arXiv subjects

Florin Adrian Radu

Publications and source records attributed to Florin Adrian Radu.

At least 19 recordsLinked to original sources

arXiv2026

Frequency Domain Biot--Allard Equations for Isotropic and Anisotropic Poroelastic Media: Two-field formulations and iterative splitting

We present a frequency-domain formulation of Biot's dynamic poroelastic equations with frequency-dependent dissipation (Biot--Allard) for anisotropic, heterogeneous media with memory effects. Two equivalent two-field representations--a displacement-pressure and a velocity-pressure-rate formulation--enable stabilized iterative splitting. While coupling operators generally lack an adjoint or skew-adjoint relationship at finite frequencies, the velocity--pressure-rate representation restores a skew-adjoint structure in the quasi-static limit. We prove continuity of the coupling operators and coercivity of the diagonal blocks, essential for convergence of the L-stabilized splitting scheme. The frequency-domain setting eliminates convolutional memory terms, incorporates attenuation and dispersion via complex-valued parameters, and reduces the time-dependent problem to a family of elliptic boundary-value problems suited for parallel computation and multi-frequency inversion. A conforming Galerkin finite element discretization preserves block structure, and numerical experiments confirm robustness and capture frequency-dependent attenuation. To illustrate discretization independence, we include a large-scale wave simulation using a pseudo-spectral method. This work provides a rigorous and efficient framework for modeling wave phenomena in complex porous media.

math.NA↗

arXiv2024

Well-posedness analysis of the Cahn-Hilliard-Biot model

We investigate the well-posedness of the recently proposed Cahn-Hilliard-Biot model. The model is a three-way coupled PDE of elliptic-parabolic nature, with several nonlinearities and the fourth order term known to the Cahn-Hilliard system. We show existence of weak solutions to the variational form of the equations and uniqueness under certain conditions of the material parameters and secondary consolidation, adding regularizing effects. Existence is shown by discretizing in space and applying ODE-theory (the Peano-Cauchy theorem) to prove existence of the discrete system, followed by compactness arguments to retain solutions of the continuous system. In addition, the continuous dependence of solutions on the data is established, in particular implying uniqueness. Both results build strongly on the inherent gradient flow structure of the model.

math.AP↗

arXiv2024

Sequential solution strategies for the Cahn-Hilliard-Biot model

This paper presents a study of solution strategies for the Cahn-Hilliard-Biot equations, a complex mathematical model for understanding flow in deformable porous media with changing solid phases. Solving the Cahn-Hilliard-Biot system poses significant challenges due to its coupled, nonlinear and non-convex nature. We explore various solution algorithms, comparing monolithic and splitting strategies, focusing on both their computational efficiency and robustness.

math.NA↗

arXiv2023

A fixed-stress type splitting method for nonlinear poroelasticity

In this paper we consider a nonlinear poroelasticity model that describes the quasi-static mechanical behaviour of a fluid-saturated porous medium whose permeability depends on the divergence of the displacement. Such nonlinear models are typically used to study biological structures like tissues, organs, cartilage and bones, which are known for a nonlinear dependence of their permeability/hydraulic conductivity on solid dilation. We formulate (extend to the present situation) one of the most popular splitting schemes, namely the fixed-stress split method for the iterative solution of the coupled problem. The method is proven to converge linearly for sufficiently small time steps under standard assumptions. The error contraction factor then is strictly less than one, independent of the Lamé parameters, Biot and storage coefficients if the hydraulic conductivity is a strictly positive, bounded and Lipschitz-continuous function.

math.NA↗

arXiv2022

A robust solution strategy for the Cahn-Larché equations

In this paper we propose a solution strategy for the Cahn-Larché equations, which is a model for linearized elasticity in a medium with two elastic phases that evolve subject to a Ginzburg-Landau type energy functional. The system can be seen as a combination of the Cahn-Hilliard regularized interface equation and linearized elasticity, and is non-linearly coupled, has a fourth order term that comes from the Cahn-Hilliard subsystem, and is non-convex and nonlinear in both the phase-field and displacement variables. We propose a novel semi-implicit discretization in time that uses a standard convex-concave splitting method of the nonlinear double-well potential, as well as special treatment to the elastic energy. We show that the resulting discrete system is equivalent to a convex minimization problem, and propose and prove the convergence of alternating minimization applied to it. Finally, we present numerical experiments that show the robustness and effectiveness of both alternating minimization and the monolithic Newton method applied to the newly proposed discrete system of equations. We compare it to a system of equations that has been discretized with a standard convex-concave splitting of the double-well potential, and implicit evaluations of the elasticity contributions and show that the newly proposed discrete system is better conditioned for linearization techniques.

