# Field-scale optimization of injection strategies for leakage mitigation
using microbially induced calcite precipitation

Svenn Tveit1 and  David Landa-Marbán1

###### Abstract

An optimization procedure for sealing leakage paths in field-scale application
of microbially induced calcite precipitation (MICP) is develop and applied to
CO2 storage. The procedure utilizes a recently developed field-scale MICP
mathematical model implemented in the industry-standard simulator Open Porous
Media (OPM) Flow. The optimization problem is solved using the ensemble-based
optimization (EnOpt) algorithm where the objective function is defined such
that maximizing calcite precipitation is done in the shortest possible MICP
operational time. An injection strategy is developed to efficiently produce
calcite in and around the leakage paths, and to avoid clogging unwanted areas
of the reservoir, e.g., the near-well area. The injection strategy consists of
combined injection of growth and cementation solutions in separate well
segments to initiate the MICP process after establishing a biofilm in the
leakage paths with an initial injection phase. The optimization procedure is
applied to three synthetic CO2 leakage scenarios. The numerical results show
that the leakage paths in all scenarios are effectively sealed while keeping
the total MICP operational time low.

1 NORCE Norwegian Research Centre AS, Nygårdsgaten 112, 5008 Bergen, Norway.
Corresponding author: Svenn Tveit (E-mail address: svtv@norceresearch.no).

#### Keywords

Carbon capture and storage $\cdot$ Ensemble-based optimization $\cdot$ Leakage
mitigation $\cdot$ Microbially induced calcite precipitation $\cdot$ Open
porous media initiative

##  1 Introduction

To reduce the emission of anthropomorphic CO2 to the atmosphere, carbon
capture and storage (CCS) in large, geological formations has been identified
as a key remediation strategy (Haszeldine et al., 2018). The advantage of
geological CCS is the huge volumetric storage potential in various saline
aquifers or depleted oil and gas reservoirs. However, the effective storage
potential of any geological formation depends, among other factors, on how
much one can inject before reaching hazardous pressure build-ups. If
supposedly closed leakage paths in the caprock, like faults, fractures, or
abandoned wells, are exposed to pressures beyond their critical threshold,
sequestrated CO2 might leak out of the storage site. Therefore, many in-depth
case studies have been conducted on potential storage sites to ensure secure
CO2 sequestration, e.g., Elenius et al. (2018), Mulrooney et al. (2020), and
Hodneland et al. (2019). Even so, in the unlikely event that CO2 leakage paths
may develop during injection, the consequences can be severe, both in terms of
impact on the nearby environment and on the public acceptance of CCS. Thus, it
is important to develop efficient and reliable leakage sealing technologies.

A promising leakage sealing technology that has gained much attention in
recent years is microbially induced calcite precipitation (MICP). The core
idea of MICP is to use microbes to catalyse the chemical production of calcium
carbonate – calcite – from urea and calcium to seal a leakage path (Phillips
et al., 2013a). Calcite is a low-permeable mineral that act as a sealing agent
by reducing pore space, and as such, reduce the permeability of the leakage
path. The technology has been proven effective both in column (Cunningham et
al., 2011) and core-scale studies (Phillips et al., 2013b) for application in
CO2 storage. Figure 1 shows a schematic representation of the MICP technology
applied to CO2 storage. It has also been used in several other applications,
such as enhanced oil recovery (Wu et al., 2017), strengthening of liquefiable
soil (Burbank et al., 2011), and concrete improvement (De Muynck et al.,
2010). Recently, Landa-Marbán et al. (2021a, b) developed a numerical model to
simulate the complex processes involved in field-scale application of MICP,
based on work done in Ebigbo et al. (2012), Hommel et al. (2015), and
Cunningham et al. (2019). The model approximates the involved chemical and
physical processes to capture the necessary field-scale behavior of MICP,
enabling much shorter simulation times than previously developed pore- and
core-scale models, e.g., Minto et al. (2019) and Nassar et al. (2018).

Figure 1: Visual representation of field-scale MICP application for CO2
leakage remediation

Sealing leakage paths with MICP generally involves injecting several
components into the reservoir: solution of pre-cultivated microbes; growth
solution to establish biofilm (i.e., community of microbes) at leakage
location; and cementation solution to initiate the MICP process in the
biofilm. Several numerical studies have been conducted to develop injection
strategies for MICP, e.g., Ebigbo et al. (2012). The results from these
studies have been applied in experiments from laboratory- to field-scale.
However, most of the strategies focus on uniform sealing from the injection
point, without completely plugging the inlet, e.g. for application of well
fracture sealing (Phillips et al., 2016). In CO2 storage applications, leakage
paths could develop tens-of-meters, or more, away from the injection well.
Initiating the MICP process at the leakage location, without negatively
impacting the rest of the storage site, is a challenging optimization task.
Adding to the challenge is sealing the leakage paths in the least amount of
time, to avoid a long shutdown of CO2 injection operation.

To perform optimization, standard gradient-based algorithms are often used.
Even though many of the algorithms have good convergence properties, they need
derivative calculations that are either computationally costly for field-scale
models, or require access to simulation code for adjoint calculations. To
address these issues, optimization methods with different stochastic
approximations of the gradient have been developed. The advantages with most
stochastic optimization methods are easy parallelization; they require only
input-output interactions with a simulator; and multiple types of uncertainty,
e.g., geological uncertainty, can be include in a straightforward manner. As a
result, significant development have been made over the last decades,
especially for the petroleum industry. In this paper, we apply the ensemble-
based optimization (EnOpt) algorithm (Chen et al., 2009) due to its simple
implementation and it has been shown to perform well in benchmark studies,
see, e.g., Chang et al. (2020).

