Production analysis for fractured vertical well in rectangularcoal reservoirsChen Li1, Zhenzhen Dong2,*, and Xiang Li1

1 College of Engineering, Peking University, Room 1010, WangKeZhen Building, Peking University, NO. 5 Yihe Road,Haidian District, Beijing, PR China

2 Petroleum Engineering Department, Xi’an Shiyou University, Room 212, 18 Dianzierlu, Yanta District, Xi’an,Shannxi Province, PR China

Received: 15 May 2018 / Accepted: 28 August 2018

Abstract. As one kind of unconventional natural gas, coalbed methane is an important energy resource that issubject to active research. Gas exists in coalbed methane reservoirs in two forms: free gas and adsorbed gas. Inthe course of coalbed methane production, the reservoir experiences pressure decrease, desorption, diffusion,and seepage. Previous models of coalbed methane production were mainly concerned with circular boundaries.However, field tests revealed that some fractured wells possess the characteristics of rectangular boundaries.For fractured rectangular coalbed methane reservoirs, it is necessary to deal with the four boundaries with mir-ror image theory, which complicates calculations. In addition, the desorption and adsorption process of coalbedmethane exerts a strong effect on the seepage process. Furthermore, the complexity of the rectangular coal seamembedded with the finite-conductivity fracture results in a significant computational challenge. For the firsttime, this paper presented a fast analytical solution for a finite-conductivity fractured vertical well model witheither rectangular closed or constant-pressure boundaries in the coal seam. On the basis of the Fick diffusionlaw and the Darcy seepage law, a mathematical model that considers diffusion in matrix and seepage withinnatural fractures was established. Then, we integrated the fracture conductivity function method with thehydraulic fracture model to greatly increase computational efficiency. The analytical solutions were validatedagainst a numerical simulation. Parameter sensitivity analysis reveals that interporosity coefficient and storagecoefficient, respectively, affect the appearance time and degree of desorption and diffusion. Desorption coeffi-cient mainly describes the capacity of desorption and diffusion. Well storage coefficient, conductivity factor,and skin factor mainly affect the early stage of production. Finally, the proposed solutions were applied to fieldhistory match. The model developed is applicable to production analysis and well testing for coalbed methanereservoirs. The new proposed model extended flow mechanism of coalbed methane, and provided a better pro-duction and pressure forecast for coalbed methane reservoirs. In addition, the analytical solutions can be usedto generate type curves for fractured vertical wells with finite conductivity and in the rectangular boundary,and provide a sound theoretical basis for well tests in the coal seam. The model is also applicable to other typesof unconventional gas reservoirs, such as gas shales, in which the same processes are present.

Nomenclature

Symbols

r Radial radius of gas reservoir, mw Pseudo pressure, MPa2/cpp Pressure, MPal Gas viscosity, mPa sZ Gas deviation factor, dimensionless/ Porosity, decimal

cg Compressibility, MPa

�1k Permeability, DT Temperature, KV Gas concentration, m3/m3

C Volumetric gas concentration in microspores,m3/m3

D Diffusion coefficient, m3/spL Langmuir pressure, MPas Image variable of Laplace transformationR External radius of matrix, m* Corresponding author: dongzz@xsyu.edu.cn

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Oil & Gas Science and Technology - Rev. IFP Energies nouvelles 73, 62 (2018) Available online at:� C. Li et al., published by IFP Energies nouvelles, 2018 www.ogst.ifpenergiesnouvelles.fr

https://doi.org/10.2516/ogst/2018055

REGULAR ARTICLEREGULAR ARTICLE

Lf Fracture half length, mDPP Dimensionless pseudo pressureDPPD Dimensionless pseudo pressure derivative

Subscripts and superscripts

D Dimensionless propertye Boundary propertyw Wellbore propertyi Initial conditionsc Standard conditionf Fracture property

