Iterative schemes for surfactant transport in porous media

In this work, we consider the transport of a surfactant in 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 five numerical examples. For these examples, the number of iterations and the condition numbers of the linear systems emerging in each iteration are presented.publishedVersio

Global random walk solvers for fully coupled flow and transport in saturated/unsaturated porous media

In this article, we present new random walk methods to solve flow and transport problems in saturated/unsaturated porous media, including coupled flow and transport processes in soils, heterogeneous systems modeled through random hydraulic conductivity and recharge fields, processes at the field and regional scales. The numerical schemes are based on global random walk algorithms (GRW) which approximate the solution by moving large numbers of computational particles on regular lattices according to specific random walk rules. To cope with the nonlinearity and the degeneracy of the Richards equation and of the coupled system, we implemented the GRW algorithms by employing linearization techniques similar to the -scheme developed in finite element/volume approaches. The resulting GRW -schemes converge with the number of iterations and provide numerical solutions that are first-order accurate in time and second-order in space. A remarkable property of the flow and transport GRW solutions is that they are practically free of numerical diffusion. The GRW solvers are validated by comparisons with mixed finite element and finite volume solvers in one- and two-dimensional benchmark problems. They include Richards’ equation fully coupled with the advection-diffusion-reaction equation and capture the transition from unsaturated to saturated flow regimes.publishedVersio

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.publishedVersio

Inside out porphyridium cruentum: Beyond the conventional biorefinery concept

Here, an unprecedented biorefinery approach has been designed to recover high-added value bioproducts starting from the culture ofPorphyridium cruentum. This unicellular marine red alga can secrete and accumulate high-value compounds that can find applications in a wide variety of industrial fields. 300 ± 67 mg/L of exopolysaccharides were obtained from cell culture medium; phycoerythrin was efficiently extracted (40% of total extract) and isolated by single chromatography, with a purity grade that allowed the crystal structure determination at 1.60 Å; a twofold increase in β-carotene yield was obtained from the residual biomass; the final residual biomass was found to be enriched in saturated fatty acids. Thus, for the first time, a complete exploitation ofP. cruentumculture was set up.P.I. would like to acknowledge ALGAE4IBD project (FROM NATURE TO BEDSIDE-ALGAE BASED BIO COMPOUND FOR PREVENTION) funded by the European Union’s Horizon 2020 Research and Innovation program under grant agreement N° 101000501. This work was supported by the Ministry of Science and Innovation of Spain (Grant No. PID2020-113050RB-I00). G.A.-R. would like to acknowledge the Ministry of Science and Innovation (MICINN) for his “Juan de la Cierva-Incorporación” postdoctoral grant IJC2019-041482-I. A.M. and G.F. thank Elettra Synchrotron of Trieste staff for their help during X-ray diffraction data collection.Peer reviewe

Iterative schemes for solving coupled, non-linear flow and transport in porous media

In this thesis we compare different iterative approaches for solving the non-linear, coupled multiphase flow and reactive transport in porous media. Especially, we consider two-phase flow and one a-half phase flow (modeled by Richards equation) coupled with an one component transport equation. Implicit and explicit iterative schemes will be compared in terms of efficiency, robustness and accuracy. The best approach seems to be a fully implicit scheme, which is a slightly modified variant of the classical splitting iterative scheme for coupled equations. We concentrate on three linearization methods: L-scheme, Modified Picard and Newton. We implement them in Matlab and tested on both an academic example, built from a manufactured analytical solution and on a realistic problem, the Salinity Problem. In the first part of the thesis we will also briefly study the generic two-phase flow plus transport equation in porous media, presenting some of the most common solving algorithms such as the IMPES method and its Fully Implicit reformulation

Robust solvers for fully coupled transport and flow in saturated/unsaturated porous media

Numerical simulations and laboratory studies are our main tools to comprehend better processes happening in the subsurface. The phenomena are modeled thanks to systems of partial differential equations (PDEs), which are extraordinarily complex to solve numerically due to their often highly nonlinear and tightly coupled character. After decades of research on new and improved solving algorithms, there is still a need for accurate and robust schemes. In this work, we investigate linearization schemes and splitting techniques for fully coupled flow and transport in porous media. A particular case of multiphase flow in porous media, the study of water flow in variably saturated porous media, modeled by the Richards equation, is studied here. An external component, e.g., a surfactant, is transported by the water phases. The resulting system of equations is fully coupled and nonlinear. In this work, we investigate three different linearization schemes, the classical Newton method, commonly used throughout the industry, the modified Picard method, and the L-scheme. Even though only linearly convergent, the latter appears to be the most robust scheme for some particular cases. The convergence of the L-scheme has also been studied theoretically. Both the Newton method and modified Picard are faster, in terms of numbers of iterations, but fail to converge in cases of unsaturated-saturated porous media or complex phenomena such as hysteresis effects. The rate of convergence of the L-scheme can be improved by combining it with the Newton method or by using the Anderson acceleration. The scheme resulting by combining the L-scheme and the Newton method appears practically to be both quadratically and globally convergent. It requires fewer iterations than the L-scheme, it is more robust than the Newton method, and it also converges for larger time steps. Using larger time steps can considerably decrease the total number of iterations over the full simulation. Alternatively, by applying the Anderson acceleration, one avoids the computation of any derivatives, and thus the implementation of the algorithm is less invasive. The rate of convergence of the L-scheme depends on user-defined parameters. Finding the optimal L values can be tedious. Optimizing the Anderson acceleration is more straightforward, and the improvements obtained are remarkable. The equations investigated in this work are not only characterized by nonlinear terms but are also fully coupled. The external component, dissolved into the water phase, directly influences the flow. We investigate two splitting approaches, the canonical nonlinear splitting, and an alternate linearized splitting. We compare them in terms of the numbers of iterations required to achieve the convergence, and the condition numbers of the systems to be solved within each iteration. After all, the latter appears to be a better alternative; it requires fewer iterations and achieves equally accurate results. A monolithic or fully implicit formulation is also investigated. The solution algorithm is computationally slower than the splitting ones but appears to be more robust. Finally, a global random walk approach for solving the coupled nonlinear problem is also studied. The scheme results in being free of numerical diffusion. Using vast numbers of particles, almost as many as the molecules involved in the reaction, it produces an intuitive representation of the process. The global random walk algorithms are explicit and thus often more straightforward than the typical finite volume/element schemes. The models investigated in this work represent a particular case of two-phase flow in porous media. Still, we are confident that similar results, concerning the linearization schemes and solving algorithms, can also be achieved in the case of more canonical models

Iterative schemes for surfactant transport in porous media

In this work, we consider the transport of a surfactant in 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 five numerical examples. For these examples, the number of iterations and the condition numbers of the linear systems emerging in each iteration are presented