diff --git a/slides/biomin.md b/slides/biomin.md
index 05e43c58ec79025e28d3ba2bee5598d9b25d9c2d..df06232b5e364cee57535a0e99784815d7e47581 100644
--- a/slides/biomin.md
+++ b/slides/biomin.md
@@ -181,14 +181,15 @@ $$
:::
::: {.column width=55%}
* Clogging: Reduction of porosity
- $$
- \phi = \phi_0 - \phi_\text{biofilm} - \phi_\text{calcite}
- $$
+$$
+\phi = \phi_0 - \phi_\text{biofilm} - \phi_\text{calcite}
+$$
+
* and reduction in permeability: Kozeny-Carman relation
- $$
- K = K_0 \left( \frac{1-\phi_0}{1-\phi} \right)^2 \left( \frac{\phi}{\phi_0} \right)^3
- $$
- or the Power Law
+$$
+K = K_0 \left( \frac{1-\phi_0}{1-\phi} \right)^2 \left( \frac{\phi}{\phi_0} \right)^3
+$$
+or the Power Law
:::
::::::
@@ -199,15 +200,14 @@ $$
:::incremental
* Mass balance equation of components
- $$
- \Sigma_\alpha \frac{\partial}{\partial t}
- (\phi \rho_\alpha x^\kappa_\alpha S_\alpha)
- + \nabla \cdot ( \rho_\alpha x^\kappa_\alpha \mathbf{v}_\alpha )
- - \nabla \cdot ( \rho_\alpha \mathbf{D}^\kappa_{\alpha;\text{pm}} \nabla x^\kappa_\alpha )
- = q^\kappa
- $$
+$$
+\Sigma_\alpha \frac{\partial}{\partial t}(\phi \rho_\alpha x^\kappa_\alpha S_\alpha)+ \nabla \cdot ( \rho_\alpha x^\kappa_\alpha \mathbf{v}_\alpha )- \nabla \cdot ( \rho_\alpha \mathbf{D}^\kappa_{\alpha;\text{pm}} \nabla x^\kappa_\alpha )= q^\kappa
+$$
+
* Mass balance for the immobile components / solid phases:
- $$\frac{\partial}{\partial t}(\rho_\varphi \phi_\varphi) = q^\varphi$$
+$$
+\frac{\partial}{\partial t}(\rho_\varphi \phi_\varphi) = q^\varphi
+$$
:::
@@ -306,20 +306,19 @@ NumEqVector source(const Element& element,
* Update porosity in dumux/material/fluidmatrixinteractions/porosityprecipitation.hh
-<section style="font-size: 0.9em">
```cpp
…
-auto priVars = evalSolution(element, element.geometry(), elemSol, scv.center());
+auto priVars = evalSolution(element, element.geometry(),
+ elemSol, scv.center());
Scalar sumPrecipitates = 0.0;
-for (unsigned int solidPhaseIdx = 0; solidPhaseIdx < numSolidPhases; ++solidPhaseIdx)
+for (int solidPhaseIdx = 0; solidPhaseIdx < numSolidPhases; ++solidPhaseIdx)
sumPrecipitates += priVars[numComp + solidPhaseIdx];
using std::max;
return max(minPoro, refPoro - sumPrecipitates);
…
```
-</section>
## Specific Implementations
diff --git a/slides/fractures.md b/slides/fractures.md
index 6774341b07d8910ecde6f550f71c522c05c6bcd0..71c31018b339fa388982ee631d6c3d43dae06224 100644
--- a/slides/fractures.md
+++ b/slides/fractures.md
@@ -61,18 +61,16 @@ Geothermal energy production
## Problem Formulation
-<font size=6>
-$\begin{equation}
- \begin{aligned}
- \mathbf{u}_i &= - \mathbf{K}_i \nabla p_i, \\
- \nabla \cdot \mathbf{u}_i &= q_i, &&\mathrm{in} \, \Omega_i, \\
- \mathbf{U}_f &=- a \mathbf{K}_{f, \tau} \nabla_\tau P_f, \\
- \nabla_\tau \cdot\mathbf{U}_f &= q_f + \left( \mathbf{u}_1 \cdot \mathbf{n}_1 + \mathbf{u}_2 \cdot \mathbf{n}_2 \right), \\
- \mathbf{u}_i \cdot \mathbf{n}_i &= - \frac{2 k_\eta}{a} ( P_f - p_i ), &&\mathrm{in} \, \gamma_N, \\
- P_f &= p_i, &&\mathrm{in} \, \gamma_D.