Few optimization studies involving MICP have been conducted in the literature.
In Tveit et al. (2020) the authors investigated a risk-aware optimization
workflow, where uncertainties on empirically determined model parameters could
be included. In this paper, we focus on developing an optimization procedure
for MICP with an injection strategy to avoid sealing unnecessary parts of the
storage site. We apply the optimization procedure on synthetic test cases to
gain insight on optimal injection strategies for field-scale applications of
MICP.

The paper is organized as follows: the different parts of the methodology is
described in Section 2. This includes the mathematical model for MICP and its
implementation in Sections 2.1 and 2.2, respectively. Section 2.3 describes
the injection strategy, while a description of the different parts of the
optimization method is given in Section 2.4. The setup of the synthetic
leakage scenarios for the numerical studies and subsequent results, together
with a short discussion are given in Section 3. Lastly, we end the paper with
some concluding remarks in Section 5.

##  2 Methodology

We consider the optimization procedure for sealing leakage paths with the MICP
technology. In the following sections, we describe the involved parts of the
optimization procedure, that is, the field-scale simulation model for MICP
processes, the injection strategy of the involved components, and the
optimization method.

###  2.1 Mathematical model

The conceptual model for field-scale application of the MICP processes is
detailed in Landa-Marbán et al. (2021a) and can be summed up as follows:
microbial solution is injected into the reservoir to attach microbes to the
rock at the leakage location; growth solution is injected to cultivate biofilm
formation; and cementation solution is injected to initiate the calcite
precipitation from the biofilm. The mathematical model only considers the
rate-limiting components of the three solutions, which are suspended microbes,
oxygen, and urea, respectively.

In this section, we give a brief overview of the mathematical model, and refer
to Landa-Marbán et al. (2021a) for a complete description. The model is based
on a preliminary study in Tveit et al. (2018), together with previous MICP
models in Ebigbo et al. (2012), Hommel et al. (2015), and Cunningham et al.
(2019). Important additions to the model in Tveit et al. (2018) were
dispersion, diffusion, and detachment by shear force. However, we recently
observe from simulation studies that dispersive effects are more relevant for
core-scale simulations, and less relevant at the field scale. Thus, we have
removed the dispersion and diffusion terms in the mathematical model in this
paper. Table 1 shows all equations in the MICP mathematical model.

Table 1: An overview of the equations in the MICP mathematical model  Name |  Equation |
---|---|---
Mass balance of water |  $\partial_{t}{\color[rgb]{0,0,0}\phi}+\nabla\cdot{\color[rgb]{0,0,0}\boldsymbol{u}_{w}}=\underbrace{q_{w}}_{\text{Source term}},$ ${\color[rgb]{0,0,0}\boldsymbol{u}_{w}}=\underbrace{-\frac{{\color[rgb]{0,0,0}\mathbb{K}}}{\mu_{w}}\left(\nabla{\color[rgb]{0,0,0}p_{w}}-\rho_{w}\boldsymbol{g}\right)}_{\text{Darcy's law}}$ |  (1)
Mass balance of suspended microbes |  $\partial_{t}({\color[rgb]{0,0,0}c_{m}}{\color[rgb]{0,0,0}\phi})+\nabla\cdot({\color[rgb]{0,0,0}c_{m}}{\color[rgb]{0,0,0}\boldsymbol{u}_{w}})=\underbrace{{\color[rgb]{0,0,0}c_{m}}q_{w}}_{\text{Source term}}+\underbrace{{\color[rgb]{0,0,0}c_{m}}{\color[rgb]{0,0,0}\phi}Y\mu\frac{{\color[rgb]{0,0,0}c_{o}}}{k_{o}+{\color[rgb]{0,0,0}c_{o}}}}_{\text{Growth}}-\underbrace{{\color[rgb]{0,0,0}c_{m}}{\color[rgb]{0,0,0}\phi}k_{a}}_{\text{Attachment}}+\underbrace{\rho_{b}{\color[rgb]{0,0,0}\phi_{b}}k_{str}({\color[rgb]{0,0,0}\phi}\|\nabla{\color[rgb]{0,0,0}p_{w}}-\rho_{w}\boldsymbol{g}\|)^{0.58}}_{\text{Detachment}}-\underbrace{{\color[rgb]{0,0,0}c_{m}}{\color[rgb]{0,0,0}\phi}k_{d}}_{\text{Death}}$ |  (2)
Mass balance of oxygen |  $\partial_{t}({\color[rgb]{0,0,0}c_{o}}{\color[rgb]{0,0,0}\phi})+\nabla\cdot({\color[rgb]{0,0,0}c_{o}}{\color[rgb]{0,0,0}\boldsymbol{u}_{w}})=\underbrace{{\color[rgb]{0,0,0}c_{o}}q_{w}}_{\text{Source term}}-\underbrace{(\rho_{b}{\color[rgb]{0,0,0}\phi_{b}}+{\color[rgb]{0,0,0}c_{m}}{\color[rgb]{0,0,0}\phi})F\mu\frac{{\color[rgb]{0,0,0}c_{o}}}{k_{o}+{\color[rgb]{0,0,0}c_{o}}}}_{\text{Consumption}}$ |  (3)
Mass balance of urea |  $\partial_{t}({\color[rgb]{0,0,0}c_{u}}{\color[rgb]{0,0,0}\phi})+\nabla\cdot({\color[rgb]{0,0,0}c_{u}}{\color[rgb]{0,0,0}\boldsymbol{u}_{w}})=\underbrace{{\color[rgb]{0,0,0}c_{u}}q_{w}}_{\text{Source term}}\underbrace{-\rho_{b}{\color[rgb]{0,0,0}\phi_{b}}\mu_{u}\frac{{\color[rgb]{0,0,0}c_{u}}}{k_{u}+{\color[rgb]{0,0,0}c_{u}}}}_{\text{Urea conversion}}$ |  (4)
Mass balance of biofilm |  $\partial_{t}(\rho_{b}{\color[rgb]{0,0,0}\phi_{b}})=\underbrace{\rho_{b}{\color[rgb]{0,0,0}\phi_{b}}Y\mu\frac{{\color[rgb]{0,0,0}c_{o}}}{k_{o}+{\color[rgb]{0,0,0}c_{o}}}}_{\text{Growth}}+\underbrace{{\color[rgb]{0,0,0}c_{m}}{\color[rgb]{0,0,0}\phi}k_{a}}_{\text{Attachment}}-\underbrace{\rho_{b}{\color[rgb]{0,0,0}\phi_{b}}k_{str}({\color[rgb]{0,0,0}\phi}\|\nabla{\color[rgb]{0,0,0}p_{w}}-\rho_{w}\boldsymbol{g}\|)^{0.58}}_{\text{Detachment}}-\underbrace{\rho_{b}{\color[rgb]{0,0,0}\phi_{b}}\left[k_{d}+\frac{{\color[rgb]{0,0,0}R_{c}}}{\rho_{c}(\phi_{0}-{\color[rgb]{0,0,0}\phi_{c}})}\right]}_{\text{Death}}$ |  (5)
Mass balance of calcite |  $\partial_{t}(\rho_{c}{\color[rgb]{0,0,0}\phi_{c}})=\underbrace{\rho_{b}{\color[rgb]{0,0,0}\phi_{b}}Y_{uc}\mu_{u}\frac{{\color[rgb]{0,0,0}c_{u}}}{k_{u}+{\color[rgb]{0,0,0}c_{u}}}}_{\text{Produced calcite}}$ |  (6)
Porosity changes |  ${\color[rgb]{0,0,0}\phi}=\underbrace{\phi_{0}-{\color[rgb]{0,0,0}\phi_{b}}-{\color[rgb]{0,0,0}\phi_{c}}}_{\text{Porosity reduction}}$ |  (7)
Permeability changes |  ${\color[rgb]{0,0,0}\mathbb{K}}=\underbrace{\begin{cases}\left[\mathbb{K}_{0}\bigg{(}\frac{{\color[rgb]{0,0,0}\phi}-\phi_{crit}}{\phi_{0}-\phi_{crit}}\bigg{)}^{\eta}+K_{\text{min}}\right]\frac{\mathbb{K}_{0}}{\mathbb{K}_{0}+K_{\text{min}}},&\phi_{\text{crit}}<{\color[rgb]{0,0,0}\phi}\\\ K_{\text{min}},&{\color[rgb]{0,0,0}\phi}\leq\phi_{\text{crit}}\end{cases}}_{\text{Porosity-permeability relationship}}$ |  (8)

