass balance and quasi-static linear momentum balance, with symmetric Cauchy stress.The formulation also defines small strain from the solid-skeleton displacement field.
- Single-Phase Poromechanics: Single-phase poromechanics relates fluid storage and pressure to porosity, compressibilities, Biot parameters, drained stiffness, and effective stress.The effective stress is defined by combining total stress with pore-pressure coupling.
- Single-Phase Poromechanics: The formulation retains volumetric strain as a separate solution variable because pressure couples to stress through volumetric deformation.This produces separate kinematic, momentum, and mass-balance relations for the PINN formulation.
- Dimensionless Relations: Dimensionless variables and relations are derived for flow velocity, displacement, stress-strain behavior, and mass balance.The dimensionless formulation is provided in both stress and strain forms.
- Sequential Splitting: In sequential iterations, Eq. (26) defines fixed-stress splitting, whereas Eq. (27) defines fixed-strain splitting.The two strategies differ in which mechanics-related quantity passes information to the flow problem.
- Two-Phase Poromechanics: Two-phase poromechanics extends the formulation using equivalent pressure, saturation, capillary pressure, phase mass conservation, and relative permeability.The bulk density combines solid and phase contributions, while phase seepage follows an extended Darcy relation.
3. Physics-Informed Neural Networks
The PINN represents poromechanics solution variables with a feed-forward neural network and minimizes a loss combining PDE, initial, boundary, and data-related terms. Training remains difficult because loss terms have competing gradients and optimization trade-offs.
- PINN Formulation: Deep neural networks approximate the poromechanics solution variables, while governing equations and initial or boundary conditions contribute separate loss terms.Network parameters are optimized over interior collocation points and boundary points.
- Neural Network: A feed-forward network composes layer transformations of space-time coordinates to produce outputs such as displacement or pressure fields.Weights and biases are optimized network parameters, and the output layer is often linear for regression.
- Loss Function: The total PINN loss combines PDE residuals with Dirichlet, Neumann, and initial-condition residuals through weighted terms.The weights act as penalties in the optimization objective.
- Optimization: Adam is scalable but requires difficult tuning of learning rates and individual loss-term weights for multi-objective PINN optimization.L-BFGS avoids some first-order limitations but lacks scalability and becomes slow for large deep networks.
- Adaptive Weights: Higher-derivative loss terms can produce larger gradients and dominate training, motivating gradient-scaling methods such as GradNorm.GradNorm adjusts weights so weighted loss-term gradient norms approach a score-adjusted average.
- Adaptive Weights: The study uses adaptive loss-term weights but does not employ adaptive spatial or temporal sampling strategies.The selected weights are evaluated using the described gradient-based strategy.
4. PINN-PoroMechanics
The PINN formulation introduces additional volumetric-strain variables and independent networks to better enforce poromechanical coupling. Because the coupled loss is difficult to optimize, the method uses sequential fixed-stress training for single- and two-phase problems.
- Single-phase formulation: The single-phase formulation treats ux, uy, p, and volumetric strain εv as unknown solution variables.Introducing εv adds a conservation PDE and helps enforce pressure–displacement coupling.
- Single-phase formulation: Explicitly imposing volumetric constraints improved training and follows a strategy with a long history in finite-element modeling of quasi-incompressible materials.The authors report this improvement based on trial and error.
- Training strategy: The total coupled loss combines multiple governing-equation, initial-value, and boundary-value terms, creating substantial optimization challenges.Meaningful Adam optimization required scored-adaptive weights, but the authors found that procedure insufficiently robust.
- Training strategy: Sequential training decouples flow and mechanics and optimizes the resulting subproblems in fixed-stress-split form.The approach is summarized in Algorithm 1 and is applied to both single-phase and two-phase poroelasticity.
- Training strategy: Sequential fixed-stress training is more robust than simultaneous optimization but requires additional operator-split iterations and computational steps.The increased cost follows from repeatedly solving the decoupled subproblems.
- Two-phase formulation: The two-phase formulation uses ux, uy, capillary pressure pc, wetting pressure pw, and a separate volumetric-strain unknown.Its alternative optimization strategy sequentially solves decoupled displacement and fluid relations following stress splitting.
5. Applications
The applications evaluate PINNs on Mandel’s consolidation, Barry-Mercer’s injection-production, and two-phase drainage problems, emphasizing training stability, coupling sensitivity, and agreement with expected solutions.
- Mandel’s problem: Mandel’s problem varies coupling through D∗, with analytical pressure and displacement responses evaluated along specified boundary lines.The study considers D∗ values of 0.13, 0.6, and 0.938 and examines pressure, horizontal displacement, and vertical displacement over time.
- Sequential solution: fixed-stress-split versus fixed-strain-split: Fixed-stress-split sequential training converges to the correct Mandel solution, whereas fixed-strain-split training diverges for D∗ = 0.934.The strain-split method can converge under a reported stability criterion, but the stress-split formulation is stable in the presented comparison.
- PINN’s D∗-dependence: For fixed hyperparameters and equal training epochs, PINN accuracy decreases as D∗ increases, indicating greater difficulty at stronger flow–mechanics coupling.The comparison includes simultaneous and fixed-stress-split sequential strategies.
- Barry-Mercer’s injection-production problem: The Barry-Mercer benchmark uses a time-dependent point injection-production source and evaluates pressure and displacement errors from fixed-stress-split sequential training.The source is Gaussian-approximated, with α = 0.04; the reported PINN solution shows great agreement with the expected solution.
- Two-phase drainage problem: The two-phase drainage results agree with the expected results using the specified relative-permeability and saturation formulation.The formulation uses λ = 3, effective saturation Se, and connate water saturation Srw = 0.2.
6. Discussions and Concluding Remarks
The study applies PINNs to fully and partially saturated coupled porous-media flow and mechanics, reporting a dimensionless formulation and fixed-stress sequential training strategy. The strategy improves convergence and robustness, but training remains slow and accuracy control remains challenging.
- The study applies PINNs to forward coupled flow and mechanics in fully and partially saturated porous media.
- A dimensionless formulation produces stable and convergent optimizer behavior for the coupled relations.
- Fixed-stress sequential training enhances convergence and robustness of the PINN formulation.
- Training remains very slow, and controlling PINN accuracy remains challenging because of multi-objective optimization and first-order optimization.
Appendix A. Analytical solutions
The appendix reports existing analytical solutions for Mandel’s and Barry-Mercer’s problems and summarizes those solutions with reference to prior work.
- Analytical solutions are reported for Mandel’s consolidation problem and Barry-Mercer’s problem.
- The appendix also discusses summaries of these analytical solutions from prior references.
Appendix A.1. Mandel’s analytical solution
Mandel’s consolidation problem is represented through an analytical solution involving the stated loading convention and roots of a defining equation.
- The analytical solution to Mandel’s consolidation problem is expressed in the appendix.
- The appendix specifies that σ0 is negative for compression and positive for tension, while αi are roots of a defining equation.
Appendix A.2. Barry-Mercer’s problem
Barry-Mercer’s problem uses an injection function and transformed coordinates to define pressure and displacement, which are then mapped back to real coordinates.
- Barry-Mercer’s problem uses an injection function in its analytical formulation.
- The formulation specifies a source location and transformed time domain for pressure and displacement definitions.
- Pressure and horizontal and vertical displacement are defined in real coordinates after the transformed-coordinate formulation.