Recently, coalbed methane (CBM) has played an increasingly important role in the energy consumption market. Researchers have focused on utilizing valid stimulation treatment, hydraulic fracturing technology, to effectively develop CBM in reservoirs, which usually consists of two components: (1) small-diameter pores, providing storage space for CBM and (2) natural fractures, not only acting as storage space, but also connecting pores in matrix and providing flow channels for fluid, which can be described by the double-porosity system [1
During hydrocarbon production, gradually decreasing pore pressure results in increasing effective stress, which further leads to decreasing permeability [3
]. Therefore, the stress sensitivity needs to be considered when modeling gas seepage in porous media during production. More specifically, Pedrosa [7
] creatively presented the definition of permeability modulus to characterize the relationship between permeability and pore pressure. Later, Zhang and Ambastha [8
] derived a numerical solution to determine the pressure responses in stress-sensitive reservoirs. Subsequently, Chen et al. [9
] proposed a mathematical model to analyze the transient BHP responses for the fractured wells with multiple radial artificial fractures in stress-sensitive CBM reservoirs. Recently, Wei et al. [10
] and Yuan et al. [12
] numerically investigated the flow mechanism for fractured horizontal wells in shale gas reservoirs, and there are many similar features between CBM reservoirs and shale gas reservoirs, such as the gas adsorption–desorption, gas diffusion, and stress sensitivity of the reservoir permeability, etc.
Numerous mathematical models have been adopted to investigate the fluid flow in CBM reservoirs with varied assumptions. King et al. [13
] numerically simulated the gas–water flow in micropores of CBM reservoirs. Later, Anbarci and Ertekin [14
] provided a novel well model in a CBM reservoir. In their model, two different flow regimes, steady state and pseudo-steady state, were considered under the effect of varied inner and outer boundary conditions. Then, Engler and Rajtar [16
] investigated the BHP responses for the horizontal well in a CBM reservoir utilizing Fourier and Laplace transformation technologies. Subsequently, Clarkson et al. [17
] analyzed the fractured well performances in terms of BHP and production rate in a CBM reservoir. In their model, the hydraulic fractures possess finite-conductivity. Recently, Nie et al. [18
] established a semi-analytical model for a horizontal well in a coal seam, in which adsorption–desorption, diffusion, and Darcy flow were taken into account. More recently, Zhao et al. [19
] obtained an analytical solution for the transient BHP response of a fractured well in a CBM reservoir. In their model, the induced fracture network in SRV were treated as an inner region with more desirable petro-physical properties.
As stimulation technologies develop, high-energy gas fracturing technology, a method employed to develop unconventional hydrocarbon reservoirs, is able to increase the number of artificial fractures and further enlarge the range of linear flow. More specifically, multi-wing radial artificial fractures can be obtained along the wellbore after the stimulation [20
], which have been verified based on core analysis and microseimic image results [9
] (Figure 1
). Some analytical and numerical models have been proposed to analyze the transient BHP and rate performances with consideration of multi-wing artificial fractures. For example, Choo and Wu [25
] derived a new numerical solution for multiple fractured vertical wells to investigate the BHP response. Later, Tiab [26
] analyzed the transient BHP response of the model, taking asymmetrically-distributed hydraulic fractures with finite conductivity into account by employing the Tiab’s Direct Synthesis Technique. Recently, Zhang et al. [20
] proposed a well-testing model considering the fractured well with multiple radial hydraulic fractures in a composite CBM reservoir in order to simulate the transient pressure and rate by means of continuous line-source function. In addition, refracturing technology is also able to generate multiple radial artificial fractures. For example, Hou et al. [27
] sketched an analytical solution to calculate the angle between adjacent hydraulic fractures initiated after refracturing. To verify the model, the authors compared the data obtained from oil wells in practical fields with the newly developed solution. However, most models proposed above are not able to consider the effect of SRV, which is a key element in enhancing gas recovery.
Usually, stimulation treatment serves two main purposes, generating fracture network and artificial fractures near the wellbore. The generated fracture network close to the wellbore in unconventional hydrocarbon reservoirs caused by stimulation treatment are generally termed as SRV. SRV in this study is considered as an inner region near the wellbore, which includes the multi-wing artificial fractures and the induced fracture network. To differentiate the SRV and un-stimulated region in CBM reservoirs, the CBM reservoir in this study is divided into two regions, each can be described by the dual-porosity system. Specifically, the inner region (SRV) possesses more desirable petro-physical properties, such as larger porosity and permeability.