In the mass balance equation of water, (1), the variables are the rock
porosity $\phi$, the discharge per unit area $\boldsymbol{u}_{w}$, and the
source term $q_{w}$. $\boldsymbol{u}_{w}$ is described by the Darcy’s law
where $\mathbb{K}$ is the rock permeability, $\mu_{w}$ the water viscosity,
$p_{w}$ the water pressure, $\rho_{w}$ the water density, and $\boldsymbol{g}$
the gravity vector. Furthermore, (2)–(6) are mass balance equations for the
suspended microbes (m), oxygen (o), urea (u), biofilm (f), and calcite (c),
respectively. The notation for mass concentrations is $c_{\xi}$
($\xi\in\\{m,o,u\\}$) and for volume fractions $\phi_{\chi}$
($\chi\in\\{b,c\\}$). The right-hand side of these equations represent
different phenomena during the MICP process, which are given under each term
in Table 1. Here, $Y$ is the growth yield coefficient, $\mu$ the maximum
specific growth rate, $k_{o}$ the half-velocity coefficient of oxygen, $k_{d}$
the microbial death coefficient, $k_{a}$ the microbial attachment coefficient,
$k_{str}$ the detachment rate, $F$ the mass ratio of oxygen consumed to
substrate used for growth, $\mu_{u}$ the maximum rate of urea utilization,
$k_{u}$ the half-velocity coefficient for urea, and $Y_{uc}$ the yield
coefficient for the produced calcite over the urea utilization. Lastly, (7)
represents the porosity change due to increasing biofilm and calcite volume
fractions, and (8) models the change in permeability due to changes in
porosity, where $\phi_{crit}$ is the critical porosity when the permeability
becomes a minimum value $K_{min}$. In Table 2 values for all model parameters
are given. Full references to the sources for these values can be found in
Landa-Marbán et al. (2021a).

Table 2: Model parameters for the mathematical model in Table 1 Parameter |  Sym. |  Value |  Unit
---|---|---|---
Density (biofilm) |  $\rho_{b}$ |  $35$ |  $\nicefrac{{\text{kg}}}{{\text{m}^{3}}}$
Density (calcite) |  $\rho_{c}$ |  $2710$ |  $\nicefrac{{\text{kg}}}{{\text{m}^{3}}}$
Density (water) |  $\rho_{w}$ |  $1045$ |  $\nicefrac{{\text{kg}}}{{\text{m}^{3}}}$
Detachment rate |  $k_{str}$ |  2.6$\times 10^{-10}$ |  $\nicefrac{{\text{m}}}{{\text{Pa s}}}$
Critical porosity |  $\phi_{\text{crit}}$ |  0.1 |  [-]
Fitting factor |  $\eta$ |  3 |  [-]
Half-velocity coefficient (oxygen) |  $k_{o}$ |  $2\times 10^{-5}$ |  $\nicefrac{{\text{kg}}}{{\text{m}^{3}}}$
Half-velocity coefficient (urea) |  $k_{u}$ |  $21.3$ |  $\nicefrac{{\text{kg}}}{{\text{m}^{3}}}$
Maximum specific growth rate |  $\mu$ |  $4.17\times 10^{-5}$ |  $\nicefrac{{1}}{{\text{s}}}$
Maximum rate of urea utilization |  $\mu_{u}$ |  $1.61\times 10^{-2}$ |  $\nicefrac{{1}}{{\text{s}}}$
Microbial attachment rate |  $k_{a}$ |  $8.51\times 10^{-7}$ |  $\nicefrac{{1}}{{\text{s}}}$
Microbial death rate |  $k_{d}$ |  $3.18\times 10^{-7}$ |  $\nicefrac{{1}}{{\text{s}}}$
Minimum permeability |  $K_{\text{min}}$ |  $10^{-20}$ |  $\text{m}^{2}$
Oxygen consumption factor |  $F$ |  $0.5$ |  [-]
Water viscosity |  $\mu_{w}$ |  $2.54\times 10^{-3}$ |  Pa s
Yield coefficient (growth) |  $Y$ |  $0.5$ |  [-]
Yield coefficient (calcite/urea) |  $Y_{uc}$ |  $1.67$ |  [-]

