biomin.md

---
title: Modelling porous medium modification </br><small>Induced Calcite Precipitation</small>
---

# SFB Project Area C: Biomineralization

## What is Induced Calcite Precipitation (ICP)?

Microbes change the chemistry in a way that promotes the precipitation of calcite.

## What is ICP?

<img src="img/biomin_intro-microbes.png"/></br>
<small>Credit: James Connolly, Montana State University.</small>

## Results of ICP

Segmented CT image of a glass-bead column mineralized by ICP

:::::: {.columns}
::: {.column width=40%}
<img src="img/biomin_introColumns.png"/>
:::
::: {.column width=60%}
<img src="img/biomin_introSegmented.png"/>
:::
::::::
<small>Credit: Johannes Hommel, University of Stuttgart</small>

## Why investigate ICP?

::::: {.columns}
::: {.column width=50%}
<img src="img/biomin_introCementedSand.jpg"/>
<small>Mineralized sand, photo by Johannes Hommel, University of Stuttgart.</small>
:::
::: {.column width=50%}
<img src="img/biomin_introPoroPermFelix.png"/>
<small>Porosity-Permeability changes observed by Felix Weinhardt, University of Stuttgart.</small>
:::
::::::

## Why investigate ICP?

Applications in which a porous medium should be cemented in-situ, e.g. sealing, leakage mitigation, creating subsurface barries, reducing erosion, or stabilizing soil.

Main desired effects of ICP in those applications are:

- reduce flow (reduce $K$ and $k_r$, increase $p_c$)
- increase mechanical strength

# Model concept

## Model concept: Relevant processes

:::::: {.columns}
::: {.column width=45%}
<img src="img/biomin_model-2p.png"/>
<small>(modified after Ebigbo et al., WRR 2012)</small>
:::
::: {.column width=55%}
- Two-phase transport
:::
::::::

## Model concept: Relevant processes

:::::: {.columns}
::: {.column width=45%}
<img src="img/biomin_model-2pnc.png"/>
<small>(modified after Ebigbo et al., WRR 2012)</small>
:::
::: {.column width=55%}
- Two-phase, multi-component transport
:::
::::::

## Model concept: Relevant processes

:::::: {.columns}
::: {.column width=45%}
<img src="img/biomin_model-microbes.png"/>
<small>(modified after Ebigbo et al., WRR 2012)</small>
:::
::: {.column width=55%}
**For this exercise:**
Neglecting microbial growth and decay, attachment and detachment

- Biomass (S. pasteurii)
  - growth / decay
  - attachment / detachment
:::
::::::

## Model concept: Relevant processes

:::::: {.columns}
::: {.column width=45%}
<img src="img/biomin_model-urea.png"/>
<small>(modified after Ebigbo et al., WRR 2012)</small>
:::
::: {.column width=55%}
- Urea hydrolysis
$$
\begin{aligned}
\underset{\text{urea}}{\mathrm{CO(NH_2)_2}} + 2 \mathrm{H_2O}
\overset{\text{urease}}{\rightarrow}
\\
\underset{\text{ammonia}}{\mathrm{2NH_3}} + \underset{\text{carbonic acid}}{\mathrm{H_2CO_3}}
\end{aligned}
$$
:::
::::::

## MICP: Main reactions

Here: Ureolytic microbes produce the enzyme urease (MICP)
$$
\mathrm{CO(NH_2)_2 + 2 H_2O + Ca^{2+} \rightarrow 2 NH_4^+ + CaCO_3}
$$

Different reactions in detail:

<!-- \! creates negative spacing to fit the two columns -->

$$
\begin{array}{lr}
\mathrm{CO(NH_2)_2 + 2 H_2O \rightarrow 2 NH_3 + H_2CO_3}       \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\!
                                                       & \text{ureolysis} \\
\mathrm{H_2CO_3 \rightleftharpoons  HCO_3^- + H^+}                 & \text{dissociation of carbonic acid} \\
\mathrm{HCO_3^- \rightleftharpoons CO_3^{2-} + H^+}                & \text{dissociation of bicarbonate ion} \\
\mathrm{2 NH_4^+ \rightleftharpoons 2 NH_3 + 2 H^+}                & \text{dissociation of ammonia} \\
\mathrm{Ca^{2+} + CO_3^{2-} \rightleftharpoons CaCO_3 \downarrow}  & \text{calcite precipitation/dissolution}
\end{array}
$$

## Model concept: Relevant processes

:::::: {.columns}
::: {.column width=50%}
<img src="img/biomin_model-precipitation.png"/>
<small>(modified after Ebigbo et al., WRR 2012)</small>
:::
::: {.column width=50%}
- Precipitation and dissolution of calcite

$$
\mathrm{
\underset{\text{calcium}}{Ca^{2+}} + \underset{\text{carbonate}}{CO_3^{2-}}
\rightleftharpoons \underset{\text{calcite}}{CaCO_3 \downarrow}
}
$$
:::
::::::

## Model concept: Relevant processes

:::::: {.columns}
::: {.column width=45%}
<img src="img/biomin_model-clogging.png"/>
<small>(modified after Ebigbo et al., WRR 2012)</small>
:::
::: {.column width=55%}
* Clogging: Reduction of porosity
$$
\phi = \phi_0 - \phi_\text{biofilm} - \phi_\text{calcite}
$$

