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Research ArticleA Model for a Multistage Fractured Horizontal Well withRectangular SRV in a Shale Gas Reservoir
Jianfa Wu, Jian Zhang, Cheng Chang, Weiyang Xie , and Tianpeng Wu
PetroChina Southwest Oil & Gas Field Co., Chengdu, Sichuan, China
Correspondence should be addressed to Weiyang Xie; [email protected]
Received 27 March 2020; Revised 12 October 2020; Accepted 20 October 2020; Published 7 December 2020
Academic Editor: Wen-Dong Wang
Copyright © 2020 Jianfa Wu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Although great success has been achieved in the shale gas industry, accurate production dynamic analyses is still a challenging task.Long horizontal wells coupling with mass hydraulic fracturing has become a necessary technique to extract shale gas efficiently. Inthis paper, a comprehensive mathematical model of a multiple fractured horizontal well (MFHW) in a rectangular drainage areawith a rectangular stimulated reservoir volume (SRV) has been established, based on the conceptual model of “tri-pores” inshale gas reservoirs. Dimensionless treatment and Laplace transformation were employed in the modeling process, while theboundary element method was used to solve the mathematical model. The Stehfest numerical inversion method and computerprograming techniques were employed to obtain dimensionless type curves, production rate, and cumulative production. Resultssuggest that 9 flow stages can be observed from the pseudopressure derivative type curve when the reservoir and the SRV arelarge enough. The number of fractures, SRV permeability, and reservoir permeability have no effect on the total productionwhen the well is abandoned. As SRV and reservoir permeability increases, the production rate is much higher in the middleproduction stage. Although the SRV scale and its permeability are very important for early and intermediate production rates,the key factors restricting the shale gas production rate are the properties of the shale itself, such as adsorbed gas content,natural fractures, and organic content. The proposed model is useful for analyzing production dynamics with stimulatedhorizontal wells in shale gas reservoirs.
1. Introduction
Shale gas, together with coalbed methane, which was oncethe source rock of oil and gas. Shale gas is a typical unconven-tional gas stored with adsorbed, free gas ([1]; Li Yong et al.,2019). Shale gas consists mainly of methane, and there aremany kinds of interlayers in shale strata, such as siltstone,silty mudstone, and politic siltstone. Currently, shale gasextraction is developing rapidly and has played an importantrole in the US energy industry since 2000. It has spreadquickly to China and Australia as a potential energy source[2]. For example, the annual production of shale gas in2007 in the US was 56.4 billion cubic meters, while it reached447.0 billion cubic meters in 2016; its proportion to the worldannual production is 12% [3]. With the success of shale gasdevelopment, shale gas has the potential to become the pri-mary energy source in the future.
Due to the special formation characteristics and poretypes of a shale gas reservoir, exploiting shale gas requireshydraulic fracturing. Specifically, i.e., horizontal wells withmassive hydraulic fracturing [4–6]. Massive hydraulic frac-turing of horizontal wells in shale gas reservoirs creates afracture network around the well called stimulated reservoirvolume (SRV). The SRV, which is the main method ofincreasing shale gas production, consists of hydraulic frac-tures and a fracture network. Therefore, it is important todevise an effective and accurate model to describe the SRVand to help understand the flow characteristics of shale gasunderground.
Kucuk and Sawyer [7] proposed a model to analyze theadsorption and desorption characteristics in fractured reser-voirs and derived a numerical solution to the model consid-ering the effects of desorption and slippage. Lancaster andGatens III [8] analyzed postfractured well testing data of
HindawiGeofluidsVolume 2020, Article ID 8845250, 18 pageshttps://doi.org/10.1155/2020/8845250
eastern Devonian shale, which was described as a dual-porosity reservoir, and those data were used to estimate frac-ture length and fracture conductivity. Devonian shale wellsshowed that fracture half-lengths calculated from conven-tional postfracture pressure-buildup tests were shorter thandesigned, based on which Johnston and Lee [9] proposed anew method to determine net pay thickness and presentedresults suggesting that conventional tests of those wells areinadequate.
Brown et al. [10] and Ozkan et al. [11] assumed that themain flow type in fractured horizontal wells is linear, andestablished a trilinear flow model for a multistage fracturedhorizontal well in a shale gas reservoir using the model pre-sented by Lee and Brockenbrough (Lee and Brockenbrough1986). Based on the linear-flow theory, a plate dual-porosity model, and the Warren-Root block dual-porositymodel, Bello and Watenbargen [12] and Al-Ahmadi et al.[13] proposed a model to analyze the production dynamicsand pressures of MFHWs in shale gas reservoirs. Nobakhtet al. (Nobakht et al., 2012) first proposed the conceptualmodel of different combinations of horizontal wells, porousmedia types, and hydraulic fractures in shale gas reservoirs,and then analyzed the different flow stages during the devel-opment process. Assuming SRV exists only in a limitedregion around the fractures and fracture half-lengths equalto the width of the reservoir, Stalgorova and Mattar [14]derived a modified trilinear flow model and obtained thesolution of this model in Laplace space. Xu et al. [15] pro-posed an unsteady seepage model of a MFHW at constantrate production in a shale gas reservoir. The model assumedthat the flow types in a fractured horizontal well are alter-nately linear and transitional, and the fracture network existsin a limited region around the hydraulic fractures.