###  2.2 Implementation

The mathematical model in Section 2.1 was implemented in the industry-
standard, open-source simulator Open Porous Media (OPM) Flow. The simulator is
part of the OPM initiative, that facilitates field-scale simulations of
different subsurface applications such as hydrocarbon recovery and geological
CO2 storage. It is a fully-implicit, finite difference simulator on corner-
point grids, with advanced well models. We refer to Rasmussen et al. (2019)
for a description of OPM Flow along with some of the implemented models, such
as the black-oil and well models.

The mathematical model for MICP described in Section 2.1 was made available in
the 2021.10 release of OPM Flow. The model in Landa-Marbán et al. (2021a) was
made available in the 2021b release of the Matlab Reservoir Simulation Toolbox
(MRST). In Landa-Marbán et al. (2021b) we show comparison between simulations
in MRST and OPM for a simple 1D horizontal system, resulting in a good
agreement between numerical results from both simulators. A description of the
OPM Flow keywords for the MICP model can be found in the OPM Flow manual
(Baxendale et al., 2021). Lastly, we have developed a Python package, py-micp,
to integrate different open-source code to perform studies of MICP treatment
and CO2 assessment. Figure 2 shows an example of the workflow on a case study.

Figure 2: Workflow example of py-micp

###  2.3 Injection strategy

In Landa-Marbán et al. (2021a), the authors developed a field-scale injection
strategy of the microbial, growth, and cementation solutions, based on earlier
experimental and numerical studies. The strategy developed there involved
injecting each solution separately, with follow-up injection of water and no-
flow periods. The numerical studies performed in Landa-Marbán et al. (2021a),
showed that even leakage paths considerably far from the injection well were
completely sealed with the proposed strategy. Furthermore, it was shown that
injecting solutions only in the top segment of the injection well, and water
in the rest of the segments, was beneficial to avoid calcite precipitation
along the whole vertical direction of the reservoir.

In this paper, we further develop the injection strategy introduced in Landa-
Marbán et al. (2021a). Let a phase be defined as an injection of one or more
solutions with subsequent water displacement and no-flow periods. Furthermore,
let $t_{1}^{p}$ be the end time of solution injection, $t_{2}^{p}$ be the end
time of water displacement, and $t_{3}^{p}$ be the end time of the no-flow
period for phase $p$. Lastly, we assume that the solutions can be injected in
different segments of the injection well, and that the injection rate is
constant.

The injection strategy proposed in this paper is then defined in Figure 3.
From the figure we see that phases $I$–$III$ involve separate injection of
microbial, growth, and cementation solutions, respectively. This follows the
strategy in Landa-Marbán et al. (2021a), with the goal of developing biofilm
and subsequently produce calcite at the leakage paths. In phases $IV$–$N_{p}$
we inject growth and cementation solutions at the same time, to utilize the
remaining microbes and biofilm at the leakage paths to produce calcite.
Moreover, the growth solution is injected in a lower segment of the injection
well than the microbial and cementation solutions. Since the permeability of
the leakage paths are higher than the rest of the reservoir, the growth
solution will be transported to the leakage location under correct flow
conditions. As a result, we limit the MICP processes to only occur in and
around the leakage paths, instead of potentially initiating the process in
unwanted areas, if all solutions are injected in the same well segment.

Figure 3: An illustration of the injection strategy where ms, gs, cs, and w
refer to injection of microbial, growth, and cementation solutions, and only
water, respectively

###  2.4 Optimization method

Let $J(\boldsymbol{x})$ denote the scalar objective function, with
$\boldsymbol{x}\in\mathbb{R}^{N_{x}}$ being the control variables. The
optimization problem is then to maximize $J(\boldsymbol{x})$ with respect to
$\boldsymbol{x}$. A widely-used method for optimization, with guarantied local
convergence, is the pre-conditioned steepest ascent method,

$\boldsymbol{x}_{k+1}=\boldsymbol{x}_{k}+\alpha_{k}\boldsymbol{\Sigma}\boldsymbol{g}_{k},$
(9)

where $k$ is the iteration number, $\alpha_{k}$ is the step size,
$\boldsymbol{\Sigma}$ is a symmetric, positive definite matrix, and
$\boldsymbol{g}_{k}$ is the gradient of $J(\boldsymbol{x}_{k})$. To solve (9),
EnOpt (Chen et al., 2009) approximates $\boldsymbol{\Sigma}\boldsymbol{g}_{k}$
by the sample cross-covariance matrix. In this paper, we use the modified
version from Do and Reynolds (2013) given by

