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. 2019 Dec 5;4(25):21381–21394. doi: 10.1021/acsomega.9b02668

Analytical Model for Leakage Detection in CO2 Sequestration in Deep Saline Aquifers: Application to ex Situ and in Situ CO2 Sequestration Processes

Mohammad Ali Ahmadi 1,*, Zhangxin Chen 1
PMCID: PMC6921673  PMID: 31867533

Abstract

graphic file with name ao9b02668_0021.jpg

One of the prominent methods for carbon dioxide sequestration is disposal into deep saline aquifers. This is mainly because deep saline aquifers provide significant capacity for storage of unwanted fluids underground for a long period. However, saline aquifers may have a leaky cap rock. The sealing capacity of a cap rock must, therefore, be evaluated to ensure the integrity and safety of its storage media; hence robust classifications of the cap rock are required even before starting the storage/disposal operations. Aqueous fluids can be injected into a target storage aquifer, and pressure changes owing to leakage can be monitored in an upper aquifer separated by a cap rock for a short period. The measurement of pressure responses in the monitoring aquifer can be used to identify and characterize any leakage path in the cap rock. This paper provides analytical models in the Laplace domain for both in situ and ex situ CO2 sequestration methods. Using the numerical Laplace inverse method called the “Stehfest” method, the analytical solution in the real domain is calculated. The analytical solutions developed can be used for determining both dimensionless pressure changes in monitoring and storage aquifers due to leakages and dimensionless leakage rates.

1. Introduction

Underground saline aquifers are widely used for the disposal and storage of waste and unwanted fluids. Due to their large capacity, deep saline aquifers and depleted oil reservoirs are now the main candidates for geological storage of CO2 as a means to cut anthropogenic CO2 emissions.1−7 There are two options for carbon dioxide sequestration in a deep saline aquifer: in situ and ex situ sequestration. In the in situ approach, carbon dioxide directly is injected to the aquifer; however, in the ex situ method, carbon dioxide converted to carbonates and then is injected to the aquifer.8−10 Also, the interaction between the rock and fluid in situ and ex situ processes is different. Cui et al.11 investigated the comprehensive study on building rock and fluid interaction models through sensitivity analysis on several operational parameters and reservoir characteristics. Unlike petroleum reservoirs, saline aquifers usually do not have competent sealing cap rocks. To have safe storage and disposal, classifying a cap rock competence is essential to detect any leaky pathway.12,13 An operator may then decide to seal the pathway or to inject far away from the leakage.14−26

There are various natural ways to facilitate the leakage phenomenon in CO2 sequestration such as fractures in carbonates.27−29 One of the examples comes to mind to highlight the existence of large fractures in carbonates is the Wabamun Lake Sequestration Project (WASP). In this project, fractures could be as large as 1 km2.30 Different methods have been developed to characterize CO2 leakage in a CO2 sequestration process. Deng et al.31 performed a numerical simulation on Rock Spring Uplift, Wyoming, to determine the leakage, injectivity, and storage capacity in different cases. They attempted to figure out the effect of heterogeneities on these variables. Shakiba and Hosseini32 employed the fast Fourier transform (FFT) method to drive a periodic pressure response; their solution only considered pressure data in the monitoring and pulsing wells. Another approach is using hydraulic controls, which provide a method for remediation of CO2 leakage from storage reservoirs. This method provides a solution in the case where the geographic location of a leakage point is uncertain. Zahasky and Benson6 evaluated three different leakage intervention strategies, including CO2 injection shutoff, CO2 production from a reservoir, and hydraulic controls. They constructed various models to investigate how residual trapping, a leakage detection time, fault permeability, and heterogeneity influenced the efficacy of different interference plans.6

Different methods can be employed for detecting CO2 leakage during the sequestration process.33−37 For instance, using the temperature distribution near the wellbore to detect the near-wellbore leakage in CO2 sequestration.33,38 Viswanathan et al.39 proposed a hybrid model considering different data from various sources to predict the overall performance of CO2 sequestration. These data sources include experimental data in the lab, power plant data, and data from the sequestration site employed in the system-level-based model to predict sequestration performance. Carroll et al. assessed the effect of water quality on the CO2 sequestration process and performed sensitivity analysis on different parameters. Jordan et al.40 developed a reduced-order model for predicting carbon dioxide leakage in a cemented wellbore through the sequestration process using a response surface method (RSM). They employed Monte Carlo to perform sensitivity analysis and figure out the most important parameters in their model. Pawar et al.41 proposed an integrated risk assessment approach to figure out the risk profile of CO2 leakage during the sequestration process and to provide a more realistic decision-making method, especially for long-term sequestration processes. Harp et al.42 developed a reduced-order model for determining leakage rate, near wellbores using a multivariable adaptive regression approach. Their model can be employed in a case of leakage in transient flow regime; this type of flow regime occurs in near-wellbore conditions.

