\documentclass{article}
\usepackage{iftex}
\usepackage{ifthen}
\usepackage{epsfig}
\usepackage[utf8]{inputenc}
\usepackage{xcolor}
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\usepackage{amsmath}

\ifPDFTeX
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  \doxynewunicodechar{²}{${}^{2}$}% Superscript two
  \doxynewunicodechar{³}{${}^{3}$}% Superscript three

\fi
\pagestyle{empty}
\begin{document}
$\kappa$
\pagebreak

$\alpha$
\pagebreak

$-\phi b_\alpha S_\alpha D \mathbf{grad}X_\alpha^\kappa$
\pagebreak

$b_\alpha$
\pagebreak

$S_\alpha$
\pagebreak

$X_\alpha^\kappa$
\pagebreak

$R_s,\;R_v,\;R_{sw},\;R_{vw}$
\pagebreak

$X_w^G=\frac{R_{sw}}{R_{sw}+\rho_w/\rho_g}$
\pagebreak

$\rho_w$
\pagebreak

$\rho_g$
\pagebreak

$-b_{w,ij}S_{w,ij}D_{w,ij}(\frac{1}{R_{sw,ij}+\rho_w/\rho_g})DT_{ij}(R_{sw,i}-R_{sw,j})$
\pagebreak

$b_{w,ij}$
\pagebreak

$S_{w,ij}$
\pagebreak

$D_{w,ij}$
\pagebreak

$R_{sw,ij}$
\pagebreak

$\frac{1}{R_{sw,ij}+\rho_w/\rho_g}$
\pagebreak

$DT_{ij}$
\pagebreak

$\frac{1}{R_{sw,ij}+(Mm_g\rho_w)/(Mm_w\rho_g)}$
\pagebreak

$-b_\alpha E||\mathrm{v}_\alpha||\mathbf{grad}X_\alpha^\kappa$
\pagebreak

$\mathrm{v}_\alpha$
\pagebreak

$-b_{w,ij}v_{w,ij}(\frac{1}{R_{sw,ij}+\rho_w/\rho_g})D_{ij}(R_{sw,i}-R_{sw,j})$
\pagebreak

$v_{w,ij}$
\pagebreak

$D_{ij}$
\pagebreak

$E_i$
\pagebreak

$E_j$
\pagebreak

$\alpha \in \{ w, g, o \}$
\pagebreak

$\kappa \in \{ W, G, O \}$
\pagebreak

\[   B_\alpha := \frac{\varrho_\alpha(1\,\text{bar})}{\varrho_\alpha(p_\alpha)}
\]
\pagebreak

$B_o$
\pagebreak

$R_s$
\pagebreak

\[   R_s := \frac{\varrho_{o}^G}{\varrho_o^O}\;.
\]
\pagebreak

\[   \sum_\alpha \frac{\partial\;\phi c_\alpha^\kappa S_\alpha }{\partial t}
   - \sum_\alpha \mathrm{div} \left\{ c_\alpha^\kappa \mathbf{v}_\alpha \right\}
   - q^\kappa = 0 \;,
\]
\pagebreak

\[ \mathbf{v}_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K}
 *\left(\mathbf{grad}\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right) \;, \]
\pagebreak

$ Re <
  1$
\pagebreak

\[ \vec{v}_\alpha =
 \left( \nabla p_\alpha - \rho_\alpha \vec{g}\right)
 \frac{\mu_\alpha}{k_{r,\alpha} K}
\]
\pagebreak

$[m^3/s / m^2]$
\pagebreak

$\mathrm{[J/(m^3]}$
\pagebreak

$\mathrm{[W/m^2 / (K/m)]}$
\pagebreak

$\mathrm{[J/m^3]}$
\pagebreak

$\mathrm{[W/m^2 /
(K/m)]}$
\pagebreak

$\approx 500$
\pagebreak

\[ \nabla p_\alpha - \rho_\alpha \vec{g} =
 - \frac{\mu_\alpha}{k_{r,\alpha}} K^{-1}\vec{v}_\alpha
 - \frac{\rho_\alpha C_E}{\eta_{r,\alpha}} \sqrt{K}^{-1}
 \left| \vec{v}_\alpha \right| \vec{v}_\alpha
\]
\pagebreak