$\boldsymbol{\Sigma}\boldsymbol{g}_{k}\approx\boldsymbol{C}_{x,J}=\frac{1}{N_{e}}\sum_{j=1}^{N_{e}}\left(\boldsymbol{x}_{k}^{j}-\boldsymbol{x}_{k}\right)\left(J(\boldsymbol{x}_{k}^{j})-J(\boldsymbol{x}_{k})\right).$
(10)

Here, $\\{\boldsymbol{x}_{k}^{j}\\}_{j=1}^{N_{e}}$ is sampled from a
multivariate normal distribution with mean $\boldsymbol{x}_{k}$ and covariance
matrix $\boldsymbol{\Sigma}$. To ensure that the approximation in (10) is
reasonable, sufficiently large $N_{e}$ and small sample perturbations when
generating the $\boldsymbol{x}_{k}^{j}$’s are required (Do and Reynolds,
2013). In practice, $N_{e}$ is chosen based on the available computational
budget, since a minimum of $N_{e}+1$ evaluations of $J(\boldsymbol{x})$ is
needed per update in (9) with (10). For a thorough theoretical discussion on
EnOpt see Stordal et al. (2016).

To enforce upper and lower bounds on the control variables, we use the log-
transform defined in Do and Reynolds (2013),

$x_{i}=\log\left(\frac{\hat{x}_{i}-\hat{x}_{i}^{low}}{\hat{x}_{i}^{up}-\hat{x}_{i}}\right),$
(11)

where $\hat{x}_{i}$ denotes the constrained control variable, and
$\hat{x}_{i}^{up}$ and $\hat{x}_{i}^{low}$ are the upper and lower bounds,
respectively. Hence, the optimization using $\boldsymbol{x}$ will be
unconstrained since $x_{i}\rightarrow-\infty$ when
$\hat{x}_{i}\rightarrow\hat{x}_{i}^{low}$, and $x_{i}\rightarrow\infty$ when
$\hat{x}_{i}\rightarrow\hat{x}_{i}^{up}$. To invert $\boldsymbol{x}$ to
$\hat{\boldsymbol{x}}$, e.g., in evaluations of $J(\boldsymbol{x})$, we can
use the inverse formula to (11),

$\hat{x}_{i}=\frac{\exp(x_{i})\hat{x}_{i}^{up}+\hat{x}_{i}^{low}}{1+\exp(x_{i})}.$
(12)

A widely-used application of optimization on field scale is production from
petroleum reservoirs. Here, the objective function $J(\boldsymbol{x})$ is
typically defined through an economic model given by the net present value
(NPV). This model has been expanded with additional terms depending on new
applications, e.g., with costs and revenues related to CO2 sequestration (Chen
and Voskov, 2020). However, to optimize the sealing of leakage paths with
MICP, a pure economic model may not be beneficial, since such operations are
merely cost. We suggest instead combining a physical property model for
calcite precipitation with a penalty term on the total number of MICP
operational days in the definition of $J(\boldsymbol{x})$. Hence, the main aim
of the optimization is to maximize the sealing of leakage paths, but at the
same time minimize the injection and no-flow periods. With the penalty term we
add an economic aspect to our optimization, since reducing operational days of
MICP remediation will reduce the cost.

To this end, let $\hat{\boldsymbol{x}}$ be the injection and no-flow periods
defined in Section 2.3. For ease of reading, we divide $\hat{\boldsymbol{x}}$
in three parts
$\hat{\boldsymbol{x}}=[\hat{\boldsymbol{x}}^{sol},\hat{\boldsymbol{x}}^{wat},\hat{\boldsymbol{x}}^{nof}]$
and define each one as follows: $\hat{\boldsymbol{x}}^{sol}=[\Delta
t^{I}_{1,3},\Delta t^{II}_{1,3},\ldots,\Delta t^{N_{p}}_{1,3}]$ where $\Delta
t^{p}_{1,3}=t^{p}_{1}-t^{p-I}_{3}$ with $t^{0}_{3}=0$;
$\hat{\boldsymbol{x}}^{wat}=[\Delta t_{2,1}^{I},\Delta
t_{2,1}^{II},\ldots,\Delta t_{2,1}^{N_{p}}]$ where $\Delta
t_{2,1}^{p}=t_{2}^{p}-t_{1}^{p}$; and $\hat{\boldsymbol{x}}^{nof}=[\Delta
t_{3,2}^{I},\Delta t_{3,2}^{II},\ldots,\Delta t_{3,2}^{N_{p}}],$ where $\Delta
t_{3,2}^{p}=t_{3}^{p}-t_{2}^{p}$. In short, $\hat{\boldsymbol{x}}^{sol}$ are
the injection periods for the microbial, growth, and cementation solutions;
$\hat{\boldsymbol{x}}^{wat}$ are the water displacement periods; and
$\hat{\boldsymbol{x}}^{nof}$ are the no-flow periods. Furthermore, let
$\boldsymbol{\phi}_{c}^{leak}(t_{3}^{p})=[\phi_{c,1}^{leak}(t_{3}^{p}),\phi_{c,2}^{leak}(t_{3}^{p}),\ldots,\phi_{c,N_{l}}^{leak}(t_{3}^{p})]$
where $\phi_{c,i}^{leak}(t_{3}^{p})$ is the calcite volume fraction in cell
$i$ inside the leakage path at time $t_{3}^{p}$. The objective function is
then defined as

$J(\boldsymbol{x})=\sum_{p}\|\boldsymbol{\phi}_{c}^{leak}(t_{3}^{p})\|_{\infty}-\gamma\|\hat{\boldsymbol{x}}\|_{1},$
(13)

where $\gamma$ is a constant weighting parameter. Hence, the first term in
(13) is the sum of the maximum calcite volume fraction in the leakage paths at
the end of a phase (indicated by $t_{3}^{p}$). Note that from (8) the maximum
value for $\phi_{c,i}^{leak}(t_{3}^{p})$ is $\phi_{0}-\phi_{crit}$.