~ Image function of Laplace transform

Intermediate variables

r pLV Lp2i qD

pLþpð Þþ pLþpið Þþ piþpð Þ

K /lcg þ pscT lzT scqDp2

i

x /lcg

K

s R2

D

k ksKL2

f

f sð Þ xsþ 1�xð Þk r

ffiffiffiffiffi

ksp

cot hffiffiffiffiffi

ksp� 1

� �

cffiffiffiffiffiffiffiffiffi

f sð Þp

Dimensionless variables

WDpipkhT sc

lizipscqscT Wi �Wð ÞtD kt

KL2f

LDLLf

1 Introduction

Coal seams are classified as unconventional gas reservoirs,together with tight gas sands and gas shales. Coalbedmethane reservoirs have a dual porosity nature with macro-spores made by a natural fracture network, and microsporeswhich exist within the coal matrix. Natural fracture systemsconsist of two fissure systems, namely, butt and face cleats.Butt cleats are less continuous and often terminate againstthe face cleat with a strike angle of 75–85 degrees (Anbarciand Ertekin, 1990).

Gas flow in a coal seam involves two mechanisms: diffu-sion and seepage. When the coal seam pressure drops to acertain degree, the methane in the matrix translates fromthe adsorbed state into the free state, diffuses into the nat-ural fracture network, and then moves through the hydrau-lic fracture into the wellbore. The relationship betweenadsorption volume and pressure can be described by the

Langmuir isothermal adsorption equation. The diffusionprocess follows the Fick diffusion law. Flow from the natu-ral fracture network to the hydraulic fracture is a Darcyflow. The seepage characteristics of coalbed methane makethe mathematical model of seepage more complex.

Since hydraulic fracturing technology is widely and effec-tively utilized to enhance production efficiency in coalbedmethane reservoirs, determining how to evaluate fracturingresults is becoming increasingly important. During the pastdecades, many scholars have contributed to analyzing pro-duction and pressure performance, as it is an essential partof reservoir evaluation. Pressure performance analysis alsoconstitutes the basis for production analysis (Mcguire andSikora, 1960; Raghavan et al., 1972; Albinali et al., 2016;de Swaan, 2016).

Pressure analysis of coalbed methane was commonlyperformed based on the dual-porosity model that was intro-duced by Warren and Root (1962), de Swaan O (1976) andKazemi et al. (1992). Anbarci and Ertekin (1990) proposedthe Langmuir isothermal adsorption equation and Fickdiffusion law to represent the adsorption and diffusionprocesses in seepage flow, and to analyze pressure and pro-duction performance. Gringarten et al. (1975) developed atransient pressure performance analysis and type curveanalysis for fractured wells. Their work identified threebasic solutions for fractured wells: infinite-conductivitysolution, uniform flux solution for vertical wells, and uni-form flux solution for horizontal wells.

However, as the number of segments and fracture half-length increase, the conductivity of the fracture cannot beignored. Cinco-Ley et al. (1978), Cinco-Ley and Meng(1988) developed a semi-analytical solution for finite-conductivity vertical fractured wells. Aminian and Ameri(2009) proposed a numerical coalbed methane model to pre-dict production behavior of coalbed methane reservoirs.Clarkson et al. (2009) made a production decline curveanalysis for fractured wells and horizontal wells in coalbedmethane reservoirs. King et al. (1986) used a numericalmethod to simulate transient pressure performance withfinite- and infinite-conductivity vertical fractures in coalseams. Nie et al. (2012) developed an analytical solutionto describe its flow characteristics in a horizontal well withmultistage fracturing in coalbed methane reservoirs.

Most of the above literature modeled the transport char-acteristics of vertical and horizontal wells with radial bound-aries. However, according to field test results, some fracturedwells exhibit the characteristics of the rectangular boundary.Thus, the objective of this paper was to establish a completesemi-analytical model for vertical wells with finite-conduc-tivity fractures in a rectangular-boundary coal seam.

In this paper, we established and solved a new mathe-matical model with rectangular boundary, adsorption, des-orption, diffusion, and finite conductivity fracture.Moreover, wellbore storage effects and skin factor can beadded to the solution conveniently. Based on the proposedmathematical model, we optimized the calculation processto improve computational efficiency. Secondly, the modelwas validated with the literature results and a numericalsimulator. Thirdly, the flow characteristics were analyzed.Using parameter sensitivity analysis, the type curves of a

C. Li et al.: Oil & Gas Science and Technology - Rev. IFP Energies nouvelles 73, 62 (2018)2

finite conductivity fracture vertical well in a rectangularcoalbed methane reservoir were established in the paper.Parameters were analyzed, including inter-porosity coeffi-cient, storage coefficient, desorption coefficient, fractureconductivity factor, wellbore storage coefficient, and skinfactor. Finally, we applied the newly established modeland calculation process to analyze the production and pres-sure performance of a well from the Qinshui Basin.