Susanto et al.43 proposed a monitoring technique to detect CO2 leakage in geological storage pilots employing hydrogen gas as a tracer. Jenkins et al.3 proposed a simple atmospheric monitoring technique for a CCS site. However, they were not able to test recovery of leak volumes thoroughly, because of uncertainty about an effective release height, but their technique was true within a factor of two, which is enough for early alarm-sounding and could be much enhanced in a certain CCS setting.3

Owing to the fast propagation of pressure, pressure monitoring can be employed as an efficient tool for detection and characterization of leakage from an injection zone to overlying zones.15,17,19,21,22,24−26 Migration through a cap rock causes a pressure change in its upper aquifers. Analytical models to determine a pressure change due to leakage through a fault have developed in the literature.32,44−51

There are lots of researches have been done for leakage characterization in toxic material disposal to the underground water supply, for instance, the study performed by Javandel et al.52 and Aller.53 Javandel et al.52 proposed a model based on the analytical solution of the diffusivity equation developed for a horizontal bed configuration; their model is useful for a case of CO2 leakage from a horizontal aquifer across an abandoned well. In their proposed model, through monitoring well the pressure at the storage aquifer could be monitored. Their developed approach considered single-phase flow with a constant density. There are various researchers, who made an effort to improve the solutions provided by Javandel et al.,52 for example, Silliman and Higgins,54 Avci,55 Avci,56 and Nordbotten et al.57

Zeidouni et al.23 developed an analytical approach to characterize and detect a leakage path in a CO2 sequestration process. In their model, they considered single-phase flow through the leakage pathway and aquifers. It should be mentioned that they examined the injection fluid as an aqueous one. Consequently, if the injected fluids are nonaqueous, their model cannot be directly employed. Also, prior to the sequestering process, contamination tests must be performed.

Zeidouni and Pooladi-Darvish24 introduced a general idea of a forward approach, where a relationship between a pressure change in the upper aquifer and leakage parameters was developed. They parameterized the corresponding inverse problem and analyzed its solution. They investigated the uniqueness of the solution via a Hessian matrix and an analytical method. Also, they studied the stability of the solution based on a correlation matrix and sensitivity coefficients. They investigated the convergence issue of several deterministic optimization approaches for leakage parameter prediction and estimated the required parameters for noise-free data. Also, they investigated the impact of noises and presented the results regarding a confidence interval.26 Zeidouni and Pooladi-Darvish25 provided an asymptotic solution to calculate real-time pressure for leakage detection. Using their advanced approach, a coefficient of leakage and a term including location parameters can be approximated explicitly. To evaluate the place of leakage, they have incorporated the data explicitly extracted by matching the pressure data.

Zeidouni)37 developed an analytical approach considering a bounded, layered reservoir in which all of the layers connected by a leaky path. He considered single-phase flow through the leakage pathway and aquifers. He presented the analytical solution in a Laplace domain regarding a real-time asymptotic solution and extended his model to multilayer reservoirs. Ahmadi and Chen14 proposed a one-dimensional linear model for calculating the leakage rate in bounded saline aquifers using pressure changes in monitoring aquifer.

In this work, the analytical models appropriate for leakage from local weakness in a cap rock are developed. The analytical solutions for pressure changes in both storage and monitoring aquifers along with a leakage rate are obtained in a Laplace domain. It should be noted that all of the developed equations are dimensionless and can be employed in both small and large cases. To evaluate the developed analytical solutions, two different synthetic cases and sensitivity analysis are performed on the most relevant parameters. Moreover, dimensionless parameters are defined to simplify the proposed analytical solution.

2. Model Description and Governing Equations

The one-dimensional physical model used for in situ CO2 sequestration is the same as those proposed by Ahmadi and Chen,14 Zeidouni et al.24Figure 1a shows the configuration of the wells as well as variables used for in situ CO2 sequestration process. In a case of in situ CO2 sequestration, there are two aquifers; the lower is considered as storage and the upper one is used as a monitoring aquifer. Also, there is a thin layer between those aquifers that the leakage happens from the leaky path on it.14,23−25 There are different assumptions made in this paper such as leakage direction is only vertical, there is no gravity effect, both injected fluid and brines in the aquifers have the same properties, both monitoring and storage aquifers are homogenous and isotropic, and the threshold pressure of the cap rock is ignored. In Figure 1, the distance between the leak location and injection well denoted by XA, Xe stands for the spacing between the injection and monitoring wells, rate of injection and monitoring are qin and qm, respectively. Permeability, height, and area of the formation denoted by K, h, and A, correspondingly. Subscript “A” stands for the storage aquifer, “l” represents the leakage path, and “m” denotes the monitoring aquifer. In the case of in situ CO2 sequestration, the injection rate is equal to q. The ratio of production to the injection rate is equal to a in the ex situ CO2 sequestration process.14,23−25Figure 1b depicts the configuration of the wells (injection and production wells) in the case of ex situ CO2 sequestration process.

