#
Effect of Computational Schemes on Coupled Flow and Geo-Mechanical Modeling of CO_{2} Leakage through a Compromised Well

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}) from industrial byproduct streams and inject it into deep geologic formations for long-term storage. Legacy wells located within the spatial domain of new injection and production activities represent potential pathways for fluids (i.e., CO

_{2}and aqueous phase) to leak through compromised components (e.g., through fractures or micro-annulus pathways). The finite element (FE) method is a well-established numerical approach to simulate the coupling between multi-phase fluid flow and solid phase deformation interactions that occur in a compromised well system. We assumed the spatial domain consists of a three-phases system: a solid, liquid, and gas phase. For flow in the two fluids phases, we considered two sets of primary variables: the first considering capillary pressure and gas pressure (PP) scheme, and the second considering liquid pressure and gas saturation (PS) scheme. Fluid phases were coupled with the solid phase using the full coupling (i.e., monolithic coupling) and iterative coupling (i.e., sequential coupling) approaches. The challenge of achieving numerical stability in the coupled formulation in heterogeneous media was addressed using the mass lumping and the upwinding techniques. Numerical results were compared with three benchmark problems to assess the performance of coupled FE solutions: 1D Terzaghi’s consolidation, Liakopoulos experiments, and the Kueper and Frind experiments. We found good agreement between our results and the three benchmark problems. For the Kueper and Frind test, the PP scheme successfully captured the observed experimental response of the non-aqueous phase infiltration, in contrast to the PS scheme. These exercises demonstrate the importance of fluid phase primary variable selection for heterogeneous porous media. We then applied the developed model to the hypothetical case of leakage along a compromised well representing a heterogeneous media. Considering the mass lumping and the upwinding techniques, both the monotonic and the sequential coupling provided identical results, but mass lumping was needed to avoid numerical instabilities in the sequential coupling. Additionally, in the monolithic coupling, the magnitude of primary variables in the coupled solution without mass lumping and the upwinding is higher, which is essential for the risk-based analyses.

## 1. Introduction

_{2}) has gradually increased, which is considered as an important criterion for the global warming. [1]. Carbon capture, utilization, and storage (CCUS) has been proposed as a technically viable approach to help reduce anthropogenic CO

_{2}emissions [2]. Once captured from industrial point sources and compressed to a dense phase, CO

_{2}can be transported for geologic storage in depleted hydrocarbon reservoirs, coal beds, deep saline aquifers, or other geologic targets [3]. Successful implementation of geologic carbon storage (GCS) requires effective containment of large volumes of injected CO

_{2}for long periods of time, and it is necessary to ensure that natural and engineered seals serve as effective barriers to unwanted fluid migration. The integrity of active and abandoned wells is also an important aspect of containment assurance [4,5].

_{2}, brine) may migrate through the annulus or crack in the compromised well. Due to fluids flow, inside the porous mediums, fluids pressure develops, which also impacts geo-mechanical aspects of wells [14]. In addition, in the compromised well, the fluids flow, fluids saturation state change, their pressure evolution, and the associated deformation flow are considered as a complex coupled problem, which is the motivation of the present research.

_{2}injection, like, the Ketzin Pilot project (Chen et al. [15]). The numerical complexities of these types of cases are similar to the Liakopoulos experiments [16]. In addition, among others, Lewis and Schrefler [17] demonstrated associated numerical issues in a homogeneous media. Three-phases (solid, liquid, and gas) finite element simulation in deformable heterogeneous mediums with fractures presents numerical convergence challenges. Helmig [18] demonstrated the computational complexities of modeling fluid flow experimental results in heterogeneous sand mediums reported by Kueper and Frind [19]. The literature reports multiple sources for the numerical instabilities in the multi-phase heterogeneous domains, including: non-linearity of coupled partial differential equations which hold the complex mediums (Binning and Celia [20]), coupling mechanisms of fluids and solids (e.g., the monolithic coupling or the sequential coupling) (Kolditz et al. [21]), and selection of fluids’ primary variables (the capillary pressure plus the gas pressure or the liquid pressure plus the gas saturation) (Helmig [18]). Additionally, for the standard FE, the mass matrices of variables are calculated at each quadrature point, while fluids flowing in porous media, the time derivative of variables are evaluated at nodes (Zienkiewicz et al. [22]). There are notable numerical complexities for the two fluids flow simulation, when (i) one fluid phase approaches to the residual saturation, (ii) the mobility term becomes zero, or (iii) the role of compressibility is substantial. For cases with three-phases heterogeneous mediums, selection of (i) numerical solvers (linear and non-linear solvers), (ii) time integration scheme (implicit, explicit, or automatic time stepping), (iii) preconditioner (Jacobi type or Incomplete Lower Upper (ILU) factorization type or without preconditioner), and (iv) memory storage (symmetry or asymmetry) are crucial issues to obtain numerical convergence (Chen et al. [23]).

