#
Semi-Analytical Solution to Assess CO_{2} Leakage in the Subsurface through Abandoned Wells

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## Abstract

**:**

_{2}) emissions at significant scales. Subsurface reservoirs with sealing caprocks can provide long-term containment for the injected fluid. Nevertheless, CO

_{2}leakage is a major concern. The presence of abandoned wells penetrating the reservoir caprock may cause leakage flow-paths for CO

_{2}to the overburden. Assessment of time-varying leaky wells is a need. In this paper, we propose a new semi-analytical approach based on pressure-transient analysis to model the behavior of CO

_{2}leakage and corresponding pressure distribution within the storage site and the overburden. Current methods assume instantaneous leakage of CO

_{2}occurring with injection, which is not realistic. In this work, we employ the superposition in time and space to solve the diffusivity equation in 2D radial flow to approximate the transient pressure in the reservoirs. Fluid and rock compressibilities are taken into consideration, which allow calculating the breakthrough time and the leakage rate of CO

_{2}to the overburden accurately. We use numerical simulations to verify the proposed time-dependent semi-analytical solution. The results show good agreement in both pressure and leakage rates. Sensitivity analysis is then conducted to assess different CO

_{2}leakage scenarios to the overburden. The developed semi-analytical solution provides a new simple and practical approach to assess the potential of CO

_{2}leakage outside the storage site. This approach is an alternative to numerical methods when detailed simulations are not feasible. Furthermore, the proposed solution can also be used to verify numerical codes, which often exhibit numerical artifacts.

## 1. Introduction

_{2}) is the main elementcontributing to the Earth’s rising temperature [6,7]. Carbon capture and storage (CCS) technologies are expected to reduce emissions from fossil fuels significantly. Carbon storage in geological formations is a promising method to store CO

_{2}in underground reservoirs at significant scales [8]. Subsurface reservoirs with sealing caprocks can provide long-term containment for the CO

_{2}. There are currently around 60 large-scale CCS facilities under operation or development worldwide, resulting in 30 million metric tons (Mt) of CO

_{2}captured and stored per year in 2020 [9]. The storage potential of CCS is different for various underground formations like saline aquifers and depleted oil and gas reservoirs [10,11,12]. Subsurface reservoirs with production history could be preferred because of their cost-effectiveness in using existing infrastructure and knowledge about the subsurface geology. However, natural and artificial permeable patterns, such as fractures and existing wells, may cause fluid leakage to the surrounding formation [13,14,15,16,17,18,19]. The related well integrity problems have been widely investigated for existing injection wells and abandoned wells in different CCS projects [20,21,22]. Among them, abandoned wells require careful attention, as they could penetrate the caprock, creating potential leakage flow-paths to the overburden. For instance, improper well abandonment may create permeable channels through the borehole, space between casing and cement, or fractures in the cement, which induce hydrodynamic communications across the penetrated geological layers [23,24,25,26,27]. Fluid leakage across geological layers is difficult to quantify because of the lack of subsurface information. Furthermore, detecting leakage from monitoring CO

_{2}concentration could be confused with the natural presence of CO

_{2}when it leaks to other connecting formations. Alternatively, fluid leakage and hydraulic communication across layers could be detected and quantified by monitoring the pressure profile in the storage site and the surrounding layers. Different methods have been employed to evaluate the leakage process, which mainly fall into two categories: numerical simulation and analytical solutions. Numerical simulations have been used for risk assessments in large-scale projects [28,29]. However, this type of approach usually requires significant computational time and detailed geological data that may not always be available. On the other hand, analytical methods are useful in providing quick evaluations with minimum input data and they are free from numerical artifacts. Furthermore, they could be used to verify and benchmark numerical methods [30,31,32]. Researchers have developed many analytical solutions and asymptotic solutions for assessing the pressure change and leakage rate [33,34,35,36,37,38,39,40,41]. The solution methods include various field scenarios corresponding to closed or constant-pressure outer boundary conditions and incomplete-sealed well [42,43,44,45]. In most methods, analytical solutions are derived in the Laplace domain, thus requiring numerical inversion from the Laplace domain to the real-time domain. Another major limitation in the existing solutions is that the fluid leakage at the abandoned well is assumed to occur simultaneously with the injection, which is not realistic. This assumption was imposed so that the Laplace transformation can be applied to the injection well and abandoned well separately but with the same initial conditions.