To summarize, there are two types of models to analyzewell production performance of MFHWs with SRVs in shalegas reservoirs. One is the linear flowmodel, which is based onthe assumption that all fractures have the same length andconductivity and are spaced uniformly along the horizontalwell ([11–21]; Zhao et al., 2016b; [22–25]), and the mainrelated schematics of linear models are listed in Table 1.Although it is common field practice to design equally spacedhydraulic fractures, the requirements of equal properties andfracture lengths of shale gas reservoirs are hard to satisfybecause of the development of natural fractures and theiranisotropy and heterogeneity. The other type of flow modelis the numerical simulation model based on finite-element,finite-difference, or other numerical methods [26–36].
Normal shale gas reservoirs have a dual-porosity nature,with macropores and micropores in matrix, and microfrac-tures made by hydraulic fracturing (Ge & Zhang, 2012;[38–40]). The output process of shale gas from the reservoircan be subdivided into 4 stages: (1) the gas desorbed fromthe matrix particle surfaces, (2) diffusion from the matrix tothe macropores, (3) flows from the macropores to the micro-fractures, and (4) Darcy flows through the fracture networkto the bottom hole.
Well testing analysis is an important dynamic monitor-ing tool for petroleum engineers to accurately determinethe parameters of oil and gas reservoir engineering, to evalu-
ate the stimulation in the shale gas reservoir, and to providemore effective plans for the oil field. In addition, a suitablemathematical model for horizontal wells with SRVs in shalegas reservoirs is vital to well testing.
As we can see from the above discussion, both analyticalmethods and linear flow models have their own deficienciesin obtaining production dynamics in shale gas reservoirs.On one side, analytical methods, such as point source andGreen’s function method, can only deal with seepage prob-lems of shale gas in regular-shaped reservoirs, such as reser-voirs with circular or infinite outer boundaries. However, thepractical boundary of shale gas reservoirs is quite complexconsidering the effects of SRV, which in consequence cannotbe handled by this traditional analytical method. On theother, linear flow models usually put the whole reservoir asa rectangular area, and subdivide the reservoir region intosome small rectangular areas, as shown in Table 1. The sub-division principle of the small rectangular sections is the gasflow capacities in formations, i.e., a bigger flow capacity in theSRV region and a weaker flow capacity as the region. In linearflowmodels, a quarter of the area around a hydraulic fractureis usually studied for simplicity, which brings into anotherassumption that all hydraulic fractures are uniformly spacedand with the same length. These assumptions are also not inaccordance with practical conditions of shale gas reservoirs.Besides, other flow patterns, such as redial flow and ellipticalflow, and interferences among fractures cannot be reflectedfrom the linear flow models. Thus, a more comprehensiveand accurate model considering both the rectangular com-posite nature of a shale gas reservoir and the fracture hetero-geneities need to be established in an effort to analyzeproduction dynamics better in shale gas reservoirs. In thispaper, we present a percolation model for a multistage hori-zontal well in a rectangular shale gas reservoir with an SRVconsidering unsteady transport theory, which cannot betackled by analytical methods or linear flow conceptions.The boundary element method is employed to solve the pro-posed mathematical model, with parameter sensitivity analy-sis conducted for production dynamics. Results show that themodel we propose should be useful for analyzing stimulatedhorizontal wells in shale gas reservoirs.
2. Transport Mechanism of Shale Gas
2.1. Characteristics of Shale Gas Reservoir. The characteristicsof a shale gas reservoir are very different from those of a con-ventional gas well, in both the structure and the flow mecha-nism. The typical classification method is that of Loucks et al.[41]. They classified the pore types of the shale matrix andthe natural microfractures; their classification is comprisedof the following 4 types: intergranular pores, intragranularpores, organic pores, and natural microfractures (Figure 1shows the main pore types of a shale gas reservoir). The first2 pore types are related to the mineral grains. The differencebetween them is that intergranular pores develop betweengrains and intragranular pores develop within grains.Organic pores are intrapores of organic matter. In shale, themain pores are micropores, nanopores, and microfractures,so approximately 20% to 80% of the gas is absorbed by the
2 Geofluids
Table 1: Schematics of the linear models.
No. Schematics of physical models References
1
Reservoir formation
SRV regionXu et al. 2013 [15]
2Homogeneous
Homogeneous
Nobakht and Clarkson 2013 [18]
3Brown et al. 2009 [10]Ozkan et al. 2013 [11]
4 Bello and Wattenbarger (2010)
5 Al-Ahmadi et al. 2010 [13]
6Nobakht et al. 2013 [19]Yao et al. 2013 [37]
3Geofluids
particles in the matrix, and the rest is free in the microfrac-tures. Gas flow behaviors in shale nanopores are quite differ-ent from that of gas flow in conventional reservoirs [42–45].The percolation mechanism includes diffusion, adsorption/-desorption, and slip flow [46–48].
2.2. Adsorption Models of Shale Gas. The physical adsorptiondefined by Brunauer et al. [49] occurs when gas contacts asolid surface: some gas molecules are captured by the solidbecause of the intermolecular force. This means that if thevolume of gas is constant, gas pressure drops; if the pressureis constant, the volume of gas decreases. The gas molecules
captured by the solid are said to be adsorbed, and the solidis called an adsorbent.
Because shale gas adsorption models are numerous, it isnecessary to choose the best one before using it. Here, weuse a nonlinear fitting method to analyze the degree of fitbetween adsorption models and some adsorption data fromsubcritical and supercritical states.
