A New Method Based on Boundary Element Method to Appraise CO2 Geological Storage Potential in Depleted Shale Gas Reservoirs

In recent years, greenhouse gases have increased in the atmosphere, and climate change concerns have triggered global efforts to find solutions for CO2 capture, separation, transport, and storage. Geological sequestration in the depleted unconventional reservoir is an effective measure to reduce the atmosphere’s CO2 content. The exact evaluation of the CO2 storage capacity can verify the feasibility of storing carbon dioxide and parameter optimization. A reasonable boundary element method to estimating the CO2 storage capacity of depleted shale gas reservoirs considering arbitrarily shaped boundaries is introduced. Firstly, the physical model with fracture networks is built based on the microseismic data. Then, the flow equation including the matrix and fracture can be obtained considering adsorption, and the star-delta transformation is used to deal with interconnected fracture segments. The point source function with an infinite boundary can be obtained after the Laplace transform method. Finally, the semianalytical flow solution is obtained by using the boundary element method in the Laplace region. Moreover, the results have a high agreement with commercial software for the regular boundary. The sensitivity of relevant parameters is analyzed by this method, and the importance of considering the boundary shape is emphasized. This method can evaluate the CO2 storage capacity of formation with the irregular boundary and is regarded as the guide of parameter optimization in CO2 storage.