Figure 1.

Figure 1

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Schematic of the problem considered in this study (a) in situ sequestration and (b) ex situ sequestration.

3. Analytical Solution

3.1. Pressure Response of the Monitoring Aquifer

To determine the response of the pressure in the monitoring aquifer in a case of in situ CO2 sequestration, the diffusivity equation is solved for pressure in this aquifer in response to the unknown leakage rate as follows14

3.1. 1

ηm stands for the diffusivity coefficient of the monitoring aquifer. The initial and boundary conditions can be expressed using the following equations

3.1. 2
3.1. 3
3.1. 4

where Pmi is the initial pressure of the monitoring aquifer, qi(t) denotes the leakage rate, B represents the formation volume factor, and Km and Am defined, as in Figure 1. Using the dimensionless variables reported in Table 1, the dimensionless form of the diffusivity equation is derived as follows

3.1. 5
3.1. 6
3.1. 7
3.1. 8

Solving eq 5 with the above-mentioned initial and boundary conditions results in the following equation for calculating the dimensionless pressure response of the monitoring aquifer due to the leakage process.

3.1. 9

The two important dimensionless groups that appear in the solution are ηD and TD. While ηD signifies the ratio of diffusivities in the monitoring to the storage aquifers, TD is the ratio of transmissivity (permeability × area) in the monitoring to the storage aquifers. Hereafter, the term “transmissivity ratio” will refer to TD and the “diffusivity ratio” will denote ηD.

Table 1. Dimensionless Variables used for Driving the Dimensionless Form of Governing Equations.

XD = X/Xe
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XAD = XA/Xe
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3.2. Pressure Response of the Storage Aquifer

To specify a pressure response throughout the storage aquifer, we have to determine a change of aquifer pressure owing to injection and the other pressure changes due to the leak. Then, via the method of superposition, we are able to calculate the pressure change in the storage aquifer. The following sections present the details of pressure change calculations.

3.2.1. Effect of Injection

To assess the influence of the injection well, we write the equations such that the storage aquifer is centered at the injection well (defined as X = 0 at the injection well) assuming no leakage. The unknown of the problem is only the pressure. The diffusivity equation under this condition is as follows14

3.2.1. 10

The initial and boundary conditions in the storage aquifer due to the injection are expressed in the below equations

3.2.1. 11
3.2.1. 12
3.2.1. 13

where Psi is the initial pressure of the storage aquifer, qin denotes the injection rate, B represents the formation volume factor, and Ks and As defined as in Figure 1. Using the dimensionless variables reported in Table 1, the dimensionless form of the diffusivity equation is derived as follows14

3.2.1. 14
3.2.1. 15
3.2.1. 16
3.2.1. 17

Using the dimensionless variables and the dimensionless boundary and initial conditions results in the following equation for calculating the dimensionless pressure change in the storage aquifer due to the injection

3.2.1. 18

It should be noted that if one is interested in calculating the effect of the injection on the storage aquifer pressure response at the leakage location, eq 18 is evaluated at XD = XAD. Using the inverse Laplace of eq 18 leads the below equation in the real domain that represents the dimensionless form of the storage aquifer pressure response owing to the injection

3.2.1. 19
3.2.1. 20

3.2.2. Effect of Leakage

In this case, the diffusivity equation is written with the center at the leakage path in the absence of any injection as follows14

3.2.2. 21

The initial and boundary conditions in the storage aquifer due to the leakage are expressed in the following equations

3.2.2. 22
3.2.2. 23
3.2.2. 24

Using the dimensionless variables reported in Table 1, the dimensionless form of the diffusivity equation for the response of the storage aquifer pressure owing to the leakage is derived as follows

3.2.2. 25
3.2.2. 26
3.2.2. 27
3.2.2. 28

Using the above-mentioned dimensionless boundary and initial conditions, eEq 25 is solved and the dimensionless pressure response of the storage aquifer owing to the leakage can be calculated as follows

3.2.2. 29

To evaluate the storage aquifer pressure response owing to leakage at the leakage location, one needs to evaluate eEq 29 when XD = XAD.