In the original state of a shale gas reservoir, adsorbed anddesorbed gasses are in dynamic pressure equilibrium. Whenthe equilibrium is disturbed after production, the gasadsorbed in the organic matter surfaces begins to desorb,and then the desorbed gas becomes free and is stored in the
Table 1: Continued.
No. Schematics of physical models References
7 Stalgorova and Mattar 2012 [14]
8 Stalgorova and Mattar 2013 [21]
(a) (b) (c)
(d) (e) (f)
Figure 1: Main pore and fracture types in a shale gas reservoir. (a) Inorganic pores (intergranular pores). (b) Inorganic pores (intercrystallinepores). (c) Inorganic pores (intragranular dissolved pores). (d) Organic pores. (e) Cleavage fractures. (f) Intergranular fractures.
4 Geofluids
reservoir until the adsorbed and free gasses are in equilibriumagain.
Congruent with the previous analysis, the Langmuir iso-therm is one of the most popular models used to describe thegas adsorption/desorption process. The amount of gasadsorbed by a solid surface is given by the Langmuir equationbelow, characterizing the desorption processes as a functionof pressure at constant temperature [33]:
V =VLpm
pL + pm, ð1Þ
where V is the volume of the adsorbed gas (cm3/g), Pm is thegas pressure (MPa), VL is the Langmuir volume (cm3/g), PLis the Langmuir pressure (1/MPa).
In addition, the expression of desorbed gas mass flux thatdesorbs from a reservoir whose volume is Vb during a unit oftime is
qdes = ρgscVb∂V∂t
, ð2Þ
where qdes is the mass flow rate of gas desorption from thereservoir of volume Vb (kg/s), ρgsc is the density of shalegas under standard conditions (kg/m3).
Incorporating equation (1) into equation (2) results in
qdes = ρgscVbVLpL
pm + pLð Þ2∂pm∂t
: ð3Þ
Introducing pseudopressure into equation (3), it can berewritten as [50]
qdes = ρgscVbVLm pLð Þ
m pmð Þ +m pLð Þð Þ2∂m pmð Þ
∂t: ð4Þ
It is assumed that the desorbed process of shale gas isinstantaneous and all the desorbed gas will become free gas,so equation (4) can be used as a continuity equation of a frac-ture network or matrix.
3. Model for Gas Flow in a Fractured ShaleGas Reservoir
3.1. Physical Model. Macropores exist in a practical shalematrix, which may lead to an error if gas storage and floware ignored in these macropores. Therefore, researchers havedevised a “tri-pore” conceptual model, where the shale gasreservoir consists of matrix pores, macropores, and micro-fractures. When the adsorbed gas desorbs from the surfaceof the matrix particles, it enters the macropores first and thenflows into the microfractures [39, 50–52]. Figure 2 shows thephysical model.
Song [39] first proposed a physical model concerningtransfusion in the macropores of the matrix. But it was onlya physical seepage-mechanism model, which was not mod-eled mathematically. In this paper, a mathematical modelfor this physical conception is proposed and solved. Theinterporosity flow of free gas between macropores andmicrofractures can be divided into 2 types: (1) a pseudos-teady interporosity flow model and (2) a transient interpor-osity flow model.
3.2. Comprehensive Mathematical Model
3.2.1. Transient Interporosity Flow Model.When the gas flowfrom macropores to a fracture network is transient inter-porosity, the continuity equation in the fracture networkis [53, 54]
1r2
∂∂r
kfμg
ρgr2 ∂pf∂r
!=∂ ϕfρg
� �∂t
− qm: ð5Þ
For unsteady interporosity and spherical matrixparticles, the continuity equation in the matrix can beexpressed as follows:
1r2m
∂∂rm
kmμg
ρgr2m∂pm∂rm
!=∂ ϕmρg
� �∂t
+ qdes: ð6Þ
Because the initial pressure is equal throughout thereservoir, the initial condition in the matrix equals to theinitial reservoir pressure pi:
Gas flow from natural fractures to artificialfractures
Desorbed gas flow from matrix to macropores
Microfracture + shale matrix
Free gas flow from macropores tomicrofractures
Figure 2: Physical model of pseudosteady flow when gas desorbs from the matrix and diffuses to a fracture [39, 50].
