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Production decline type curves analysis of a finite conductivity fractured well in coalbed methane reservoirs

  • Mingqiang Wei EMAIL logo , Ming Wen , Yonggang Duan , Quantang Fang and Keyi Ren
Published/Copyright: April 14, 2017
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Open Physics
From the journal Open Physics Volume 15 Issue 1

Abstract

Production decline type curves analysis is one of the robust methods used to analyze transport flow behaviors and to evaluate reservoir properties, original gas in place, etc. Although advanced production decline analysis methods for several well types in conventional reservoirs are widely used, there are few models of production decline type curves for a fractured well in coalbed methane (CBM) reservoirs. In this work, a novel pseudo state diffusion and convection model is firstly developed to describe CBM transport in matrix systems. Subsequently, based on the Langmuir adsorption isotherm, pseudo state diffusion and convection in matrix systems and Darcy flow in cleat systems, the production model of a CBM well with a finite conductivity fracture is derived and solved by Laplace transform. Advanced production decline type curves of a fractured well in CBM reservoirs are plotted through the Stehfest numerical inversion algorithm and computer programming. Six flow regimes, including linear flow regime, early radial flow in cleat systems, interporosity flow regime, late pseudo radial flow regime, transient regime and boundary dominated flow regime, are recognized. Finally, the effect of relevant parameters, including the storage coefficient of gas in cleat systems, the transfer coefficient from a matrix system to the cleat system, the modified coefficient of permeability, dimensionless fracture conductivity and dimensionless reservoir drainage radius, are analyzed on type curves. This paper does not only enrich the production decline type curves model of CBM reservoirs, but also expands our understanding of fractured well transport behaviors in CBM reservoirs and guides to analyze the well's production performance.

Keywords: Coal bed methane; Langmuir adsorption isotherm; production; type curves; fractured well
PACS: 47.56.+r

Nomenclature

βmmodify coefficient of permeability
γfree gas storage coefficient
λtransfer coefficient from matrix systems to cleat systems
μgas viscosity (mPa·s)
ωstorage coefficient of gas in cleat systems
ψ¯fDdimensionless pressure in cleat systems in Laplace space
ψ¯mDdimensionless pressure in matrix systems in Laplace space
q¯Ddimensionless rate in Laplace space
ψfDdimensionless pressure in cleat systems
ψmDdimensionless pressure in matrix systems
ρfgas density in cleat systems under the reservoir condition (kg/m3)
ρmgas density under the reservoir condition (kg/m3)
ϕfporosity in cleat systems (decimal)
bmslippage factor
cgmgas compressibility in matrix systems (MPa−1)
Cmgas molar concentration (mol/m3)
ctmtotal compressibility in matrix systems (MPa−1)
Dgas diffusion coefficient (m2/s)
FcDdimensionless fracture conductivity
hreservoir thickness (m)
kfhpermeability in cleat systems (m2)
kffracture permeability (μm2)
Lreference length (m)
Mgas relative molecular mass (kg/mol)
pDiddimensionless pressure integral derivative function
pDidimensionless pressure integral function
pfpressure in cleat system (MPa)
pmpressure in the matrix system (MPa)
qDdiddimensionless rate integral derivative function
qDdidimensionless rate integral function
qDdimensionless decline rate
qfDdimensionless production rate of the fracture
qscproduction rate (m3/day)
rradial distance (m)
rDdimensionless radius
reDdimensionless reservoir drainage radius
rmDdimensionless radial distance in matrix system
Rmspherical radius of matrix elements (m)
rmspherical radius for a matrix element (m)
sffracture wall skin
tDAdimensionless time with A area
tDddimensionless decline time
tDdimensionless time
vmflow velocity in matrix (m2/s)
vmcvelocity caused by concentration gradient (m2/s)
vmpvelocity caused by pressure gradient (m2/s)
Wfracture width (m)
xDi+1end of the ith segment
xDibeginning of the ith segment
xDdimensionless x distance
xfhalf length of fracture (m)
yDdimensionless y distance
ztime in Laplace space

1 Introduction

Coal bed methane (CBM) reservoirs are considered as an important unconventional gas reservoirs since these ones have made a significant contribution to the world gas production[1–3]. Coal beds are naturally fractured reservoirs with cleat network and matrix blocks. Compared with conventional gas reservoirs, CBM reservoirs have complex reservoirs characteristics, such as adsorption/desorption in matrix systems[4–6]. Meanwhile, different coal seams have different compositions (e.g. thermal maturity, pressure and adsorption/desorption capacity, etc.). Therefore, each CBM reservoir has different transport behaviors. As it's known, transient pressure analysis (well testing analysis) and production decline analysis are practical and powerful tools to characterize the unsteady flow behaviors of reservoirs[7–9] and to recognize formation parameters. In recent years, production analysis for reservoirs characterization is approaching the popularity of transient pressure analysis, but there are limited production decline type curves models and methods for CBM reservoirs [10].