_{2}leakage through observation points including in the fracture zone, (ii) monitor coupled geo-mechanical responses due to CO

_{2}injection, and (iii) investigate the effect of coupling scheme (e.g., monolithic and sequential coupling) with and without the mass lumping and the upwinding algorithms on the coupled finite element solutions of CO

_{2}leakage.

## 2. Governing Equations

#### 2.1. Momentum Conservation Equations

#### 2.2. Mass Balance Equation

^{−3}and 1 × 10

^{−5}m, respectively, while a similar magnitude for the pressure is 1 Pa. For the convergence, we consider the relative error criterion (see also Chen et al. [23]).

## 3. Coupled Finite Element Formulations

## 4. Benchmark Solutions

#### 4.1. Terzaghi’s 1D-Consolidation

#### 4.2. Liakopoulos Test

#### 4.3. Kueper and Frind Test

## 5. Numerical Modeling of Compromised Well

#### 5.1. Geometry and Properties

_{2}storage reservoir (6.0 m), and (ii) a relatively impermeable caprock (20.0 m) and stress above the caprock. We also assume that the storage reservoir is located 900 m below the surface. It is worth mentioning that we consider a simplified geometry for the coupled FE modeling to reduce the computational time. Thereby, in the numerical model, above the CO

_{2}storage reservoir, we assume the stress state as an overburden pressure (see also Figure 10b). In addition, in Figure 10b, we consider fracture in the left wall of the steel casing, while its right wall is intact. Hence, we also assume that fluid will not flow through the steel wall’s right impervious wall. As a result, in Figure 10b, possible fluid flow paths are compromised fracture zone and the porous medium. Moreover, in Figure 10b, to minimize the right end-boundary effect in the numerical simulation, we extend the longitudinal dimension to 50.0 m on the other side of the CO

_{2}injection face. Theoretically, depending on the injection rate and geophysical properties of the study domain (see Figure 10b), coupled hydro-mechanical responses to any points are proportionate from the fluid injection phase distance. However, such responses in coupled finite element modeling also depend on the selection of the coupled algorithms (see also Figure 8 and Figure 9), which we investigate herein for a compromised well. Therefore, considering the above-mentioned assumptions, we use a total of four observation points in Figure 10b to monitor the coupled hydro-mechanical behavior.

^{3}. In addition, we consider the domain’s temperature is 34 °C and in the isothermal state. With such a stress state and temperature condition, we obtain corresponding fluid properties from the National Institute of Standards Technology webbook (NIST [46]), and we present them in Table 7. Additionally, to account for the transition of the wetting phase to the non-wetting phase, we consider the Brooks and Corey model, which we demonstrate in Table 8.

#### 5.2. Initial Conditions and Boundary Conditions

_{2}injection boundary on the left side. We introduce fluids saturation in terms of the capillary pressure. To maintain the quasi-static condition, we assume small-time incremental and a constant CO

_{2}injection rate, so that we may avoid any disturbance during CO

_{2}injection. Additionally, at the top and the bottom of the domain, we restrict the vertical displacement. We also consider that along the horizontal direction (x-axis), the CO

_{2}injection face (see Figure 10b) is movable. In contrast, we assume another front of the horizontal direction is the fixed boundary by restricting the displacement.

## 6. Results and Discussions of a Compromised Well

#### 6.1. Effect of Coupling Mechanisms

#### 6.2. Effect of the Injection Rate

_{2}in a storage reservoir; it is desirable to calculate the CO

_{2}injection rate so that the stress state does not exceed the failure criterion, which can be obtained from the laboratory experiments. In addition, it is to note that herein, we use finite element model parameters for the compromised well as an arbitrary value. Therefore, to present a comparison of the experimental results are beyond the scope of this paper.

_{2}injection, which results from dropping the stress state. Moreover, the required time to change the stress state from the lowest values to the increase of stress may be related to the energy dissipation of the porous media. It is worth mentioning that during and after the CO

_{2}injection, the dilative and contractive nature of the stress components are crucial for the stability of the compromised well. If such a stress reversal exceeds the limiting stress condition of geomaterial, it may result in a failure state.

