Jakub Both

University of Bergen | UiB · Department of Mathematics

Flow in variably saturated porous media is typically modelled by the Richards equation, a nonlinear elliptic-parabolic equation which is notoriously challenging to solve numerically. In this paper, we propose a robust and fast iterative solver for Richards' equation. The solver relies on an adaptive switching algorithm, based on rigorously derived...

Understanding porous media flow is inherently a multi-scale challenge, where at the core lies the aggregation of pore-level processes to a continuum, or Darcy-scale, description. This challenge is directly mirrored in image processing, where pore-scale grains and interfaces may be clearly visible in the image, yet continuous Darcy-scale parameters...

We perform a series of repeated CO2 injections in a room-scale physical model of a faulted geological cross-section. Relevant parameters for subsurface carbon storage, including multiphase flows, capillary CO2 trapping, dissolution and convective mixing, are studied and quantified. As part of a validation benchmark study, we address and quantify si...

We present a framework for integrated experiments and simulations of tracer transport in heterogeneous porous media using digital twin technology. The physical asset in our setup is a meter-scale FluidFlower rig. The digital twin consists of a traditional physics-based forward simulation tool and a correction technique which compensates for mismatc...

Successful deployment of geological carbon storage (GCS) requires an extensive use of reservoir simulators for screening, ranking and optimization of storage sites. However, the time scales of GCS are such that no sufficient long-term data is available yet to validate the simulators against. As a consequence, there is currently no solid basis for a...

Carbon, capture, and storage (CCS) is an important bridging technology to combat climate change in the transition towards net-zero. The FluidFlower concept has been developed to visualize and study CO$_2$ flow and storage mechanisms in sedimentary systems in a laboratory setting. Meter-scale multiphase flow in two geological geometries, including n...

We perform a series of repeated CO2 injections in a room-scale physical model of a faulted geological cross-section. Relevant parameters for subsurface carbon sequestration, including multiphase flows, capillary CO2 trapping, dissolution, and convective mixing, are studied and quantified. As part of a forecasting benchmark study, we address and qua...

Understanding porous media flow is inherently a multi-scale challenge, where at the core lies the aggregation of pore-level processes to a continuum, or Darcy-scale, description. This challenge is directly mirrored in image processing, where grains and interfaces may be clearly visible, yet continuous parameters are desirable to measure. Classical...

Flow in variably saturated porous media is typically modelled by the Richards equation, a nonlinear elliptic-parabolic equation which is notoriously challenging to solve numerically. In this paper, we propose a robust and fast iterative solver for Richards' equation. The solver relies on an adaptive switching algorithm, based on rigorously derived...

We present a framework for integrated experiments and simulations of tracer transport in heterogeneous porous media using digital twin technology. The physical asset in our setup is a meter-scale FluidFlower rig. The digital twin consists of a traditional physics-based forward simulation tool and a correction technique which compensates for mismatc...

In this paper we propose a solution strategy for the Cahn-Larch\'e 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 no...

This lecture has the ambition to give an overall flavour of splitting algorithms for coupled problems, their interpretations, aiding their design and analysis. This is combined with direct applications for poroelasticity; but similar concepts should be applicable to a greater class of coupled problems. One key result will be the fixed-stress split...

In this paper, the convergence of the fundamental alternating minimization is established for non-smooth non-strongly convex optimization problems in Banach spaces, and novel rates of convergence are provided. As objective function a composition of a smooth, and a block-separable, non-smooth part is considered, covering a large range of application...

We address numerical solvers for a poromechanics model particularly adapted for soft materials, as it generally respectsthermodynamics principles and energy balance. Considering the multi-physics nature of the problem, which involves solid andfluid species, interacting on the basis of mass balance and momentum conservation, we decide to adopt a sol...

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 evolve according to a generalized Ginzburg–Landau energy functional, with additional impact from both elastic and fluid effects, and the coupling between flow...

We study unsaturated poroelasticity, i.e., coupled hydro-mechanical processes in variably saturated porous media, here modeled by a non-linear extension of Biot's well-known quasi-static consolidation model. The coupled elliptic-parabolic system of partial differential equations is a simplified version of the general model for multi-phase flow in d...

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 b...

There is currently an increasing interest in developing efficient solvers for variational phase-field models 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...

Sequential block-partitioned solvers have in the recent past been quite popular for multi-physics and in particular poroelasticity models. Such enable tailored solver technology for the respective single-physics problems via iterative coupling, as well as suggest suitable block-preconditioners for monolithic solvers.In this talk, we focus on a ther...

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 el...

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...

We address numerical solvers for a poromechanics model particularly adapted for soft materials, as it generally respects thermodynamics principles and energy balance. Considering the multi-physics nature of the problem, which involves solid and fluid species, interacting on the basis of mass balance and momentum conservation, we decide to adopt a s...

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 mon...

