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?

Credit: James Connolly, Montana State University.

Results of ICP

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

:::::: {.columns} ::: {.column width=40%} ::: ::: {.column width=60%} ::: :::::: Credit: Johannes Hommel, University of Stuttgart

Why investigate ICP?

::::: {.columns} ::: {.column width=50%} Mineralized sand, photo by Johannes Hommel, University of Stuttgart. ::: ::: {.column width=50%} Porosity-Permeability changes observed by Felix Weinhardt, University of Stuttgart. ::: ::::::

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%} (modified after Ebigbo et al., WRR 2012) ::: ::: {.column width=55%}

• Two-phase transport ::: ::::::

Model concept: Relevant processes

:::::: {.columns} ::: {.column width=45%} (modified after Ebigbo et al., WRR 2012) ::: ::: {.column width=55%}

• Two-phase, multi-component transport ::: ::::::

Model concept: Relevant processes

:::::: {.columns} ::: {.column width=45%} (modified after Ebigbo et al., WRR 2012) ::: ::: {.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%} (modified after Ebigbo et al., WRR 2012) ::: ::: {.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:

\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

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• 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%} (modified after Ebigbo et al., WRR 2012) ::: ::: {.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:

\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

…
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

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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• 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%}

::: ::::::

Exercise tasks

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 $\mathrm{CO_2}$

Exercise

First step: Go to the Biomineralization Exercise and check out the README