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>

## What is ICP?

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<video controls>
<source src="img/biomin_MIP-timelapse.ogg"/>
</video>
</br>
<small>Credit: James Connolly, Montana State University.</small>
:::
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* Biofilm (green) is alive
* Flow is induced, biofilm moves
* Reactions occur
  * Calcite precipitates (white)
  * Biofilm dies slowly
:::
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## Results of ICP

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

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<img src="img/biomin_introColumns.png"/>
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<img src="img/biomin_introSegmented.png"/>
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<small>Credit: Johannes Hommel, University of Stuttgart</small>

## Why investigate ICP?

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<img src="img/biomin_introCementedSand.jpg"/>
<small>Mineralized sand, photo by Johannes Hommel, University of Stuttgart.</small>
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<img src="img/biomin_introPoroPermFelix.png"/>
<small>Porosity-Permeability changes observed by Felix Weinhardt, University of Stuttgart.</small>
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## 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

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<img src="img/biomin_model-2p.png"/>
<small>(modified after Ebigbo et al., WRR 2012)</small>
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- Two-phase transport
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## Model concept: Relevant processes

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<img src="img/biomin_model-2pnc.png"/>
<small>(modified after Ebigbo et al., WRR 2012)</small>
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- Two-phase, multi-component transport
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## Model concept: Relevant processes

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

- Biomass (S. pasteurii)
  - growth / decay
  - attachment / detachment
:::
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## Model concept: Relevant processes

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

## MICP: Main reactions

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

Different reactions in detail:

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

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

## Model concept: Relevant processes

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<img src="img/biomin_model-precipitation.png"/>
<small>(modified after Ebigbo et al., WRR 2012)</small>
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- Precipitation and dissolution of calcite

$$
\mathrm{
\underset{\text{calcium}}{Ca^{2+}} + \underset{\text{carbonate}}{CO_3^{2-}}
\leftrightarrow \underset{\text{calcite}}{CaCO_3 \downarrow}
}
$$
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## Model concept: Relevant processes

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<img src="img/biomin_model-clogging.png"/>
<small>(modified after Ebigbo et al., WRR 2012)</small>
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* 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
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# 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 -->

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

## Supplementary Equation:

* Updating porosity and permeability

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

## Specific Implementations

* Implement reactive sources and sinks, calling a seperate chemistry file

```cpp
#include "chemistry/simplebiominreactions.hh" // chemical reactions
…
using Chemistry = typename Dumux::SimpleBiominReactions<NumEqVector,
                                                        VolumeVariables>;
…
```

## Specific Implementations

* Implement reactive sources and sinks, calling a seperate chemistry file

```cpp
NumEqVector source(const Element& element,
                   const FVElementGeometry& fvGeometry,
                   const ElementVolumeVariables& elemVolVars,
                   const SubControlVolume& scv) const
{
	NumEqVector source(0.0);
	Chemistry chemistry;

	const auto& volVars = elemVolVars[scv];
	chemistry.reactionSource(source, volVars);

	return source;
}
```

## 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 (unsigned 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

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Academic problem setup

* 2 aquifers with sealing aquitard
  * Upper aquifer: "drinking water"
  * Lower aquifer: "$CO_2$ storage"
* Problem:
  * Leakage pathway
  * Stored $CO_2$ would migrate to drinking water aquifer!
* Biomineralization injection could "seal" the leakage pathway
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<img src="img/biomin_exercise-setup.png"/>
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## Exercise tasks

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 $CO_2$

## Exercise

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