3.2.3. Effect of Production

To assess the influence of the production well, we define a well location X = Xe as the production well location, assuming no leakage. Also, we consider that the production rate is constant, and it can be written as a ratio to the injection rate. So, in ex situ condition, we haveInline graphic. The unknown of the problem is only the pressure. The diffusivity equation under this condition is as follows

3.2.3. 30

The initial and boundary conditions in the storage aquifer due to the production are expressed in the following equations

3.2.3. 31
3.2.3. 32
3.2.3. 33

where, qprod denotes the production rate, and all of the parameters are the same as previous ones. Using the dimensionless variables reported in Table 1, the dimensionless form of the diffusivity equation is derived as follows

3.2.3. 34
3.2.3. 35
3.2.3. 36
3.2.3. 37

Using the dimensionless variables and the dimensionless boundary and initial conditions results in the below equation for evaluating the dimensionless storage aquifer pressure response owing to the production

3.2.3. 38

3.3. Superposition Approach

3.3.1. In Situ CO2 Sequestration Case

In this section, the solution derived for a case of in situ CO2 sequestration is presented in the following.

3.3.1.1. Pressure Response at the Monitoring Well

As mentioned before, to examine the dimensionless pressure response at the monitoring well, we have to calculate eq 9 when XD = 1

3.3.1.1. 39

To calculate the dimensionless leakage rate, one needs to have both dimensionless pressures in the storage and monitoring aquifers at the leakage location (XAD). Using Eeq 9 and the combination of Eeqs 18 and 29 at the leakage location (XAD) (the storage aquifer pressure response at the leakage location due to the injection and leakage) results in the dimensionless leakage rate as follows

3.3.1.1. 40

where

3.3.1.1. 41

Rearrangement of Eeq 40 yields the following equation

3.3.1.1. 42

To calculate the leakage rate at the leakage location, we have to determine the inverse Laplace of Eeq 42 as follows

3.3.1.1. 43

However, determining an analytical inverse Laplace of the above equation is impossible; we used the Stehfest approach to find the solution behavior in a real-time domain.

3.3.1.1. 44

Rearrangement of Eeq 44 yields the following equation

3.3.1.1. 45

To calculate the monitoring aquifer pressure response at the location of the monitoring well, we have to determine the inverse Laplace of Eeq 45 as follows

3.3.1.1. 46
3.3.1.2. Pressure Response in the Storage Aquifer

As mentioned before, the pressure response at the leakage path in the storage aquifer can be determined using the superposition of the pressure changes in the storage aquifer owing to the leakage and injection. Using the superposition approach yields the following equation for determining the storage aquifer pressure response

3.3.1.2. 47

Rearranging Eeq 47 yields the following equation

3.3.1.2. 48

Recalling Eeq 42 and substituting it into Eeq 48 result in the following equation

3.3.1.2. 49

Equation 49 is employed for evaluating the storage aquifer pressure response; however, the inverse Laplace of the equation above cannot be determined. Consequently, to calculate the pressure change using Eeq 49, we have to use the numerical Laplace inversion methods, i.e., the Stehfest method.

3.3.2. Ex Situ CO2 Sequestration

The only difference between the in situ and ex situ CO2 sequestration studied in this paper is using the production well that completed in the storage aquifer. The following section provides the solution for the pressure response owing to the production from the storage aquifer. It is worth to mention that all of the other sections are the same for both in situ and ex situ carbon dioxide sequestration.

3.3.2.1. Pressure Change at the Production Well

As mentioned before, to evaluate the dimensionless pressure response at the production well, we have to calculate Eeq 9 when XD = 1

3.3.2.1. 50

To calculate the dimensionless leakage rate, one needs to have both dimensionless pressures in the storage and monitoring aquifers at the leakage location (XAD). Based on the Darcy equation, we have14,23,24

3.3.2.1. 51

From the dimensionless pressure definition, we have

3.3.2.1. 52
3.3.2.1. 53

Substitution of Eeqs 52 and 53 into Eeq 51 results in

3.3.2.1. 54

where

3.3.2.1. 55

We assumed that the second term of Eeq 54 will be zero because of a long period of leakage process and we have

3.3.2.1. 56

Replacing PDS and PDm when XD = XAD leads the equation below

3.3.2.1. 57

where

3.3.2.1. 58

Rearranging eq 57 results in the following equation

3.3.2.1. 59

To calculate the leakage rate at the leakage location, we have to determine the inverse Laplace of Eeq 59 as follows

3.3.2.1. 60

However, determining an analytical inverse Laplace of the above equation is impossible; same as previous sections, we used the Stehfest approach for evaluating the solutions in a real-time domain.

3.3.2.1. 61

To determine the monitoring aquifer pressure response at the monitoring well location, we have to determine the inverse Laplace of Eeq 61 as follows

3.3.2.1. 62
3.3.2.2. Pressure Response in the Storage Aquifer

As mentioned before, the pressure change at the leakage path in the storage aquifer can be calculated using the superposition of the pressure changes in the storage aquifer owing to the leakage, production, and injection. Using the superposition approach yields the following equation for evaluating the storage aquifer pressure response

3.3.2.2. 63

Rearranging Eeq 63 results in the equation below

3.3.2.2. 64

Equation 64 is employed for calculating the pressure change in the storage aquifer; however, the inverse Laplace of the equation above cannot be determined. Consequently, to calculate the pressure change using Eeq 64, we have to use the numerical Laplace inversion methods, i.e., the Stehfest method.