$C_E$
\pagebreak

$\eta_{r,\alpha}$
\pagebreak

$\frac{k_{r_alpha}}{mu} K$
\pagebreak

\[ \frac{k_{r_alpha}}{mu} K \left( \nabla p_\alpha - \rho_\alpha \vec{g}\right) =
 - \vec{v}_\alpha
 - \frac{\rho_\alpha C_E}{\eta_{r,\alpha}}  \frac{k_{r_alpha}}{mu} \sqrt{K}
 \left| \vec{v}_\alpha \right| \vec{v}_\alpha
\]
\pagebreak

$k_{r,\alpha}$
\pagebreak

$[m^2]$
\pagebreak

$\mathrm{[K]}$
\pagebreak

$\mathrm{[m/s^2]}$
\pagebreak

$\boldsymbol{g} = ( 0,\dots,\ -9.81)^T $
\pagebreak

$\boldsymbol{g} = ( 0,\dots, 0)^T $
\pagebreak

$M > 0$
\pagebreak

$\mathrm{[m^3]}$
\pagebreak

\[    \frac{\partial f(x)}{\partial x} \approx \frac{f(x) - f(x - \epsilon)}{\epsilon}
\]
\pagebreak

\[    \frac{\partial f(x)}{\partial x} \approx
         \frac{f(x + \epsilon) - f(x - \epsilon)}{2 \epsilon}
\]
\pagebreak

\[    \frac{\partial f(x)}{\partial x} \approx
         \frac{f(x + \epsilon) - f(x)}{\epsilon}
\]
\pagebreak

$ f $
\pagebreak

$x$
\pagebreak

$\epsilon$
\pagebreak

\[    \frac{\partial f(x)}{\partial x} \approx \frac{f(x + \epsilon) - f(x)}{\epsilon}
\]
\pagebreak

\[ u^{k+1} = u^k - \Delta u^k \]
\pagebreak

$1 m$
\pagebreak

$1m \times 1m$
\pagebreak

$M \geq 1$
\pagebreak

$N \geq
   M$
\pagebreak

\[   \mathbf{v}_\alpha =
   - \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K}
   \left(\mathbf{grad}\, p_\alpha
         - \varrho_{\alpha} \mathbf{g} \right) \;,
\]
\pagebreak

\[   \sum_\alpha \frac{\partial\;\phi c_\alpha^\kappa S_\alpha }{\partial t}
   - \sum_\alpha \mathrm{div} \left\{ c_\alpha^\kappa \mathbf{v}_\alpha \right\}
   - q^\kappa = 0 \;.
\]
\pagebreak

$c^\kappa = \sum_\alpha S_\alpha x_\alpha^\kappa \rho_{mol, \alpha}$
\pagebreak

\[   \mathbf{v}_\alpha =
   - \frac{k_{r\alpha}}{\mu_\alpha}
   \mathbf{K}\left(\mathbf{grad}\, p_\alpha -
                   \varrho_{\alpha} \mathbf{g} \right) \;,
\]
\pagebreak

\[   \frac{\partial\;\phi S_\alpha \rho_\alpha }{\partial t}
   - \mathrm{div} \left\{ \rho_\alpha \mathbf{v}_\alpha  \right\}
   - q_\alpha = 0 \;.
\]
\pagebreak

$p_0$
\pagebreak

$M - 1$
\pagebreak

$T$
\pagebreak

$[kg/(m^2\,s)]$
\pagebreak

$[mol/(m^2\,s)]$
\pagebreak

$A \in \{m^2, m^3\}$
\pagebreak

$[J/(m^2\,s)]$
\pagebreak

$[m^3/(m^2\,s)]$
\pagebreak

$M$
\pagebreak

$N$
\pagebreak

$\alpha \in \{ 1, \dots, M \}$
\pagebreak

$N \geq M - 1$
\pagebreak

$\kappa \in \{ 1, \dots, N \} $
\pagebreak

\[     \mathbf{v}_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K}
     \left(\mathbf{grad}\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right) \;,
\]
\pagebreak

$1$
\pagebreak

\[ \forall \alpha: S_\alpha = 0 \implies \sum_\kappa
   x_\alpha^\kappa \leq 1 \]
\pagebreak

\[ \forall \alpha: \sum_\kappa x_\alpha^\kappa = 1 \implies S_\alpha \geq 0
 *\]
\pagebreak

\[   \forall \alpha: S_\alpha \left( \sum_\kappa x_\alpha^\kappa - 1 \right) = 0
\]
\pagebreak

