| Abstract | U ovom radu smo izveli model dvostruke poroznosti koji opisuje gibanje fluida kroz frakturiranu poroznu sredinu. Frakturirana porozna sredina \(\Omega\) je porozna sredina koja se sastoji od matrice poroznih blokova \(\Omega_m^\varepsilon\), koja je okruzena sustavom fraktura \(\Omega_f^\varepsilon\), gdje parametar \(\varepsilon\) predstavlja veličinu matričnog bloka. U radu smo najprije opisali model jednofaznog toka fluida na mikroskopskoj skali. Dobili smo sljedeću inicijalno rubnu zadaću \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*} gdje \(\rho^\varepsilon\) predstavlja gustoću fluida unutar \(\Omega\), a \(\Phi^\varepsilon\) i \(K^\varepsilon\) su funkcije poroznosti i propusnosti fluida \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*} Pomoću teorije homogenizacije izveli smo model na makroskopskoj skali gdje smo funkcije \(\Phi^\varepsilon\) i \(K^\varepsilon\) zamijenili sljedećim funkcijama \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*} te smo dobili model dvostruke poroznosti \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*} Prostornu i vremensku diskretizaciju modela smo proveli primjenom metode konačnih elemenata, odnosno implicitne Eulerove metode. Numeričku usporedbu modela proveli smo pomoću DUNE-a, software-a za rješavanje parcijalnih diferencijalnih jednadžbi. Došli smo do zaključka da je model na makroskopskoj skali manje zahtjevan za numeričko rješavanje, jer zahtjeva manji broj elemenata u triangulaciji domene \(\Omega\). Modele smo testirali za različite vrijednosti parametra \(\varepsilon\), te smo primijetili da se relativna greska numeričkih rješenja makroskopskog i mikroskopskog modela smanjuje sa smanjenjem parametra \(\varepsilon\). |
| Abstract (english) | 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. |