4. Results and Discussion

4.1. In Situ CO2 Sequestration Problem

To examine the pressure responses of both the storage and monitoring aquifers along with the leakage rate, a numerical Laplace inverse method called the Stehfest method is employed. Two different cases are considered to determine the effects of the most important parameters on the pressure changes in both the storage and monitoring aquifers accompanied by the leakage rate. Moreover, a sensitivity analysis of the leakage path location and dimensionless diffusivity is performed in both cases. The following sections present the details in each case.

4.2. Synthetic Case 1

The analytical solutions are applied to this synthetic case in which the dimensionless storage aquifer pressure response at the monitoring well and the dimensionless leakage rate are calculated. In this case, we consider the injection rate of 0.02 m3/s. The thickness of the storage aquifer, monitoring aquifer, and the impermeable layer is equal to 30, 45, and 16 m, respectively. The permeability of the monitoring aquifer is equal to 2 × 10–13 m2, and the permeability of the storage aquifer and the leaky path is 5 × 10–13 and 2.5 × 10–15 m2, respectively. All of the parameters for the synthetic case 1 are reported in Table 2.

Table 2. Properties of the Storage and Monitoring Aquifers in the Synthetic Case 1.

parameter value parameter value parameter value parameter value
hm 30 Xe 100 km 2 × 10–13 ηm 2.67
hl 16 TD 0.026667 kl 2.5 × 10–15 ηs 5
hs 45 μ 0.0005 ks 5 × 10–13 ηD 0.533
q 0.02 ϕm 0.15 cs 1 × 10–9    
XA 50 Φs 0.2 cm 1 × 10–9    
XB 50 Bw 1 As 4500    
XAD 0.5 Am 3000 Al 1600    
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Figure 2 demonstrates the behavior of the dimensionless leakage rate using eq 43. As depicted in this figure, changing the dimensionless diffusivity coefficient (ηD) affects the dimensionless leakage rate considerably. In this case, the important factor is the ratio of Km/Ks because the order of magnitude for porosity values is the same. Hence, an increase or decrease in the dimensionless diffusivity corresponds to the role of each permeability value, which contributes directly or inversely to the leakage. Increasing the permeability value of the storage aquifer reduces the dimensionless diffusivity coefficient (ηD). As shown in Figure 2, the lower the values of the dimensionless diffusivity coefficient (ηD), the higher the dimensionless leakage rate. The story is the same as monitoring aquifer, the higher the permeability of monitoring aquifer, the higher the value of dimensionless diffusivity coefficient (ηD). The higher the values of dimensionless diffusivity coefficient (ηD), the lower the value of dimensionless leakage rate. It should be noted that in this case, we consider that the location of the leakage has the same distance from both the injection and monitoring wells.

Figure 2.

Figure 2

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Dimensionless leakage rate (QD) vs dimensionless time (tD) in a case of in situ CO2 sequestration considering different dimensionless diffusivity coefficients (ηD) [0.01:0.01:0.1] when XAD = 0.5.

Figure 3 depicts the behavior of the dimensionless leakage rate against the corresponding dimensionless time (tD) considering the different dimensionless leakage location (XAD) when the dimensionless diffusivity coefficient is equal to 0.533. As illustrated in this figure, increasing the XAD results in delaying the leakage. This is mainly because the time required for the pressure front to reach the leaky path depends on the leakage location. Consequently, increasing the distance between the leaky path and the injection well results in the pressure front to take more time to reach the leakage well.

Figure 3.

Figure 3

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Dimensionless leakage rate (QD) vs dimensionless time (tD) in a case of in situ CO2 sequestration considering different dimensionless distance between leakage path and injection well (XAD) [0.1:0.1:0.9] when ηD = 0.533.

Figure 4 demonstrates the behavior of the PDs at the monitoring well in the storage aquifer vs the dimensionless time (tD) considering different ηD when XAD is equal to 0.5. As shown in this figure, increasing the ηD does not have an impact on the dimensionless pressure in the storage aquifer at the earlier time. However, at the late time, the effect of the ηD on the PDs is recognizable. Increasing the ηD decreases the PDs at the late time.

Figure 4.

Figure 4

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Dimensionless pressure (PDs) at the location of monitoring well in the storage aquifer vs dimensionless time (tD) in a case of in situ CO2 sequestration considering different dimensionless diffusivity coefficients (ηD) [0.01:0.01:0.1] when XAD = 0.5.

Figure 5 shows the behavior of the PDs at the monitoring well in the storage aquifer vs the dimensionless time (tD) considering a different XAD when the ηD is equal to 0.533. As illustrated in this figure, increasing the XAD results in no considerable change in the PDs at the location of the monitoring well in the storage aquifer.