math.NA↗

arXiv2022

A posteriori error estimates for hierarchical mixed-dimensional elliptic equations

Mixed-dimensional elliptic equations exhibiting a hierarchical structure are commonly used to model problems with high aspect ratio inclusions, such as flow in fractured porous media. We derive general abstract estimates based on the theory of functional a posteriori error estimates, for which guaranteed upper bounds for the primal and dual variables and two-sided bounds for the primal-dual pair are obtained. We improve on the abstract results obtained with the functional approach by proposing four different ways of estimating the residual errors based on the extent the approximate solution has conservation properties, i.e.: (1) no conservation, (2) subdomain conservation, (3) grid-level conservation, and (4) exact conservation. This treatment results in sharper and fully computable estimates when mass is conserved either at the grid level or exactly, with a comparable structure to those obtained from grid-based a posteriori techniques. We demonstrate the practical effectiveness of our theoretical results through numerical experiments using four different discretization methods for synthetic problems and applications based on benchmarks of flow in fractured porous media.

math.NA↗

arXiv2021

A Cahn-Hilliard-Biot system and its generalized gradient flow structure

In this work, we propose a new model for flow through deformable porous media, where the solid material has two phases with distinct material properties. The two phases of the porous material follow a Cahn-Hilliard type evolution, with additional impact from both elastic and fluid effects, and the coupling between flow and deformation is governed by Biot's theory. This results in a three-way coupled system which can be seen as an extension of the Cahn-Larché equations with the inclusion of a fluid flowing through the medium. The model covers essential coupling terms for several relevant applications, including solid tumor growth, biogrout, and wood growth simulation. Moreover, we show that this coupled set of equations follow a generalized gradient flow framework. This opens a toolbox of analysis and solvers which can be used for further study of the model. Additionally, we provide a numerical example showing the impact of the flow on the solid phase evolution in comparison to the Cahn-Larché system.

math-ph↗

arXiv2021

Free energy diminishing discretization of Darcy-Forchheimer flow in poroelastic media

In this paper, we develop a discretization for the non-linear coupled model of classical Darcy-Forchheimer flow in deformable porous media, an extension of the quasi-static Biot equations. The continuous model exhibits a generalized gradient flow structure, identifying the dissipative character of the physical system. The considered mixed finite element discretization is compatible with this structure, which gives access to a simple proof for the existence, uniqueness, and stability of discrete approximations. Moreover, still within the framework, the discretization allows for the development of finite volume type discretizations by lumping or numerical quadrature, reducing the computational cost of the numerical solution.

math.NA↗

arXiv2020

Efficient Solvers for Nonstandard Models for Flow and Transport in Unsaturated Porous Media

We study several iterative methods for fully coupled flow and reactive transport in porous media. The resulting mathematical model is a coupled, nonlinear evolution system. The flow model component builds on the Richards equation, modified to incorporate nonstandard effects like dynamic capillarity and hysteresis, and a reactive transport equation for the solute. The two model components are strongly coupled. On one hand, the flow affects the concentration of the solute; on the other hand, the surface tension is a function of the solute, which impacts the capillary pressure and, consequently, the flow. After applying an Euler implicit scheme, we consider a set of iterative linearization schemes to solve the resulting nonlinear equations, including both monolithic and two splitting strategies. The latter include a canonical nonlinear splitting and an alternate linearized splitting, which appears to be overall faster in terms of numbers of iterations, based on our numerical studies. The (time discrete) system being nonlinear, we investigate different linearization methods. We consider the linearly convergent L-scheme, which converges unconditionally, and the Newton method, converging quadratically but subject to restrictions on the initial guess. Whenever hysteresis effects are included, the Newton method fails to converge. The L-scheme converges; nevertheless, it may require many iterations. This aspect is improved by using the Anderson acceleration. A thorough comparison of the different solving strategies is presented in five numerical examples, implemented in MRST, a toolbox based on MATLAB.