##  3 Numerical experiments

In this section, we apply the optimization procedure to seal leakage paths in
three different scenarios. A synthetic reservoir was set up similar to the 3D
system studied in Landa-Marbán et al. (2021a), which was based on the CO2
benchmark study in Ebigbo et al. (2007) and Class et al. (2009). The reservoir
consisted of two aquifers separated by a caprock, where we in this study had
three scenarios with different leakage paths through the caprock. To reduce
the computational time, we only considered a quarter of the full system; a
common approximation in the literature, see, e.g., Zhang and Agarwal (2012).
Examples considering the full system can be found in py-micp, and they confirm
that the quarter-system approximation is valid in the three leakage scenarios.
Thus, the computational domain was a 37$\times$37$\times$30 grid, with
equidistant 1m$\times$1m$\times$1m cells close to the injection wells and
exponentially increasing cell sizes in $x$\- and $y$-directions towards the
boundaries. Note that the caprock is modeled with non-active cells, thus the
computational domain consisted of only 13 710 active cells. The injection well
is located in the lower left corner of the domain, which had a constant rate
of $2\cdot 10^{-2}$ m/s during the injection periods, and a production well
was placed in the upper, right corner to simulate open boundaries in OPM Flow,
since it has fixed no-flow boundary conditions. A summary of the reservoir and
fluid properties is given in Table 3, and an illustration of the full system
is shown in Figure 4, where the upper right quarter was used in the scenarios.
Note that we assume constant permeability in each cell, thus the tensor
$\mathbb{K}_{0}$ in (8) reduces to a scalar $K_{0}$.

Table 3: Reservoir and fluid properties  Parameter |  Sym. |  Value |  Unit
---|---|---|---
Height domain |  $H$ |  $30$ |  m
Height aquifers |  $h_{l}/h_{t}$ |  $5$ |  m
Width/length |  $W/L$ |  $100$ |  m
Initial permeability aquifer |  $K_{0}^{A}$ |  $1\cdot 10^{-14}$ |  $\text{m}^{2}$
Initial permeability leakage |  $K_{0}^{L}$ |  $2\cdot 10^{-14}$ |  $\text{m}^{2}$
Initial porosity |  $\phi_{0}$ |  0.15 |  [-]
Microbial concentration |  $c_{m}$ |  0.01 |  $\text{kg}/\text{m}^{3}$
Oxygen concentration |  $c_{o}$ |  0.04 |  $\text{kg}/\text{m}^{3}$
Urea concentration |  $c_{u}$ |  60.0 |  $\text{kg}/\text{m}^{3}$
Figure 4: Illustration of the full 3D system with two aquifers separated by a
caprock (not shown explicitly). The quarter domain considered in the numerical
experiments is outlined by the dashed lines

The three scenarios were set up as follows: (i) a single leakage path, similar
to the leaky well scenario considered in Landa-Marbán et al. (2021a); (ii) two
separate leakage paths; and (iii) a wide, connected leakage path, similar to
the one considered in Landa-Marbán et al. (2021b). For convenience, we label
scenarios (i), (ii), and (iii) as Single leak, Double leak, and Diagonal leak,
respectively. In Figure 5, illustrations of the three scenarios are shown,
with properties for each scenario summarized in Table 4. Due to computational
constraints, we considered leakage paths that are relatively large in size,
especially compared to typical fractures sizes. Thus, the scenarios can be
seen as mimicking three types of damaged zones in the caprock, with a slightly
higher permeability ($K_{0}^{L}=2\cdot K_{0}^{A}$; see Table 3).

(a)  (b)  (c)

Figure 5: Illustrations of the 3D domains for the (a) Single, (a) Double, and (c) Diagonal leak scenarios Table 4: Domain properties for the three leakage scenarios. All values are in meters  Single leak |  | Double leak |  | Diagonal leak
---|---|---|---|---
$a$ | $l_{x}$ | $l_{y}$ |  | $a_{1}$ | $a_{2}$ | $l_{x,1}$ | $l_{y,1}$ | $l_{x,2}$ | $l_{y,2}$ |  | $a$ | $l_{x}$ | $l_{y}$
1 | 13 | 14 |  | 1 | 1 | 10 | 11 | 17 | 18 |  | 1 | 11 | 19

We applied the injection strategy described in Section 2.3 with $N_{p}=5$.
Hence, we followed up phases $I$–$III$ with two phases, $IV$ and $V$, of
combined injection of growth and cementation solutions. Note that more follow-
up phases to $I$–$III$ could be considered in some applications to ensure
complete sealing of leakage paths. Examples of ad-hoc injection strategies
using additional phases to fully seal the leakage paths in the three scenarios
can be found in py-micp. However, that could also lead to a lot of wasted
solution if there are no microbes and/or biofilm at the leakage locations.
Furthermore, the extended MICP operation time with more phases can lead to
little benefit in terms of further sealing the leakage paths, and thus be
unsound from an economical perspective.

In all scenarios, the control variables for phases $I$–$V$, i.e., the
injection and no-flow periods in each phase, were constrained with lower
bounds all equal to 0 d, and upper bounds equal to 5 d for
$\hat{\boldsymbol{x}}^{sol}$ and $\hat{\boldsymbol{x}}^{wat}$, and 6 d for
$\hat{\boldsymbol{x}}^{nof}$. For the objective function, $J(\boldsymbol{x})$,
we define the cells in $\boldsymbol{\phi}_{c}^{leak}$ as the first cells in
the leakage paths immediately above the lower aquifer. Hence, we want to seal
the leakage paths efficiently by maximizing the calcite precipitation in and
around the entry of the leakage paths. We evaluated
$\boldsymbol{\phi}_{c}^{leak}(t_{3}^{p})$ at $p=III,IV,V$. Furthermore,
$\gamma$ was chosen in all scenarios to weight the first term in (13) higher
than the second term. In preliminary studies, we have seen that too much
weight on the second term may lead to optimization results with little or no
calcite precipitation due to very short injection and no-flow periods.