2 Methodology

Some important assumptions were made for mathematicmodel development:

1. the well is in the center of a rectangular reservoir thatis bounded by upper and lower impermeable layers;

2. the inner boundary has either constant flow rate orconstant pressure (Fig. 1);

3. adsorption in the coal matrix obeys the Langmuir iso-therm adsorption equation, diffusion from the matrixto the natural fracture network obeys the Fick law,flow from the natural fracture network to the hydrau-lic fracture and flow to the wellbore obey the Darcylaw (Fig. 2);

4. the well is intercepted by a fully penetrating fracture,and the flow rate in the fracture is different along thefracture (Fig. 3);

5. the reservoir is isotropic and homogeneous with a con-stant height, porosity, and permeability;

6. the initial reservoir is uniform;7. gas flow takes place only through fractures;8. there is no pressure loss along the wellbore;9. pressure gradients are so small that the gravity effect

is negligible; and10. wellbore storage effect and skin effect are non-

negligible.

2.1 Model establishment and solution

Anbarci and Ertekin (1990) obtained a series of solutionsfor pressure transient behavior in coal seams for variousouter- and inner-boundary conditions. However, their

solutions only work for vertical, unfractured, and radialflow. In this article, to establish a sound continuity equationfor coalbed methane reservoirs, we considered the flowmechanism of free gas and adsorbed gas separated fromthe matrix and fractures. According to mirror image theory,the seepage condition of the one well with one fault in thereservoir is completely identical to that of two wells in thereservoir. One well is the same as the original well, andthe other well is a reflected duplication of the original wellthat appears almost identical, but is reversed in the direc-tion perpendicular to the fault surface. Using the Laplacetransformation and the mirror image theory, we can obtainthe solution of the continuity equation from Ozkan andRaghavan (1988, 1991) source solution.

2.1.1 Modeling flow in the reservoir

In the material balance equation of coalbed methane reser-voirs, besides free gas flow, adsorbed gas separates from thematrix and participates in the flow, which is different fromconventional gas. The continuity equation is defined asequation (1):

1r@

@rr

plz@p@r

� �

¼ ucgpkz

@p@tþ pscT

kT sc

@V@t: ð1Þ

Gas seepage from matrix to natural fracture networkwas usually expressed by the Fick diffusion law:

1r2

@

@rr2D

@C@r

� �

¼ @C@t: ð2Þ

The Langmuir isothermal adsorption equation describesthe desorption from adsorbed gas to free gas:

C ¼ V LppL þ p

: ð3Þ

Matrix geometry is assumed to be composed of sphericalblocks with radius R. The unsteady-state sorption equationfor the coal matrix can be written as:

@V@t¼ 3D

R@C@ri

�

�

�

�

ri¼R

: ð4Þ

Inner boundary condition in the dimensionless form forthe microspore equation is:

@CD

@rDirDi ¼ 0; tDð Þ ¼ 0: ð5Þ

Outer boundary condition in the dimensionless form forthe microspore equation is:

CD rDi ¼ 1; tDð Þ ¼ CDjpDa rDa;tDð Þ: ð6Þ

The solution for equation (2):

d ~CD

drDirDi ¼ 1ð Þ ¼ ~CDa

ffiffiffiffiffi

ksp

cot hffiffiffiffiffi

ksp� 1

�

: ð7Þ

Combining with equation (4), one can linearizeequation (1).

FractureWell

Constant pressure

Constant flow rate

Closed

Constant pressure

Fig. 1. Reservoir model with rectangular boundary.

