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1. Fluid-poroviscoelastic structure interaction problem with nonlinear coupling

   Jeffrey Kuan, Suncica Canic (December 2024, Jorunal de Mathematiques Pures and Appliques)

   We prove the existence of a weak solution to a fluid-structure interaction (FSI) problem between the flow of an incompressible, viscous fluid modeled by the Navier-Stokes equations, and a poroviscoelastic medium modeled by the Biot equations. The two are nonlinearly coupled over an interface with mass and elastic energy, modeled by a reticular plate equation, which is transparent to fluid flow. The existence proof is constructive, consisting of two steps. First, the existence of a weak solution to a regularized problem is shown. Next, a weak-classical consistency result is obtained, showing that the weak solution to the regularized problem converges, as the regularization parameter approaches zero, to a classical solution to the original problem, when such a classical solution exists. While the assumptions in the first step only require the Biot medium to be poroelastic, the second step requires additional regularity, namely, that the Biot medium is poroviscoelastic. This is the first weak solution existence result for an FSI problem with nonlinear coupling involving a Biot model for poro(visco)elastic media. 

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2. An augmented fully mixed formulation for the quasistatic Navier–Stokes–Biot model

   https://doi.org/10.1093/imanum/drad036  

   Li, Tongtong; Caucao, Sergio; Yotov, Ivan (June 2023, IMA Journal of Numerical Analysis)

   Abstract We introduce and analyze a partially augmented fully mixed formulation and a mixed finite element method for the coupled problem arising in the interaction between a free fluid and a poroelastic medium. The flows in the free fluid and poroelastic regions are governed by the Navier–Stokes and Biot equations, respectively, and the transmission conditions are given by mass conservation, balance of fluid force, conservation of momentum and the Beavers–Joseph–Saffman condition. We apply dual-mixed formulations in both domains, where the symmetry of the Navier–Stokes and poroelastic stress tensors is imposed in an ultra-weak and weak sense. In turn, since the transmission conditions are essential in the fully mixed formulation, they are imposed weakly by introducing the traces of the structure velocity and the poroelastic medium pressure on the interface as the associated Lagrange multipliers. Furthermore, since the fluid convective term requires the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin-type terms. Existence and uniqueness of a solution are established for the continuous weak formulation, as well as a semidiscrete continuous-in-time formulation with nonmatching grids, together with the corresponding stability bounds and error analysis with rates of convergence. Several numerical experiments are presented to verify the theoretical results and illustrate the performance of the method for applications to arterial flow and flow through a filter. 

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3. Induced seismicity in the Dallas-Fort Worth Basin: Enhanced seismic catalogue and evaluation of fault slip potential

   https://doi.org/10.1190/segam2020-3428222.1  

   Li, Bing Q.; Avouac, Jean-Philippe; Ross, Zachary E.; Du, Jing; Rebel, Estelle (September 2020, SEG Technical Program Expanded Abstracts 2020)

   null (Ed.)

   We present an updated catalogue of seismicity in the Dallas-Fort Worth basin from 2008 to the end of 2019 using state-of-the-art phase picking and association methods based on machine learning. We then calculate the pore pressure and poroelastic stress changes on a monthly basis between 2000 and 2020 for the whole basin, incorporating fluid injection/extraction histories at 104 saltwater injection and 20576 production wells. These pore pressure and poroelastic stress changes are calculated using coupled analytical solutions for a point source injection in a 3D homogeneous isotropic medium, and are superposed for all wells. We suggest that the poroelastic effects of produced gas and water contribute significantly to fault instability. 