In the past decades, much progresses have been made. Considering the adsorption/desorption, Anbarci and Ertekin[11] investigated the transient flow behaviors of vertical well for single phase gas flow in CBM reservoirs. Pinzon and Patterson[12], and Clarkson et al. [13] analyzed production data for a dry gas CBM well and taken radial flow type curves of CBM data to account for gas desorption, respectively. Assuming a constant pressure inner boundary condition, Mohaghegh and Ertekin[14] obtained radial flow type curves using a numerical simulator. Later, Guo et al. [15] investigated numerical model of a CBM reservoirs. Aminian et al. [16] also applied a numerical simulator to generate production type curves by defining a new set of dimensionless variables. Aminian and Ameri[17] developed a numerical reservoir model of CBM to analyze the well's performance. Later, Clarkson et al. [13] studied the production decline of a fractured and horizontal CBM well by using Fetkovich type curves method to evaluate reservoir information quantitatively. Nie et al. [18] and Guo et al. [15] investigated the transient transport behavior and production decline type curves of a horizontal well in CBM reservoirs by using the semi-analytical method.

But the aforementioned transient transport models of CBM reservoirs only considered diffusion mechanism in matrix [19–25]. However, King and Ertekin[26] and Li et al. [27] stated that the more rigorous model of CBM should also take pressure transient (convection) in matrix systems into consideration. In addition, although most aforementioned studies have utilized radical flow type curves of CBM vertical wells and horizontal wells to match the production data, there are few attempts to study the production decline type curves for CBM fractured well.

Thus, this paper focuses on generating production decline type curves for a CBM well with a finite conductivity vertical fracture by considering diffusion and convection (Darcy flow) in matrix systems at same time. Firstly, the CBM transport model coupled with pseudo state diffusion and convection in matrix system is developed. Secondly, a production decline model for a single phase CBM fractured well is derived and solved by using the semi-analytical method. Thirdly, the model solution is verified by a simplified model, then production decline type curves are plotted, their characteristics and the effects of related parameters are analyzed. The proposed model of a finite conductive fracture well in a CBM reservoirs does not only expand our understanding of fractured well behaviors in CBM reservoirs, but also enriches the production decline type curves model in CBM reservoirs.

2 Model construction

CBM reservoirs are known as a typical dual porosity media, which is consisted of cleat networks and matrix systems (Figure 1a). The physical properties of the matrix and cleat of the CBM formulation are independent. Each matrix element is assumed as spherical in shape. The absorbed gas will be desorbed from matrix blocks. Darcy flow in cleats, pseudo-steady state flow, including diffusion driven by concentration gradient, and Darcy flow driven by the pressure gradient in the matrix elements’ micropores are considered (Figure 1b).

Figure 1 a) Plan view of a physical model in a CBM b) Transport scheme in CBM reservoir (modified from Nie et al. [18])
Figure 1

a) Plan view of a physical model in a CBM b) Transport scheme in CBM reservoir (modified from Nie et al. [18])

Based on flow mechanisms in coal matrixs, the flow velocity is dominated by both the pressure gradient and the concentration gradient. Referencing the research result of Ertekin et al. [28], the velocity in the matrix can be added by the velocity of the pressure gradient and the concentration gradient linearly, which can be expressed as:

(1)vm=vmp+vmc=kmμ∂pm∂rm+MDρm∂Cm∂rm

where vm is the flow velocity in the matrix, m2/s; vmp is the velocity caused by the pressure gradient, m2/s; vmc is the velocity caused by the concentration gradient, m2/s; rm is the spherical radius for a matrix element, m; μ is the gas viscosity, mPa.s; ρm is the gas density under the reservoir condition, kg/m3; pm is the pressure in the matrix system, MPa; Cm is the gas molar concentration, mol/m3; M is the gas relative molecular mass, kg/mol; D is the gas diffusion coefficient, m2/s.