math.NA↗

arXiv2020

An efficient numerical scheme for fully coupled flow and reactive transport in variably saturated porous media including dynamic capillary effects

In this paper, we study a model for the transport of an external component, e.g., a surfactant, in variably saturated porous media. We discretize the model in time and space by combining a backward Euler method with the linear Galerkin finite elements. The Newton method and the L-Scheme are employed for the linearization and the performance of these schemes is studied numerically. A special focus is set on the effects of dynamic capillarity on the transport equation.

math.NA↗

arXiv2020

Iterative schemes for surfactant transport in porous media

In this work we consider the transport of a surfactant in a variably saturated porous media. The water flow is modelled by the Richards equations and it is fully coupled with the transport equation for the surfactant. Three linearization techniques are discussed: the Newton method, the modified Picard and the L-scheme. Based on these, monolithic and splitting schemes are proposed and their convergence is analyzed. The performance of these schemes is illustrated on four numerical examples. For these examples, the number of iterations and the condition numbers of the linear systems emerging in each iteration are presented.

math.NA↗

arXiv2020

Robust linear domain decomposition schemes for reduced non-linear fracture flow models

In this work, we consider compressible single-phase flow problems in a porous media containing a fracture. In the latter, a non-linear pressure-velocity relation is prescribed. Using a non-overlapping domain decomposition procedure, we reformulate the global problem into a non-linear interface problem. We then introduce two new algorithms that are able to efficiently handle the non-linearity and the coupling between the fracture and the matrix, both based on linearization by the so-called L-scheme. The first algorithm, named MoLDD, uses the L-scheme to resolve the non-linearity, requiring at each iteration to solve the dimensional coupling via a domain decomposition approach. The second algorithm, called ItLDD, uses a sequential approach in which the dimensional coupling is part of the linearization iterations. For both algorithms, the computations are reduced only to the fracture by pre-computing, in an offline phase, a multiscale flux basis (the linear Robin-to-Neumann co-dimensional map), that represent the flux exchange between the fracture and the matrix. We present extensive theoretical findings and in particular, the stability and the convergence of both schemes are obtained, where user given parameters are optimized to minimise the number of iterations. Examples on two important fracture models are computed with the library PorePy and agree with the developed theory.

math.NA↗

arXiv2020

An accelerated staggered scheme for phase-field modeling of brittle fracture

There is currently an increasing interest in developing efficient solvers for phase-field modeling of brittle fracture. The governing equations for this problem originate from a constrained minimization of a non-convex energy functional, and the most commonly used solver is a staggered solution scheme. This is known to be robust compared to the monolithic Newton method, however, the staggered scheme often requires many iterations to converge when cracks are evolving. The focus of our work is to accelerate the solver through a scheme that sequentially applies Anderson acceleration and over-relaxation, switching back and forth depending on the residual evolution, and thereby ensuring a decreasing tendency. The resulting scheme takes advantage of the complementary strengths of Anderson acceleration and over-relaxation to make a robust and accelerating method for this problem. The new method is applied as a post-processing technique to the increments of the solver, hence, the implementation can be done with minor modifications to already available software. Moreover, the cost of combining the two acceleration schemes is negligible. The robustness and efficiency of the method are demonstrated through numerical examples.

math.NA↗

arXiv2019

Mathematical Modeling, Laboratory Experiments, and Sensitivity Analysis of Bioplug Technology at Darcy Scale

In this paper we study a Darcy-scale mathematical model for biofilm formation in porous media. The pores in the core are divided into three phases: water, oil, and biofilm. The water and oil flow are modeled by an extended version of Darcy's law and the substrate is transported by diffusion and convection in the water phase. Initially there is biofilm on the pore walls. The biofilm consumes substrate for production of biomass and modifies the pore space which changes the rock permeability. The model includes detachment of biomass due to water flux and death of bacteria, and is implemented in MRST. We discuss the capability of the numerical simulator to capture results from laboratory experiments. We perform a novel sensitivity analysis based on sparse-grid interpolation and multi-wavelet expansion to identify the critical model parameters. Numerical experiments using diverse injection strategies are performed to study the impact of different porosity-permeability relations in a core saturated with water and oil.