## 7. Conclusions

_{2}) are mobile, like, CO

_{2}injection, the Richards solutions may not be appropriate for its assumptions.

_{2}storage reservoir, cement sheath between steel casing and host rock, and a horizontal fracture. Numerical simulation results of this compromised well scenario for the monolithic coupling, with and without the mass lumping and the upwinding techniques. Considering this heterogeneous problem, using upwinding alone in the sequential coupling did not address convergence issues. We also found that using both mass lumping and the upwinding for the long-term prediction yielded results that were significantly lower than the monolithic coupling case. In contrast, to the Kueper and Frind benchmark experiment where the convergence was achieved, computational time was reduced, and fidelity was maintained, the mass lumping and the upwinding techniques failed to work on the long-term compromised legacy well scenario, and should be avoided on problems of this complexity.

_{2}injection rate is needed to avoid the mechanical failure condition in the compromised well. Considering three different types of injection rates we found that long-term predictions showed gradually increasing radial displacement of the solid phase. After reaching peak radial displacement energy dissipated and the displacement magnitude began to decrease. At higher injection rate, stress shock resulted in reduction to the stress component (inversely proportional to the injection rate), with subsequent increase in stress component. In addition, for the initial phase of the CO

_{2}injection due to the developed extension at the well surface, the stress component starts to decline. Then, in the domain, the fluid phase displacement results to increase the fluid pressure progressively and the stress component also increases. These types of dilative and contractive geo-mechanical behaviors are critical for the compromised legacy well. The gradual increase of the CO

_{2}pressure in the storage reservoir resulted in changing the hydro-mechanical behavior of the system. It is important that CO

_{2}injection operations should be controlled such that maximum magnitude of the stress component should be lower than the failure condition to ensure safe CO

_{2}injection.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Constitutive Relations

Equation Name | References | Number of Equations |
---|---|---|

Linear momentum equation for solid | Equation (1) | 3 |

Linear momentum equation for liquid | 3 | |

Linear momentum equation for gas | 3 | |

Mass conservation equation for solid | Equation (2) | 1 |

Mass conservation equation for liquid | 1 | |

Mass conservation equation for gas | 1 | |

Stress–strain equation | Equation (A18) | 6 |

Strain displacement equation | Equation (A16) | 6 |

Darcy’s law for liquid | Equation (A8) | 3 |

Darcy’s law for gas | 3 | |

Capillary pressure and saturation relation | Equation (A22) | 1 |

Density equation for solid | Equation (A4) | 1 |

Density equation for liquid | Equation (A4) | 1 |

Density equation for gas | Equation (A4) | 1 |

Total | 34 |

Variables | $\stackrel{\xb4}{\mathit{\sigma}}$ | $\mathit{\epsilon}$ | ${\mathit{v}}_{\mathit{s}}$ | ${\mathit{v}}_{\mathit{l}}$ | ${\mathit{v}}_{\mathit{g}}$ | ${\mathit{p}}_{\mathit{l}}$ | ${\mathit{p}}_{\mathit{g}}$ | ${\mathit{S}}_{\mathit{e}}$ | ${\mathit{\rho}}^{\mathit{s}}$ | ${\mathit{\rho}}^{\mathit{l}}$ | ${\mathit{\rho}}^{\mathit{g}}$ | ${\mathit{w}}_{\mathit{l}}$ | ${\mathit{w}}_{\mathit{g}}$ | $\mathit{\phi}$ | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Number of unknowns | 6 | 6 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 34 |

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**Figure 4.**Comparison between the Terzaghi’s 1D consolidation analytical solution and finite element results at the bottom.

**Figure 6.**Finite element results and Liakopoulos experiments: (

**a**) water pressure, (

**b**) outflow rate, (

**c**) water saturation, and (

**d**) vertical displacement.

**Figure 7.**Configuration of heterogeneous sand lenses in parallel plate cell (see Kueper and Frind [19]).

**Figure 8.**Comparison of observed (redrawn) and predicted tetrachloroethylene distribution in terms of the liquid at 34 s (