Further, inserting the gas state equation into Eq. (1), Eq. (1) can be rewritten as:

(2)vm=kmμ1+μcgmDkm∂pm∂rm=kmμ1+bmpm∂pm∂rm=kmμβm∂pm∂rm

where cgm is the gas compressibility in matrix systems, MPa−1; bm is the slippage factor, and βm is the modified coefficient of permeability, defined as:

(3)βm=1+bmpm

2.1 Physical conceptual model and its assumptions

Due to the features of low permeability and porosity in CBM reservoirs, a fractured well is one of the common well types to develop CBM reservoirs. Thereby, a fractured well in a CBM reservoir with a radial coordinate system is considered in this paper (Figure 2), and assumptions of the physical model are as follows:

  1. The CBM reservoir is assumed as a spherical dual porosity model (Swaan [29]) with uniform thickness, initial pressure, permeability, Langmuir volume, Langmuir pressure and porosity;

  2. A fractured well is located in the center of the reservoir, and produces gas at a constant rate or at a constant wellbore flowing pressure, whose external boundaries are closed;

  3. The hydraulic fracture penetrates the formation vertically, and hhas a half fracture length χf a permeability kf, a finite conductivity FcD and a width W;

  4. The desorption occurring at the matrix surface can be described by the Langmuir adsorption isotherm equation, and both pseudo steady diffusion and convection are considered in matrix systems; while the flow in cleat systems obeys Darcy's law[18, 24];

  5. The influence of gas gravity is neglected, and the reservoir is isothermal.

Figure 2 Theoretical model of a fractured well in a coal bed methane reservoir
Figure 2

Theoretical model of a fractured well in a coal bed methane reservoir

2.2 Mathematical modeling

According to the above assumptions, the mathematical models can be established and derived as follows:

(0)In matrix systems

Considering the pseudo steady diffusion and convection from matrix systems to cleat systems and the gas desorption from the matrix surface, the governing equation in spherical matrix systems can be written based on Eq. (2) [30]:

(4)−15kmβmRm2μρ0pm−pf=∂ρmφm∂t+ρsc∂∂tVLpmPL+pm

where Rm is the spherical radius of matrix elements, m; pf is the pressure in a cleat system, MPa.

Substituting the equation of gas state into Eq. (4), the following equation can be obtained:

(5)−15kmβmRm2p0μzpm−pf=φmcgmpmz∂pm∂t+pscTTscVLpLpL+pm2∂pm∂t

Due to variation of physical properties of gas with pressure, Ertekin and Sung[20] and King and Ertekin[26] used pseudo-pressure function to simplify the transport mathematical model.

(6)ψ=2∫pscppμzdp

Substituting Eq. (6) into Eq. (5), Eq. (5) can be rewritten as:

(7)−15R2ψm−ψf=φmμkmβmcgm+ρscρmφmVLpLpL+pm2∂ψm∂t

Furthermore, to simplify the derivation, the mathematical model can be derived in dimensionless form. Dimensionless definitions of all variables in the mathematical model proposed in this paper are shown in the Appendix.

With the definitions in the Appendix and the Laplace transform, Eq. (7) can be written in the following form:

(8)1−ωγγβm∂ψ¯mD∂tD+λzψ¯mD−ψ¯fD=0

where z is the time in Laplace space, λ is the transfer coefficient from matrix systems to cleat systems; ω is the storage coefficient of gas in cleat systems. ctm is the total compressibility in matrix systems, MPa−1; γ is the free gas storage coefficient.

(9)λ=15kmkfhL2R2ω=φfμcgfμφmcgm+φfcgfγ=φmcgm+φfcgfφmctm+φcgfctm=cgm+ρscρmφmVLpLpL+pm2

where kfh is the permeability in cleat systems, m2; L is the reference length, m.

(0)In cleat systems

Based on the Darcy flow and material balance law, the governing equation in cleat systems can be written as:

(10a)1r∂∂rrρfkfhμ∂pf∂r+15Rm2kmβmμρ0pm−pf=∂ρfφf∂t

where r is the radial distance, m; ρf is the gas density in cleat systems under the reservoir condition, kg/m3; φf is the porosity in cleat systems, decimal.

Using the initial condition:

(10b)pft=0=pi

In the cleat systems, the inner boundary condition at constant rate production is:

(10c)r∂pf∂rr→0=qscμB2πkfhh

where qsc is the production rate, m3/day; h is the reservoir thickness, m.

The external boundary (closed boundary) is:

(10d)∂pf∂rr=re=0

Furthermore, combined dimensionless definitions with the Laplace transform, Eqs. (10a)–(10d) can be written as:

(11a)∂2ψ¯fD∂rD2+1rD∂ψ¯fD∂rD−λβzψ¯fD−ψ¯mD=ωz∂ψ¯fD

Inner boundary:

(11b)limrD→0∂ψ¯fD∂rD=−1z

External boundary:

(11c)∂ψ¯fD∂rDreD,z=0

Combining Eq. (8) with Eqs. (11a)–(11c), the point source solution of transient pressure response in Laplace space can be obtained.

(12)ψ¯fDrD,z=1zK0rDzfz+K1reDzfzI1reDzfzI0rDzfz

where rD=xD2+yD2,fz=λβm+ω1−ωγzγλβm+1−ωγz

(0)Flow model of hydraulic fracture

According to the details derivation of a finite conductivity vertical fracture in reservoirs[31, 32], the pressure response of a finite conductivity fractured well in CBM reservoirs in Laplace space can be obtained.