- \end{aligned}
-\end{equation}$
-</font>
+$$
+\begin{aligned}
+ \mathbf{u}_i &= - \mathbf{K}_i \nabla p_i, \\
+ \nabla \cdot \mathbf{u}_i &= q_i, &&\mathrm{in} \, \Omega_i, \\
+ \mathbf{U}_f &=- a \mathbf{K}_{f, \tau} \nabla_\tau P_f, \\
+ \nabla_\tau \cdot\mathbf{U}_f &= q_f + \left( \mathbf{u}_1 \cdot \mathbf{n}_1 + \mathbf{u}_2 \cdot \mathbf{n}_2 \right), \\
+ \mathbf{u}_i \cdot \mathbf{n}_i &= - \frac{2 k_\eta}{a} ( P_f - p_i ), &&\mathrm{in} \, \gamma_N, \\
+ P_f &= p_i, &&\mathrm{in} \, \gamma_D.
+\end{aligned}
+$$
# Fracture exercise
diff --git a/slides/intro.md b/slides/intro.md
index 225eefa977024caefc85e9413806bcc6caf099c5..732d3ab18dfa4892f63c4459be560ac32c39589a 100644
--- a/slides/intro.md
+++ b/slides/intro.md
@@ -3,7 +3,6 @@ title: Introduction to DuMu^x^
subtitle: Overview and Available Models
---
-
# Table of Contents
## Table of Contents
@@ -138,13 +137,15 @@ Preimplemented models:
* Describes the advective flux in porous media on the macro-scale
* Single-phase flow
-
- $$\mathbf{v} = - \frac{\mathbf{K}}{\mu} \left(\textbf{grad}\, p - \varrho \mathbf{g} \right)$$
+$$
+\mathbf{v} = - \frac{\mathbf{K}}{\mu} \left(\textbf{grad}\, p - \varrho \mathbf{g} \right)
+$$
* Multi-phase flow (phase $\alpha$)
-
- $$\mathbf{v}_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right)$$
- where $k_{r\alpha}(S_\alpha)$ is the relative permeability, a function of saturation $S_\alpha$.
+$$
+\mathbf{v}_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_\alpha \mathbf{g} \right)
+$$
+where $k_{r\alpha}(S_\alpha)$ is the relative permeability, a function of saturation $S_\alpha$.
* For non-creeping flow, Forchheimer's law is available as an alternative.
@@ -152,8 +153,9 @@ Preimplemented models:
* Uses standard Darcy approach for the conservation of momentum by default
* Mass continuity equation
-
- $$\frac{\partial\left( \phi \varrho \right)}{\partial t} + \text{div} \left( \varrho \mathbf{v} \right) = q$$
+$$
+\frac{\partial\left( \phi \varrho \right)}{\partial t} + \text{div} \left( \varrho \mathbf{v} \right) = q
+$$
* Primary variable: $p$
@@ -163,8 +165,9 @@ Preimplemented models:
* Uses standard Darcy approach for the conservation of momentum by default
* Transport of component $\kappa \in \{\text{H2O}, \text{Air}, ...\}$
-
- $$\frac{\partial\left( \phi \varrho X^\kappa \right)}{\partial t} + \text{div} \left( \varrho X^\kappa \mathbf{v} - \varrho D^\kappa_\text{pm} \textbf{grad} X^\kappa \right) = q$$
+$$
+\frac{\partial\left( \phi \varrho X^\kappa \right)}{\partial t} + \text{div} \left( \varrho X^\kappa \mathbf{v} - \varrho D^\kappa_\text{pm} \textbf{grad} X^\kappa \right) = q
+$$
* Closure relation: $\sum_\kappa X^\kappa = 1$
* Primary variables: $p$ and $X^\kappa$
@@ -174,13 +177,14 @@ Preimplemented models:
## 1pncmin -- with Fluid-Solid Phase Change
* Transport equation for each component $\kappa \in \{\text{H2O}, \text{Air}, ...\}$
-
- $$\frac{\partial \left( \varrho_f X^\kappa \phi \right)}{\partial t}
- + \text{div} \left( \varrho_f X^\kappa \mathbf{v} - \mathbf{D_\text{pm}^\kappa} \varrho_f \textbf{grad}\, X^\kappa \right) = q_\kappa$$
+$$
+\frac{\partial \left( \varrho_f X^\kappa \phi \right)}{\partial t} + \text{div} \left( \varrho_f X^\kappa \mathbf{v} - \mathbf{D_\text{pm}^\kappa} \varrho_f \textbf{grad}\, X^\kappa \right) = q_\kappa