EnOpt was set up with $N_{e}=20$ and
$\boldsymbol{\Sigma}=\sigma\boldsymbol{I}$, where $\sigma=0.01$. The
optimization procedure was terminated when one or more of the following
criteria were reached:

  1. (A1)

An uphill direction (i.e, $J(\boldsymbol{x}_{k+1})>J(\boldsymbol{x}_{k})$),
could not be found with at least ten cuts to the step size, $\alpha_{k}$

  2. (A2)

$\lvert J(\boldsymbol{x}_{k+1})-J(\boldsymbol{x}_{k})\rvert/\lvert
J(\boldsymbol{x}_{k})\rvert\leq 10^{-6}$

  3. (A3)

$\|\boldsymbol{x}_{k+1}-\boldsymbol{x}_{k}\|_{2}\leq 10^{-6}$

  4. (A4)

Total number of iterations, including step-size cuts, reaches 50

To assess the quality of the optimization results, we ran simulations of CO2
injection in the reservoir before and after MICP using the final control
variables from the optimization. We injected CO2 for 400 d with injection rate
equal to $1\cdot 10^{-4}$ m3/s. To quantify the reduction in leakage of CO2
through the leakage paths before and after MICP, we calculated the percentage
reduction in accumulated CO2 mass in the upper aquifer, given in Table 5.

Table 5: Optimization results for all scenarios. For convenience, the injection and no-flow periods in $\boldsymbol{x}$ have been converted to times $t_{1}$, $t_{2}$, and $t_{3}$ for each phase in hours; see Section 2.3 Scenario | Phase | Times [h] | Leaked CO2 [%]
---|---|---|---
|  | $t_{1}^{p}\;$ | $t_{2}^{p}\;$ | $t_{3}^{p}$ | before | after
| $I$ | 14.93 | 15.35 | 17.87 |  |
| $II$ | 33.24 | 34.97 | 36.38 |  |
Single leak | $III$ | 36.78 | 37.91 | 43.92 | 4.30 | 6.67$\cdot 10^{-12}$
| $IV$ | 155.11 | 156.04 | 157.88 |  |
| $V$ | 159.11 | 160.27 | 161.52 |  |
| $I$ | 13.70 | 14.16 | 22.84 |  |
| $II$ | 53.87 | 54.48 | 71.28 |  |
Double leak | $III$ | 85.25 | 87.18 | 102.84 | 5.29 | 2.76$\cdot 10^{-10}$
| $IV$ | 183.11 | 184.12 | 191.44 |  |
| $V$ | 196.33 | 198.83 | 207.65 |  |
| $I$ | 4.97 | 6.83 | 23.70 |  |
| $II$ | 58.30 | 59.01 | 74.58 |  |
Diagonal leak | $III$ | 83.92 | 85.25 | 86.74 | 7.04 | 1.28$\cdot 10^{-9}$
| $IV$ | 191.26 | 191.83 | 194.23 |  |
| $V$ | 195.40 | 196.98 | 199.50 |  |

###  3.1 Single leak

In this scenario, the initial control variables were 1 d for
$\hat{\boldsymbol{x}}^{sol}_{0}$, 0.05 d for $\hat{\boldsymbol{x}}^{wat}_{0}$,
and 2 d for $\hat{\boldsymbol{x}}^{nof}_{0}$, and $\gamma=7.14\cdot 10^{-3}$
d-1. The final optimization results, converted to times $t_{1}^{p}$,
$t_{2}^{p}$, and $t_{3}^{p}$, are shown in Table 5. We see that the total
time, $t_{3}^{V}$, is 6.73 d (or 161.52 h), which is a significant reduction
from the total time of the initial control variables, 15.25 d (or 366 h). A
cross section of the calcite distribution at the final time is shown in Figure
6a. We see that most calcite has precipitated in and around the entry of the
leakage path, with a maximum value of 0.0483. From Figure 6b we see that
$J(\boldsymbol{x})$ have gone from negative to positive, indicating an
increase in calcite precipitation, along with a decrease in injection and no-
flow periods, during the iterations.

(a)  (b)

Figure 6: Single leak: (a) vertical cross-section plot at y = 14 m of the
calcite distribution using the final optimization results and (b) objective
function values versus iterations

Comparing Figure 7a and Figure 7b, we see the impact of the final calcite
distribution on the CO2 injection simulation. From Figure 7c and the
percentage reduction in CO2 mass in the upper aquifer in Table 5, we see that
the leakage path is essentially sealed.

(a)

(b)

(c)

Figure 7: Single leak: CO2 saturation after 400 d injection (a) before and (b)
after MICP using the final optimization results, and (c) percentage of
accumulated CO2 mass in upper aquifer over the injection period for both cases

###  3.2 Double leak

In this scenario, the initial control variables were the same as for Single
leak (cf. Section 3.1), but $\gamma=5.36\cdot 10^{-3}$ d-1. From Table 5, we
see that the final optimization time, $t_{3}^{V}$, was 8.65 d (or 207.65 h).
This is a significant reduction from the initial 15.25 d (or 366 h), but
slightly higher than the final optimization time for Single leak. From the
cross-section plot in Figure 8a we see that the calcite has precipitated in
and around the entry of both leakage paths, with maximum value of 0.0487.
Figure 8b shows again that $J(\boldsymbol{x})$ have gone from negative to
positive, thus calcite precipitation have increased along with decrease of
total injection and no-flow time.

