Model dvostruke poroznosti

Radišić, Ivana (2014) Model dvostruke poroznosti. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

In this paper we derived double porosity model of flow in fractured porous media. Fractured porous media $\Omega$ contains system of fracture planes $\Omega_f^\varepsilon$ dividing the porous rock into collection of blocks $\Omega_m^\varepsilon$. In this paper we first have described model of single phase flow on a microscopic scale. Single phase flow is described by following equations \begin{align*} \begin{cases} \Phi^\varepsilon \frac{\partial}{\partial t}\rho^\varepsilon - div \left( \frac{K^\varepsilon}{\mu c} \nabla \rho^\varepsilon \right)=f &\text{u } \Omega \times (0,T)\\ \rho^\varepsilon=\rho_{bd} &\text{na} \: \Gamma_D \times (0,T)\\ \frac{K\varepsilon}{\mu c} \nabla \rho^\varepsilon \cdot \nu_\Omega=h &\text{na } \Gamma_N \times (0,T)\\ \rho^\varepsilon=\rho^{init} &\text{na } \Omega \times \{0\}, \end{cases} \end{align*} where $\rho^\varepsilon$ denotes fluid density. $\Phi^\varepsilon$ and $K^\varepsilon$ are porosity and permeability functions \begin{align*} \Phi^\varepsilon(x)= \begin{cases} \Phi^\ast &x \in \Omega_f^\varepsilon\\ \phi^\varepsilon(x) &x \in \Omega_m^\varepsilon, \end{cases}\\ K^\varepsilon(x)= \begin{cases} K^\ast &x \in \Omega_f^\varepsilon\\ \varepsilon^2 k^\varepsilon(x) &x \in \Omega_m^\varepsilon, \end{cases} \end{align*} Macroscopic model is derived from homogenization theory. We have replaced $\Phi^\varepsilon$ and $K^\varepsilon$ by following functions \begin{align*} \Phi^H= &\frac{\left|Y_f\right|}{\left|Y\right|}\Phi^\ast \\ K^H= &K^\ast \frac{1}{\left|Y\right|}\int_{Yf}(\nabla_y\omega(y)+\mathbb{I})dy,\\ \end{align*} Double porosity model is given by \begin{align*} \begin{cases} \Phi^H\partial_t\rho_f^0 - div \left( \frac{K^H}{\mu c} \nabla \rho_f^0 \right)=f- \frac{1}{\left|Y\right|} \int_{Y_m}\phi(y)\partial_t\rho_m^0dy &\text{u } \Omega \times (0,T)\\ \rho_f^0=\rho_{bd} &\text{na }\Gamma_D \times (0,T)\\ \frac{K^H}{\mu c}\nabla \rho_f^0 \cdot \nu_\Omega=h &\text{na }\Gamma_N \times (0,T)\\ \rho_f^0=\rho_f^{init} &\text{na } \Omega \times \{0\}, \end{cases} \end{align*} \begin{align*} \begin{cases} \phi(y)\partial_t\rho_m^0(x,y,t)-div_y \left( \frac{k(y)}{\mu c} \nabla_y \rho_m^0(x,y,t) \right) =f(x,t) &\text{u } \Omega \times Y_m \times (0,T)\\ \rho_m^0(x,y,t)=\rho_f^0(x,t) &\text{na } \Omega \times \partial Y_m \times (0,T)\\ \rho_m^0(x,y,0)=\rho_m^{init}(x) &\text{na } \Omega \times Y_m. \end{cases} \end{align*} We applied finite element method for spatial discretization and backward Euler for discretization in time. Numerical model comparison is done by DUNE, modular toolbox for solving partial differential equations. We have concluded that double porosity model is much easier to approximate computationaly because it requires less grid elements. Models have been tested for different $\varepsilon$ values. We have noticed that relative error decreases as $\varepsilon$ decreases.

Item Type: Thesis (Diploma thesis) Jurak, Mladen 2014 66 NATURAL SCIENCES > Mathematics Faculty of Science > Department of Mathematics Iva Prah 06 Jul 2015 14:11 24 Jan 2017 12:45 http://digre.pmf.unizg.hr/id/eprint/4084