5Geofluids
pm t = 0, rmð Þ = pi: ð7Þ
Because seepage of the gas in the macropores is sym-metrical, the inner boundary condition is
∂pm∂rm
t, rm = 0ð Þ = 0: ð8Þ
Because the outer boundary of the spherical matrixparticles is connected to the fracture system, the outerboundary pressure of the matrix is equal to the pressureof the fracture system:
pm t, rm = Rmð Þ = pf : ð9Þ
When the flow between the macropores and themicrofractures is unsteady, the interporosity flow rate is
qm = −3ρgmRm
kmμg
∂pm∂rm
�����rm=Rm
: ð10Þ
By the Langmuir isotherm equation, the desorptionrate of shale gas qdes can be written as
qdes = ρg 1 − ϕf − ϕmð Þ VLm pLð Þm pLð Þ +m pmð Þ½ �2
∂m pmð Þ∂t
: ð11Þ
By introducing dimensionless variables into the abovemathematical models and then reformatting them by sev-eral simple transformations (a detailed derivation processis described in Appendix A, while the dimensionless vari-ables are shown in Appendix B), the dimensionless seep-age equation of the fracture system can be obtained as
1r2D
∂∂rD
r2D∂Δ�mf∂rD
� �= f sð ÞΔ�mf , ð12Þ
where the expression of f ðsÞ is
f sð Þ = ωf s +λ
5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi15 1 − ωf + ωdð Þs
λ
rcoth
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi15 1 − ωf + ωdð Þs
λ
r− 1
" #:
ð13Þ
3.2.2. Pseudosteady Interporosity Flow Model. For pseudos-teady gas flow from the macropores to the fractures,incorporating pseudosteady interporosity flow rate andpseudopressure into the seepage equation of the fracturesystem, the seepage equation of fracture system can bewritten as
1r∂∂r
r2∂mf∂r
� �=ϕfμf cfgi
kf
∂mf∂t
−αkmkt
mm −mf½ �: ð14Þ
For pseudosteady gas flow from the matrix to the frac-tures, the seepage equation of the macropores is
−αkmkf
mm −mf½ � = ϕmμgicmgikf
∂mm∂t
+ϕmμgicd
kf
∂mm∂t
, ð15Þ
where cd is an additional compressibility that considers theeffect of desorption. The expression of cd is
cd =2Tpsc
ϕmμgiTsc
1 − ϕf − ϕmð ÞVLm pLð Þm pLð Þ +m pmð Þ½ �2 : ð16Þ
Incorporating the above dimensionless variables intothe seepage equation of the matrix and the fracture sys-tem, the dimensionless seepage equations of the matrixand the fracture system are
1rD
∂∂rD
r2D∂Δmf∂rD
� �= ωf
∂Δmf∂tD
− λ Δmm − Δmf½ �, ð17Þ
−λ Δmm − Δmf½ � = 1 − ωfð Þ ∂Δmm∂tD
+ ωd∂Δmm∂tD
: ð18Þ
Using the Laplace transform method, equations (17)and (18) can be rewritten as
1rD
∂∂rD
r2D∂Δ�mf∂rD
� �= ωf sΔ�mf − λ Δ�mm − Δ�mf½ �s, ð19Þ
−λ Δ�mm − Δ�mf½ � = 1 − ωf + ωdð ÞsΔ�mm: ð20Þ
According to equations (19) and (20), the seepageequation of the fracture system is
1r2D
∂∂rD
r2D∂Δ�mf∂rD
� �= f sð ÞΔ�mf , ð21Þ
where
f sð Þ = λ 1 + ωdð Þ + ωf 1 − ωf + ωdð Þsλ + 1 − ωf + ωdð Þs s: ð22Þ
4. MFHWs in a Rectangular Composite ShaleGas Reservoir with an SRV
4.1. Physical Model. In a shale gas or tight-gas reservoir,hydraulic fracturing is widely used to improve the pro-ductivity of the reservoir. A MFHW can make the crackscontact the matrix completely, form flow channels, andensure efficient extraction from an unconventional reser-voir [52, 55–58].
In 2008, Mayerhofer et al. [57] studied the change ofcracks in Barnett shale by microseismic technology and pro-posed the SRV concept. The SRV refers to an effective reser-voir that is made by hydraulic fracturing, with a composite
6 Geofluids
fracture network formed as a consequence. SRV can maxi-mize the contact surface between matrix and fractures so thatthe seepage distance of oil and gas in any direction frommatrix to cracks will be shortened, which improves the effec-tive permeability of the entire reservoir. According to themicroseismic monitoring map for an MFHW in Barnettshale reported by Fisher et al. [56], the physical model(Figure 3: left) can be approximated by the rectangularcomposite shale gas reservoir conceptual model (Figure 3:right) with enough accuracy, which is better than the cir-cular composite conceptual model for long horizontal wellcases.
To simplify the analysis, we use the following basicassumptions for this model: the MFHW is located at thecenter of a closed rectangular reservoir; the reservoir length,width, and height are xe, ye, and h, respectively; the frac-tures are perpendicular to the horizontal well and randomlydistributed along it; the effective well length is L (from theleftmost to the rightmost fractures); each fracture is sym-metrical or asymmetrical, and the fracture half-lengths areLfui and Lfli; the length and width of the SRV region arexm and ym, respectively; the number of fractures is M,and every fracture has infinite conductivity; no flow fromthe matrix to the wellbore and the pressure drop causedby the gas flow through the wellbore are ignored.
4.2. Boundary Element Model of a Composite Rectangular GasReservoir considering the SRV. According to the physicalmodel described above, the governing equation for gas flowin an inner fracture network system is ([59–62], 2019)
∂Δ�mf1∂x2D
+ ∂Δ�mf1∂y2D
+ ∂Δ�mf1∂z2D
+ 2Tpsckf1Tsch
qv1sδ
� xD − xs1D′ x, yD − ys1D′ , zD − zs1D′� �
= f1 sð ÞΔ�mf1:
ð23Þ
For the outer region
∂Δ�mf2∂x2D
+ ∂Δ�mf2∂y2D
+ ∂Δ�mf2∂z2D
+ 2Tpsckf2Tsch
qv2sδ
� xD − xs1D′ , yD − ys1D′ , zD − zs1D′� �
= f2 sð ÞΔ�mf2:
ð24Þ
The cohesive conditions on the surface between the innerand outer regions are
Δ�mf1 = Δ�mf2, xD, yD, zDð Þ ∈ Sinterface,∂Δ�mf1∂nD
= −1
M12
∂Δ�mf2∂nD
, xD, yD, zDð Þ ∈ Sinterface:ð25Þ
Because the outer boundary of our model is closed, theouter boundary condition is
∂Δ�mf2∂n
= 0, xD, yD, zDð Þ ∈ Sinterface: ð26Þ
Incorporating the dimensionless variables defined inAppendix B into equations (23) and (24), the mathematicalmodel in the Laplace space becomes
∂Δmf1D∂x2D
+ ∂Δmf1D∂y2D
+ ∂Δmf1D∂z2D
= f1 sð ÞΔ�mf1D −2πqv1DsM12
δ xD − xs1D′ , yD − ys1D′ , zD − zs1D′� �
,
ð27Þ
West to east (ft)–1000
Observationwell
–3000
–2500
–2000
–1500
–1000
Sout
h to
nor
th (ft
)
–500
0
500
1000
1500
–500 0 500 1000 1500 2000 2500 3000
Outer re
servo
ir regi
on
SRV re
gion
Hydrau
lic fra
ctures
Figure 3: Physical model schematic of fractured wells considering SRV [56].