Unlike conventional natural gas, unconventional natural gas such as CBM and shale gas are generally absorbed on the mineral particle surface in coal seam matrix and the CBM flow in reservoirs is subject to multiple transport mechanisms [28
]. More specifically, as reservoir pressure decreases, the absorbed CBM molecules are able to desorb from the surface of mineral particles. Then, the gas molecules can be driven towards fracture under the effect of concentration difference (diffusion). Finally, due to pressure difference, the flow of CBM from coal seam matrix can be observed in fractures, which can be characterized by Darcy law, see Figure 2
As is well known, transient pressure analysis (TPA) is suitable to determine key reservoir parameters and to monitor the transient BHP performance of gas wells. The research interest of this study is to propose a well-testing model, considering multiple transport mechanisms of CBM in the reservoir, to investigate the transient BHP response of a fractured well with multiple artificial fractures in a stress-sensitive CBM reservoir. The main characteristics of type curves obtained in this work are discussed. The model proposed in the study can be useful in well testing interpretations and production transient analyses of unconventional gas reservoirs.
5. Results and Discussion
5.1. Type Curves for the Proposed Model
According the description in Section 3
, since there are (
) algebraic equations with (
) unknowns, the combination of Equation (18) to Equation (23) and the utilization of computer programming could result in the time-dependent PPR and PPD, see Figure 6
. Based on Figure 6
, the type curves resulting from the proposed model can be divided into the following eight flow regimes:
Stage 1: Wellbore storage. The PPR and PPD both are straight lines with unit slope during this flow regime.
Stage 2: Short-time transition flow regime. An obvious hump can be observed in the PPD.
Stage 3: Fracture linear flow regime in SRV. The linear fluid flow from SRV to fractures occurs during this regime and the PPR curves are the straight lines with a slope of 1/2.
Stage 4: Pseudo-radial flow in SRV. As the pressure wave propagates, the pseudo-radial flow in SRV can be observed during this regime and the curve of the PPR exhibits a horizontal line. It is worth noting that this flow regime is subject to the radial fracture length–SRV radius ratio.
Stage 5: Short-time transition flow regime from pseudo-radial flow in the inner region to radial fluid flow in the outer region.
Stage 6: Radial flow regime in the natural fracture system of the outer region. As the pressure wave propagates farther, the SRV-centered radial flow occurs in the outer region. This flow regime is characterized as a flat trend in the PPD curve during this period.
Stage 7: Diffusive flow regime (matrix-dominated flow regime). The gas concentration difference between the natural fracture system and matrix can be expected with the production of CBM residing in the natural fracture system in the outer region, CBM molecules start to desorb from the surfaces of mineral particles in the matrix and diffuse into the natural fracture system under the effect of concentration difference between the natural fracture system and coal seam matrix. For the pseudo-steady state diffusion model, an obvious “dip” appears in the PPD curve during this flow regime, however, for the unsteady state diffusion model, a less obvious “dip” can be found in the PPD curve during this flow regime.
Stage 8: Pseudo-radial flow in the unstimulated regime. A dynamic balance is achieved for the gas transfer between the natural fracture system and the coal seam matrix in the un-stimulated region and the PPD curve exhibits a horizontal line, whose vertical-axis value is 0.5.
It is worth noting that the upward trend both in the PPR and PPD at late flow periods can be observed if the stress-sensitive effect of the fracture network in SRV is considered, representing that more pressure depletion is required for the production of CBM in stress-sensitive reservoirs, see Figure 6
Additionally, because the un-steady state diffusion model is more practical in most cases, the un-steady state diffusion model is applied here to perform the following PPR/PPD-sensitive analyses. Figure 7
illustrates the three main flow regimes resulting from the proposed model.
5.2. Effect of the Ratio of Permeability in the Inner Region on that in the Outer Region
In the following subsections, sensitivity analyses are performed in terms of several key parameters for the development of CBM based on the developed model and programming. Usually, SRV can be generated after stimulation treatment for the cost-effective development of unconventional hydrocarbon. As a result, the determination of permeability of SRV is of great significance [37
]. Figure 8
demonstrates the impacts of the ratio of the permeability in the SRV to that in the un-stimulated region, M12
, on the PPR/PPD for a rate-constant production. The PPR/PPD decreases with increasing M12
in both linear and pseudo-radial flow regimes, denoting that the higher permeability of SRV is able to decrease energy consumption for the rate-constant production. Therefore, the initialization of SRV is of great importance for the effective development of CBM.
