[slides][biomin] fixup formatting, power law, tasks

slides/biomin.md

@@ -122,10 +122,10 @@ Neglecting microbial growth and decay, attachment and detachment
 - Urea hydrolysis
 $$
 \begin{aligned}
-\underset{\text{urea}}{CO(NH_2)_2} + 2 H_2O
+\underset{\text{urea}}{\mathrm{CO(NH_2)_2}} + 2 \mathrm{H_2O}
 \overset{\text{urease}}{\rightarrow}
 \\
-\underset{\text{ammonia}}{2NH_3} + \underset{\text{carbonic acid}}{H_2CO_3}
+\underset{\text{ammonia}}{\mathrm{2NH_3}} + \underset{\text{carbonic acid}}{\mathrm{H_2CO_3}}
 \end{aligned}
 $$
 :::
@@ -135,7 +135,7 @@ $$
 
 Here: Ureolytic microbes produce the enzyme urease (MICP)
 $$
-CO(NH_2)_2 + 2 H_2O + Ca^{2+} \rightarrow 2 NH_4^+ + CaCO_3
+\mathrm{CO(NH_2)_2 + 2 H_2O + Ca^{2+} \rightarrow 2 NH_4^+ + CaCO_3}
 $$
 
 Different reactions in detail:
@@ -144,12 +144,12 @@ Different reactions in detail:
 
 $$
 \begin{array}{lr}
-CO(NH_2)_2 + 2 H_2O \rightarrow 2 NH_3 + H_2CO_3       \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\!
+\mathrm{CO(NH_2)_2 + 2 H_2O \rightarrow 2 NH_3 + H_2CO_3}       \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\!
                                                        & \text{ureolysis} \\
-H_2CO_3 \leftrightarrow  HCO_3^- + H^+                 & \text{dissociation of carbonic acid} \\
-HCO_3^- \leftrightarrow CO_3^{2-} + H^+                & \text{dissociation of bicarbonate ion} \\
-2 NH_4^+ \leftrightarrow 2 NH_3 + 2 H^+                & \text{dissociation of ammonia} \\
-Ca^{2+} + CO_3^{2-} \leftrightarrow CaCO_3 \downarrow  & \text{calcite precipitation/dissolution}
+\mathrm{H_2CO_3 \rightleftharpoons  HCO_3^- + H^+}                 & \text{dissociation of carbonic acid} \\
+\mathrm{HCO_3^- \rightleftharpoons CO_3^{2-} + H^+}                & \text{dissociation of bicarbonate ion} \\
+\mathrm{2 NH_4^+ \rightleftharpoons 2 NH_3 + 2 H^+}                & \text{dissociation of ammonia} \\
+\mathrm{Ca^{2+} + CO_3^{2-} \rightleftharpoons CaCO_3 \downarrow}  & \text{calcite precipitation/dissolution}
 \end{array}
 $$
 
@@ -166,7 +166,7 @@ $$
 $$
 \mathrm{
 \underset{\text{calcium}}{Ca^{2+}} + \underset{\text{carbonate}}{CO_3^{2-}}
-\leftrightarrow \underset{\text{calcite}}{CaCO_3 \downarrow}
+\rightleftharpoons \underset{\text{calcite}}{CaCO_3 \downarrow}
 }
 $$
 :::
@@ -242,7 +242,7 @@ $$
 \begin{aligned}
 \qquad\qquad & \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\!
 \text{Precipitation rate:} \\
-r_\text{precip} &= f\; \left( A_\text{interface}, \Omega = \frac{\left[\mathrm{Ca}^{2+}\right]\left[CO_3^{2-}\right]}{K_\text{sp}}, T \right)
+r_\text{precip} &= f\; \left( A_\text{interface}, \Omega = \frac{\left[\mathrm{Ca}^{2+}\right]\left[\mathrm{CO_3}^{2-}\right]}{K_\text{sp}}, T \right)
 \\
 \qquad\qquad & \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\!
 \text{For this exercise:} \\
@@ -266,7 +266,7 @@ K &= K_0 \left(\frac{1-\phi_0}{1-\phi}\right)^2 \left(\frac{\phi}{\phi_0}\right)
 \\
 \text{or}&
 \\
-K &= K_0 \left( \frac{1-\phi_0}{1-\phi} \right)^\eta
+K &= K_0 \left( \frac{\phi}{\phi_0} \right)^\eta
 \end{aligned}
 $$
 
@@ -348,10 +348,10 @@ Academic problem setup
 
 * 2 aquifers with sealing aquitard
   * Upper aquifer: "drinking water"
-  * Lower aquifer: "$CO_2$ storage"
+  * Lower aquifer: "$\mathrm{CO_2}$ storage"
 * Problem:
   * Leakage pathway
-  * Stored $CO_2$ would migrate to drinking water aquifer!
+  * Stored $\mathrm{CO_2}$ would migrate to drinking water aquifer!
 * Biomineralization injection could "seal" the leakage pathway
 :::
 ::: {.column width=55%}
@@ -363,12 +363,11 @@ Academic problem setup
 
 1. Get familiar with the code
 2. Implement the simplified chemical reactions
-   * Add kinetic reaction rates to chemistry-file
-   * Use source()-function to link chemistry-file to problem
-3. Vary parameters, so that leakage pathway is "sealed" (porosity <0.07)
-4. Implement new boundary condition for $CO_2$-injection in lower aquifer
-5. Exchange the permeability law from Kozeny-Carman to a Power Law
-6. Use tabulated values for $CO_2$
+3. Use source()-function to link chemistry-file to problem
+4. Vary parameters, so that leakage pathway is "sealed" (porosity $<0.07$)
+5. Implement new boundary condition for $\mathrm{CO_2}$-injection in lower aquifer
+6. Exchange the permeability law from Kozeny-Carman to a Power Law
+7. Use tabulated values for $\mathrm{CO_2}$
 
 ## Exercise