## 2. Background

_{2}injection well penetrating the caprock of a reservoir that injects fluid into the storage site (Figure 1). An abandoned well is assumed to be connecting the storage site with an upper geological layer. The upper layer and storage site (lower layer) are assumed to be homogeneous and isotropic with uniform thicknesses. The layers are also assumed to be infinite-acting in the horizontal directions. We assume that the injected fluid has the same properties as the host fluid in both layers, and the fluids are compressible. Note that we are not attempting to model the transport front of CO

_{2}, but rather the pressure wave resulted from the water displaced from the zone of injection. The diffusivity equation in single-phase flow is, therefore, sufficient to capture the pressure propagation, which travels faster than the mass transport of CO

_{2}. Other mechanisms, such as CO

_{2}solubility in brine, are also ignored. The leaky path created by the abandoned well is approximated with a permeable channel with constant resistance to flow. The injection well operates at a constant rate, and the flow in porous media obeys Darcy’s law, where mass conservation can be modeled by the diffusivity equation.

## 3. Governing Equations

_{t}[psi−] is the total rock/fluid compressibility, k[mD] is the permeability, α

_{1}= 0.000264 is a constant used for unit conversion, and t[h] is the time.

_{2}= 0.001127 is a unit conversion factor, B is the volume formation factor given in reservoir barrel per stock tank barrel [rb/STB], and h [ft] is the layer’s thickness.

## 4. Proposed Solution

_{i}is the exponential integral, defined by

_{L}

_{,1}[psi] is the pressure change in the lower layer caused by the injection well (that is, only by well 1), as shown in Figure 2.

## 5. Solution with the Abandoned Well

_{2}injection. The time-varying leakage rate can be expressed by the difference between the bottom hole pressure between the lower and upper layers and the flow resistance factor within the wellbore [34], that is,

_{l}[STB/D] is the leakage rate, r

_{w}2[ft] is the radius of the abandoned well, ${P}_{L}(r={r}_{{w}_{2}},t)$[psi] is the bottom hole pressure for the abandoned well at the lower layer, ${P}_{U}(r={r}_{{w}_{2}},t)$[psi] is the bottom hole pressure for the abandoned well at the upper layer, and Ω[psi/(STB/D)] is the flow resistance within the leaky well.

_{b}. Using the superposition of time, we subdivide the leakage rate into a series of constant rates, q

_{i}. For instance, the induced leakage rate, q

_{1}, occurs during the time interval t

_{1}− t

_{b}. The bottom-hole pressure change caused by leakage at this time interval becomes

_{L}

_{,2}[psi] is pressure change at lower layer caused by the abandoned well (well 2) only, and a

_{L}[psi/(STB/D)] and b

_{L}[h/ft2] are constant parameters related to the reservoir and fluid properties of the lower layer.

## 6. Total Pressure Change and Leakage Rate

_{L}

_{,1}, and the leakage well, p

_{L}

_{,2}, are determined from Equations (7) and (9), respectively.

_{L,i}[psi] is the initial pressure in the lower layer.

_{U,i}[psi] is the initial pressure in the upper layer.

- 1.
- Define the time intervals in terms of the total time.
- 2.
- Compute the pressure change caused by injection at the abandoned well location p
_{L}_{,1}(r_{l}_{1}= R) for all time intervals. - 3.
- For the time intervals, when pressure change p
_{L}_{,1}(r_{l}_{1}= R, t = t_{i}) is zero, the leakage rates q_{l}(t = t_{i}) are also recorded as zero. - 4.
- From the time interval, when the pressure change p
_{L}_{,1}(r_{l}_{1}= R, t = t_{b}) is not zero, the rest of leakage rates can be solved based on the abandoned well constraint (Equation (8)). - 5.

## 7. Results

#### 7.1. Sensitivity of Leakage Rate Discretization

#### 7.2. Solution Verification

_{2}flow problem [45,50]. We used the IMEX simulator in CMG for comparison. The details of the simulation grids are provided in Table 2. The pore volumes of the simulation grid blocks at the boundary of the simulation model were increased to one billion times that of the other blocks to mimic infinite boundary conditions. The abandoned well is located at the center of the reservoir, and the injection well is assigned to another block located at 60 ft radial distance from the abandoned well. The injected fluid is set the same as host fluid in the reservoir with the same physical properties, as discussed in the previous section (see Table 1).

#### 7.3. Sensitivity Analysis

_{i}, which is a parameter reflecting the relationship between breakthrough time and the rock/fluid physical properties. It is a measure of the extent of the reservoir that has been influenced by the pressure disturbance [51], that is,

_{i}is the radius of investigation.