7Geofluids
∂Δmf2D∂x2D
+ ∂Δmf2D∂y2D
+ ∂Δmf2D∂z2D
= f2 sð ÞΔ�mf2D −2πqv2D
sδ xD − xs1D′ , yD − ys1D′ , zD − zs1D′� �
,
ð28Þmf1D =mf2D, xD, yD, zDð Þ ∈ Sinterface, ð29Þ
∂mf1D∂nD
= −1
M12
∂mf2D∂nD
, xD, yD, zDð Þ ∈ Sinterface: ð30Þ
When using the boundary element method to solve themodel above, the fundamental solutions of differential equa-tions are needed. It is assumed that the fundamental solu-tions of the inner and outer regions in the shale gasreservoir are E1ðRD, RD′ , sÞ and E2ðRD, RD′ , sÞ, respectively,which satisfy
∇2E1 RD, RD′ , s� �
− f1 sð ÞE1 RD, RD′ , s� �
+ 2πδ RD, RD′� �
= 0,
ð31Þ
∇2E2 RD, RD′ , s� �
− f2 sð ÞE1 RD, RD′ , s� �
+ 2πδ RD, RD′� �
= 0:
ð32ÞSince the fractures are assumed completely open in this
paper, the expressions of fundamental solutions E1 and E2are
E1 = K0ffiffiffiffiffiffiffiffiffiffif1 sð Þ
pRD − RD′� �h i
,
E2 = K0ffiffiffiffiffiffiffiffiffiffif2 sð Þ
pRD − RD′� �h i
:ð33Þ
Both sides of equations (31) and (32) are multiplied bymf1D and mf2D, respectively, and the equations can beobtained by
mf1D∇2E1 − f1 sð Þmf1DE1 + 2πmf1Dδ RD, RD′
� �= 0, ð34Þ
mf2D∇2E2 − f2 sð Þmf2DE1 + 2πmf2Dδ RD, RD′
� �= 0: ð35Þ
Both sides of equations (23) and (28) are multiplied by E1and E2, respectively, and we have
E1∇2mf1D = f1 sð ÞE1Δ�mf1D − E1
2πqv1DsM12
δ RD − RwD′� �
, ð36Þ
E2∇2mf1D = f1 sð ÞE2Δ�mf1D − E2
2πqv2Ds
δ RD − RwD′� �
:
ð37Þ
Using equations (34) and (35) minus equations (36) and(37), respectively, the equations can be transformed as
mf1D∇2E1 − E1∇
2mf1D + 2πmf1Dδ RD, RD′� �
= E12πqv1DsM12
δ RD − RwD′� �
,ð38Þ
mf2D∇2E2 − E2∇
2mf2D + 2πmf2Dδ RD, RD′� �
= E22πqv2D
sδ RD − RwD′� �
:ð39Þ
Integrating equations (38) and (39) on the inner regionΩ1 and outer region Ω2, respectively, we have
ðΩ1
mf1D∇2E1 − E1∇
2mf1D + 2πmf1Dδ RD, RD′� �
− E12πqv1DsM12
�
� δ RD − RwD′� �
dΩ1 = 0,
ð40Þ
2356789 14
Inner SRV region
Outer reservoir region
Figure 4: Grid of the inner and outer boundaries and the sequence of the discrete elements in a rectangular composite gas reservoir.
8 Geofluids
ðΩ2
mf2D∇2E2 − E2∇
2mf2D + 2πmf2Dδ RD, RD′� �
− E22πqv2D
s
�
� δ RD − RwD′� �
dΩ2 = 0:
ð41ÞWe use Green’s second identity as follows [61, 62]:
ðΩ
μ∇2ν − ν∇2μ �
dΩ′ =ðsμ∇ν − ν∇μð ÞdS′, ð42Þ
where SðS =∑iSiÞ is the entire boundary of region Ω.Based on the characteristics of equation (42) and the
delta function, the differential equation of the inner and outerregions can be changed into an integral equation on theboundary, and the boundary integral equations of equations(40) and (41) are
mf1D RDð Þ = 12π
ðS1
E1∇mf1D −mf1D∇E1½ �dS1′
+ 1M12s
ðΩ1
qv1D Rs1D′� �
E1dΩ1′ ,ð43Þ
mf2D RDð Þ = 12π
ðS2
E2∇mf2D −mf2D∇E2½ �dS2′
+ 1s
ðΩ2
qv2D Rs2D′� �
E2dΩ2′:ð44Þ
There are two kind of boundaries (the outer and theinner) in the gas reservoir. It is assumed that the outerboundary can be discretized into NO elements and the innerboundary can be discretized into NI elements, so the totalnumber of the elements for the boundaries is NO + NI. Theserial number of inner boundary elements is clockwise, whileit is counterclockwise for outer boundary elements, as shownin Figure 4.