Figure 5.

Figure 5

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Dimensionless pressure (PDs) at the location of monitoring well in the storage aquifer vs dimensionless time (tD) in a case of in situ CO2 sequestration considering the different dimensionless distance between the leakage path and injection well (XAD) [0.1:0.1:0.9] when ηD = 0.533.

Figure 6 demonstrates the behavior of PDs at the monitoring well against the dimensionless time (tD) considering different ηD when XAD is equal to 0.5. As mentioned before, the ηD is defined as the ratio of the diffusivity coefficient in the monitoring aquifer over the diffusivity coefficient in the storage aquifer. This means that increasing the diffusivity coefficient in the monitoring aquifer increases the ηD. As a result, as clearly seen from Figure 6, increasing the ηD increases the PDm.

Figure 6.

Figure 6

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Dimensionless pressure (PDm) at the monitoring well vs dimensionless time (tD) in a case of in situ CO2 sequestration considering different dimensionless diffusivity coefficients (ηD) [0.01:0.01:0.1] when XAD = 0.5.

Figure 7 illustrates the analytical solutions of the PDm at the monitoring well against the dimensionless time (tD) considering various XAD when ηD is equal to 0.533. As demonstrated in this figure, at the late time, increasing the XAD increases slightly the PDm.

Figure 7.

Figure 7

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Dimensionless pressure (PDm) at the monitoring well vs dimensionless time (tD) in a case of in situ CO2 sequestration considering the different dimensionless distance between the leakage path and injection well (XAD) [0.1:0.1:0.9] when ηD = 0.533.

4.3. Synthetic Case 2

In this case, the only changed parameter is the permeability of the monitoring aquifer, which is equal to 2 × 10–12 m2. As clearly seen, the ηD, in this case, is 10 times that in the previous case. This is because we will determine the effect of the parameters in both cases on the matter whether the ηD is large or small. All of the parameters for the synthetic case 2 are reported in Table 3.

Table 3. Properties of the Storage and Monitoring Aquifers in the Synthetic Case 2.

parameter value parameter value parameter value parameter value
hm 30 Xe 100 km 2 × 10–12 ηm 26.67
hl 16 TD 0.26667 kl 2.5 × 10–15 ηs 5
hs 45 μ 0.0005 ks 5 × 10–13 ηD 5.333
q 0.02 ϕm 0.15 cs 1 × 10–9    
XA 50 Φs 0.2 cm 1 × 10–9    
XB 50 Bw 1 As 4500    
XAD 0.5 Am 3000 Al 1600    
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Figure 8 demonstrates the behavior of the dimensionless leakage rate in the synthetic case 2 using eq 34. As depicted in this figure, changing the ηD from 0.000533 to 5.333 affects the dimensionless leakage rate considerably. Increasing ηD decreases the dimensionless leakage rate. In this case, there is a competitive effect between the permeability of the monitoring aquifer and storage aquifer because all of the other parameters are kept constant, and values for the porosity have the same order of magnitude. As a result, the permeability of monitoring and storage aquifers inversely affected the dimensionless leakage rate. The higher the permeability of the monitoring and storage aquifers, the lower the dimensionless leakage rate.

Figure 8.

Figure 8

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Dimensionless leakage rate (QD) vs dimensionless time (tD) in a case of in situ CO2 sequestration considering different dimensionless diffusivity coefficients (ηD) [0.000533 0.00533 0.0533 0.5333 5.333] when XAD = 0.5.

Figure 9 shows the behavior of the dimensionless leakage rate vs the corresponding dimensionless time (tD) considering a different XAD when the dimensionless diffusivity coefficient is equal to 5.33. As illustrated in this figure, increasing the XAD results in delaying the leakage. Moreover, the comparison between the results presented in Figures 3 and 9 reveals that the delay period when ηD = 5.33 is greater than the case in which ηD = 0.533.

Figure 9.

Figure 9

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Dimensionless leakage rate (QD) vs dimensionless time (tD) in a case of in situ CO2 sequestration considering different dimensionless distance between the leakage path and injection well (XAD) [0.1:0.1:0.9] when ηD = 5.33.

Figure 10 demonstrates the behavior of the PDs at the monitoring well in the storage aquifer vs the dimensionless time (tD) considering different ηD when XAD is equal to 0.5. As shown in this figure, increasing the ηD affects the dimensionless pressure noticeably in the storage aquifer at the late time.

Figure 10.

Figure 10

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Dimensionless pressure (PDs) at the location of monitoring well in the storage aquifer vs dimensionless time (tD) in a case of in situ CO2 sequestration considering different dimensionless diffusivity coefficients (ηD) [0.000533 0.00533 0.0533 0.5333 5.333] when XAD = 0.5.