$\Phi(a, b)$
\pagebreak

\[\Phi(a,b) = 0 \iff a \geq0 \land b \geq0  \land a \cdot b = 0 \]
\pagebreak

\[ \Phi(a,b) = a + b - \sqrt{a^2 + b^2} \;. \]
\pagebreak

\[ \Phi(a,b) = \min \{a,  b \}\;, \]
\pagebreak

$p_1$
\pagebreak

$f^1, \dots, f^{N}$
\pagebreak

$M-1$
\pagebreak

$S_1, \dots, S_{M-1}$
\pagebreak

\[    \mathbf{v}_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K}
    \left(\mathbf{grad}\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right) \;,
\]
\pagebreak

\[   0 \stackrel{!}{=}
    f_\alpha = \left\{
    \begin{array}{cl}
      S_\alpha&  \quad \text{if phase }\alpha\text{ is not present} \    \
      1 - \sum_\kappa x_\alpha^\kappa&  \quad \text{else}
    \end{array}
    \right.
\]
\pagebreak

$S_\alpha < 0$
\pagebreak

$\sum_\kappa
       x_\alpha^\kappa > 1$
\pagebreak

$PV_0 =
        p_0$
\pagebreak

$S_\alpha = PV_i$
\pagebreak

$PV_i =
       x_{\alpha^\star}^\alpha$
\pagebreak

$\alpha^\star$
\pagebreak

$N-M$
\pagebreak

$x_{\alpha^\star}^\kappa$
\pagebreak

\[\frac{\partial\;\phi S_\alpha \rho_\alpha}{\partial t}
-
\mathrm{div} \left\{
\rho_\alpha \frac{k_{r\alpha}}{\mu_\alpha}\; \mathbf{K}\;
\mathbf{grad}\left[
p_\alpha - g\rho_\alpha
\right]
\right\}
=
q_\alpha,
\]
\pagebreak

$\alpha \in \{w, n\}$
\pagebreak

$\rho_\alpha$
\pagebreak

$\phi$
\pagebreak

$k_{r\alpha}$
\pagebreak

$\mu_\alpha$
\pagebreak

$\mathbf{K}$
\pagebreak

$p_\alpha$
\pagebreak

$g$
\pagebreak

$1\%$
\pagebreak

$\frac{k_{r\alpha}}{\mu_\alpha}$
\pagebreak

$\frac{k_{rn}}{\mu_n}$
\pagebreak

$p_w$
\pagebreak

\[S_w = p_c^{-1}(p_n - p_w)
\]
\pagebreak

$p_n$
\pagebreak

$\mathrm{[s]}$
\pagebreak

\[ |a - b| \leq \text{tol} = \text{abstol} + \text{reltol} \times \max(|a|, |b|) \]
\pagebreak

$ 10^{-15} \times 10^9 = 10^{-6} $
\pagebreak

\[ e^k = \max_i\{
\left| r^k_i \right| \}\;, \]
\pagebreak

$r^k = \mathbf{A} x^k - b $
\pagebreak

$x^k$
\pagebreak

\[ e^k = \max_i\{ \left| w_i \delta^k_i \right| \}\;, \]
\pagebreak

$\delta = x^k - x^{k + 1} $
\pagebreak

$x^{k + 1}$
\pagebreak

$w_i$
\pagebreak

$i$
\pagebreak

$U_{ii} = U_{ii} +$
\pagebreak

$U_{ii} = U_{ii} + sign(U_{ii}) * $
\pagebreak

$U_{ii} = U_{ii} sign(U_{ii}) * $
\pagebreak

$U_{ii} = U_{ii} + (U_{ii}>0?1:0) * $
\pagebreak

$ y = A(x) $
\pagebreak

$ y = y + \alpha A(x) $
\pagebreak

\[ e^k = \frac{\left| A x_k - b \right|}{\left| A x_0 - b \right|}\;, \]
\pagebreak

\[ e^k = \max_i\{ \left| w_i r^k_i \right| \}\;, \]
\pagebreak

\end{document}