(13)ψ¯wD−12∫01q¯fDα,zK0xD−α2+yD2⋅zfzdα+12zzf(z)K1zfzreDI1zfzreD∫0zfz1−xDI0αdα+∫0zfz1+xDI0αdα−sfq¯fDxD,z+πFcD∫0xD∫0x′qfD(x″,z)dx″dx′=πxDFcDz

where FcD is the dimensionless fracture conductivity; sf is the fracture wall skin.

To solve Eq. (13), the fracture should firstly be discreted in several segments. If we discretize the half length of fracture in n segments with equal length, then Eq. (13) can be rewritten as:

(14)ψ¯wD−12∑i=1nq¯fDz∫xDixDi+1K0xD−x′+K0xD+x′dx′+12z1zfzK1zfzreDI1zfzreD∫0zfz1−xDI0αdα+∫0zfz1+xDI0αdα−sfq¯fDxD,z=πzFcDxD−z∫0xD∫0vqfDu,zdudv

where χDi and χDi + 1 are the beginning and the end of the ith segment. χD is the center of a segment.

If we write Eq. (14) for every fracture segment, n equations with n+1 unknowns (ψ¯wDz,q¯fDiz,i=1,⋯n) will be obtained. According to the constraints of the flow rate, the additional equation is:

(15)∑i=1nqfDiz=1z

Combining Eq. (14) with Eq. (15), the n+1 unknowns including q¯fDiz,i=1,⋯n can be calculated by solving the above equations systems.

3 Production decline type curves theory

Furthermore, the production solution in Laplace space for modeling a finite conductive fractured well with constant pressure production, can be obtained by[33–35]:

(16)q¯D=1z2ψ¯wD

where q̅D is the dimensionless rate in Laplace space.

To analyze production data of a fractured well in low permeability and diagnose reservoirs properties, Pratikno et al. [36] developed the production decline type curves theory, which has been widely used in field production dynamic analysis. The general definition of production decline type curves as given are[9, 36]:

Dimensionless decline rate:

(17)qDd=qDbDpss

Dimensionless decline time:

(18)tDd=2πbDpsstDA

For a well with a finite conductivity vertical fracture, the bDpss is a function of reD and FcD [36]:

(19)bDpss=lnreD−0.049298+0.43464reD2+a1+a2u+a3u2+a4u3+a5u41+b1u+b2u2+b3u3+b4u4

where

  1. u = ln (FcD)

  2. a1 = 0.93626800

  3. b1 = −0.38553900

  4. a2 = −1.00489000

  5. b2 = −0.06988650

  6. a3 = 0.31973300

  7. b3 = −0.0484653

  8. a4 = −0.04235320

  9. b4 = −0.00813558

  10. a5 = 0.00221799

Dimensionless rate integral function qDdi:

(20)qDdi=NpDdtDd=1tDd∫0tDdqDd(α)dα

Dimensionless rate integral derivative function qDdid:

(21)qDdid=−dqDdidln⁡(tDd)=−tDddqDdidtDd

4 Results and discussions

4.1 Model Validation

For this paper, the CBM reservoir is assumed as a dual porosity one which contains cleat systems and matrix systems. If some parameters of Eq. (14) are set to satisfy some conditions, the model proposed in this paper can be converted into other models. If the function f(z) = 1, this new model can be simplified as a well with a finite conductivity fracture in the homogenous reservoir. To validate our results, the production decline type curves data of the simplified model reported in literature[36] are chosen. Figure 3 shows the comparison results of production decline type curves of a finite conductivity vertical fracture well with FcD = 0.1π, reD = 500 and sf = 0.

Figure 3 Comparison of production decline type curves of a finite conductivity vertical fracture well in homogenous reservoirs with FcD = 0.1π, reD = 500 and sf = 0
Figure 3

Comparison of production decline type curves of a finite conductivity vertical fracture well in homogenous reservoirs with FcD = 0.1π, reD = 500 and sf = 0

As seen from Figure 3, the type curves of the simplified model show excellent agreement with the work of Pratikno et al. [36]. Therefore, the type curves of the model proposed in this paper are reliable.