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4. A Robin–Robin strongly coupled partitioned method for fluid–poroelastic structure interaction

   https://doi.org/10.1515/jnma-2024-0057  

   Parrow, Connor; Bukač, Martina (February 2025, Journal of Numerical Mathematics)

   Abstract This work focuses on the development of a novel, strongly-coupled, second-order partitioned method for fluid–poroelastic structure interaction. The flow is assumed to be viscous and incompressible, and the poroelastic material is described using the Biot model. To solve this problem, a numerical method is proposed, based on Robin interface conditions combined with the refactorization of the Cauchy’s one-legged ‘ϑ-like’ method. This approach allows the use of the mixed formulation for the Biot model. The proposed algorithm consists of solving a sequence of Backward Euler–Forward Euler steps. In the Backward Euler step, the fluid and poroelastic structure problems are solved iteratively until convergence. Then, the Forward Euler problems are solved using equivalent linear extrapolations. We prove that the iterative procedure in the Backward Euler step is convergent, and that the converged method is stable whenϑ∈ [1/2, 1]. Numerical examples are used to explore convergence rates with varying parameters used in our scheme, and to compare our method to a monolithic method based on Nitsche’s coupling approach. 

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5. Three-dimensional wave propagation and earthquake dynamic rupture simulations in complex poroelastic media

   https://doi.org/10.1093/gji/ggaf184  

   Wolf, Sebastian; Gabriel, Alice-Agnes; Galis, Martin; Moczo, Peter; Gregor, David; Bader, Michael (June 2025, Geophysical Journal International)

   SUMMARY Numerical simulations of earthquakes and seismic wave propagation require accurate material models of the solid Earth. In contrast to purely elastic rheology, poroelasticity accounts for pore fluid pressure and fluid flow in porous media. Poroelastic effects can alter both the seismic wave field and the dynamic rupture characteristics of earthquakes. For example, the presence of fluids may affect cascading multifault ruptures, potentially leading to larger-than-expected earthquakes. However, incorporating poroelastic coupling into the elastodynamic wave equations increases the computational complexity of numerical simulations compared to elastic or viscoelastic material models, as the underlying partial differential equations become stiff. In this study, we use a Discontinuous Galerkin solver with Arbitrary High-Order DERivative time stepping of the poroelastic wave equations implemented in the open-source software SeisSol to simulate 3-D complex seismic wave propagation and 3-D dynamic rupture in poroelastic media. We verify our approach for double-couple point sources using independent methods including a semi-analytical solution and a finite-difference scheme and a homogeneous full-space and a poroelastic layer-over-half-space model, respectively. In a realistic carbon capture and storage reservoir scenario at the Sleipner site in the Utsira Formation, Norway, we model 3-D wave propagation through poroelastic sandstone layers separated by impermeable shale. Our results show a sudden change in the pressure field across material interfaces, which manifests as a discontinuity when viewed at the length scale of the dominant wavelengths of S or fast P waves. Accurately resolving the resulting steep pressure gradient dramatically increases the computational demands, requiring high-resolution modelling. We show that the Gassmann elastic equivalent model yields almost identical results to the fully poroelastic model when focusing solely on solid particle velocities. We extend this approach using suitable numerical fluxes to 3-D dynamic rupture simulations in complex fault systems, presenting the first 3-D scenarios that combine poroelastic media with geometrically complex, multifault rupture dynamics and tetrahedral meshes. Our findings reveal that, in contrast to modelling wave propagation only, poroelastic materials significantly alter rupture characteristics compared to using elastic equivalent media since the elastic equivalent fails to capture the evolution of pore pressure. Particularly in fault branching scenarios, the Biot coefficient plays a key role in either promoting or inhibiting fault activation. In some cases, ruptures are diverted to secondary faults, while in others, poroelastic effects induce rupture arrest. In a fault zone dynamic rupture model, we find poroelasticity aiding pulse-like rupture. A healing front is induced by the reduced pore pressure due to reflected waves from the boundaries of the poroelastic damage zone. Our results highlight that poroelastic effects are important for realistic simulations of seismic waves and earthquake rupture dynamics. In particular, our poroelastic simulations may offer new insights on the complexity of multifault rupture dynamics, fault-to-fault interaction and seismic wave propagation in realistic models of the Earth’s subsurface. 

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