A multiscale flux basis for mortar mixed discretizations of reduced Darcy-Forchheimer fracture models

In this paper, a multiscale flux basis algorithm is developed to efficiently solve a flow problem in fractured porous media. Here, we take into account a mixed-dimensional setting of the discrete fracture matrix model, where the fracture network is represented as lower-dimensional object. We assume the linear Darcy model in the rock matrix and the non-linear Forchheimer model in the fractures. In our formulation, we are able to reformulate the matrix-fracture problem to only the fracture network problem and, therefore, significantly reduce the computational cost. The resulting problem is then a non-linear interface problem that can be solved using a fixed-point or Newton-Krylov methods, which in each iteration require several solves of Robin problems in the surrounding rock matrices. To achieve this, the flux exchange (a linear Robin-to-Neumann co-dimensional mapping) between the porous medium and the fracture network is done offline by pre-computing a multiscale flux basis that consists of the flux response from each degree of freedom on the fracture network. This delivers a conserve for the basis that handles the solutions in the rock matrices for each degree of freedom in the fractures pressure space. Then, any Robin sub-domain problems are replaced by linear combinations of the multiscale flux basis during the interface iteration. The proposed approach is, thus, agnostic to the physical model in the fracture network. Numerical experiments demonstrate the computational gains of pre-computing the flux exchange between the porous medium and the fracture network against standard non-linear domain decomposition approaches

Shema usrednjenja prijenosa za zakone sačuvanja

Zakoni sačuvanja opisuju brojne prirodne pojave, između ostalog tok u poroznoj sredini (Buckley-Leverettove jednadžbe). U ovom diplomskom radu opisat ćemo nedavno uvedenu numeričku metodu za zakone sačuvanja te dokazati konvergenciju metode prema entropijskom rješenju. Metoda povezuje suvremene s poznatim tehnikama iz tog područja, poput kinetičke formulacije i entropijskih uvjeta. Opisno govoreći, osnova metode je pretvaranje nelinearne jednadžbe skalarnog zakona sačuvanja u linearnu (kinetičku) jednadžbu. Usrednjavanjem rješenja linearne jednadžbe dobiva se dopustivo entropijsko rješenje inicijalnog problema. Na kraju dajemo numeričku implementaciju metode na poznatim problemima Burgersove i Buckley-Leverettove jednadžbe.Scalar conservation laws are key in modelling different natural phenomena, for example two-phase flow in porous media (Buckley-Leverett equations). In this thesis a new numerical method for conservation laws is described and it is proven that the solution given by this method converges to the entropy solution. This method includes both modern and classic methods for solving conservation laws, such as entropy conditions in method of characteristics and kinetic formulation. Basic idea is to make a linear (kinetic) equation from a nonlinear homogeneous conservation law and then average out the solution to get the entropy admissible solution of the initial problem. We also provide numerical examples of implementing the described method on Burgers’ equation and Buckley-Leverett equation

Shema usrednjenja prijenosa za zakone sačuvanja

Zakoni sačuvanja opisuju brojne prirodne pojave, između ostalog tok u poroznoj sredini (Buckley-Leverettove jednadžbe). U ovom diplomskom radu opisat ćemo nedavno uvedenu numeričku metodu za zakone sačuvanja te dokazati konvergenciju metode prema entropijskom rješenju. Metoda povezuje suvremene s poznatim tehnikama iz tog područja, poput kinetičke formulacije i entropijskih uvjeta. Opisno govoreći, osnova metode je pretvaranje nelinearne jednadžbe skalarnog zakona sačuvanja u linearnu (kinetičku) jednadžbu. Usrednjavanjem rješenja linearne jednadžbe dobiva se dopustivo entropijsko rješenje inicijalnog problema. Na kraju dajemo numeričku implementaciju metode na poznatim problemima Burgersove i Buckley-Leverettove jednadžbe.Scalar conservation laws are key in modelling different natural phenomena, for example two-phase flow in porous media (Buckley-Leverett equations). In this thesis a new numerical method for conservation laws is described and it is proven that the solution given by this method converges to the entropy solution. This method includes both modern and classic methods for solving conservation laws, such as entropy conditions in method of characteristics and kinetic formulation. Basic idea is to make a linear (kinetic) equation from a nonlinear homogeneous conservation law and then average out the solution to get the entropy admissible solution of the initial problem. We also provide numerical examples of implementing the described method on Burgers’ equation and Buckley-Leverett equation