Introduction
Greenhouse gases such as CO 2 have increased in the atmosphere leading to climate change concerns [1]. Global efforts have been made to reduce atmospheric CO 2 concentration by carbon sequestration, including geological sequestration, mineral carbonation, and ocean storage [2]. Furthermore, the CO 2 geological sequestration is a widely applied measure. Industrial analogs, including natural gas and acid gas injection projects, have proved that CO 2 can be safely injected and stored [3].
The gas injection into the reservoirs is widely used in petroleum engineering to enhance oil recovery, and it is fea-sible to store CO 2 in geological formations as a CO 2 reduction option [4]. In the beginning, most research on gas injection for improving oil recovery takes a typical block as an example, which is based on the black oil numerical simulation with less consideration on the mechanism. In 2009, Shoaib and Hoffman first conducted mathematical modeling studies on enhanced recovery by CO 2 flooding and CO 2 injection huff-n-puff based on the Elm Coulee field (0.01-0.04 mD) block Bakken unconventional reservoir [5]. Wan et al. studied the influence of the stimulated reservoir volume (SRV) characteristics on the gas injection and huff-npuff by numerical simulation, including the size of the SRV, fracture space, and stress sensitivity [6].
The research combined with the gas flooding mechanism in the experimental study and the component model was introduced to consider the miscibility based on the actual block data. Alharthy et al. [7] established a numerical model of gas injection stimulation and predicted the production rate of a horizontal well after volume fracturing in the field, which is based on the flow mechanism of CO 2 injection in the tight matrix proposed by Hawthorne et al., including the diffusion effect caused by crude oil expansion, viscosity reduction, and concentration difference [8]. Wan et al. [9] and Pu [10] established the component model about the gas huff-n-puff model considering the miscibility of crude oil and injected gas and carried out a numerical simulation study on enhanced oil recovery in the SRV region. Yu et al. [11] and Zhu et al. [12] used the component model to analyze the sensitivity of each influencing factor of gas flooding. Many numerical methods coupling business software are increasingly used due to advantages considering mechanism in the study of CO 2 -EOR simulation [13]. Zuloaga-Molero et al. [14] applied an EDFM (embedded discrete fracture model) to build a compositional reservoir model investigating the effects on CO 2 huff-n-puff and CO 2 continuous injection, including complex fracture geometries, CO 2 diffusion, permeability, and natural fractures. Sun et al. [15] applied the proposed fracture discretization approach to CO 2 huff-n-puff in complex fracture and obtained more accurate simulation results than the dual continuum approach.
Some scholars proposed that saline aquifers can be a good storage site because of their vast volumes. However, oil and gas reservoirs also are suitable for storage, and the geological conditions and storage scale are explicit in the oil and gas exploration. On the other hand, the production well and injection well can be used as carbon dioxide injection devices, which can significantly reduce the cost of remake injection equipment [16]. The depleted shale reservoir is very promising compared to the deep saline aquifer [17]. The depleted shale gas reservoir is a good choice for CO 2 storage because of its tight pore space as both a storage reservoir and a seal [18]. First, the CO 2 could sorb into the nanoscale pore structure and be immobilized within the shale reservoir rock [19], which is an additional trapping mechanism compared to those in the deep saline aquifers. Besides, large volumes of CO 2 could be emplaced in the fracture network formation [20]. Moreover, their extremely tight pore in the shale formations indicates that they are less likely than saline aquifers to leak CO 2 [18]. Godec et al. [21] suggested that the economic estimate is predicted to be 71 Tcm of enhanced methane recovery, facilitating 280 Gt of CO 2 storage in gas shales globally. In addition, some physical simulation results also show that the depleted shale reservoir for CO 2 storage is effective and feasible. Busch et al. [22] provide evidence that shale has significant CO2 storage capacity through diffusion experiments and adsorption experiments. Arif et al. [23] made wettability experiments. The conclusions are that the shale with low TOC is suitable to be used as a typical caprock, and when the TOC content is higher than 1.1 wt%, the shale reservoir can trap a large amount of CO 2 gas by adsorption. As a potential CO 2 stor-age sites, it is meaningful and essential to precisely evaluate the depleted shale gas reservoir capacity. CO 2 storage projects include depleted oil and gas reservoirs and saline aquifers, unmineable coal seams [17]. The research about depleted oil and gas reservoir evaluation method can be divided into the volume-or productionbased calculation method, numerical simulation method, and semianalytical model method [24]. The volume-or production-based calculation method proposes the method based on original gas in place to calculate the storage volume previously occupied by gas and oil, which can be replaced by CO 2 [25,26]. Kang et al. [27] and Tian et al. [28] estimated the reservoir capacity through rock samples' storage capacity and the overall reservoir volume. Abuov et al. [29] estimated effective CO 2 storage capacities in oil reservoirs, gas reservoirs, and saline aquifers for basins of Kazakhstan using the Carbon Sequestration Leadership Forum (CSLF) and United States Department of Energy (USDOE) methods. For the numerical simulation method, Ren and Duncan [28] discussed the effects of both injection strategies and reservoir heterogeneity on CO 2 storage by reservoir simulation of carbon storage associated with CO 2 EOR in residual oil zones in San Andres formation of West Texas Evgeniy. Their research showed that hydraulically fractured shale formations can be regarded as reservoirs for carbon dioxide sequestration. Andersen and Nilsen [30] evaluated CO 2 storage numerical simulation with thermal effects considering fluid properties and geomechanical stresses and the rate of geochemical reactions caused by temperature change around the injection well. Lashgari et al. [31] used CMG-GEM software to build a field-scale numerical simulation model to analyze the important physical mechanisms (miscibility and CO 2 adsorption, along with gas diffusion) of CO 2 storage and EOR. Liu et al. [32] evaluated the CO 2 sequestration ability of shale under three-dimensional complex geological conditions using a projection-based embedded discrete fracture model. For the semianalytical model method, many scholars study the CO 2 storage process using the semianalytical method by Laplace transformation, source function, and numerical discretion. Chen et al. [33,34] developed a quick analytical method for estimating the CO 2 storage capacity of depleted shale gas based on pressure transient analysis (PTA). Xiao et al. [24,35] quantitatively determined the abandonment pressure and analyze related impacts of dynamic parameters on it. Volume-based calculation methods simplified the CO 2 flow behavior and injection rate. Furthermore, the numerical simulation method needs modeling and mesh subdivision to apply to the field scale, which is time consuming. Semianalytical methods are suitable for rectangular or circular boundaries. However, there are few works that discussed the irregular boundary of formation. Therefore, it is vital to find efficient and convenient methods to consider the irregular boundary of formation.
Water in the shale will reduce the storage capacity due to gas-liquid two-phase flow and the occupied water volume. Moreover, the CO 2 -water-shale reaction has implications for CO 2 flow [36]. Chen et al. [33,34] evaluate the depleted shale CO 2 storage capacity by a single-phase flow for more convenient and quick evaluation. Similarly, Myshakin et al.

Geofluids
[37] only consider CO 2 flow, single-phase flow, to evaluate flow regimes and storage efficiency. So the model is simplified to derive a semianalytical single-phase flow solution with the complex boundary in this paper. The semianalytical solution evaluation is carried out under relatively ideal conditions. The boundary element method (BEM) has been used widely in fluid mechanics, fracture mechanics, and contact mechanics to solve linear partial differential equations formulated as integral equations. Many scholars have applied and modified the BEM into petroleum and environmental engineering [38][39][40]. The production and pressure calculation in forming the irregular boundary can be solved by BEM [41,42].
A quick and reasonable boundary element method for estimating CO 2 storage capacity of depleted shale gas formation considering arbitrarily shaped boundaries is introduced in this paper. This physical model is built based on the microseismic data, and BEM solves the mathematical model. The star-delta transformation deals with interconnected fracture segments, and the results have a high agreement with commercial software for the regular boundary. Furthermore, a real case with a horizontal well after large-scale hydraulic fracturing in depleted shale gas reservoirs is discussed. The influence of injection pressure, stress sensitivity coefficient, fracture permeability, and shale matrix permeability on the storage capacity was studied.