## 8. Conclusions

_{2}storage in subsurface geological formations. This problem is challenging because of the lack of detailed subsurface data. Pressure transient analysis is a promising method that can be used to quantify the leakage process with limited information. In this work, we introduced a new semi-analytical solution to estimate the leakage rate and transient pressure behavior within the storage site and the overburden geological layer, hydraulically connected by an abandoned well. Fluid and rock compressibilities are taken into consideration, which overcame the restrictive assumption in the conventional solution. We subdivide the leakage rate into a piecewise constant function and applied a superposition-of-time principle. The proposed semi-analytical solution is easy to implement to assess the leakage rate and pressure change distribution. The solution was first examined to assess its convergence as a function of the time discretization. It showed that the ultimate solution in the long term is insensitive to the discretization level. Verification with numerical simulation was then conducted with an industrial simulator by CMG. The results showed excellent agreements between the semi-analytical and the numerical solutions. We demonstrate the potential of this semi-analytical solution to be applied efficiently for sensitivity analysis to understand the subsurface uncertainties in the context of CO

_{2}storage leakage.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

a | Constant parameters related to the reservoir and fluid properties |

b | Constant parameters related to the reservoir and fluid properties |

B | Volume formation factor |

C | Rock/fluid compressibility |

h | Layer thickness |

k | Permeability |

p | Fluid pressure |

q | Fluid rate |

r | Radial distance |

t | Time |

ϕ | Porosity |

Ω | Flow resistance within the leaky well |

α | Unit conversion constant |

μ | Fluid viscosity |

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**Figure 1.**Schematic of the CO

_{2}storage site, where the leakage problem consisting of a CO

_{2}injection well into the storage site and an abandoned well leaking with hydraulic conductivity to an upper geological layer.

**Figure 2.**Illustration of the lower and upper layers with the variables used in the solution method, where the abandoned well is located at distance R from the injection well.

**Figure 4.**The leakage rate and pressure change solutions with different discretization intervals; dt = 0.01, 0.1, and 0.2 h (

**A**,

**B**), and dt = 1, 10, and 20 h (

**C**,

**D**) show that the ultimate solution is not impacted significantly by the discretization steps.

**Figure 5.**Comparison between the semi-analytical and the numerical solutions for the pressure change in the lower and upper layers (

**A**) and the leakage rates (

**B**).

**Figure 6.**Pressure change profile versus radial distance between the injector and the leakage wells in the lower layer.

**Figure 7.**Comparison of the pressure distribution maps obtained by the semi-analytical and numerical methods in the lower layer after 24 h (

**A**,

**B**) and 240 h (

**C**,

**D**).

**Figure 8.**Leakage rate profiles for three scenarios where the permeability in the lower layer corresponds to k = 10, 20, and 30 mD.

**Figure 9.**Comparison of leakage rate profiles with different flow resistances in the abandoned well (

**A**) and different well spacing (

**B**).

Porosity for both layers (dimensionless) | 0.3 |

Rock compressibility (1/psi) | 9 × 10^{−6} |

Fluid compressibility (1/psi) | 3 × 10^{−6} |

Fluid density (lb/ft^{3}) | 62 |

Permeabilities of both layers (mD) | 10 |

Viscosity (cp) | 1 |

Injection rate (STB/D) | 100 |

Well distance (ft) | 60 |

Wellbore radius of the abandoned well (ft) | 0.5 |

Resistance of abandoned well (psi/(STB/D)) | 1 |

Layer thickness (ft) for both layers | 100 |

Reservoir thickness (ft) | 300 |

Reservoir length (ft) | 10,100 |

Reservoir width (ft) | 10,100 |

Number of grid blocks in the x-direction | 101 |

Number of grid blocks in the y-direction | 101 |

Number of grid blocks in the z-direction | 10 |

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**MDPI and ACS Style**

Qiao, T.; Hoteit, H.; Fahs, M. Semi-Analytical Solution to Assess CO_{2} Leakage in the Subsurface through Abandoned Wells. *Energies* **2021**, *14*, 2452.
https://doi.org/10.3390/en14092452

**AMA Style**

Qiao T, Hoteit H, Fahs M. Semi-Analytical Solution to Assess CO_{2} Leakage in the Subsurface through Abandoned Wells. *Energies*. 2021; 14(9):2452.
https://doi.org/10.3390/en14092452

**Chicago/Turabian Style**

Qiao, Tian, Hussein Hoteit, and Marwan Fahs. 2021. "Semi-Analytical Solution to Assess CO_{2} Leakage in the Subsurface through Abandoned Wells" *Energies* 14, no. 9: 2452.
https://doi.org/10.3390/en14092452