A multilinear flow model for multistage fractured horizontal wells in shale reservoirs
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Abstract
This paper presents an analytical multilinear flow model for shale gas reservoirs with multistage fractured horizontal wells (MFHW). It has been proved that the hydraulic fractures are branched rather than simple biwing shape, and the seepage flow of shale gas reservoirs is more complicated than conventional gas reservoirs due to the gas occurrence characteristics and fracture networks. Based on the published trilinear flow models, a developed fiveregion model considering effective fractured volume and adsorption effect was established. Laplace transformation method and Stehfest numerical algorithm were used to obtain typical pressure response curves. In addition, the presented model was validated by the actual production data, different flow regimes were divided, and the prediction of presented model was compared with the results of Eclipse simulator. Effects of some factors such as stimulated reservoir volume, storativity ratio, and Langmuir volume on the performance were analyzed. The results showed that the presented model considering stimulated volume and adsorbed gas could predict the productivity of MFHW better. The linear flow of stimulated region was the main contribution to gas production, and the duration of formation linear flow was influenced by different parameters. So the selection of optimal combination is very important in the development of shale gas reservoirs.
Keywords
Shale gas reservoir Adsorption and desorption Multistage fractured horizontal well Multilinear flow model Semianalytical solutionAbbreviations
 p
The gas pressure (MPa)
 ρ_{g}
The shale gas density (kg/m^{3})
 ρ_{gsc}
The shale gas density in standard condition (kg/m^{3})
 k
The permeability (m^{2})
 ϕ
The porosity, fraction
 μ
The gas viscosity (mPa/s)
 p_{L}
Langmuir pressure (MPa)
 V_{L}
Langmuir volume (m^{3})
 V_{E}
The volume of gas adsorbed per unit volume of the reservoir in equilibrium at pressure p (m^{3}/m^{3})
 Z
The compressibility factor of shale gas, dimensionless
 M
The apparent molecular weight (g/mol)
 R
The universal gas constant, 8.314 J/(mol K)
 T
The reservoir temperature (K)
 C_{D}
The dimensionless wellbore storage coefficient
 m
The pseudopressure (MPa/s)
 η
The pressure transitivity
 x
Reservoir size in xdirection (m)
 y
Reservoir size in ydirection (m)
 x_{f}
The length of the hydraulic fracture (m)
 n_{F}
The numbers of hydraulic fractures along a horizontal well
 s
Laplace transform parameter with respect to dimensionless time
Subscript
 D
Dimensionless
 m
Matrix
 f
Fracture
 i
Initial condition
Superscript
 −
Laplace transform
Introduction
The rapid economic growth results in the increased demand of energy and development of shale gas reservoirs. Due to the extremely ultralow permeability of the reservoirs, it is a great challenge to develop shale reservoir commercially. With technical innovation in the past decades, massive stimulation has been broadly applied into the field and proved effectively, especially the application of multistage fractured horizontal well (MFHW) achieves the commercial exploitation. However, modeling fluid flow in the complex fracture networks remains challenging (Ezulike and Dehghanpour 2014).
In many cases, a fracture propagation can create a branch pattern and a complex fracture networks around the hydraulic fractures (Ali Daneshy 2003), which were defined as stimulated reservoir volume (SRV) (Mayerhofer et al. 2010; Wang et al. 2014). The high conductivity of SRV makes liquids flow into the well easily and benefits the well production (Stalgorova and Mattar 2012a; Clarkson 2013). Most of shale gas reservoirs in Eagle ford, Barnett, and Marcellus (Suliman et al. 2013; Agboada and Ahmadi 2013; Mayerhofer et al. 2006) have obtained high production due to SRV. In addition, carbonrich components lead to the existence of the adsorbed gas and free gas phase in the shale formations (Juan and Aquiles 2012). With the pressure dropping down, the adsorbed gas will desorb from the surface of matrix during the development process, which also has a significant influence on gas production and could not be ignored in mathematical models (Bumb and McKee 1988).