4.2 Type curves of a fractured well in CBM reservoirs

Based on the proposed model outlined above, production decline type curves of a finite conductivity vertical fracture well in CBM reservoirs can be obtained by solving Eqs. (16), (17), (20) and (21). To explain the transient transport process of a fractured well in CBM reservoirs more clearly, the following curves are integrated in Figure 4, including type curves of qDd, qDdi, and qDdid, pDi and pDid versus tDd. The definition of pDi and pDid are given in the Appendix. According to the type curves behavior (Figure 4), six main flow regimes are recognized:

Figure 4 Dimensionless pressure integral function, dimensionless pressure integral derivative function and production decline type curves of a finite conductivity fracture well in CBM reservoirs (FcD=10, reD=260, λ=0.001, ω=0.05, βm=3, sf=0)
Figure 4

Dimensionless pressure integral function, dimensionless pressure integral derivative function and production decline type curves of a finite conductivity fracture well in CBM reservoirs (FcD=10, reD=260, λ=0.001, ω=0.05, βm=3, sf=0)

Regime I: linear flow regime (Figure 5a). The curves of dimensionless pressure integral function and dimensionless pressure integral derivative function are two parallel sloping lines. This flow regime represents fluid flowing linearly from the formation into hydraulic fractures and from hydraulic fractures into the wellbore.

Figure 5 Schematic diagrams of the typical flow regimes of a finite conductivity fracture well in CBM reservoirs
Figure 5

Schematic diagrams of the typical flow regimes of a finite conductivity fracture well in CBM reservoirs

Regime II: early radial flow in cleat systems (Figure 5b). The slope of the dimensionless rate integral derivative function is constant, and the slope of dimensionless pressure integral derivative function is zero.

Regime III: interporosity flow regime. Both of the curves of the dimensionless rate integral derivative function and dimensionless pressure integral derivative function are concave, which reflect the transfer of CBM from the matrix system to cleat systems.

Regime IV: the second radial flow (late pseudo radial flow) regime (Figure 5c). The curves of the dimensionless rate and dimensionless rate integral derivative converge to a constant slope line, and the slope of dimensionless pressure integral derivative is zero. This regime indicates that fluid is flowing radially from the formation into the wellbore.

Regime V: transient regime.

Regime VI: boundary dominated flow regime (Figure 5d). The dimensionless decline rate converges to a straight line with −1 slope, while the slope of the dimensionless pressure integral function and pressure integral derivative function is 1.

4.3 Parameter sensitivity analysis

Varying parameters can affect the transient production decline type curves significantly. Therefore, the effect of these parameters on production decline type curves will be discussed in detail.

Figure 6 shows the type curves characteristic of production decline affected by the gas storage coefficient of gas in the cleat (ω). The ω represents the relative storage capacity of CBM in cleat systems. It can be observed that ω mainly affects the type curves in regime I-III. Meanwhile a bigger ω leads to a larger value of type curves and a shallower concavation of dimensionless rate integral derivative function in the interporosity flow regime.

Figure 6 Effect of the storage coefficient of gas in the cleat (ω) on production decline type curves
Figure 6

Effect of the storage coefficient of gas in the cleat (ω) on production decline type curves

Figure 7 shows the production decline type curves of different transfer coefficient from a matrix to the cleat (λ). The parameter λ reflects the relative flow capacity from matrix systems to cleat systems. As is shown in Figure 6, parameter λ has a dominant effect on the interporosity flow (regime III). A larger λ means a stronger flow capacity in matrix systems, which results in an earlier emergence time of regime III. When λ equals 0.001 in Figure 6, the second radial flow of dimensionless rate integral derivative function curve does not emerge (Figure 7). Therefore, if λ is too small, then regime IV will be concealed.

Figure 7 Effect of the transfer coefficient from the matrix to the cleat (λ) on production decline type curves
Figure 7

Effect of the transfer coefficient from the matrix to the cleat (λ) on production decline type curves

The effect of the modified coefficient of permeability (βm) on production decline type curves is nearly the same as that of the storage coefficient of gas in the cleat (ω), but is less significant. As is shown in Figure 6 and Figure 7, the period of regime will be start earlier as λ and βm increases. This is because a bigger λ and βm will lead to a higher relative flow capacity of matrix systems.

Figure 9 shows that the dimensionless fracture conductivity (FcD) has a great effect on regime – of production decline type curves. It can be seen that the value of production decline type curves increases with increasing value of FcD during the four regimes. The main reason for this is that the larger FcD is, the smaller the resistance of flow in a hydraulic fracture is. Notably, we can obtain more insight from the detail in Figure 8. As FcD increases from 1 to 10, the improved value of production decline type curves is much less than the one when FcD increases from 10 to 100. In addition, there is an interesting phenomenon that type curves are nearly coincident when FcD = 100 and FcD = 1000. This is observed when the dimensionless fracture conductivity increases to 100, and the production decline type curves of a finite conductivity vertical fracture well (FcD = 0.1π, reD = 500, sf=0) is nearly the same as that of an infinite conductivity vertical fracture well. Therefore, when the dimensionless fracture conductivity is more than 100, the fracture can be considered as an infinitely conductive one.