The fixed-stress splitting scheme is a popular method for iteratively solving the Biot equations. The method successively solves the flow and mechanics 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...

We present an iterative coupling scheme for the numerical approximation of the mixed hyperbolic-parabolic system of fully dynamic poroelasticity. We prove its convergence in the Banach space setting for an abstract semi-discretization in time that allows the application of the family of diagonally implicit Runge–Kutta methods. Recasting the semi-di...

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 el...

We present an iterative coupling scheme for the numerical approximation of the mixed hyperbolic-parabolic system of fully dynamic poroelasticity. We prove its convergence in the Banach space setting for an abstract semi-discretization in time that allows the application of the family of diagonally implicit Runge-Kutta methods. Recasting the semi-di...

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...

In this paper, the convergence of alternating minimization is established for non-smooth convex optimization in Banach spaces, and novel rates of convergence are provided. As objective function a composition of a smooth and a non-smooth part is considered with the latter being block-separable, e.g., corresponding to convex constraints or regulariza...

In this paper, we consider unsaturated poroelasticity, i.e., coupled hydro-mechanical processes in unsaturated porous media, modeled by a non-linear extension of Biot's quasi-static consolidation model. The coupled, elliptic-parabolic system of partial differential equations is a simplified version of the general model for multi-phase flow in defor...

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 an...

In this work we are interested in efficiently solving the quasi‐static, linear Biot model for poroelasticity. We consider the fixed‐stress splitting scheme, which is a popular method for iteratively solving Biot's equations. It is well‐known that the convergence properties of the method strongly depend on the applied stabilization/tuning parameter....

We study the numerical solution of the quasi-static linear Biot equations solved iteratively by the fixed-stress splitting scheme. In each iteration the mechanical and flow problems are decoupled, where the flow problem is solved by keeping an artificial mean stress fixed. This introduces a numerical tuning parameter which can be optimized. We inve...

Mathematical models for flow and reactive transport in porous media often involve non-linear, degenerate parabolic equations. Their solutions have low regularity, and therefore lower order schemes are used for the numerical approximation. Here the backward Euler method is combined with a mixed finite element method, which results in a stable and lo...

In this work we are interested in effectively solving the quasi-static, linear Biot model for poromechanics. We consider the fixed-stress splitting scheme, which is a popular method for iteratively solving Biot's equations. It is well-known that the convergence of the method is strongly dependent on the applied stabilization/tuning parameter. In th...

In this paper, we study the robust linearization of nonlinear poromechanics of unsaturated materials. The model of interest couples the Richards equation with linear elasticity equations, generalizing the classical Biot equations. In practice a monolithic solver is not always available, defining the requirement for a linearization scheme to allow t...

In this paper, we study the robust linearization of nonlinear poromechanics of unsaturated materials. The model of interest couples the Richards equation with linear elasticity equations, employing the equivalent pore pressure. In practice a monolithic solver is not always available, defining the requirement for a linearization scheme to allow the...

We study the numerical solution of the quasi-static linear Biot's equations solved iteratively by the fixed-stress splitting scheme. In each iteration the mechanical and flow problems are decoupled, where the flow problem is solved by keeping an artificial mean stress fixed. This introduces a numerical tuning parameter which can be optimized. We in...

We study the numerical solution of the quasi-static linear Biot's equations solved iteratively by the fixed-stress splitting scheme. In each iteration the mechanical and flow problems are decoupled, where the flow problem is solved by keeping an artificial mean stress fixed. This introduces a numerical tuning parameter which can be optimized. We in...

Mathematical models for flow and reactive transport in porous media often involve non-linear, degenerate parabolic equations. Their solutions have low regularity, and therefore lower order schemes are used for the numerical approximation. Here the backward Euler method is combined with a mixed finite element method scheme, which results in a stable...

This work concerns the linearization of a three-field discretization of generalized Biot’s equations describing coupled fluid flow and mechanical deformation in unsaturated porous media. The model of interest employs the effective stress based on the so-called equivalent pore pressure and can be interpreted as linear mechanics nonlinearly coupled w...

We study the iterative solution of coupled flow and geomechanics in heterogeneous porous media, modeled by a three-field formulation of the linearized Biot’s equations. We propose and analyze a variant of the widely used Fixed Stress Splitting method applied to heterogeneous media. As spatial discretization, we employ linear Galerkin finite element...

Interaction between CO2 and oil exhibits complex phase behavior that results in gravity-driven convection and enhanced mixing of CO2 in the oil zone. We have shown that the density of oil increases between 4% and 6% at CO2 concentrations between 65% and 75% by mass. Different cubic EoS give different values for density of CO2-oil mixtures. The incr...

We present a space-time finite element method capable of dealing with flows in multiple co-rotating reference frames. Since equal order interpolation is used for all degrees of freedom, Galerkin/Least–Squares stabilization is applied. We give a detailed derivation of the equations involved, introduce the variational form, present the stabilization...