Shema usrednjenja prijenosa za zakone sačuvanja

Zakoni sačuvanja opisuju brojne prirodne pojave, između ostalog tok u poroznoj sredini (Buckley-Leverettove jednadžbe). U ovom diplomskom radu opisat ćemo nedavno uvedenu numeričku metodu za zakone sačuvanja te dokazati konvergenciju metode prema entropijskom rješenju. Metoda povezuje suvremene s poznatim tehnikama iz tog područja, poput kinetičke formulacije i entropijskih uvjeta. Opisno govoreći, osnova metode je pretvaranje nelinearne jednadžbe skalarnog zakona sačuvanja u linearnu (kinetičku) jednadžbu. Usrednjavanjem rješenja linearne jednadžbe dobiva se dopustivo entropijsko rješenje inicijalnog problema. Na kraju dajemo numeričku implementaciju metode na poznatim problemima Burgersove i Buckley-Leverettove jednadžbe.Scalar conservation laws are key in modelling different natural phenomena, for example two-phase flow in porous media (Buckley-Leverett equations). In this thesis a new numerical method for conservation laws is described and it is proven that the solution given by this method converges to the entropy solution. This method includes both modern and classic methods for solving conservation laws, such as entropy conditions in method of characteristics and kinetic formulation. Basic idea is to make a linear (kinetic) equation from a nonlinear homogeneous conservation law and then average out the solution to get the entropy admissible solution of the initial problem. We also provide numerical examples of implementing the described method on Burgers’ equation and Buckley-Leverett equation

Preconditioning for Flow in Fractured Porous Media

The key to reliable simulations of flow in fractured porous media is the proper design, analysis and implementation of numerical methods. These methods should take into account the specific properties of the underlying model, while at the same time be flexible enough to handle variability of the model's components. The particular features of fractured porous media we concern ourselves with are the complex geometry of the fracture network and the disparity in scales in the model parameters. The model we study is based on interpretation of fractures and the porous rock as a mixed-dimensional geometry, and the resulting system of partial differential equations is highly coupled and parameter-dependent. In this thesis, we build upon the common approaches to discretization of the flow problem and deliver a numerical solution by constructing efficient numerical solvers and preconditioners. The two main topics of our research are the design of preconditioners to finite element discretization of the linear flow model and the development of linearization methods to the non-linear model. In the first part, we consider the fact that our flow problem reveals the saddle-point structure. This motivates to see how some established approaches to preconditioning saddle-point problems work under the mixed-dimensional complexity. We construct the preconditioners to the classical solving approaches, such as Krylov subspace methods, based on the well-posedness of our saddle-point system. As the goal of any preconditioner is to approximate the inverse of the coefficient operator of the system, the principal idea of our approach is to find that inverse mapping that is equivalent in terms of norms on the given function spaces. In this way, we can ensure that the preconditioned numerical solvers will converge more rapidly, independently of values of the discretization and physical parameters. In our case, we are able to derive two such preconditioners by identifying two different topologies on the given discrete finite-element spaces. In fact, one of the approaches leads to a general framework to preconditioning mixed-dimensional elliptic problems that can be applied to other problems with a similar hierarchical structure as the model of flow in fractured porous media. Finally, we study a choice of non-linear and time-dependent flow models that appear in cases of enhanced conductivity of the fractures and compressibility of the fluid. The development of the iterative solution methods for our problem considers the natural domain decomposition setting imposed by the fracture network and the standard linearization methods are adapted to the mixed-dimensional setting. By using non-matching grids, we can employ a multiscale method to the interface problem to handle the dominating computational cost in each iteration of the non-linear solver, namely the repeated solving process on the rock matrix subdomains. The flexibility of the method is showcased by successfully applying it to several non-linear flow models

Country of production as a determinant of the perception of brand luxury

Zemlja proizvodnje marki kao i percepcija luksuznosti koju potrošači imaju o određenoj zemlji utječu na donošenje odluke o kupovini marki te o stvaranju lojalnosti prema markama. U uvjetima današnje globalizacije zemlja u kojoj se određena marka proizvodi postaje determinanta koja uvelike utječe na svijest potrošača i na stvaranje imidža proizvoda. Ovisno o razini luksuza pojedinih marki, potrošači sve više traže informacije o proizvodima poznatih marki, cijenama, baš kao i o samoj zemlji proizvodnje istih te se u očima potrošača mijenja percepcija luksuznosti pojedine marke proizvoda, ovisno je li on proizveden u razvijenim zemljama ili pak zemljama u razvoju

Country of production as a determinant of the perception of brand luxury

Zemlja proizvodnje marki kao i percepcija luksuznosti koju potrošači imaju o određenoj zemlji utječu na donošenje odluke o kupovini marki te o stvaranju lojalnosti prema markama. U uvjetima današnje globalizacije zemlja u kojoj se određena marka proizvodi postaje determinanta koja uvelike utječe na svijest potrošača i na stvaranje imidža proizvoda. Ovisno o razini luksuza pojedinih marki, potrošači sve više traže informacije o proizvodima poznatih marki, cijenama, baš kao i o samoj zemlji proizvodnje istih te se u očima potrošača mijenja percepcija luksuznosti pojedine marke proizvoda, ovisno je li on proizveden u razvijenim zemljama ili pak zemljama u razvoju