Figure 8 Effect of the modified coefficient of permeability (βm) on production decline type curves
Figure 8

Effect of the modified coefficient of permeability (βm) on production decline type curves

Figure 9 Effect of the dimensionless fracture conductivity (FcD) on production decline type curves
Figure 9

Effect of the dimensionless fracture conductivity (FcD) on production decline type curves

Figure 10 Comparison of production decline type curves of a finite conductivity vertical fracture well (FcD = 0.1π, reD = 500, sf = 0) and of an infinite conductivity vertical fracture well
Figure 10

Comparison of production decline type curves of a finite conductivity vertical fracture well (FcD = 0.1π, reD = 500, sf = 0) and of an infinite conductivity vertical fracture well

Figure 11 shows the effect of the dimensionless reservoir drainage radius (reD) on production decline type curves. It can be seen that four regimes including linear flow, early radial flow in cleat systems, interporosity flow regime, late pseudo radial flow regime, are governed by reD. In addition, the value of type curves increases with a decrease in the value of reD during the four regimes.

Figure 11 Effect of dimensionless reservoir drainage radius (reD) on production decline type curves
Figure 11

Effect of dimensionless reservoir drainage radius (reD) on production decline type curves

According to the results discussed above, the production decline type curves for CBM reservoirs in this work provides a useful tool for analysis of actual field production data of a fractured well in CBM reservoirs.

5 Conclusions

This paper investigates the production decline type curves of a finite conductivity fracture well in CBM reservoirs, and analyzes the effects of the characteristic parameters. The main conclusions can be drawn:

  1. A novel production decline type curves analysis model of a finite conductivity fractured well in CBM reservoirs, which considers the pseudo steady diffusion and convection from matrix systems to cleat systems, is established.

  2. Standard production decline type curves are plotted, and divided into six regimes including linear flow regime, early radial flow in cleat systems, interporosity flow regime, the second radial flow regime, transient regime, and boundary dominated flow regime.

  3. The effects of five parameters ω, λ, βm, reD, and FcD on production decline type curves are analyzed in detail. The results show the following: A bigger ω leads to a shallower concavation of dimensionless rate integral derivative function in interporosity flow regime. The period of interporosity flow regime will start earlier as λ and βm increases. During linear flow, early radial flow in cleat systems, interporosity flow regime, and late pseudo radial flow regime, the value of production decline type curves increases with a decrease in reD while the value of type curves increases with an increase in FcD.

Acknowledgement

This article was supported by Sichuan youth science and technology innovation research team project. The authors would also like to thank the reviewers and editors. They thoroughly reviewed the manuscript and their critical comments were very helpful in preparing this paper.

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Appendix

Dimensionless radial distance in the matrix system:

rmD=rmRm

Dimensionless radius:

rD=rxf

Dimensionless reservoir drainage radius:

reD=rexf

Dimensionless χ distance:

xD=xxf

Dimensionless y distance:

yD=yxf

Dimensionless time (A):

tDA=kftμφmcgm+φfcgfA2

Dimensionless time (xf):

tD=kftμφmcgm+φfcgfxf2

Dimensionless pressure in cleat systems:

ψfD=πkfhTscpscqscTψi−ψf

Dimensionless pressure in matrix systems:

ψmD=πkfhTscpscqscTψi−ψm

Dimensionless fracture conductivity:

FcD=kfWkfhxf

Dimensionless production rate of the fracture:

qfD=qfqsc:

Dimensionless pressure integral function:

pDi=1tDa∫0tDapDdtDa

Dimensionless pressure integral derivative function:

pDid=dpDIdIntDa=tDadpDidtDa
Received: 2016-9-30
Accepted: 2016-12-21
Published Online: 2017-4-14