**a**–

**c**) and 313 s (

**d**–

**f**).

**Figure 9.**Comparison of predicted tetrachloroethylene distribution in terms of the liquid at 313 Sec: (

**a**–

**c**) with the mass lumping and the upwinding and (

**d**–

**f**) without the mass lumping and the upwinding.

**Figure 10.**(

**a**) Schematic diagram of the cross-section of the compromised well and (

**b**) schematic dimensions of the cross-section and locations of observation points (see also Ravi et al. [45]).

**Figure 11.**Comparisons of the displacement along the X-axis: (

**a**) the monolithic coupling and the sequential coupling and (

**b**) with and without the mass lumping and the upwinding.

**Figure 12.**Comparisons of the stress along the X-axis: (

**a**) the monolithic coupling and the sequential coupling and (

**b**) with and without the mass lumping and the upwinding.

**Figure 13.**Comparisons of the gas phase saturation: (

**a**) the monolithic coupling and the sequential coupling and (

**b**) with and without the mass lumping and the upwinding.

**Figure 14.**Comparisons of the gas phase velocity: (

**a**) the monolithic coupling and the sequential coupling and (

**b**) with and without the mass lumping and the upwinding.

Parameters | Symbol | Unit | Value |
---|---|---|---|

Young’s Modulus | E | Pa | 3.0 × 10^{4} |

Poisson’s ratio | ν | --- | 0.20 |

Porosity | φ | --- | 0.50 |

Permeability | k | m^{2} | 1.0 × 10^{−10} |

Liquid density | ${\rho}^{l}$ | kg/m^{3} | 1000 |

Liquid viscosity | ${\mu}^{l}$ | Pa-s | 1.0 × 10^{−3} |

Parameters | Symbol | Unit | Value |
---|---|---|---|

Young’s Modulus * | E | Pa | 1.3 × 10^{6} |

Poisson’s ratio * | ν | --- | 0.40 |

Porosity | φ | --- | 0.2975 |

Solid’s density | ${\rho}^{s}$ | kg/m^{3} | 2000 |

Permeability | K | m^{2} | 4.5 × 10^{−13} |

Liquid density | ${\rho}^{l}$ | kg/m^{3} | 1000 |

Liquid viscosity | ${\mu}^{l}$ | Pa-s | 1.0 × 10^{−3} |

Air density | ${\rho}^{g}$ | kg/m^{3} | Ideal Gas Law’s |

Air viscosity | ${\mu}^{g}$ | Pa-s | 1.8 × 10^{−5} |

Atmospheric reference pressure | ${p}_{g}$ | Pa | 0 |

Capillary pressure ^{§} | ${p}_{c}$ | Pa | ${S}^{l}=1-1.9722\times {10}^{11}{p}_{c}^{2.4279}$ |

Liquid phase relative permeability ^{§} | ${k}_{rel}^{l}$ | --- | ${k}_{rel}^{l}=1-2.207{\left(1-{S}^{l}\right)}^{1.0121}$ |

Gas phase relative permeability ^{‡} | ${k}_{rel}^{g}$ | --- | ${k}_{rel}^{g}={\left(1-{S}_{e}\right)}^{2}\left(1-{S}_{e}^{\frac{5}{3}}\right)$ ${S}_{e}=\left({S}^{l}-0.2\right)/\left(1-0.2\right)$ |

Fluid Properties | Symbol | Unit | Water | Tetrachloroethylene |
---|---|---|---|---|

Density | ${\rho}^{\beta}$ | kg/m^{3} | 1 × 10^{3} | 1.63 × 10^{3} |

Viscosity | ${\mu}^{\beta}$ | Pa-s | 1 × 10^{−3} | 0.90 × 10^{−3} |

Sand | ${\mathit{P}}_{\mathit{d}}\left(\mathbf{Pa}\right)$ | $\mathit{\lambda}$ (-) | S | k (m^{2}) | n (-) |
---|---|---|---|---|---|

#16 Silica | 369.73 | 3.86 | 0.078 | 5.04 × 10^{−10} | 0.40 |

#25 Ottawa | 434.45 | 3.51 | 0.069 | 2.05 × 10^{−10} | 0.39 |

#50 Ottawa | 1323.95 | 2.49 | 0.098 | 5.26 × 10^{−11} | 0.39 |

#70 Silica | 3246.15 | 3.30 | 0.189 | 8.19 × 10^{−12} | 0.41 |

Time Step | Computational Time (Hours) for the PP Scheme | |
---|---|---|

With the Mass Lumping and the Upwinding | Without the Mass Lumping and the Upwinding | |