physics.app-ph↗

arXiv2019

The fixed-stress splitting scheme for Biot's equations as a modified Richardson iteration: Implications for optimal convergence

The fixed-stress splitting scheme is a popular method for iteratively solving the Biot equations. The method successively solves the flow and mechanic subproblems while adding a stabilizing term to the flow equation, which includes a parameter that can be chosen freely. However, the convergence properties of the scheme depend significantly on this parameter and choosing it carelessly might lead to a very slow, or even diverging, method. In this paper, we present a way to exploit the matrix structure arizing from discretizing the equations in the regime of impermeable porous media in order to obtain a priori knowledge of the optimal choice of this tuning/stabilization parameter.

math.NA↗

arXiv2019

The gradient flow structures of thermo-poro-visco-elastic processes in porous media

In this paper, the inherent gradient flow structures of thermo-poro-visco-elastic processes in porous media are examined for the first time. In the first part, a modelling framework is introduced aiming for describing such processes as generalized gradient flows requiring choices of physical states, corresponding energies, dissipation potentials and external work rates. It is demonstrated that various existing models can be in fact written within this framework. Ultimately, the particular structure allows for a unified well-posedness analysis performed for different classes of linear and non-linear models. In the second part, the gradient flow structures are utilized for constructing efficient discrete approximation schemes for thermo-poro-visco-elasticity -- in particular robust, physical splitting schemes. Applying alternating minimization to naturally arising minimization formulations of (semi-)discrete models is proposed. For such, the energy decrease per iteration is quantified by applying abstract convergence theory only utilizing convexity and Lipschitz continuity properties of the problem -- a fairly simple but flexible machinery. By this approach, e.g., the widely used undrained and fixed-stress splits for the linear Biot equations are derived and analyzed. By application of the framework to more advanced models, novel splitting schemes with guaranteed theoretical convergence rates are naturally derived. Moreover, based on the minimization character of the (semi-)discrete equations, relaxation of splitting schemes by line search is proposed; numerical results show a potentially great impact on the acceleration of splitting schemes for both linear and nonlinear problems.

math.NA↗

arXiv2019

An upscaled model for permeable biofilm in a thin channel and tube

In this paper, we derive upscaled equations for modelling biofilm growth in porous media. The resulting macro-scale mathematical models consider permeable multi-species biofilm including water flow, transport, detachment and reactions. The biofilm is composed of extracellular polymeric substances (EPS), water, active bacteria and dead bacteria. The free flow is described by the Stokes and continuity equations and the water flux inside the biofilm by the Brinkman and continuity equations. The nutrients are transported in the water phase by convection and diffusion. This pore-scale model includes variations of the biofilm composition and size due to reproduction of bacteria, production of EPS, death of bacteria and shear forces. The model includes a water-biofilm interface between the free flow and the biofilm. Homogenization techniques are applied to obtain upscaled models in a thin channel and a tube, by investigating the limit as the ratio of the aperture to the length $\varepsilon$ of both geometries approaches to zero. As $\varepsilon$ gets smaller, we obtain that the percentage of biofilm coverage area over time predicted by the pore-scale model approaches the one obtained using the effective equations, which shows a correspondence between both models. The two derived porosity-permeability relations are compared to two empirical relations from the literature. The resulting numerical computations are presented to compare the outcome of the effective (upscaled) models for the two mentioned geometries.

physics.flu-dyn↗

arXiv2019

Iterative solvers for Biot model under small and large deformation

We consider L-scheme and Newton based solvers for Biot model under small or large deformation. The mechanical deformation follows the Saint Venant-Kirchoff constitutive law. Further, the fluid compressibility is assumed to be nonlinear. A Lagrangian frame of reference is used to keep track of the deformation. We perform an implicit discretization in time (backward Euler) and propose two linearization schemes for solving the nonlinear problems appearing within each time step: Newton's method and L-scheme. The linearizations are used monolithically or in combination with a splitting algorithm. The resulting schemes can be applied for any spatial discretization. The convergences of all schemes are shown analytically for cases under small deformation. Illustrative numerical examples are presented to confirm the applicability of the schemes, in particular, for large deformation.

math.NA↗