For the integral equation on the boundary, RD is an arbi-trary point on the boundary, while RD′ is an arbitrary point inthe reservoir region. RS1D and RS2D are the dimensionlesscoordinates of source points in the inner and outer regions,respectively. If RD′ is not only in the reservoir region but also
on the boundaries, equations (43) and equations (44) can berewritten as
θmf1D RDð Þ = 12π
ðS1
E1∇mf1D −mf1D∇E1½ �dS1′
+ 1M12s
ðΩ1
qv1D Rs1D′� �
E1dΩ1′ ,ð45Þ
θmf2D RDð Þ = 12π
ðS2
E2∇mf2D −mf2D∇E2½ �dS2′
+ 1s
ðΩ2
qv2D Rs2D′� �
E2dΩ2′ ,ð46Þ
where θ is a constant related to the geometric shape of RD′ ,and the expression of θ is [62]
θ = α
2π , ð47Þ
where α is the angle of the boundary tangent line at RD′ .Incorporating the fundamental solutions E1 and E2 into
equations (45) and (46), a series of linear equation sets canbe obtained by solving the reservoir points RD′ along theboundary points and fracture points, after which the dimen-sionless pseudopressure response curves of the fractured wellcan be obtained by solving the equation sets above. Due tothe complexity of the process, its details are not shown in thisarticle; interested readers may consult these papers fordetails: [52, 59, 60, 63–71].
4.3. Pressure and Production Type Curve Analysis. Theboundary element method is employed to acquire typecurves of dimensionless bottomhole pressures at constantrate, and the production rate and cumulative production ata constant pressure are based on the parameters in Table 2.
Figure 5 shows the well-test type curves of reservoirswith different sizes and SRVs. The following flow stagescan be observed when the reservoir and SRV sizes arelarge enough:
(a) Stage 1: early wellbore storage
(b) Stage 2: transition flow between wellbore storage andearly linear flow of a fracture system. A hump is inthe pseudopressure derivative curve for this stage,
Table 2: Reservoir properties.
Reservoir property Value Reservoir property Value
Permeability of SRV, kf1 (mD) 0.05 Porosity of SRV, ∅f1 (fraction) 0.05
Permeability of outer reservoir, kf2 (mD) 0.01 Porosity of SRV, ∅f2 (fraction) 0.002
Length of SRV, xm (m) 1600 Width of SRV, ym (m) 400
Inner matrix permeability, km1 (mD) 0.0001 Inner matrix porosity, ∅m1 0.01
Outer matrix permeability, km2 (mD) 0.0001 Outer matrix porosity, ∅m2 0.01
Wellbore storage coefficient, CD (dimensionless) 1 × 10−4 Gas relative density, γg (fraction) 0.65
9Geofluids
and the height and width of the hump depend on thewellbore storage and skin factor
(c) Stage 3: early linear flow of the fracture system. Inthis flow stage, the pseudopressure derivativecurve is a straight line, and the slope is 0.5. Theposition of the straight line is controlled by thepermeability of the SRV, and the length of this
straight line is related to the permeability of theSRV, the fracture length, and the distance betweenthe fractures
(d) Stage 4: early radial flow of fracture system. Forthis stage, the pseudopressure derivative curve isa portion of a horizontal straight line. In addition,when the height of the SRV is close to the length
1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
1E-3 1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8
mW
D´ m
WD⁎ t D
/CD
tD/CD
xe = 6000 m, ye = 1500 m, xm = 1600 m, ym = 400 mxe = ye = 60000 m, xm = ym = 7000 m
1
2
3
45
6
7
89
Figure 5: Effect of reservoir size on type curves.
1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
1E-3 1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6
M = 4M = 6M = 8
mW
D´ m
WD⁎ t D
/CD
tD/CD
Figure 6: Effect of fractures number on well-test type curves.
10 Geofluids
of the fracture, this line may be covered or offsetunder the effect of outer reservoir properties
(e) Stage 5: transition flow between early radial flow ofthe fracture system and radial flow of the SRV
(f) Stage 6: radial flow of the SRV. For this stage, thepseudopressure derivative curve is a section of a hor-izontal straight line. It is found that the higher the
permeability of the SRV, the lower the position of thishorizontal line
(g) Stage 7: transition flow between radial flow of theSRV and radial flow of the outer region
(h) Stage 8: radial flow of the outer region. In this stage,the pseudopressure derivative curve is a straight linewhose slope is 0.5
M = 4M = 6M = 8
0
0.2
0.4
0.6
0.8
1
1.2
0
10
20
30
40
50
60
70
80
90
100
0.1 1 10 100 1000 10000t(d)
0123456789
10
100 1000
qsc
(104 m
3 /d)
Qsc
(108 m
3 )
Figure 7: Effects of number of fractures on production rate and cumulative production curves.
1E-4
1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
mW
D´ m
WD⁎ t D
/CD
Kf1 = 0.05Kf1 = 0.25Kf1 = 1.25
1E-3 1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6tD/CD
Figure 8: Effect of SRV permeability on well-test type curves.