C. Li et al.: Oil & Gas Science and Technology - Rev. IFP Energies nouvelles 73, 62 (2018) 3

It is required for ~pD to vanish as rD !1 for tD � 0 andat tD ¼ 0 for rD > � �! 0þð Þ. These conditions areexpressed by:

~pD tD � 0; rD !1ð Þ ¼ 0 ð8Þ

~pD tD ¼ 0; rD > � �! 0þð Þð Þ ¼ 0: ð9Þ

Normalized pseudo-pressure defined by Meunier et al.(1984) is:

w ¼ lizi

pi

Z p

0

plz

dp ð10Þ

The final continuity equation in the Laplace domain is:

1rD

@

@rDrD@ ~wD

@rD

!

¼ f sð Þ ~wD: ð11Þ

The closed boundary or constant pressure conditions ina rectangular coordinate system are:

@ ~wD

@xDþ @

~wD

@yD

!�

�

�

�

�

xD¼xeD;yD¼yeDð Þ

¼ 0 ð12Þ

~wD

�

�

xD¼xeDð Þ ¼ 0 ð13Þ

~wD

�

�

yD¼yeDð Þ ¼ 0: ð14Þ

Translating a radial coordinate system to a rectangularcoordinate system as:

rD ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2D þ y2

D

q

: ð15Þ

Ozkan and Raghavan (1991) presented solutions for arectangular closed and constant-pressure outer boundaryin the conventional reservoir as follows.

Rectangular closed outer boundary:

~pD xD; yDð Þ ¼ p2xeD

~qfD

"

cos hffiffi

sp

yD1ð Þ þ cos hffiffi

sp

yD2ð Þffiffi

sp

sin hffiffi

sp

yeDð Þ

þ 2X

1

k¼1

1xeD

cos kpxwD

xeD

� �

cos kpxD

xeD

� �

cos h ekyD1ð Þ þ cos h ekyD2ð Þek sin h ekyeDð Þ

þ2X

1

n¼1

cos npzD

hD

� �

cos npzwD

hD

� �

cos h enyD1ð Þ þ cos h enyD2ð Þen sin h enyeDð Þ

þ2X

1

k¼1

cos kpxD

xeD

� �

cos kpxwD

xeD

� �

cos h ek;nyD1ð Þ þ cos h ek;nyD2ð Þek;n sin h ek;nyeDð Þ

!#

:

ð16ÞRectangular constant-pressure outer boundary:

~pD xD; yDð Þ ¼ pxeD

~qfD

X

1

k¼1

sin kpxD

xeD

� �

sin kpxwD

xeD

� �

"

cos h ekyD1ð Þ � cos h ekyD2ð Þek sin h ekyeDð Þ

þ 2X

1

n¼1

cos npzD

hD

� �

cos npzwD

hD

� �

X

1

k¼1

sin kpxD

xeD

� �

sin kpxwD

xeD

� �

cos h ek;nyD1ð Þ � cos h ek;nyD2ð Þek;n sin h ek;nyeDð Þ

#

:

ð17Þ

Different from conventional gas reservoirs, coalbedseams have strong adsorption and desorption with the seep-age. So, we must reflect this phenomenon in the controlequation (Eq. (1)). The new parameter c (the specificexpression shown in Intermediate Variables) will reveal this

Nature fracture network MicroporesCoal grain

Diffusion in micropores Darcy flow in fractures

Flow in the nature fracture networkDesorption from a coal frame Flow in the fracture network

Desorption from matrix

Fig. 2. Desorption, diffusion, and Darcy flow in the matrix and fracture.

WellfractureFracture

Well

Fig. 3. Flow in the finite-conductivity fracture.

C. Li et al.: Oil & Gas Science and Technology - Rev. IFP Energies nouvelles 73, 62 (2018)4

specific phenomenon in the coalbed seam. Combining withit, we obtain the solutions in the coal seam as follows.

Rectangular closed outer boundary:

~pD xD; yDð Þ ¼ p2xeD

~qfDcos h cyD1ð Þ þ cos h cyD2ð Þ

c sin h cyeDð Þ

�

þ 2X

1

k¼1

1xeD

cos kpxwD

xeD

� �

cos kpxD

xeD

� �

cos h ekyD1ð Þ þ cos h ekyD2ð Þek sin h ekyeDð Þ

!