+$$
* Mass balance solid phases
-
- $$\frac{\partial \left(\varrho_\lambda \phi_\lambda \right)}{\partial t} = q_\lambda$$
+$$
+\frac{\partial \left(\varrho_\lambda \phi_\lambda \right)}{\partial t} = q_\lambda
+$$
* Primary variables: $p$, $X^k$ and $\phi_\lambda$
@@ -190,8 +194,9 @@ Preimplemented models:
* Uses standard multi-phase Darcy approach for the conservation of momentum by default
* Conservation of the phase mass of phase $\alpha \in \{\text{w}, \text{n}\}$
-
- $$\frac{\partial \left( \phi \varrho_\alpha S_\alpha \right)}{\partial t} + \text{div} \left(\varrho_\alpha \mathbf{v}_\alpha \right) = q_\alpha$$
+$$
+\frac{\partial \left(\phi \varrho_\alpha S_\alpha \right)}{\partial t} + \text{div} \left(\varrho_\alpha \mathbf{v}_\alpha \right) = q_\alpha
+$$
* Constitutive relations: $p_\text{c} := p_\text{n} - p_\text{w} = p_\text{c}(S_\text{w})$, $k_{r\alpha}$ = $k_{r\alpha}(S_\text{w})$
* Physical constraint (void space filled with fluid phases): $S_\text{w} + S_\text{n} = 1$
@@ -202,8 +207,11 @@ Preimplemented models:
## 2pnc -- Two-Phase Compositional
* Transport equation for each component $\kappa \in \{\text{H2O}, \text{Air}, ...\}$ in phase $\alpha \in \{\text{w}, \text{n}\}$
-
- $$\begin{aligned}\frac{\partial \left( \sum_\alpha \varrho_\alpha X_\alpha^\kappa \phi S_\alpha \right)}{\partial t} &+ \sum_\alpha \text{div} \left( \varrho_\alpha X_\alpha^\kappa \mathbf{v}_\alpha - \mathbf{D_{\alpha, pm}^\kappa} \varrho_\alpha \textbf{grad}\, X^\kappa_\alpha \right) = \sum_\alpha q_\alpha^\kappa \end{aligned}$$
+$$
+\begin{aligned}
+\frac{\partial \left( \sum_\alpha \varrho_\alpha X_\alpha^\kappa \phi S_\alpha \right)}{\partial t} &+ \sum_\alpha \text{div}\left( \varrho_\alpha X_\alpha^\kappa \mathbf{v}_\alpha - \mathbf{D_{\alpha, pm}^\kappa} \varrho_\alpha \textbf{grad}\, X^\kappa_\alpha \right) = \sum_\alpha q_\alpha^\kappa
+\end{aligned}
+$$
* Constitutive relation: $p_\text{c} := p_\text{n} - p_\text{w} = p_\text{c}(S_\text{w})$, $k_{r\alpha}$ = $k_{r\alpha}(S_\text{w})$
* Physical constraints: $S_\text{w} + S_\text{n} = 1$ and $\sum_\kappa X_\alpha^\kappa = 1$
@@ -214,12 +222,16 @@ Preimplemented models:
## 2pncmin
* Transport equation for each component $\kappa \in \{\text{H2O}, \text{Air}, ...\}$
-
- $$\begin{aligned}\frac{\partial \left( \sum_\alpha \varrho_\alpha X_\alpha^\kappa \phi S_\alpha \right)}{\partial t} &+ \sum_\alpha \text{div} \left( \varrho_\alpha X_\alpha^\kappa \mathbf{v}_\alpha - \mathbf{D_{\alpha, pm}^\kappa} \varrho_\alpha \textbf{grad}\, X^\kappa_\alpha \right) = \sum_\alpha q_\alpha^\kappa \end{aligned}$$
+$$
+\begin{aligned}
+\frac{\partial \left( \sum_\alpha \varrho_\alpha X_\alpha^\kappa \phi S_\alpha \right)}{\partial t} &+ \sum_\alpha \text{div}\left( \varrho_\alpha X_\alpha^\kappa \mathbf{v}_\alpha - \mathbf{D_{\alpha, pm}^\kappa} \varrho_\alpha \textbf{grad}\, X^\kappa_\alpha \right)= \sum_\alpha q_\alpha^\kappa
+\end{aligned}
+$$
* Mass balance solid phases
-
- $$\frac{\partial \left(\varrho_\lambda \phi_\lambda \right)}{\partial t} = q_\lambda \quad \forall \lambda \in \Lambda$$
+$$
+\frac{\partial \left(\varrho_\lambda \phi_\lambda \right)}{\partial t} = q_\lambda \quad \forall \lambda \in \Lambda
+$$
for a set of solid phases $\Lambda$ each with volume fraction $\varrho_\lambda$