(a)  (b)

Figure 8: Double leak: (a) Vertical cross-section plot from (x, y) = (0.0
m,1.0 m) to (x, y) = (90.5 m, 0.0 m) of the calcite distribution using the
final optimization result (the x-axis, r, indicates the cross-section length)
and (b) objective function values versus iterations

Figures 9a and 9b show that the CO2 leakage have been reduced significantly.
From Figure 9c and the percentage reduction in CO2 mass in the upper aquifer
in Table 5, we see that the leakage path is essentially sealed.

(a)

(b)

(c)

Figure 9: Double leak: CO2 saturation after 400 d injection (a) before and (b)
after MICP using the final optimization results, and (c) percentage of
accumulated CO2 mass in upper aquifer over injection period for both cases

###  3.3 Diagonal leak

In this scenario, the initial control variables were changed from the Single
and Double leak scenarios, to 0.5 d for $\hat{\boldsymbol{x}}^{sol}_{0}$, 0.05
d for $\hat{\boldsymbol{x}}^{wat}_{0}$, and 1 d for
$\hat{\boldsymbol{x}}^{nof}_{0}$, in addition to $\gamma=1.79\cdot 10{-3}$
d-1. Hence, we start with a short total time of 7.75 d (or 186 h). We
compensate this by having a low $\gamma$, to put even more weight on the first
term in (13) compared to the Single and Double leak scenarios. From Table 5 we
see that the final optimization time is 8.31 d (or 199.5 h), which is not a
significant increase from the total time of the initial control variables.
Furthermore, the cross-section plot in Figure 10a show that the calcite
distribution in and around the entry of the leakage path is high, with a
maximum value of 0.0484. Thus, the optimization have increased the calcite
precipitation in the leakage path while keeping the total injection and no-
flow time low, which is also what the objective function plot in Figure 10b
indicates.

(a)  (b)

Figure 10: Diagonal leak: (a) Vertical cross-section plot from (x, y) = (0.0
m, 1.0 m) to (x, y) = (90.5 m, 0.0 m) of the calcite distribution using the
final optimization result (the x-axis, r, indicates the cross-section length)
and (b) objective function values versus iterations

In Figures 11a and 11b we see that the CO2 in the upper aquifer have been
reduced significantly. From Figure 11c and the percentage reduction in CO2
mass in the upper aquifer in Table 5, we see that the leakage path is
essentially sealed.

(a)

(b)

(c)

Figure 11: Diagonal leak: CO2 saturation after 400 d injection (a) before and
(b) after MICP using the final optimization results, and (c) percentage of
accumulated CO2 mass in upper aquifer over injection period for both cases

##  4 Discussion

From Table 5 we see that the injection period with the combined growth and
cementation solutions in phase $IV$ (i.e. $\Delta t_{1,3}^{IV}$) is the
longest in all three scenarios. This indicates that when the biofilm has
successfully been established in phases $I$–$III$, a long-time injection of
growth and cementation solutions is needed for maximizing the sealing of the
leakage paths. Furthermore, we see that longer no-flow periods are needed in
phases $I$–$III$ compared to phases $IV$ and $V$. This confirms the
observation that establishing a biofilm in phases $I$–$III$ is important.
Lastly, we see that for the Single and Diagonal leak scenarios phase $V$ is
short, indicating four phases may be enough for efficient sealing, while a
reasonable amount of time was spent in Double leak phase $V$, thus indicating
the need for five phases.

Simulating radial injection from a vertical well necessarily means that a lot
of the microbial, growth, and cementation solutions are wasted during the
injections, which would be even more prominent if we had simulated the full
domain. Even though it might be possible from an engineering perspective to
direct well flow in a specific direction in the reservoir, it is complex from
a simulation perspective. A preliminary study simulating directional radial
wells can be found in py-micp. When simulation tools for directional injection
have been developed, economic terms for efficient use of the three injected
solutions may be included in the objective function, such as upper limits on
injected mass of expensive components.

##  5 Conclusions

In this paper, we have presented and applied an optimization procedure for
sealing leakage paths with MICP. The field-scale mathematical model for MICP
processes were simulated using the industry-standard simulator OPM Flow, while
the gradient-based EnOpt algorithm was used for the optimization. An injection
strategy for field-scale application of MICP was developed, where growth and
cementation solutions were injected at the same time but in different well
segments, to efficiently initiate the MICP process at the leakage paths.
Furthermore, we defined the objective function such that maximizing calcite
precipitation at the leakage paths are weighted with total injection and no-
flow time during the optimization. Hence, the economic aspect of having the
shortest possible MICP operational time is considered in the optimizations.

The optimization procedure was applied to three synthetic CO2 leakage
scenarios. The numerical results showed that the optimization procedure was
able to essentially close the leakage path in the Single leak scenario, and
almost completely seal the two leakage paths and wide, diagonal leakage path
in the Double and Diagonal leak scenarios, respectively. In the optimizations,
the total injection and no-flow time was low while the calcite distribution in
and around the entry of the leakage paths were maximized. Hence, including
economic aspects, like total operational time, in the objective function is
recommended for MICP, and similar leakage sealing, optimization procedures.

#### Acknowledgements:

The authors are grateful for the financial support from Research Council of
Norway through the project Efficient models for microbially induced calcite
precipitation as a seal for CO2 storage (MICAP) (grant 268390). Open Access
funding provided by NORCE Norwegian Research Centre AS. The authors thank Tor
Harald Sandve for useful discussions and guidance for the implementation of
the MICP model in OPM Flow.

#### Data availability:

OPM Flow, the numerical simulator with MICP functionality used in this study,
can be obtained at https://github.com/OPM/opm-simulators. The MATLAB/GNU
Octave code MRST, which also includes the MICP module ad-micp, was used to
generate the grids and can be obtained at
https://www.sintef.no/projectweb/mrst/. The Python package py-micp can be
found at https://github.com/daavid00/py-micp.git.

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