Figure 11 shows the behavior of the PDs at the monitoring well in the storage aquifer vs the dimensionless time (tD) considering a different XAD when the ηD is equal to 5.33. As illustrated in this figure, increasing the XAD results in no considerable change in the PDs at the location of the monitoring well in the storage aquifer.

Figure 11.

Figure 11

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Dimensionless pressure (PDs) at the location of monitoring well in the storage aquifer vs dimensionless time (tD) in a case of in situ CO2 sequestration considering the different dimensionless distance between the leakage path and injection well (XAD) [0.1:0.1:0.9] when ηD = 5.33.

Figure 12 illustrates the behavior of the PDm at the monitoring well vs the dimensionless time (tD) considering various ηD when XAD is equal to 0.5. As clearly seen from this figure, increasing the ηD from 0.000533 to 5.333 increases the PDm at the monitoring well considerably at the late time.

Figure 12.

Figure 12

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Dimensionless pressure (PDm) at the monitoring well vs dimensionless time (tD) in a case of in situ CO2 sequestration considering different dimensionless diffusivity coefficients (ηD) [0.000533 0.00533 0.0533 0.5333 5.333] when XAD = 0.5.

Figure 13 depicts the analytical solutions of the PDm at the monitoring well against the dimensionless time (tD) considering a different XAD when ηD is equal to 5.33. As demonstrated in this figure, at the late time, increasing the XAD results in a very small change in the PDm at the monitoring well.

Figure 13.

Figure 13

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Dimensionless pressure (PDm) at the monitoring well vs dimensionless time (tD) in a case of in situ CO2 sequestration considering the different dimensionless distance between the leakage path and injection well (XAD) [0.1:0.1:0.9] when ηD = 5.33.

4.4. Ex Situ CO2 Sequestration Problem

In the case of ex situ carbon dioxide sequestration, we consider the constant ratio between the production rate and injection rate a = 0.2. All of the parameters are the same as the synthetic case 1 for in situ CO2 sequestration. Figure 14 demonstrates the behavior of the dimensionless leakage rate in the case of ex situ CO2 sequestration using eq 63. As depicted in this figure, changing the ηD affects the dimensionless leakage rate considerably. Increasing ηD decreases the dimensionless leakage rate. It should be noted that in this case, we consider that the location of the leakage has the same distance from both the injection and production wells.

Figure 14.

Figure 14

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Dimensionless leakage rate (QD) vs dimensionless time (tD) for in situ CO2 sequestration considering different dimensionless diffusivity coefficients (ηD) [0.000533 0.00533 0.0533 0.5333 5.333 53.333] when XAD = 0.5.

Figure 15 shows the trend of the dimensionless leakage rate change against dimensionless time at the different XAD in a case of ex situ CO2 sequestration. As shown in this figure, changing the leakage path location (XAD) affects the dimensionless leakage rate noticeably. Increasing XAD results in delaying the leakage; the ultimate leakage rates in all leakage locations are the same. It reveals that the ultimate leakage rate is independent of the leakage location.

Figure 15.

Figure 15

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Dimensionless leakage rate (QD) vs dimensionless time (tD) in a case of ex situ CO2 sequestration considering different dimensionless distance between the leakage path and injection well (XAD) [0.1:0.1:0.5] when ηD = 0.533.

Figure 16 demonstrates the trend of the dimensionless leakage rate in a case of ex situ CO2 sequestration against dimensionless time at different leakage path permeabilities. As depicted in this figure, changing the leakage path permeability (kl) affects the dimensionless leakage rate noticeably. Increasing the leakage path permeability (kl) increases the dimensionless leakage rate.

Figure 16.

Figure 16

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Dimensionless leakage rate (QD) vs dimensionless time (tD) in a case of ex situ CO2 sequestration considering different leakage path permeabilities [2.5 × 10–19 2.5 × 10–18 2.5 × 10–17 2.5 × 10–16] when ηD = 0.533.

Figure 17 demonstrates the behavior of the dimensionless leakage rate in a case of ex situ CO2 sequestration against dimensionless time at different thicknesses of monitoring aquifer. As depicted in this figure, changing the monitoring aquifer thickness (hm) affects the dimensionless leakage rate noticeably. Increasing the monitoring aquifer thickness (hm) increases the dimensionless leakage rate.

Figure 17.

Figure 17

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Dimensionless leakage rate (QD) vs dimensionless time (tD) in a case of ex situ CO2 sequestration considering different monitoring aquifer thicknesses (hm) [1 10 100] when ηD = 0.533.

Figure 18 depicts the behavior of the PDm at the production well vs the dimensionless time (tD) in a case of ex situ CO2 sequestration considering different ηD when XAD is equal to 0.5. As clearly seen from this figure, increasing the ηD from 0.000533 to 5.333 increases the PDm at the production well considerably at the late time.

Figure 18.