:

ð18Þ

Rectangular constant-pressure outer boundary:

~pD xD; yDð Þ ¼ pxeD

~qfD

X

1

k¼1

sin kpxD

xeD

� �

sin kpx0wD

xeD

� �

cos h ekyD1ð Þ � cos h ekyD2ð Þek sin h ekyeDð Þ ð19Þ

where

�n ¼ c2þn2p2=h2D

� �1=2; yD1 ¼ yeD � jyD � ywDj; yD2

¼ yeD � jyD þ ywDj; yD ¼ ywD ¼ yeD=2:

2.1.2 Modeling flow in the fracture

Taking equation (18) to integral along the fracture direc-tion yields:

~wD xD; yDð Þ ¼ p2xeD

Z xwDþ1

xwD�1~qfD

cos h cyD1ð Þ þ cos h cyD2ð Þc sin h cyeDð Þ da

þ pxeD

Z xwDþ1

xwD�1~qfD

X

1

k¼1

1xeD

� cos kpxwD

xeD

� �

cos kpxD

xeD

� �

cos h ekyD1ð Þ � cos h ekyD2ð Þek sin h ekyeDð Þ da: ð20Þ

In this equation, ~qfD is a function of the location in thefracture. To integral the equation along the fracture, weuse:

a ¼ xwD � x; da ¼ �dx; x ¼ xwD � a;Z xwDþ1

xwD�1~qfDda

¼Z 1

�1~qfDxdx:

Using this definition, equation (20) can be translated as:

~wD xD; yDð Þ ¼ 2pxeD

Z 1

0~qfD

X

1

k¼1

cos kpxD

xeD

� �

coskpaxeD

� �

coskp2

� �

cos h ekyD1ð Þ þ cos h ekyD2ð Þek sin h ekyeDð Þ da: ð21Þ

Assuming that the fracture can be divided into nsegments. Discretizing equation (21) with Figure 4 yields:

~wD xD; yDð Þ ¼ 4X

n

i¼1

~qfD

X

1

k¼1

1k

cos kpxD

xeD

� �

coskp2

� �

cos kpxDiþ1 þ xDi

2xeD

� �

sinkp�x2xeD

� cos h ekyD1ð Þ þ cos h ekyD2ð Þek sin h ekyeDð Þ : ð22Þ

In the Laplace domain, the sum of all of the flow rates ineach part is 1/s. So, it can be expressed as:

�xX

n

i¼1

~qfDi sð Þ ¼ 1s: ð23Þ

In the process of discretization, we divided the fractureinto n parts. We assumed that the flow rate of each part isdifferent, but the flow rate is constant in the same part. Foreach part, there is a pressure equation. Combined with theflow rate normalization equation, we can obtain an equa-tion set with n + 1 equations and n + 1 unknown numbersas ~wD; ~qDhf 1; ~qDhf 2; ~qDhf 3; . . . ; ~qDhfn. Solving this equation set,the solution of ~wD can be determined.

Duhamel’s theory can be utilized to incorporate thewellbore storage coefficient and skin factor into wellresponse. So, the final solution with wellbore storage andskin in the Laplace domain can be written as:

~wswD ¼1

s2CD þ s s~wwD sð Þ þ Sk

h i : ð24Þ

2.1.3 Calculation process optimization

Three factors lead to the complex calculation of finite-con-ductivity fractured wells in rectangular coal reservoirs:(1) solution of the rectangular reservoir from the mirrorimage theory, whichproducesmirrorwells; (2) diffuse gas flo-wed in the coal seam will increase complexity; and (3) finite-conductivity fracture requires solving an equation set. All ofthese reasons eventually result in slower computing for thenormal solution that is not suitable for software develop-ment. The objective was to obtain a quick calculationmethod to determine the pressure or production for finite-conductivity fractured wells in a rectangular coal seam.

We introduced the fracture conductivity function totranslate the infinite-conductivity fracture solution to the

xD1 xD2 xD3 xDj

xD1 xD2 xD3 xD4 xDn xDn+1

xD1 xD2 xD3 xDj

xD1 xD2 xD4xD3 xDn xDn+1

Fig. 4. Discretizing the equation along the fracture.