© 2017 M. Wei et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/phys-2017-0015
Keywords for this article
Coal bed methane; Langmuir adsorption isotherm; production; type curves; fractured well
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  56. Numerical investigation of magnetohydrodynamic slip flow of power-law nanofluid with temperature dependent viscosity and thermal conductivity over a permeable surface
  57. Regular Articles
  58. Charge conservation in a gravitational field in the scalar ether theory
  59. Regular Articles
  60. Measurement problem and local hidden variables with entangled photons
  61. Regular Articles
  62. Compression of hyper-spectral images using an accelerated nonnegative tensor decomposition
  63. Regular Articles
  64. Fabrication and application of coaxial polyvinyl alcohol/chitosan nanofiber membranes
  65. Regular Articles
  66. Calculating degree-based topological indices of dominating David derived networks
  67. Regular Articles
  68. The structure and conductivity of polyelectrolyte based on MEH-PPV and potassium iodide (KI) for dye-sensitized solar cells
  69. Regular Articles
  70. Chiral symmetry restoration and the critical end point in QCD
  71. Regular Articles
  72. Numerical solution for fractional Bratu’s initial value problem
  73. Regular Articles
  74. Structure and optical properties of TiO2 thin films deposited by ALD method
  75. Regular Articles
  76. Quadruple multi-wavelength conversion for access network scalability based on cross-phase modulation in an SOA-MZI
  77. Regular Articles
  78. Application of ANNs approach for wave-like and heat-like equations
  79. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  80. Study on node importance evaluation of the high-speed passenger traffic complex network based on the Structural Hole Theory
  81. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  82. A mathematical/physics model to measure the role of information and communication technology in some economies: the Chinese case
  83. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  84. Numerical modeling of the thermoelectric cooler with a complementary equation for heat circulation in air gaps
  85. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  86. On the libration collinear points in the restricted three – body problem
  87. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  88. Research on Critical Nodes Algorithm in Social Complex Networks
  89. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  90. A simulation based research on chance constrained programming in robust facility location problem
  91. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  92. A mathematical/physics carbon emission reduction strategy for building supply chain network based on carbon tax policy
  93. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  94. Mathematical analysis of the impact mechanism of information platform on agro-product supply chain and agro-product competitiveness
  95. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  96. A real negative selection algorithm with evolutionary preference for anomaly detection
  97. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  98. A privacy-preserving parallel and homomorphic encryption scheme
  99. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  100. Random walk-based similarity measure method for patterns in complex object
  101. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  102. A Mathematical Study of Accessibility and Cohesion Degree in a High-Speed Rail Station Connected to an Urban Bus Transport Network
  103. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  104. Design and Simulation of the Integrated Navigation System based on Extended Kalman Filter
  105. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  106. Oil exploration oriented multi-sensor image fusion algorithm
  107. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  108. Analysis of Product Distribution Strategy in Digital Publishing Industry Based on Game-Theory
  109. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  110. Expanded Study on the accumulation effect of tourism under the constraint of structure
  111. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  112. Unstructured P2P Network Load Balance Strategy Based on Multilevel Partitioning of Hypergraph
  113. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  114. Research on the method of information system risk state estimation based on clustering particle filter
  115. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  116. Demand forecasting and information platform in tourism
  117. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  118. Physical-chemical properties studying of molecular structures via topological index calculating
  119. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  120. Local kernel nonparametric discriminant analysis for adaptive extraction of complex structures
  121. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  122. City traffic flow breakdown prediction based on fuzzy rough set
  123. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  124. Conservation laws for a strongly damped wave equation
  125. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  126. Blending type approximation by Stancu-Kantorovich operators based on Pólya-Eggenberger distribution
  127. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  128. Computing the Ediz eccentric connectivity index of discrete dynamic structures
  129. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  130. A discrete epidemic model for bovine Babesiosis disease and tick populations
  131. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  132. Study on maintaining formations during satellite formation flying based on SDRE and LQR
  133. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  134. Relationship between solitary pulmonary nodule lung cancer and CT image features based on gradual clustering
  135. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  136. A novel fast target tracking method for UAV aerial image
  137. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  138. Fuzzy comprehensive evaluation model of interuniversity collaborative learning based on network
  139. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  140. Conservation laws, classical symmetries and exact solutions of the generalized KdV-Burgers-Kuramoto equation
  141. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  142. After notes on self-similarity exponent for fractal structures
  143. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  144. Excitation probability and effective temperature in the stationary regime of conductivity for Coulomb Glasses
  145. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  146. Comparisons of feature extraction algorithm based on unmanned aerial vehicle image
  147. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  148. Research on identification method of heavy vehicle rollover based on hidden Markov model
  149. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  150. Classifying BCI signals from novice users with extreme learning machine
  151. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  152. Topics on data transmission problem in software definition network
  153. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  154. Statistical inferences with jointly type-II censored samples from two Pareto distributions
  155. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  156. Estimation for coefficient of variation of an extension of the exponential distribution under type-II censoring scheme
  157. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  158. Analysis on trust influencing factors and trust model from multiple perspectives of online Auction
  159. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  160. Coupling of two-phase flow in fractured-vuggy reservoir with filling medium
  161. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  162. Production decline type curves analysis of a finite conductivity fractured well in coalbed methane reservoirs
  163. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  164. Flow Characteristic and Heat Transfer for Non-Newtonian Nanofluid in Rectangular Microchannels with Teardrop Dimples/Protrusions
  165. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  166. The size prediction of potential inclusions embedded in the sub-surface of fused silica by damage morphology