Figure 18

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Dimensionless pressure (PDm) at the production well vs dimensionless time (tD) in a case of ex situ CO2 sequestration considering different dimensionless diffusivity coefficients (ηD) [0.000533 0.00533 0.0533 0.5333 5.333] when XAD = 0.5.

Figure 19 depicts the behavior of the PDm at the production well vs the dimensionless time (tD) in a case of ex situ CO2 sequestration considering different XAD when ηD is equal to 0.533. As clearly seen from this figure, increasing the XAD from 0.1 to 0.5 increases the PDm at the production well considerably at the late time.

Figure 19.

Figure 19

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Dimensionless pressure (PDm) at the production well vs dimensionless time (tD) in a case of ex situ CO2 sequestration considering the different dimensionless distance between the leakage path and injection well (XAD) [0.1:0.1:0.5] when ηD = 0.533.

Figure 20 depicts the behavior of the PDm at the production well vs the dimensionless time (tD) in a case of ex situ CO2 sequestration considering different monitoring aquifer thickness (hm) when XAD is equal to 0.5. As clearly seen from this figure, increasing the monitoring aquifer thickness (hm) from 1 to 100 does not have any effect on the PDm at the production well. This means that the PDm at the production well is independent of the thickness of the monitoring aquifer.

Figure 20.

Figure 20

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Dimensionless pressure (PDm) at the production well vs dimensionless time (tD) in a case of ex situ CO2 sequestration considering different monitoring aquifer thicknesses (hm) [1 10 100] when XAD = 0.5.

Figure 21 demonstrates the behavior of the PDs at the location of the production well in the storage aquifer vs the dimensionless time (tD) in a case of ex situ CO2 sequestration considering different storage aquifer thickness (hs) when ηD is equal to 0.533. As illustrated in this figure, increasing the storage aquifer thickness (hs) increases the dimensionless pressure in the storage aquifer. It reveals that the dimensionless pressure of the storage aquifer at the location of production well clearly depends on storage aquifer thickness (hs).

Figure 21.

Figure 21

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Dimensionless pressure (PDs) at the location of a production well in the storage aquifer vs dimensionless time (tD) in a case of ex situ CO2 sequestration considering different storage aquifer thicknesses (hs) [1 10 100] when ηD = 0.533.

Figure 22 demonstrates the behavior of the PDs at the location of the production well in the storage aquifer vs the dimensionless time (tD) in a case of ex situ CO2 sequestration considering the XAD when ηD is equal to 0.533. As depicted in this figure, increasing the XAD does not have an impact on the PDs. It reveals that PDs at the production well is independent of leakage location.

Figure 22.

Figure 22

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Dimensionless pressure (PDs) at the location of a production well in the storage aquifer vs dimensionless time (tD) in a case of ex situ CO2 sequestration considering the dimensionless distance between the leakage path and injection well (XAD) [0.1:0.1:0.5] when ηD = 0.533.

Figure 23 demonstrates the behavior of the PDs at the location of the production well in the storage aquifer vs the dimensionless time (tD) in a case of ex situ CO2 sequestration considering different ηD when XAD is equal to 0.5. As shown in this figure, increasing the ηD does not have a clear consequence on the PDs at the earlier time. However, at the late time, the effect of the ηD on the dimensionless pressure of the storage aquifer is significant. Increasing the ηD increases the PDs at the late time.

Figure 23.

Figure 23

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Dimensionless pressure (PDs) at the location of production well in the storage aquifer versus dimensionless time (tD) in a case of ex situ CO2 sequestration considering dimensionless diffusivity coefficients (ηD) [0.000533 0.00533 0.0533 0.5333 5.333 53.333] when XAD = 0.5.

Those semianalytical models presented above can give practical and helpful information for both in situ and ex situ CO2 capture schemes in saline aquifers. Those models are capable of calculating the CO2 leakage rate due to induced microfractures or leaky paths. It is worth to stress that the outputs of such analytical and/or semianalytical just useable as a screening tool to determine the possible CO2 leakage rate in such aquifers.

5. Conclusions

Analytical models are developed to determine a dimensionless leakage rate and a dimensionless pressure response owing to leakage from the storage aquifer toward the monitoring aquifer for both in situ and ex situ CO2 sequestration processes. These models are obtained by solving the dimensionless form of the flow equations in the storage and monitoring aquifers, which are joined by a dimensionless flow rate at the leakage path. The principle of superposition is employed to calculate the storage aquifer pressure response at the leakage location. The exact analytical solutions are obtained in a Laplace domain, and the Stehfest method is employed to evaluate the analytical solutions in the real-time domain numerically. Two different cases in in situ process and one ex situ example have been considered to evaluate the behavior of the analytical solutions for both the leakage rate and the pressure response in the storage and monitoring aquifers.

The authors declare no competing financial interest.

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