Two-scale models pose a promising approach in simulating reactive flow and transport in evolving porous media. Classically, homogenized flow and transport equations are solved on the macroscopic scale, while effective parameters are obtained from auxiliary cell problems on possibly evolving reference geometries (micro-scale). Despite their perspective success in rendering lab/field-scale simulations computationally feasible, analytic results regarding the arising two-scale bilaterally coupled system often restrict to simplified models. In this paper, we first derive smooth-dependence results concerning the partial coupling from the underlying geometry to macroscopic quantities. Therefore, alterations of the representative fluid domain are described by smooth paths of diffeomorphisms. Exploiting the gained regularity of the effective space- and time-dependent macroscopic coefficients, we present local-in-time existence results for strong solutions to the partially coupled micro-macro system using fixed-point arguments. What is more, we extend our results to the bilaterally coupled diffusive transport model including a level-set description of the evolving geometry.

The paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion-convection-reaction equations and boundary conditions of mixed type. Since neither conformity nor consistency properties are assumed, the method is called completely discrete. We investigate two different stabilized discretizations and obtain stability and optimal error estimates in energy-type norms and, by generalizing the Aubin-Nitsche technique, optimal error estimates in weaker norms.

We formulate a hybridizable discontinuous Galerkin method for parabolic equations with non-linear tensor-valued coefficients and jump conditions (Henry’s law). The analysis of the proposed scheme indicates the optimal convergence order for mildly non-linear problems. The same order is also obtained in our numerical studies for simplified settings. A series of numerical experiments investigate the effect of choosing different order approximation spaces for various unknowns.

Transport and reaction of dissolved species in hierarchically structured complex media, e.g. porous media, can be described by micro-macro models (models with distributed micro-structure, FE2 models). These models offer advantages compared to pure micro- or pure macroscale models but exhibit, compared to pure macroscale models, a strongly increased numerical complexity. We show how parallel computing together with Schur complement techniques can make those models numerically feasible.

In this paper, we introduce a pore-scale model for reactive flow and transport in evolving porous media exhibiting two competing mineral phases. By formal two-scale asymptotic expansion in a level-set framework an effective micro-macro model is derived. As such, our approach comprises flow and transport equations on the macroscopic scale including effective hydrodynamic parameters calculated from representative unit cells. Conversely, the macroscopic solutes' concentrations alter the unit cells' geometrical structure by triggering dissolution or precipitation processes. The numerical implementation of such micro-macro models poses several challenges, especially in terms of geometry representation and computational complexity. In this research, the Voronoi implicit interface method is applied to characterize and evolve the two-mineral structure and first-order convergence is obtained in a test case where analytical solutions are available. Furthermore, we present a sophisticated overall solution strategy for the introduced fully coupled nonlinear micro-macro problem and conduct numerical simulations. In doing so, the significant performance enhancements arising from machine learning techniques are evaluated. To this end, a convolutional neural network is trained on (realistic) unit cell geometries for permeability prediction and deployed in a micro-macro simulation. The outcome is compared to the respective results obtained by classical methods in terms of predictive power and computational effort.

We investigate reactive flow and transport in evolving porous media. Solute species that are transported within the fluid phase are taking part in mineral precipitation and dissolution reactions for two competing mineral phases. The evolution of the three phases is not known a-priori but depends on the concentration of the dissolved solute species. To model the coupled behavior, phase-field and level-set models are formulated. These formulations are compared in three increasingly challenging setups including significant mineral overgrowth. Simulation outcomes are examined with respect to mineral volumes and surface areas as well as derived effective quantities such as diffusion and permeability tensors. In doing so, we extend the results of current benchmarks for mineral dissolution/precipitation at the pore-scale to the multiphasic solid case. Both approaches are found to be able to simulate the evolution of the three-phase system, but the phase-field model is influenced by curvature-driven motion.

The finite element method, frequently abbreviated by FEM, was developed in the fifties in the aircraft industry, after the concept had been independently outlined by mathematicians at an earlier time.

An essential step in the implementation of the finite element method (and also of the finite volume method as described in Chapter 8) is to create an initial “geometric discretization” of the domain \(\Omega .\) This part of a finite element code is usually included in the so-called preprocessor (see also Section 2.4.1). In general, a stand-alone finite element code consists further of the intrinsic kernel (assembling of the finite-dimensional system of algebraic equations, rearrangement of data (if necessary), solution of the algebraic problem) and the postprocessor (editing of the results, extraction of intermediate results, preparation for graphic output, a posteriori error estimation).