  167. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  168. Research on carbonate reservoir interwell connectivity based on a modified diffusivity filter model
  169. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  170. The method of the spatial locating of macroscopic throats based-on the inversion of dynamic interwell connectivity
  171. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  172. Unsteady mixed convection flow through a permeable stretching flat surface with partial slip effects through MHD nanofluid using spectral relaxation method
  173. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  174. A volumetric ablation model of EPDM considering complex physicochemical process in porous structure of char layer
  175. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  176. Numerical simulation on ferrofluid flow in fractured porous media based on discrete-fracture model
  177. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  178. Macroscopic lattice Boltzmann model for heat and moisture transfer process with phase transformation in unsaturated porous media during freezing process
  179. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  180. Modelling of intermittent microwave convective drying: parameter sensitivity
  181. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  182. Simulating gas-water relative permeabilities for nanoscale porous media with interfacial effects
  183. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  184. Simulation of counter-current imbibition in water-wet fractured reservoirs based on discrete-fracture model
  185. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  186. Investigation effect of wettability and heterogeneity in water flooding and on microscopic residual oil distribution in tight sandstone cores with NMR technique
  187. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  188. Analytical modeling of coupled flow and geomechanics for vertical fractured well in tight gas reservoirs
  189. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  190. Special Issue: Ever New "Loopholes" in Bell’s Argument and Experimental Tests
  191. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  192. The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  193. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  194. Erratum to: The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  195. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  196. Rhetoric, logic, and experiment in the quantum nonlocality debate
  197. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  198. What If Quantum Theory Violates All Mathematics?
  199. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  200. Relativity, anomalies and objectivity loophole in recent tests of local realism
  201. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  202. The photon identification loophole in EPRB experiments: computer models with single-wing selection
  203. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  204. Bohr against Bell: complementarity versus nonlocality
  205. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  206. Is Einsteinian no-signalling violated in Bell tests?
  207. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  208. Bell’s “Theorem”: loopholes vs. conceptual flaws
  209. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  210. Nonrecurrence and Bell-like inequalities
  211. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  212. Three-dimensional computer models of electrospinning systems
  213. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  214. Electric field computation and measurements in the electroporation of inhomogeneous samples
  215. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  216. Modelling of magnetostriction of transformer magnetic core for vibration analysis
  217. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  218. Comparison of the fractional power motor with cores made of various magnetic materials
  219. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  220. Dynamics of the line-start reluctance motor with rotor made of SMC material
  221. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  222. Inhomogeneous dielectrics: conformal mapping and finite-element models
  223. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  224. Topology optimization of induction heating model using sequential linear programming based on move limit with adaptive relaxation
  225. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  226. Detection of inter-turn short-circuit at start-up of induction machine based on torque analysis
  227. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  228. Current superimposition variable flux reluctance motor with 8 salient poles
  229. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  230. Modelling axial vibration in windings of power transformers
  231. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  232. Field analysis & eddy current losses calculation in five-phase tubular actuator
  233. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  234. Hybrid excited claw pole generator with skewed and non-skewed permanent magnets
  235. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  236. Electromagnetic phenomena analysis in brushless DC motor with speed control using PWM method
  237. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  238. Field-circuit analysis and measurements of a single-phase self-excited induction generator
  239. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  240. A comparative analysis between classical and modified approach of description of the electrical machine windings by means of T0 method
  241. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  242. Field-based optimal-design of an electric motor: a new sensitivity formulation
  243. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  244. Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors
  245. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  246. Virtual reality as a new trend in mechanical and electrical engineering education
  247. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  248. Holonomicity analysis of electromechanical systems
  249. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  250. An accurate reactive power control study in virtual flux droop control
  251. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  252. Localized probability of improvement for kriging based multi-objective optimization
  253. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  254. Research of influence of open-winding faults on properties of brushless permanent magnets motor
  255. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  256. Optimal design of the rotor geometry of line-start permanent magnet synchronous motor using the bat algorithm
  257. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  258. Model of depositing layer on cylindrical surface produced by induction-assisted laser cladding process
  259. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  260. Detection of inter-turn faults in transformer winding using the capacitor discharge method
  261. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  262. A novel hybrid genetic algorithm for optimal design of IPM machines for electric vehicle
  263. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  264. Lamination effects on a 3D model of the magnetic core of power transformers
  265. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  266. Detection of vertical disparity in three-dimensional visualizations
  267. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  268. Calculations of magnetic field in dynamo sheets taking into account their texture
  269. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  270. 3-dimensional computer model of electrospinning multicapillary unit used for electrostatic field analysis
  271. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  272. Optimization of wearable microwave antenna with simplified electromagnetic model of the human body
  273. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  274. Induction heating process of ferromagnetic filled carbon nanotubes based on 3-D model
  275. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  276. Speed control of an induction motor by 6-switched 3-level inverter
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