Methodology for Estimating CO 2
Storage Potential 2.1. Physical Model. Microseismic detection is used widely to analyze the underground situation, which is shown in Figure 1(a). There is a microseismic map, which uses acoustic signals to determine the underground fractures. The same color points represent the same fracture, and the dense colored dots indicate fractures. Therefore, fracture networks consist of hydraulic and secondary fractures, explained and derived by microseismograms shown as Figure 1(b). The black straight line represents the horizontal well, and the area within the black curve represents the stimulated reservoir volume (SRV). It can be seen that the SRV with irregular boundaries has hydrofractures to easily inject gas, which can be used as a CO 2 storage place.

Mathematics
Model and Solution. Figure 1(b) provides a schematic diagram of the horizontal well after large-scale hydraulic fracturing. The stimulated reservoir volume region shape is irregular. The following assumptions are provided: (1) Only isothermal single-phase flow exists

Horizontal well
Fracture segment 3 Geofluids (2) The SRV is a dual porosity continuum medium. Gas flows only from the matrix system into the fracture system (3) The gas flowing into the well from the formation is only caused by hydraulic fractures (4) CO 2 adsorbs in the reservoirs following the Langmuir isothermal adsorption law. The adsorption only exists in the matrix system [33,34].
(5) Hydraulic fractures and secondary fractures are infinite conductivity fractures (6) Gravity and capillary forces were ignored in the study (7) The stress sensibility exists in the fracture system leading to permeability changing with pressure [43].
The velocity equation and flow equation based on the law of conservation of mass can be written as follows: Matrix system: Fracture system: Table 1: Reservoir parameters and production parameters (data from the literature [24]).

Geofluids
Basic equations for CO 2 can be obtained from the following assumptions: CO 2 adsorption equation: Permeability-pressure equation The flow model, which consists of the matrix system and fracture system after dimensionless transformation, can be written as follows: Matrix system: The boundary condition can be written as Fracture system: The boundary condition can be written as A point source function is obtained by using Laplace transform, Pedrosa's substitution [38], and solution of the CO 2 flow model [33,34]. Moreover, the dimensionless definitions of this article are given in the appendix.
where   Figure 6: The cumulative CO 2 injection volume (storage volume) after 720 days.

Geofluids
The integral equation of the dimensionless pressure using Green's secondary identity can be written as The fracture segments can be considered as point sources written as Equation (9). And then, the second term on the right side of Equation (11) can be simplified as where the source function for the jth segment is given by The boundary is divided into many boundary elements by BEM, and the number of the element on a boundary Γ is N b , as shown in Figure 2(b). For the irregular zone, the discretized form of Equation (11) is Equation (14).
where ΔΓ DL is the Lth element on the boundary Γ D .
The finite difference method (FDM) is used to discretize the fracture flow formula, as shown in Figure 2(a). The boundary can be discredited into N b segments, and the fracture can be discredited into N F segments. So 2 × N b + 2 × N F variable, including the pressure and saturation of every segment, can be solved by 2 × N b + 2 × N F equations, which consists of N b + N F equations of the boundary condition. And N F equations from BEM are shown as Equation (15), and N b equations from FDM are shown as Equation (16).
where T, R, and b are corresponding coefficient matrixes and vector, respectively.
where A, B, and C are the corresponding coefficient matrixes. The star-delta transformation is an effective method to deal with the flow of multiple adjacent fracture units. The transmissibility between the two adjacent segments with four fracture segments is given by Karimi-Fard et al. [44] and Slough et al. [45].
Assuming that another segment connected to segment 4 is labeled 5. Thus, the flow equation of segment 4 can be expressed as

CO 2 Storage Potential Calculation.
Based on the injection well transient pressure solution, the downhole pressure and the injection gas rate can be obtained. When downhole pressure reaches the target pressure, we can get the corresponding injection time t in . Therefore, the CO 2 storage capacity equals the total injected CO 2 volume, which can be written as:

Verification
In this section, the developed model is validated by commercial software CMG (2012) and semianalytical models from Chen et al. [34] and Xiao et al. [24]. The parameters are shown in Table 1 and Figure 3. In total, the targeted model has a grid system of 55 * 55 * 1, among which each grid is 50 m in the x and y directions, while it is 15 m in the z direction. The selected region covers an area of 2750 × 2750 m. The model validation study is conducted by comparing the cumulative injection volume under different constrained pressures, based on the same condition with regular boundary. For constant injection rate and constrained injection pressure, we compare calculated results. The new model has a high agreement with commercial software so that the new model can be regarded as an effective method, shown in Figure 4. In this case, the conventional rectangleboundary-formation storage capacity is evaluated to verify the new method's accuracy. However, commercial software evaluation needs to use local grid refinement or irregular grid subdivision for irregular boundary reservoirs. The advantages of the new method are mainly in dealing with reservoirs with complex boundaries, without subdivision grid. The semianalytical method is adopted to evaluate the buried storage capacity according to reservoir parameters. Therefore, the semianalytical method has higher computational efficiency than the numerical simulation method.