C. Li et al.: Oil & Gas Science and Technology - Rev. IFP Energies nouvelles 73, 62 (2018) 5

finite-conductivity fracture. The specific equation translatesas Wang et al., (2014):

s~pwD ¼ s~pwD inf þ s~f CfD

� �

: ð25Þ

From Riley et al. (1991), ~f CfD

� �

expresses the fractureconductivity as:

s~f CfD

� �

¼ 2pX

1

n¼1

1

n2p2CfD þ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2p2 þ sp

þ 0:4063p

p CfD þ 0:8997� �

þ 1:6252s: ð26Þ

2.1.4 Model verification

To verify our model and solutions, we compared it withAnbarci and Ertekin’s work (1992) and a reservoirsimulator- the UNConventional Oil and Gas simulator(UNCONG) (Li et al., 2015). Anbarci and Ertekin (1992)set the model with an infinite reservoir boundary and infi-nite conductivity fracture. Corresponding to their model,we set a large enough boundary and fracture conductivityfactor. UNCONG incorporates comprehensive flowingmechanisms for unconventional reservoirs (Li et al., 2015).The same as Anbarci et al.’s work, basic data used to per-form the calculation are listed in Table 1.

Compared the normal solution with Anbarci’s analyti-cal solution, the solution presented in this paper fits betterwith the numerical solution (Fig. 5). It can thus be con-cluded that the new method to solve the fracture model ismore accurate.

From Figure 6, we can see that using the fracture con-ductivity function method can improve computationspeed greatly. In fact, it is 488 times faster with a closedboundary, and 599 times faster with a constant-pressureboundary. For the closed boundary, the relative error was1.25%, and for the constant pressure boundary, the relativeerror was 3.23%. So, this function can be used for rectangu-lar boundaries to dramatically increase calculation speed.

3 Results and discussion

3.1 Flow characteristics analysis of type curves

Because of the two different sets of borders, pressurereaches the boundary twice. Under this condition, the flowstate is different from square or circle boundaries. It can bedivided into seven stages (Fig. 7):

1. Wellbore storage stage. This stage mainly reflects theeffect of wellbore storage at an early time. The linearslope of both dimensionless pseudo-pressure and itsderivative is 1.

2. Transition stage of wellbore storage. In this stage, thewellbore storage influence decreases, and the gas inthe reservoir flows into the wellbore gradually. Thepressure is influenced by the reservoir and the well-bore storage, and the curve of the dimensionlesspseudo-pressure derivative decreases.

3. Linear flow stage from natural fracture to hydraulicfracture. This stage can be divided into two sub-stages: bilinear flow and linear flow. The slope of thedimensionless pseudo-pressure derivative curvechanges from 1/4 to 1/2. Fracture conductivity deter-mines the length of bilinear flow. Specifically, the lar-ger the conductivity, the longer the duration time.The dimensionless pseudo-pressure decreases slightlyin the reservoir in this stage; the diffusion phe-nomenon is not obvious.

4. Desorption and diffusion stage. With the pressuredecreasing, desorption becomes increasingly obvious.The adsorbed gas desorbed from the matrix becomes

Table 1. Basic data to verify results.

s (hr) 32 8990 rw (ft) 0.5l (cp) 0.01082 c (psia�1) 0.002234T (R) 530 z 0.9404qsc (MMscf/d) 0.2 VL (scf/ft3) 18.632pic (psia) 447.7 k (md) 26u 0.01 h (ft) 6xf (ft) 100 PL (psi) 167.58

360

380

400

420

440

460

0.001 0.01 0.1 1 10 100

Pre

ssur

e (p

sia)

Production day (day)

analytical solution of this paper

numerical solution of UNCONG

analytical solution of Anbarci

Fig. 5. Comparison between Anbarci analytical and numericalsolution.

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E-05 1.00E-03 1.00E-01 1.00E+01 1.00E+03 1.00E+05

Dim

ensi

onle

ss P

ress

ure

Dimensionless Production Time

constant pressure boundary with normal modelclosed boundary with acceleration modelconstant pressure boundary with acceleration modelclosed boundary with normal model

Elapsed time(ms)12479201486080

25572482

Fig. 6. Comparison between normal and conductivity functioncalculations.