* Source term models **dissolution/precipiation/phase transition** fluid ↔ solid
@@ -230,8 +242,9 @@ for a set of solid phases $\Lambda$ each with volume fraction $\varrho_\lambda$
* Uses standard multi-phase Darcy approach for the conservation of momentum by default
* Conservation of the phase mass of phase $\alpha \in \{\text{w}, \text{g}, \text{n}\}$
-
- $$\frac{\partial \left( \phi \varrho_\alpha S_\alpha \right)}{\partial t} - \text{div} \left( \varrho_\alpha \mathbf{v}_\alpha \right) = q_\alpha$$
+$$
+\frac{\partial \left( \phi \varrho_\alpha S_\alpha \right)}{\partial t} - \text{div} \left( \varrho_\alpha \mathbf{v}_\alpha \right) = q_\alpha
+$$
* Physical constraint: $S_\text{w} + S_\text{n} + S_g = 1$
* Primary variables: $p_\text{g}$, $S_\text{w}$, $S_\text{n}$
@@ -241,9 +254,11 @@ for a set of solid phases $\Lambda$ each with volume fraction $\varrho_\lambda$
## 3p3c -- Three-Phase Compositional
* Transport equation for each component $\kappa \in \{\text{H2O}, \text{Air}, \text{NAPL}\}$ in phase $\alpha \in \{\text{w}, \text{g}, \text{n}\}$
-
- $$\begin{aligned}\frac{\partial \left( \phi \sum_\alpha \varrho_{\alpha,\text{mol}} x_\alpha^\kappa S_\alpha \right)}{\partial t}
- &+ \sum_\alpha \text{div} \left( \varrho_{\alpha,\text{mol}} x_\alpha^\kappa \mathbf{v}_\alpha - D_\text{pm}^\kappa \frac{1}{M_\kappa} \varrho_\alpha \textbf{grad} X^\kappa_{\alpha} \right) = q^\kappa \end{aligned}$$
+$$
+\begin{aligned}
+\frac{\partial \left( \phi \sum_\alpha \varrho_{\alpha,\text{mol}} x_\alpha^\kappa S_\alpha \right)}{\partial t}&+ \sum_\alpha \text{div} \left( \varrho_{\alpha,\text{mol}} x_\alpha^\kappa \mathbf{v}_\alpha - D_\text{pm}^\kappa \frac{1}{M_\kappa} \varrho_\alpha \textbf{grad} X^\kappa_{\alpha} \right) = q^\kappa
+\end{aligned}
+$$
* Physical constraints: $\sum_\alpha S_\alpha = 1$ and $\sum_\kappa x^\kappa_\alpha = 1$
* Primary variables: depend on the locally present fluid phases
@@ -269,9 +284,11 @@ for a set of solid phases $\Lambda$ each with volume fraction $\varrho_\lambda$
* Local thermal equilibrium assumption
* One energy conservation equation for the porous solid matrix and the fluids
- $$\begin{aligned}\frac{\partial \left( \phi \sum_\alpha \varrho_\alpha u_\alpha S_\alpha \right)}{\partial t} &+ \frac{\partial \left(\left(1 - \phi \right)\varrho_s c_s T \right)}{\partial t}
+ $$
+ \begin{aligned}\frac{\partial \left( \phi \sum_\alpha \varrho_\alpha u_\alpha S_\alpha \right)}{\partial t} &+ \frac{\partial \left(\left(1 - \phi \right)\varrho_s c_s T \right)}{\partial t}
+ \sum_\alpha \text{div} \left( \varrho_\alpha h_\alpha \mathbf{v}_\alpha \right)
- - \text{div} \left(\lambda_\text{pm} \textbf{grad}\, T \right) = q^h \end{aligned}$$
+ - \text{div} \left(\lambda_\text{pm} \textbf{grad}\, T \right) = q^h \end{aligned}
+ $$
* Specific internal energy $u_\alpha = h_\alpha - p_\alpha / \varrho_\alpha$
* Can be added to other models, additional primary variable temperature $T$
@@ -288,12 +305,16 @@ for a set of solid phases $\Lambda$ each with volume fraction $\varrho_\lambda$
* Momentum balance equation for a single-phase, isothermal RANS model
- $$\frac{\partial \left(\varrho \textbf{v} \right)}{\partial t} + \nabla \cdot \left(\varrho \textbf{v} \textbf{v}^{\text{T}} \right) = \nabla \cdot \left(\mu_\textrm{eff} \left(\nabla \textbf{v} + \nabla \textbf{v}^{\text{T}} \right) \right)