11Geofluids
(i) Stage 9: boundary-dominated flow. For this stage,the pseudopressure and pseudopressure derivativecurves overlap, and this section is a straight line witha slope of 1. Because xe = ye = 60000m, there is nolinear flow of the reservoir
When the sizes of the reservoir and the SRV are reason-able, the pseudopressure and pseudopressure derivativecurves are shown in Figures 5–11 as the plots with yellowpellets. After the early linear flow of the fracture system, thepressure reaches the region close to the reservoir boundary,
so the curves turn upward, and then the pseudopressureand pseudopressure derivative curves are almost parallel toeach other as a straight line, but the slope is not 0.5. Whenthe pressure reaches the reservoir boundary, the 2 curvesoverlap.
Figures 6 and 7 show the effects of the fracture numberon well-test type curves (dimensionless pseudopressure andits derivative) and production curves (production rate andcumulative production), respectively. As shown in Figure 6,for a constant reservoir size and SRV, the number of hydrau-lic fractures affects mainly the early flow periods following
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
50
100
150
200
250
300
0.1 1 10 100 1000 10000
0
2
4
6
8
10
12
1.00E+02 1.00E+03
t(d)
qsc
(104 m
3 /d)
Qsc
(108 m
3 )
Kf1 = 0.05Kf1 = 0.25Kf1 = 1.25
Figure 9: Effects of SRV permeability on production rate and cumulative production curves.
1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
mW
D´ m
WD⁎ t D
/CD
Kf2 = 0.01Kf2 = 0.05Kf2 = 0.25
1E-3 1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6tD/CD
Figure 10: Effect of reservoir permeability on well-test type curves.
12 Geofluids
the wellbore storage flow period. When the pressure wavetransports to the outer region, the effects will vanish and allpressure and pressure derivative curves will coincide. Whena well produces at a constant rate, the higher the fracturenumber, the higher the early production; this effect lasts forjust a few years.
Figures 8 and 9 show the effects of different flow regionpermeabilities on well-test type curves and productioncurves, respectively. The effects occur in the middle flowstage. The position of the well-test type curves is lower whenthe SRV permeability is higher, but the difference betweenthe well-test type curves becomes smaller and smaller as thepressure reaches the reservoir boundary. Figure 9 shows thatwhen SRV permeability increases, the early production rate ishigher than the previous one, the production rate is muchhigher in the middle stage, and the SRV permeability hasno effect on the final cumulative production.
Figures 10 and 11 show the effects of reservoir perme-ability on type curves and production curves, respectively.Because the reference permeability in our model is the res-ervoir permeability, the reservoir permeability has theopposite effect on type curves compared to that of SRV.The higher the reservoir permeability, the higher the loca-
tion of the type curves (as shown in Figure 10), whichdoes not mean that the drawdown pressure is higher.When the well is producing at constant pressure, the gassupply is timely due to the high permeability of the reser-voir, and gas production rate increases with reservoir per-meability increasing (as shown in Figure 11). Even in thelater production period when the well has been producingfor 1000d and 2000 d, the well production rate for a reser-voir with a higher permeability is still much higher, asshown in Table 3. Therefore, a desert area is very impor-tant for the exploitation and development of unconven-tional shale gas reservoirs.
5. Conclusions
We established a comprehensive mathematical model todescribe a multistage fractured horizontal well in a rectangu-lar composite shale gas reservoir with an SRV, and the solu-tion was also obtained by a boundary element method. Themain conclusions we derived are as follows:
(1) A composite rectangular model of a fractured hori-zontal well using the “tri-pores” conceptual modelis proposed in this paper, and the complex modelwas solved by BEM, which has not been comprehen-sively reported before
(2) The number of fractures affects the well productionrate at primarily the early flow period; the effect onthe later flow period is weak
(3) The permeability in the SRV region has a substantialeffect on the early well production rate; the higher thepermeability, the greater the rate will be
0
0.5
1
1.5
2
0
20
40
60
80
100
120
140
0.1 1 10 100 1000 10000
Qsc
(108 m
3 )
qsc
(104 m
3 /d)
t(d)
Kf2 = 0.01Kf2 = 0.05Kf2 = 0.25
0
2
4
6
8
10
100 1000 10000
Figure 11: Effects of reservoir permeability on production rate and cumulative production curves.
Table 3: Well production rates for different reservoir permeabilitiesat different times.