C. Li et al.: Oil & Gas Science and Technology - Rev. IFP Energies nouvelles 73, 62 (2018)6

free gas, leading to concentration increase around thematrix and diffusion to the natural fracture. Thisstage reflects the dimensionless pseudo-pressureappearing in a ‘‘V’’ shape; its depth and appearancetime are influenced by the wellbore storage coefficient,interporosity coefficient, storage coefficient, anddesorption coefficient. This stage constitutes theimportant flow stage that reflects the characteristicsof coalbed methane.

5. Pseudo-radial flow stage. In this stage, the gas dif-fused from the matrix has dynamic equilibrium withthe gas flowing into the wellbore. The speed of pres-sure decline becomes slow.

6. First boundary control flow stage. At this time, thepressure wave reaches the first boundary. With theconstant-pressure boundary, the pressure reaches sta-bilization quickly. However, the closed boundary has atransition, and so the dimensionless pseudo-pressureand derivative reflect gradually. This is because theconstant-pressure boundary has a sufficient energysupply to enable the pressure to reach a balance in ashort time.

7. Second boundary control flow stage. At this time, theconstant-pressure boundary has already reachedbalance in the first boundary control flow stage, butthe closed boundary must reach the pseudo-steadystate again.

In these seven stages, the linear flow stage mainlyreflects the characteristics of fractured vertical wells andthe conductivity of a hydraulic fracture. The desorptionand diffusion stage mainly reflect the characteristics of anunconventional gas reservoir that has the phenomenon ofdesorption and diffusion, such as coalbed methane andshale gas. The two boundary control flows mainly reflectthe characteristics of a rectangular reservoir. In this picture,C represents the closed boundary type, CP represents the

constant pressure boundary type, IN represents the infiniteboundary type.

3.2 Sensitivity analysis

Based on the definition above, traditional factors, such aspermeability, porosity and compressibility, were regroupedas new factors, such as interporosity coefficient, storagecoefficient, and desorption coefficient (shown in Intermedi-ate Variables). Combined with the fracture conductivityfactor, wellbore storage coefficient and skin factor, we per-formed a sensitivity analysis on those six factors (Tab. 2).

Interporosity coefficient k is the direct ratio of perme-ability and time constant, and the inverse ratio of porosity,viscosity, fracture half-length, and gas compressibility.Figure 8a gives the effect of k on coal seams with closedboundary conditions. It can be seen from the curves thata greater k will lead to a smaller time of concave-downcharacteristic appearance. The k slightly influences thedegree of desorption and diffusion.

Storage coefficient x is the direct ratio of porosity; itreflects the storage ability of the fracture system. It hasan inverse effect compared with the conventional dualporosity medium model reflecting the matrix storage capac-ity. Figure 8b gives the effect of x on coal seams with closedboundary conditions. The larger the x, the greater theporosity of the fracture and the gas flow rate from the frac-ture, the smaller the gas flow rate from the matrix diffusion.Pressure maintenance due to desorption is more obviouslymarked as x becomes smaller. However, the effect is inverse

Table 2. Basic factors for sensitivity analysis.

k 50 r 500 x 0.5Sk 0.1 CD 0.0001 CfD 10p

Fig. 7. Flow stage divided for boxed coal reservoirs.

C. Li et al.: Oil & Gas Science and Technology - Rev. IFP Energies nouvelles 73, 62 (2018) 7

at an early time due to the flow occurring in the naturalfractures.

Desorption coefficient r mainly describes the capacity ofdesorption and diffusion. It constitutes a direct ratio ofLangmuir volume, flow rate, and original pressure. Figure 8cgives the effect of r on coal seams with closed boundary con-ditions. It can be seen from the curves that a greater r willlead to more gas that the rock can adsorb and produce. Inother words, a quicker change of gas concentration leadsto an enhanced desorption and diffusion effect, so as to makethe concave-down phenomenon become obvious and early.

Well storage coefficient mainly affects the early stage ofproduction. Figure 8d gives the effect of well storage coeffi-cient on coal seams with closed boundary conditions. It can

be seen from the curves that a larger coefficient leads to amore obvious well storage effect and longer well storagestage. If the well storage coefficient reaches a certain value,it covers up the linear flow phase, making the flow movedirectly into the matrix methane diffusion phase.