The production of MFHW in shale reservoirs is mainly affected by the fluids in matrix, in fracture network, and in hydraulic fractures. Brown et al. (2009, 2011) proposed the trilinear flow model to research the MFHW performance in unconventional gas reservoirs. In their model, pressure transient analysis was obtained. Considering the limited width of the simulated region, fiveregion model was defined to simulate the SRV to extend the trilinear model (Stalgorova and Mattar 2012b). Dehghanpour and Shirdel (2011) improved the Ozkan’s dualporosity model (Ozkan et al. 2010) to explain the unexpected high gas production in shale gas reservoirs based on the pseudosteady model of Warren and Root (Warren and Root 1963). Then, equivalent flow model (Ketineni S and Ertekin 2012) was used to describe the SRV and solved the elliptical flow problem by Mathieu modified functions. Based on this model, Su et al. (2015) characterized the SRV using a circular region in shale gas reservoir and analyzed the pressure performance considering the SRV. In addition to the analytical models, there are some numerical models to simulate the seepage flow of MFHW in unconventional reservoirs (Mediros et al. 2007; Mayerhofer et al. 2006; Meyer and Bazan 2011). All these models have some drawbacks, such as the complex computational process, relationships of parameters, and difficult application, so the simplifications of the flow models have to be considered.
At present, few models can simulate the performance behavior including pressure and rate transient analysis (PTA and RTA) successfully for MFHW with SRV in shale gas reservoirs. Bumb and Mckee (1988) took the desorption compressibility into account for shale gas reservoirs. Then, based on this model, lots of work (Gerami et al. 2008; Guo et al. 2012) have been done for unconventional gas reservoirs. These studies mainly focused on vertical well or horizontal well and few about the MFHW. Although Ozkan et al. (2010) and Brown et al. (2009) developed a trilinear flow model to simplify the complex process and get good results in unconventional gas reservoirs, desorption and adsorption mechanism, which is the key mechanism of shale gas reservoirs, was ignored. Sang et al. (2014) presented a new mathematical model considering adsorption and desorption process, which made up the disadvantage of the trilinear flow model for MFHW in shale gas reservoirs. Zhang et al. (2015) then presented a numerical fiveregion model with multinonlinearity to study the production of shale gas, which was difficultly to solve. In this paper, we extended the linear flow model considering effective stimulated volume and adsorption effect to multistage fractured horizontal wells in shale gas reservoirs. The bottomhole pressure and production formulas are established, and effects of several key parameters are analyzed. The duration of formation linear flow under different parameters is studied, which helps to understand the flow mechanism of multiple hydraulic fractures in shale gas reservoirs.
Model description
Physical model and its assumptions
Mathematical model

Liner flow in matrix
In the region 2, the adsorption–desorption process as well as the flux from the region 4 is taken into account.

Linear flow in stimulated region

Linear flow in hydraulic fracture region
Model solutions
Model verification
Field example
Shale reservoir and horizontal well parameters
Parameters  Unit  Value 

Matrix porosity, ϕ _{m}  %  8.0 
Fracture porosity, ϕ _{f1}  %  0.6 
Hydraulic fracture porosity, ϕ _{f}  %  1.0 
Matrix compressibility, c _{tm}  1/MPa  2e−5 
Fracture compressibility, c _{tf1}  1/MPa  3.5e−4 
Hydraulic fracture compressibility, c _{tf}  1/MPa  4.5e−4 
Matrix permeability, k _{m}  mD  1e−6 
Fracture permeability, k _{f1}  mD  1e−3 
Hydraulic fracture permeability, k _{f}  mD  300 
Formation thickness, h  m  46 
Horizontal section length, L  m  1200 
Reservoir size in ydirection, y _{2}  m  200 
Reservoir size in xdirection, x _{2}  m  100 
Hydraulic fracture halflength, x _{f}  m  75 
Hydraulic fracture width, w _{f}  m  0.1 
Numbers of hydraulic fractures, n _{F}  –  10 
Langmuir volume, V _{L}  m^{3}/m^{3}  4 
Langmuir pressure, P _{L}  MPa  12 
Well depth, H  m  1500 
Initial reservoir pressure, p _{i}  MPa  21 
Initial reservoir temperature, T _{i}  K  338 
Comparison of the analytic model and Eclipse simulator
The free gas in fractures and inorganic matrix pores is produced first. Then, the absorbed gas on the organic matrix desorbs into the fractures gradually. Therefore, the production rate is relatively high in the first 2 years, which is above 10,000 m^{3}/day. There is a sharp decline, and the production rate remains a slower pace afterward. In addition, it is apparent that lower flowing bottomhole pressure leads to larger drawdown pressure, which makes the production rate higher, such as the production rate under p _{wf} = 4 MPa is four times that under p _{wf} = 16 MPa at the corresponding time.