5.3. Effect of the Radius of SRV Region
illustrates the influence of the varied SRV radius (r1
) on the PPR/PPD while the rate production keeps constant. The parameter r1D
is the dimensionless radius of SRV in the composite CBM reservoirs, including the induced micro-fracture network and multiple fracture wings. As can be seen in Figure 9
, the SRV radius dramatically affects the duration time of the pseudo-radial flow in SRV. More specifically, larger dimensionless radius of SRV corresponds to the longer duration time of the pseudo-radial flow regime in SRV, denoting that more pressure (or energy) is required for production.
5.4. Effect of Permeability Modulus
Since stress-sensitive permeability of the micro-fracture network in SRV is considered in this study, Figure 10
illustrates the impact of the permeability modulus (
) on the PPR/PPD for the rate-constant production. Based on Figure 10
, a smaller permeability modulus corresponds to the lower PPR/PPD (less obvious upward trend) in late flow regimes, which represents that the existence of stress sensitivity leads to larger pressure depletion in reservoirs compared with the no-stress-sensitivity case.
5.5. Effect of Radial Fracture Angle Symmetry
illustrates the impact of radial fracture wing angle symmetry on the PPR/PPD. As shown in Figure 11
, the PPR/PPD increases with the increasing θ3
and with the decreasing θ1
in the later period of linear flow. This can be explained by the fact that the fracture interference becomes more severe as the angle between adjacent radial fractures decreases. That is to say, uniformly distributed radial fracture wings are able to weaken the fracture interference, and, as a result, reduce the energy consumption.
5.6. Effect of the Number of Fracture Wings
presents the effect of the number of radial fracture wings on the PPR/PPD. Based on Figure 12
, the number of radial fracture wings impacts the PPR/PPD drastically. The PPR/PPD decreases with the increasing number of radial fracture wings from two to six. Therefore, the generation of multiple radial fracture wings is able to reduce the pressure depletion (energy consumption) required for production. The results obtained here are of great importance for the practical stimulation design.
5.7. Effect of the Length of Hydraulic Fracture Wings
illustrates the impact of the length of radial fracture wings on the PPR/PPD for a rate-constant production. The linear flow regime and pseudo-radial flow regime in SRV can be affected by the length of radial fracture wings for a constant SRV radius, see Figure 13
. The PPR/PPD decreases with the increasing length of radial fracture wings from 10m to 30m and the duration time of the pseudo-radial flow in SRV becomes shorter as the length of radial fracture wing increases. More specifically, the radial fracture wings with smaller length increase the pressure depletion during production.
5.8. Effect of the Storativity Ratio of the Outer Region
Since the double-porosity system is considered in this study, Figure 14
shows the effect of the storativity ratio of the outer region on the PPR/PPD. The storativity ratio of the outer region mainly affects the diffusive flow regime in the outer region, see Figure 14
. A lower storativity ratio of the outer region corresponds to a wider and deeper concave during this flow regime.
5.9. Effect of the Inter-Porosity Flow Parameter
The other important parameter resulting from the double-porosity system is the inter-porosity flow parameter. Figure 15
demonstrates the effect of the inter-porosity flow parameter on the PPR/PPD for the rate-constant production. Based on the results provided by Figure 15
, the inter-porosity flow parameter affects the diffusion flow regime drastically: the diffusion flow regime occurs later as the inter-porosity flow parameter decreases.
5.10. Effect of the Adsorption–Desorption Constant
shows the effect of the adsorption–desorption constant on the PPR/PPD. The adsorption–desorption constant mainly affects the diffusion flow regime, as can be seen in Figure 16
. More specifically, a higher adsorption–desorption constant corresponds to a deeper and wider concave, which represents the diffusion regime. The adsorption–desorption constant is adopted in this study to represent the amount of gas adsorbed at the surface of mineral surfaces and a higher adsorption–desorption constant denotes more adsorbed gas existing in the coal seam matrix. Therefore, more adsorbed gas is able to desorb and diffuse into the fractures for a larger adsorption–desorption constant during production.