* 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
:::
::::::

# Equations

## Balance Equations

:::incremental

* Mass balance equation of components
$$
\Sigma_\alpha \frac{\partial}{\partial t}(\phi \rho_\alpha x^\kappa_\alpha S_\alpha)+ \nabla \cdot ( \rho_\alpha x^\kappa_\alpha \mathbf{v}_\alpha )- \nabla \cdot ( \rho_\alpha \mathbf{D}^\kappa_{\alpha;\text{pm}} \nabla x^\kappa_\alpha )= q^\kappa
$$

* Mass balance for the immobile components / solid phases:
$$
\frac{\partial}{\partial t}(\rho_\varphi \phi_\varphi) = q^\varphi
$$

:::

## Overall procedure of implementing chemical reactions in DuMu$^\mathsf{x}$:

1. Chemical equation    calculate equilibrium/kinetic reaction rate e.g. $r_\text{urea}$
2. Reaction rate        set component source/sink term e.g. $q^\varphi$ depending on and chemical reaction

## Sources & Sinks:

**For this exercise:**

* Neglecting microbial growth and decay, attachment and detachment as we assume a fixed biofilm for simplicity!
* We also assume that the rate of precipitation is equal to the rate of ureolysis, saving the work of detailed geochemistry calculations for the sake of both simplicity and faster run times.

## Sources & Sinks:

$$
\begin{aligned}
&\text{Urea:}        &   q^{\text{urea}} &= -r_\text{urea}     \\
&\text{Calcium:}     &   q^{\mathrm{Ca}^2+} &= -r_\text{precip}   \\
&\text{Total carbon:}&   q^{\mathrm{C}_\text{tot}^+} &= r_\text{urea} - r_\text{precip} \\
&\text{Calcite:}     &   q^{\mathrm{C}} &= r_\text{precip}
\end{aligned}
$$

## Sources & Sinks:

<!-- use creates negative spacing from \! to align equations -->

$$
\begin{aligned}
\qquad\qquad & \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\!
\text{Precipitation rate:} \\
r_\text{precip} &= f\; \left( A_\text{interface}, \Omega = \frac{\left[\mathrm{Ca}^{2+}\right]\left[\mathrm{CO_3}^{2-}\right]}{K_\text{sp}}, T \right)
\\
\qquad\qquad & \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\!
\text{For this exercise:} \\
r_\text{precip} &= r_\text{urea}
\\
\qquad\qquad & \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\! \!\!\!\!\!\!
\text{Ureolysis rate:} \\
r_\text{urea} &= k_\mathrm{urease}^\mathrm{m} k_{\mathrm{urease}, \text{biofilm}} \left(\rho_\text{biofilm} \phi_\text{biofilm}\right) \frac{m_\text{urea}}{K_\text{urea} + m_\text{urea}}
\end{aligned}
$$

## Supplementary Equation:

* Updating porosity and permeability

$$
\begin{aligned}
\phi &= \phi_0 - \Sigma_\varphi \phi_\varphi
\\
K &= K_0 \left(\frac{1-\phi_0}{1-\phi}\right)^2 \left(\frac{\phi}{\phi_0}\right)^3
\\
\text{or}&
\\
K &= K_0 \left( \frac{\phi}{\phi_0} \right)^\eta
\end{aligned}
$$

## Specific Implementations

* Update porosity in dumux/material/fluidmatrixinteractions/porosityprecipitation.hh

```cpp
…
auto priVars = evalSolution(element, element.geometry(),
                            elemSol, scv.center());
Scalar sumPrecipitates = 0.0;

for (int solidPhaseIdx = 0; solidPhaseIdx < numSolidPhases; ++solidPhaseIdx)
    sumPrecipitates += priVars[numComp + solidPhaseIdx];

using std::max;
return max(minPoro, refPoro - sumPrecipitates);
…
```

## Specific Implementations

* Update permeability in /material/fluidmatrixinteractions/permeabilitykozenycarman.hh
```cpp
template<class Scalar>
PermeabilityType evaluatePermeability(PermeabilityType refPerm,
                                      Scalar refPoro,
                                      Scalar poro) const
{
	using std::pow;
	auto factor = pow((1.0 - refPoro)/(1.0 - poro), 2)
        * pow(poro/refPoro, 3);
	refPerm *= factor;
	return refPerm;
}
```

# Biomineralization exercise

## Exercise

:::::: {.columns}
::: {.column width=50%}
* 2 aquifers with sealing aquitard
  * Upper aquifer: "drinking water"
  * Lower aquifer: "$\mathrm{CO_2}$ storage"
* Problem:
  * Leakage pathway
  * Stored $\mathrm{CO_2}$ would migrate to drinking water aquifer!
* Biomineralization to "seal" the leakage
:::
::: {.column width=50%}
<img src="img/biomin_exercise-setup.png"/>
:::
::::::

## Exercise tasks

<div style="text-align: left; margin-left: 200px">
<small>
1. Get familiar with the code  
2. Implement the simplified chemical reactions  
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 $\mathrm{CO_2}$
</small>
</div>

## Exercise

First step: Go to the
[Biomineralization Exercise](https://git.iws.uni-stuttgart.de/dumux-repositories/dumux-course/tree/master/exercises/exercise-biomineralization)
and check out the README