- - \nabla p + \varrho \textbf{g} - \textbf{f}$$
+ $$
+ \frac{\partial \left(\varrho \textbf{v} \right)}{\partial t} + \nabla \cdot \left(\varrho \textbf{v} \textbf{v}^{\text{T}} \right) = \nabla \cdot \left(\mu_\textrm{eff} \left(\nabla \textbf{v} + \nabla \textbf{v}^{\text{T}} \right) \right)
+ - \nabla p + \varrho \textbf{g} - \textbf{f}
+ $$
* The effective viscosity is composed of the fluid and the eddy viscosity
- $$\mu_\textrm{eff} = \mu + \mu_\textrm{t}$$
+ $$
+ \mu_\textrm{eff} = \mu + \mu_\textrm{t}
+ $$
* Various turbulence models are implemented
diff --git a/slides/materialsystem.md b/slides/materialsystem.md
index b8dbc75a6c8c34b56acfd9b197286dde85d6df07..d061b5603b5c59f20bfd8c263da81e0269959528 100644
--- a/slides/materialsystem.md
+++ b/slides/materialsystem.md
@@ -177,6 +177,7 @@ _Specifying a solid system is only necessary if you work with a non-isothermal o
* Effective diffusivity after _Millington and Quirk_
## Van-Genuchten
+
$\begin{equation}
p_c = \frac{1}{\alpha}\left(S_e^{-1/m} -1\right)^{1/n}
\end{equation}$
@@ -186,6 +187,7 @@ p_c = \frac{1}{\alpha}\left(S_e^{-1/m} -1\right)^{1/n}
$\rightarrow$ the empirical parameters $\alpha$ and $n$ have to be specified
## Brooks-Corey
+
$\begin{equation}
p_c = p_d S_e^{-1/\lambda}
\end{equation}$
diff --git a/slides/problem.md b/slides/problem.md
index 86148e3f4f5148404e8cdc12ee2918556602beae..3ab8baade6d12ec3683f52342bc4913a119a52c4 100644
--- a/slides/problem.md
+++ b/slides/problem.md
@@ -7,30 +7,34 @@ title: DuMu^x^ applications
## Gas injection / immiscible two phase flow
Mass balance equations for two fluid phases:
-
-$\begin{aligned}
+$$
+\begin{aligned}
\frac{\partial \left(\phi \varrho_\alpha S_\alpha \right)}{\partial t}
-
\nabla \cdot \boldsymbol{v}_\alpha
-
q_\alpha = 0, \quad \alpha \in \lbrace w, n \rbrace.
-\end{aligned}$
+\end{aligned}
+$$
Momentum balance equations (multiphase-phase Darcy's law):
-
-$\begin{aligned}
+$$
+\begin{aligned}
\boldsymbol{v}_\alpha = \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\nabla\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right), \quad \alpha \in \lbrace w, n \rbrace.
-\end{aligned}$
+\end{aligned}
+$$
## Gas injection / immiscible two phase flow
-$\begin{aligned}
+$$
+\begin{aligned}
\frac{\partial \left(\phi \varrho_\alpha S_\alpha \right)}{\partial t}
-
\nabla \cdot \left( \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\nabla\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right) \right)
-
q_\alpha = 0
-\end{aligned}$
+\end{aligned}
+$$
* $p_w$, $p_n$: wetting and non-wetting fluid phase pressure
* $\varrho_\alpha$, $\mu_\alpha$: fluid phase density and dynamic viscosity
@@ -40,13 +44,15 @@ $\begin{aligned}
## Gas injection / immiscible two phase flow
-$\begin{aligned}
+$$
+\begin{aligned}
\frac{\partial \left(\phi \varrho_\alpha S_\alpha \right)}{\partial t}
-
\nabla \cdot \left( \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\nabla\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right) \right)
-
q_\alpha = 0
-\end{aligned}$
+\end{aligned}
+$$
* Constitutive relations: $p_n := p_w + p_c$, $p_c := p_c(S_w)$, $k_{r\alpha}$ = $k_{r\alpha}(S_w)$
* Physical constraint (no free space): $S_w + S_n = 1$