08:30
Modelling Porous Media (Part 1)
Chair: Iuliu Sorin Pop
08:30
25 mins
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Comparison between equilibrium and kinetic reactions in the parameterization framework
Stephan de Hoop, Denis Voskov
Abstract: Increasing demand for cleaner energy sources, e.g. geothermal energy, has led to the comprehensive investigation of high-enthalpy carbonate reservoirs. These reservoirs are often chemically altered and hence contain a large uncertainty in the spatial distribution of the reservoir parameters. A successful development of these cleaner energy resources and effective management of the associated risks requires an evaluation of the large ensemble of multiphase reactive flow and transport simulations. This has led to the development of an efficient element based reduction technique which can significantly decrease the number of conservation equations and thereby reduce the computational time. In this approach, components or chemical species are translated to their constituent elements using chemical and thermodynamic equilibrium relations. Next, a finite-volume unstructured discretization in space is applied together with a fully-implicit approximation in time. The resulting complex nonlinear system is parameterized using the Operator Based Linearization (OBL).
The OBL framework transfer the governing nonlinear Partial Differential Equations into a linearized operator-form where the Jacobian is constructed as a product of a matrix of derivatives with respect to state variables and discretization operators. The state-dependent operators are only evaluated adaptively at vertices of the mesh introduced in the parameter-space. The continuous representation of state-dependent operators as well as their derivatives is achieved by using a multi-linear interpolation in parameter-space. This means that the usually time-consuming phase and chemical equilibrium computations, performed on each nonlinear iteration and in every control volume, are only executed when evaluating the operators in the new supporting points, thereby significantly reducing both the linearization time and the number of nonlinear iterations.
Equilibrium reactions in these advection-diffusion problems are usually valid when the rate of the reaction is far greater than characteristic velocity or diffusion. This is typically not true when considering heterogeneous reactions such as complex dissolution/precipitation processes. Therefore, a kinetic reaction module has also been developed within the OBL approach. The newly added module in the OBL framework for multiphase reactive transport and flow is validated for several dissolution regimes at different combination of Damkohler and Peclet numbers. Finally, the element based reduction framework, using equilibrium reactions, is compared with the component kinetic reaction framework at different kinetic rates.
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08:55
25 mins
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Finite element analysis of a coupled Stokes-Biot system
Elisa A. Bergkamp, Clemens V. Verhoosel, Joris J.C. Remmers, David M.J. Smeulders
Abstract: Understanding the interaction of a free flow and a deformable porous medium is essential for geomechanical applications such as Enhanced Geothermal Systems and the controlled extraction of oil and gas from fractured reservoirs. To study this interaction we developed a coupled model. In the model, the free flow is described by the Stokes equations, the porous formation is described by Biot’s equations, and the fluid flow in the formation is described by Darcy’s law. Both the free flow region and the formation are saturated with fluid.
One of the challenges addressed in this contribution is the choice and implementation of suitable coupling conditions on the shared interface of the free flow and the poroelastic medium. To fully couple the Stokes equations to Biot’s equations, both stress and flow conditions are prescribed. The slip encountered by the fluid flowing along the saturated formation is enforced by the Beavers-Joseph-Saffman condition [1,2]. Furthermore, the Darcy flux over the interface is driven by a pressure jump condition, as introduced by Showalter [3].
We solve the coupled problem using a staggered FEA approach. The numerical model is shown to fully couple a free flow and a saturated deformable porous medium in one-, two-, and three-dimensional cases and for various geometries. Results of the coupled model are validated using analytical results and a literature benchmark [4].
REFERENCES
[1] G.S. Beavers and D.D. Joseph, “Boundary conditions at a naturally permeable wall”, J. Fluid Mech., Vol. 30, pp. 197-207, 1967.
[2] P.G. Saffman, “On the boundary condition at the surface of a porous medium”, Stud. Appl. Math., Vol. 1, pp. 93-101, 1971.
[3] R.E. Showalter, “Poroelastic filtration coupled to Stokes flow”, in: Lecture Notes in Pure and Applied Mathematics, Chapman & Hall, Boca Raton, Vol. 242, pp. 229-241, 2005.
[4] I. Ambartsumyan, E. Khattatov, I. Yotov and P. Zunino, “A Lagrange multiplier method for a Stokes-Biot fluid-poroelastic structure interaction model”, Numer. Math., Vol. 40, 2, pp. 513-553, 2018.
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09:20
25 mins
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Bayesian network PDEs for multiscale representations of porous materials
Eric Hall, Kimoon Um, Markos Katsoulakis, Daniel Tartakovsky
Abstract: Microscopic (pore-scale) properties of porous media affect and often determine their macroscopic (Darcy- or continuum-scale) counterparts. Understanding the relationship between processes on these two scales is essential to both the derivation of macroscopic models of, e.g., transport phenomena in natural porous media, and the design of novel materials, e.g., for energy storage. Most microscopic properties exhibit complex statistical correlations and geometric constraints, which presents challenges for the estimation of macroscopic quantities of interest (QoIs), e.g., in the context of global sensitivity analysis (GSA) of macroscopic QoIs with respect to microscopic material properties. We present a systematic way of building correlations into stochastic multiscale models through Bayesian networks. The proposed framework allows us to construct the joint probability density function (PDF) of model parameters through causal relationships that are informed by domain knowledge and emulate engineering processes, e.g., the design of hierarchical nanoporous materials. These PDFs also serve as input for the forward propagation of parametric uncertainty. To assess the impact of correlations and causal relationships between microscopic parameters on macroscopic material properties, we use a moment-independent GSA based on the differential mutual information that leverages the structure of the Bayesian network and accounts for both correlated inputs and complex non-Gaussian QoIs. The global sensitivity indices are used to rank the effect of uncertainty in microscopic parameters on macroscopic QoIs and to quantify the impact of causality on the multiscale model's predictions. Our findings from numerical experiments indicate two practical outcomes. Firstly, the inclusion of correlations through structured priors based on causal relationships impacts predictions of QoIs which has important implications for decisions support and engineering design. Secondly, we observe the inclusion of structured priors with non-trivial correlations yields different effect rankings than independent priors and moreover these rankings are more consistent with the anticipated physics of a model problem.
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