Auto Time Step | 27.06 | 34.70 |

$\Delta t=0.1$ | 15.41 | 17.03 |

$\Delta t=0.5$ | 7.14 | 8.04 |

$\Delta \mathrm{t}=1.0$ | 5.76 | Convergence problem |

Section | Properties | Symbol | Unit | Value | |
---|---|---|---|---|---|

Casing | Mechanical | Young’s Modulus | E_{casing} | Pa | 1.7 × 10^{11} |

Poisson’s Ratio | ${v}_{casing}$ | 0.27 | |||

Density | ${\rho}_{casing}$ | kg/m^{3} | 8000 | ||

Porosity | ϕ_{casing} | --- | --- | ||

Flow | Permeability | k_{casing} | m^{2} | 1.0 × 10^{−21} | |

Reservoir | Mechanical | Young’s Modulus | E_{rock} | Pa | 2.0 × 10^{11} |

Poisson’s Ratio | ${v}_{rock}$ | 0.30 | |||

Density | ${\rho}_{rock}$ | kg/m^{3} | 2650 | ||

Porosity | ϕ_{rock} | --- | 0.30 | ||

Flow | Permeability | k_{rock} | m^{2} | 1.0 × 10^{−13} | |

Cement | Mechanical | Young’s Modulus | E_{cement} | Pa | 8.3 × 10^{9} |

Poisson’s Ratio | ${v}_{cement}$ | 0.25 | |||

Density | ${\rho}_{cement}$ | kg/m^{3} | 2000 | ||

Porosity | ϕ_{cement} | --- | 0.132 | ||

Flow | Permeability | k_{cement} | m^{2} | 1 × 10^{−20} | |

Fracture | Mechanical | Young’s Modulus | E_{fracture} | kPa | 2.0 × 10^{8} |

Poisson’s Ratio | ${v}_{fracture}$ | 0.20 | |||

Density | ${\rho}_{fracture}$ | kg/m^{3} | 1600 | ||

Porosity | ϕ_{fracture} | --- | 0.80 | ||

Flow | Permeability | k_{fracture} | m^{2} | 3.25 × 10^{−8} | |

Geometry | Thickness | b | Micro-m | 625 |

**Table 7.**Fluid properties (From NIST webbook NIST [46]).

FluidProperties | Symbol | Unit | Fluids | |
---|---|---|---|---|

Brine | CO_{2} | |||

Density | ${\rho}^{\beta}$ | kg/m^{3} | 1004.6 | 895.39 |

Viscosity | ${\mu}^{\beta}$ | Pa-s | 7.3446 × 10^{−4} | 9.0285 × 10^{−5} |

Items | Unit | Symbol | Reservoir | Cement | Fracture | Casing |
---|---|---|---|---|---|---|

Values | ||||||

Residual CO_{2} saturation | --- | ${S}^{rg}$ | 0.03 | 0.20 | 0.05 | N/A |

Maximum CO_{2} saturation | --- | ${S}^{mg}$ | 0.60 | 0.70 | 0.90 | |

Residual brine saturation | --- | ${S}^{rl}$ | 0.40 | 0.30 | 0.10 | |

Maximum brine saturation | --- | ${S}^{ml}$ | 0.97 | 0.80 | 0.95 | |

Brooks-Corey parameter | --- | λ | 2 | 2 | 2 | |

Air entry pressure | Pa | ${p}_{d}$ | 10^{4} | 10^{4} | 10^{4} |

Input | Values | Unit |
---|---|---|

Overburden pressure $\left({\sigma}_{y}\right)$ | −23.40 × 10^{6}–25,996.5 × y | Pa |

Maximum horizontal stress $\left({\sigma}_{x}\right)$ | −18.72 × 10^{6}–20,797.2 × y | Pa |

Minimum horizontal stress $\left({\sigma}_{z}\right)$ | −18.72 × 10^{6}–20,797.2 × y | Pa |

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**MDPI and ACS Style**

Islam, M.; Huerta, N.; Dilmore, R. Effect of Computational Schemes on Coupled Flow and Geo-Mechanical Modeling of CO_{2} Leakage through a Compromised Well. *Computation* **2020**, *8*, 98.
https://doi.org/10.3390/computation8040098

**AMA Style**

Islam M, Huerta N, Dilmore R. Effect of Computational Schemes on Coupled Flow and Geo-Mechanical Modeling of CO_{2} Leakage through a Compromised Well. *Computation*. 2020; 8(4):98.
https://doi.org/10.3390/computation8040098

**Chicago/Turabian Style**

Islam, Mohammad, Nicolas Huerta, and Robert Dilmore. 2020. "Effect of Computational Schemes on Coupled Flow and Geo-Mechanical Modeling of CO_{2} Leakage through a Compromised Well" *Computation* 8, no. 4: 98.
https://doi.org/10.3390/computation8040098