As we have seen in the introductory Chapter 0, the modelling of transport and reaction processes in porous media results in differential equations of the form $$ \partial _t u - \nabla \cdot (\boldsymbol{K}\nabla u-\boldsymbol{c}u) = f\,, $$which is a special case of the form ( 0.33), and just the time-dependent version of the formulation in divergence form ( 3.36) with \(b=1\).

It is well known that the generalized Darcy law describing multiphase flow in porous media has some shortcomings. In particular, it cannot explain hysteresis effects in the capillary pressure--saturation curve which have been observed in measurements. In this work, we derive a numerically tractable micro-macro model including coupled generalized Darcy's laws that still includes the microscale dynamics which are responsible, e.g., for hysteresis effects. For this purpose, we extend the two-scale expansion approach of periodic homogenization to include different time scales which allows us to start from a fully instationary Navier--Stokes--Cahn--Hilliard model at the pore scale as microscale. Identifying and separating the time scales allows us to derive local fast scale equations describing the microscale dynamics and global slow-scale equations giving rise to the macroscopic Darcy law.

Despite the concentration of most of the book on linear boundary value problems and their discretizations, many real-world models are rather nonlinear elliptic or parabolic boundary value problems with various types of nonlinearities. To be specific, we restrict ourselves to the formulation in divergence form ( 3.36) and its time-dependent counterpart and the classical FEM based on Chapter 2 and Chapter 3. Let \(a, \ell \) be the forms from ( 3.36) to ( 3.39), a first nonlinear (here: semilinear) boundary value problem reads.

This chapter illustrates the scientific context in which differential equation models may occur, in general, and also in a specific example. Section 0.1 reviews the fundamental equations, for some of them discretization techniques will be developed and investigated in this book. In Sections 0.2–0.4 we focus on reaction and transport processes in porous media. These sections are independent of the remaining parts and may be skipped by the reader. Section 0.5, howevershould be consulted because it fixes some notation to be used later on.

We now continue the definition and analysis of the “correct” function spaces that we began in ( 2.19)–( 2.22).

Until now, the general framework has been as follows: Let V be a Hilbert space with the scalar product \(\langle \cdot , \cdot \rangle _{V}\) and induced norm \(\Vert \cdot \Vert _{V}\), \(a:\; V \times V \rightarrow \mathbb {R}\) a bilinear form, and \(\ell \in V'\).

The FEM studied in Chapter 3 and the node-oriented FVM in Chapter 8 are all conforming with a variational formulation based on \(V\subset H^1(\Omega )\) and thus exhibit inter-element continuity (see Theorem 3.21).

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern

In this section we want to introduce the finite difference method, frequently abbreviated as FDM, using the Poisson equation on a rectangle as an example. By means of this example and generalizations of the problem, advantages and limitations of the approach will be elucidated. Also, in the following section the Poisson equation will be the main topic, but then on an arbitrary domain.

In this section, initial boundary value problems for the linear case of the differential equation ( 0.33) are considered. We choose the form ( 3.13) together with the boundary conditions ( 3.19)–( 3.21), which have already been discussed in Section 0.5.

We consider again the system of linear equations with nonsingular matrix \(A \in \mathbb {R}^{m,m}\), right-hand side \(b \in \mathbb {R}^m\), and solution \(x\in \mathbb {R}^m\). As shown in Chapters 2 and 3, such systems of equations arise from finite element discretizations of elliptic boundary value problems. The matrix A is the stiffness matrix and thus sparse, as can be seen from ( 2.41).

We consider a macroscale model of transport and reaction of chemical species in a porous medium with a special focus on mineral precipitation–dissolution processes. In the literature, it is frequently proposed that the reaction rate should depend on the reactive mineral surface area, and so on the amount of mineral. We point out that a frequently used model is ill posed in the sense that it admits non-unique solutions. We investigate what consequences this non-uniqueness has on the numerical solution of the model. The main novelty in this article is our proposal of a certain substitution which removes the ill-posedness from the system and which leads to better numerical results than some “ad hoc methods.” We think that the proposed substitution is a rather elegant way to get rid of the non-uniqueness and the numerical difficulties and is much less technical than other ideas. As a proof of concept, we present some numerical tests and simulations for the new model.