Case Study
Based on an actual case from the shale gas formation, capacity evaluation is studied in this section. Furthermore, the effects of critical parameters are investigated.

Background.
A depleted shale gas formation has been selected as the targeted formation, which is located in China. China has enormous development potential for shale gas, and it can be a prospective location for CO 2 storage projects to reduce greenhouse gases. The case's payzone depth ranges from 1730 m to 1745 m with a shale gas formation, and the parameters are shown in Table 2. The microseismic detection and the scale of the reservoir are shown in Figure 1.

Simulation
Results. Based on this method, the CO 2 geological storage capacity can be evaluated in the actual case for the irregular boundary. After 60, 90, 180, and 360 days, the system pressure distribution is shown as Figure 5.

Effects of Boundary
Shape. The innovation of the method is that it can effectively deal with the actual shale reservoir's irregular geological boundary and have faster calculation speed than the numerical solution. The boundary shape of the reservoir is usually simplified to a circular or rectangular shape. We compared the cumulative CO 2 injection under different boundary shapes. We simulated the model with regular boundary; one is a rectangular boundary with a length of 810 m and a width of 470 m, and the other is a circular boundary with a radius of 410 m. And the cumulative CO 2 injection volume after 720 days is shown in Figure 6. Pressure distribution after 720 days with different  Figure 7. It can be seen that the shape of the reservoir boundary affects the calculation of the CO 2 storage capacity.

Sensitivity Analysis
4.4.1. Injection Pressure. Figure 8 shows the cumulative injection volume (storage volume) of CO 2 under different injection pressures. Obviously, the greater the injection pressure, the more cumulative injection amount of CO 2 ( Figure 8). Figure 9 can optimize constrained injection pressure of CO 2 storage. Figure 10 shows the cumulative injection volume of CO 2 after 360 days under different stress sensitivity coefficients. It can be seen that the increase of fracture permeability caused by stress sensitivity has a significant influence on shale storage potential, especially when a is greater than 0.3. Therefore, the premise of accurately evaluating the CO 2 storage potential of shale reservoirs correctly considers and characterizes stress sensitivity.

Fracture and Shale Matrix
Permeability. The cumulative injection volume of CO 2 after 360 days with different fracture permeability conditions without considering stress sensitivity is shown in Figure 11(a). It can be seen from Figure 11(a) that when the fracture conductivity is minimal, the CO 2 storage potential of the reservoir is meager. With the increase of fracture conductivity, the CO 2 reservoir potential increases sharply. It then tends to be stable, indicating that we need to find abandoned wells with better fracturing effects on the one hand. On the other hand, it is not necessary to blindly seek the high conductivity of fractures. The same law is reflected in the cumulative injection volume of CO 2 injected after 360 days with different matrix permeabilities (Figure 11(b)). CO 2 sequestration is a long-term and complex process considering not only the storage capacity of the reservoir but also the possibility of leakage after storage. It is necessary to comprehensively consider the effectiveness and safety to finding suitable CO 2 reservoirs. It should be noted that the purpose of this method is only to provide quick and effective reference for researchers.

Conclusions
This study appraises CO 2 geological storage potential based on the boundary element method, evaluating the storage capacity considering arbitrarily shaped boundaries. Furthermore, this paper discusses the influence of relevant parameters on the storage capacity of a horizontal well undergoing volumetric fracturing in depleted shale gas reservoirs.
(1) The evaluation results based on the boundary element method have good agreement with commercial software so that the method can be regarded as an effective method (2) The boundary shape of the reservoir is critical to evaluate the CO 2 geological storage capacity of depleted reservoirs accurately. The method proposed in this paper can deal with any complex boundary and simulate the CO 2 storage process effectively and efficiently (3) The CO 2 geological storage capacity under different injection pressures, stress sensitivity coefficients, fracture permeabilities, and shale matrix permeabilities were analyzed. CO 2 geological storage design optimization should be carried out to achieve maximum benefits The main advantage of the method proposed in this paper is that it offers quick and efficient access to CO 2 8 Geofluids sequestration-potential estimation considering the irregular boundary, which helps select wells and monitor facilities for storing CO 2 in residual depleted shale reservoirs. Of course, some complex mechanisms were ignored, which is our subsequent work.