Results and discussion
Analysis of type curve
As shown in Fig. 5, the type curve of MFHW considering SRV and adsorption and desorption process can be divided into the following eight regimes: (1) the early wellbore storage characterized by a slope of 1 in pressure and derivative curves. (2) the first transition flow stage between wellbore storage and the early linear flow. The dimensionless pressure curve and derivative curve are separate. (3) the duallinear flow stage, which is characterized by a slope of 0.25 in pressure derivative curve. (4) the linear flow stage of the formation. Pressure differential is proportional to the square root of dimensionless time, and the slope of pressure derivative in log–log curve is 0.5. (5) the adsorption and desorption process, which affects a depression in the pressure derivative curve. (6) the second transition flow stage between the linear flow in inner formation and outer formation. (7) the second linear flow stage of formation characterized by a slope of 0.5 in pressure derivative curve. With the influence of the nonflow boundary, the typical radial flow does not show on the graph. (8) The quasisteady flow stage occurs in the last period when the whole system reaches steady state. Furthermore, the curves coincide again and go up together due to the nonflow boundary condition.
Analysis of pressure transient responses
Analysis of production performance
The linear flow region is affected by fracture halflength, stimulated volume, and the permeability in different regions. The log–log pressure and pressure derivative plots were used to identify flow regimes,while the linear flow analysis (ratenormalized pressure vs. square root of time) was used to obtain the parameters by producing data of wells (Kurtoglu et al. 2013). The time tef was defined to present the over time of formation linear flow, and the effects of different parameters on linear flow analysis were studied.
Conclusions
 1.
Mathematical models for conventional reservoirs cannot be used to represent the fluid flow in unconventional reservoirs. The coupling modified mathematical model could describe the comprehensive gas flow in both multistage fractured horizontal well and formation. Specifically, the stimulated reservoir volume and the shale gas properties are introduced into the model in this paper compared to the Eclipse simulator. The transient pressure curves are divided into eight regimes, and these regimes change with different formation properties and MFHW properties.
 2.
There are two occurrence modes for shale gas in shale formation, adsorbed gas mainly existing in matrix and free gas mainly existing in natural fractures. The comparison of the composite model predictions using the field data and actual production data demonstrates that the fiveregion model considering the stimulated reservoir volume and adsorption and desorption process is able to describe the gas flow of multistage fractured horizontal well in shale gas reservoir. In addition, the simulation results of Eclipse simulator without the adsorption and desorption process show that desorption of the adsorbed gas in shale formation matrix should not be neglected in mathematical models.
 3.
Besides adsorption and desorption process, wellbore storage coefficient, SRV, and bottomhole pressure have significant effects on the well pressure and production performance of multistage fractured horizontal well in shale gas reservoirs. The wellbore storage coefficient mainly affects the early production stage. The SRV has significant effects on the late flow stage in the transient pressure and production rate curves. Larger SRV leads to lower transient pressure at constant production and longer tef at constant flowing pressure for the same reason. That means the development of shale gas reservoirs depends on the stimulated reservoir volume. Large Langmuir volume means the adsorption and desorption abilities are strong, which is important to maintain stable gas production rate in late flow stage. The bottomhole pressure is also a significant parameter for developing the shale gas reservoirs. In view of the simplification of this model, some further work is required to make the model approximate to the practice.
Notes
Acknowledgments
This work was supported by the National Basic Research Program of China (2014CB239103) and the Natural Science Foundation of Shandong Province, China (ZR2014EL014).
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