Reservoir permeability (mD)Production rate (104m3/d)1000 (d) 2000 (d)
0.25 5.42 2.78
0.05 3.41 2.01
0.01 1.99 1.19
13Geofluids
(4) During the intermediate flow period, the effect will bethe opposite, which can be explained by mass conser-vation for a closed constant-volume gas reservoir
Appendix
A. Transformations
Let mf =mðpf Þ and mm =mðpmÞ; then the pseudopressureand gas density expression are introduced into equation (5),so the continuity equation in the fracture network is
1r2
∂∂r
r2∂mf∂r
� �=μgiϕf cfgi
kf
∂mf∂t
+ 3Rm
kmkf
∂mm∂rm
����rm=Rm
: ðA:1Þ
The shape factor of the spherical matrix particles is
α = 15R2m: ðA:2Þ
Inserting the shape factor into equation (A.1) yields
1r2
∂∂r
r2∂mf∂r
� �=μgiϕf cfgi
kf
∂mf∂t
+ αRm5
kmkf
∂mm∂rm
����rm=Rm
:
ðA:3Þ
Inserting equation (11) into equation (6) yields
1r2m
∂∂rm
kmμg
ρgrm2 ∂pm∂rm
!=ϕmμgicmgi
km
∂mm∂t
+ϕmμgicdkm
∂mm∂t
,
ðA:4Þ
where cd is an additional compressibility taking into accountthe effect of desorption; the expression is given in the follow-ing formula:
cd =2Tpsc
ϕmμgiTsc
1 − ϕf − ϕmð ÞVLm pLð Þm pLð Þ +m pmð Þ½ �2 : ðA:5Þ
The initial condition of the spherical matrix is
mmj t = 0, rmð Þ =m pið Þ: ðA:6Þ
The inner boundary condition of the spherical matrix is
∂mm∂rm
���� t, rm = 0ð Þ = 0: ðA:7Þ
The outer boundary condition of the matrix is
mmj t, rm = Rmð Þ =mf : ðA:8Þ
All the dimensionless variables are defined as
rmD = rmRm
,
rD = rLref
,
tD = kf t
ϕmcmgi + ϕf cfgi �
μgiL2ref
,
ωf =ϕf cfgi
ϕmcmgi + ϕf cfgi,
ωd =ϕmcd
ϕmcmgi + ϕf cfgi,
λ = αkmkf
L2ref ,
Δmf =m pið Þ −m pfð Þ,Δmm =m pið Þ −m pmð Þ:
ðA:9Þ
Inserting the dimensionless variables above into the seep-age model of the matrix and fracture network, we can rewritethe model as
1r2D
∂∂rD
r2D∂Δmf∂rD
� �= ωf
∂Δmf∂tD
+ λ
5∂Δmm∂rmD
����rmD=1
,
1r2mD
∂∂rmD
r2mD∂Δmm∂rmD
� �= 15 1 − ωfð Þ
λ
∂Δmm∂tD
+ 15ωdλ
∂Δmm∂tD
,
Δmmj tD = 0, rmDð Þ = 0,∂Δmm∂rmD
���� tD, rmD = 0ð Þ = 0,
Δmmj tD, rmD = 1ð Þ = Δmf :
ðA:10Þ
Using the Laplace transform method, we can get
Δ�m =ð∞0Δme−stDdtD: ðA:11Þ
And the model above can be written as
1r2D
∂∂rD
r2D∂Δ�mf∂rD
� �= ωf sΔ�mf +
λ
5∂Δ�mm∂rmD
����rmD=1
, ðA:12Þ
1r2mD
∂∂rmD
r2mD∂Δ�mm∂rmD
� �= 15s 1 − ωf + ωdð Þ
λΔ�mm, ðA:13Þ
Δ�mmj tD = 0, rmDð Þ = 0, ðA:14Þ∂Δ�mm∂rmD
���� tD, rmD = 0ð Þ = 0, ðA:15Þ
Δ�mmj tD, rmD = 1ð Þ = Δ�mf : ðA:16Þ
14 Geofluids
Do the following variable substitution
W = rmDΔ�mm: ðA:17Þ
Inserting equation (A.17) into equation (A.13), we have
∂2W∂r2mD
= 15s 1 − ωf + ωdð Þλ
W: ðA:18Þ
According to the characteristics of the above equation,the general solution of equation (A.18) can be obtained easilyby
W = A sinh ffiffiffig
prmDð Þ + B cosh ffiffiffi
gp
rmDð Þ, ðA:19Þ
where
g = 15 1 − ωf + ωdð Þsλ
: ðA:20Þ
Inserting equation (A.19) into equation (A.17),
Δ�mm = A sinh ffiffiffig
prmD
�+ B cosh ffiffiffi
gp
rmD �
rmD: ðA:21Þ
B can be determined by the inner boundary condition,and A can be determined by the outer boundary condition.A and B have the following values:
B = 0, ðA:22Þ
A = Δ�mf /sinh ð ffiffiffig
p Þ:Incorporating the value of A and B into equation (A.21)
yields
Δ�mm = Δ�mfsinh ffiffiffi
gp � sinh ffiffiffi
gp
rmD �rmD
: ðA:23Þ
The derivative of equation (A.23) is
∂Δ�mm∂rmD
����rmD=1
= ffiffiffig
p coth ffiffiffig
pð Þ − 1½ �Δ�mf : ðA:24Þ
Incorporating equation (A.24) into equation (A.12) andthen simplifying it, we can then obtain the following:
1r2D
∂∂rD
r2D∂Δ�mf∂rD
� �= f sð ÞΔ�mf , ðA:25Þ
where the expression of f ðsÞ is
f sð Þ = ωf s +λ
5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi15 1 − ωf + ωdð Þs
λ
rcoth
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi15 1 − ωf + ωdð Þs
λ
r− 1
" #:
ðA:26Þ
B. Dimensionless variables
All the dimensionless variables are defined as
rD = rLref
,
tD = kf t
ϕmcmgi + ϕf ccgi �
μgiL2ref
,
ωf =ϕf cfgi
ϕmcmgi + ϕf cfgi,
ωd =ϕmcd
ϕmcmgi + ϕf cfgi,
λ = αkmkf
L2ref ,
Δmf =m pið Þ −m pfð Þ,Δmm =m pið Þ −m pmð Þ:
ðB:1Þ
Data Availability
The production data used to support the findings of thisstudy are included within the article.
Conflicts of Interest
The authors declare that they have not conflicts of interests.
Acknowledgments
This work was supported by the Demonstration Projectof the National Science and Technology of China (GrantNo. 2016ZX05062), the Science and Technology MajorProject of PetroChina (Grant No. 2016E-0611), and theNational Science and Technology Major Project of theMinistry of Science and Technology of China (GrantNo. 2017ZX05037-001)
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