Skin factor mainly affects pressure performance forwellbore storage and its transition stage, primarily influenc-ing the end time of this stage and dimensionless pseudo-pressure decrease. Figure 8e gives the effect of skin factoron coal seams with closed boundary conditions. It can beseen from the curves that a larger skin factor will prolongthe time of the wellbore storage stage. This is because, atthe same flow rate conditions, the higher skin factor willlead to more energy consumed and pressure decreased.

Fig. 8. Influence of parameters on type curve. (a) Influence of interporosity coefficient; (b) Influence of storage coefficient;(c) Influence of desorption coefficient; (d) Influence of well storage; (e) Influence of skin factor; (f) Influence of fracture conductivity.

C. Li et al.: Oil & Gas Science and Technology - Rev. IFP Energies nouvelles 73, 62 (2018)8

After this stage, the skin factor does not influence pressureperformance.

Fracture conductivity factor mainly influences the tran-sition stage of wellbore storage and the linear flow stage; ithas no influence on the wellbore storage stage or after thelinear flow stage. This is because those stages have lowgas flow coming from the natural fracture to the hydraulicfracture. At this time, the finite-conductivity fracture canlead these gas flows to the wellbore quickly, and can beequivalent to the infinite-conductivity fracture. In addition,with the fracture conductivity factor increasing to fourtimes of Pi (Pi = 3.14), the finite-conductivity fracture getsclose to the infinite-conductivity fracture (Fig. 8f). This willguide us to not blindly improve fracture conductivity toimprove output; this will not only not influence output,but will increase fracturing cost.

3.3 Production analysis

Duhamel convolution is used to construct complex solutions(such as variable rates or pressure) with simple solutions(such as constant rates or pressure). Superposition uses thetheory that a rate that changes from q1 at time t to a new rateq2 is equivalent toq1 continuing forever, superposed, or addedon (q2–q1) starting at time t and continuing forever (Fig. 9).

The steps of the Duhamel convolution are as follows:(1) determine the average pressure through the materialbalance equation; (2) normalize the production time; (3)calculate the dimensionless pressure or rates through theanalytical model and time superposition; and (4) obtainthe rate or bottomhole pressure through a lookup table.

3.4 Field application

3.4.1 Basic data

The proposed model was used to analyze production datafrom a fractured vertical well in a coal methane reservoirlocated in the Qinshui Basin in Shanxi Province of China.Daily production data, flowing wellhead pressure measure-ments (converted to bottomhole pressure for analysis), andsome reservoir parameters were collected (Tab. 3). Fromthe seismic data analysis, the well located in the closed rect-angle boundary. We used the proposed model in this paperto match the reservoir pressure and the flow rate.

3.4.2 History match result

Using the model in this paper, and combining with the prin-ciple of time superposition and the dimensional result fromthe model, we can obtain the match result. Figure 10ashowed the pressure matching. The average error was18%. Some additional factor may have resulted in the largeerror at the end of the match.

Figure 10b presented the flow rate matching. The aver-age error was 17%. The same as the pressure match, someadditional factor may have resulted in the large error atthe end of the match. Figure 10c showed the cumulative

q1q2

q3

t1 t2

qp

pi

Δ =

Δ = − −

Δ = − −

t

Δ = − −+

+

Fig. 9. The principle diagram of Duhamel convolution.

Table 3. Basic data of actual field well.

pic (psia) 340 s 0.37k (md) 0.05 qb (g/cm 3) 1.47xe (ft) 1027 ye (ft) 2200h (ft) 30 Ct (psi�1) 0.0031Rw (ft) 0.3 T (F) 57.2u 0.2 VL (scf/ton) 321Lf (ft) 365 PL (psi) 167.58

(a) History match of pressure

(b) History match of flow rate

(c) History match of cumulative gas production

0

5

10

15

20

25

30

35

40

45

50

0

50

100

150

200

250

300

2016/4/20 2016/6/9 2016/7/29 2016/9/17 2016/11/6 2016/12/26 2017/2/14

Flow

Rat

e (m

scf/d

)

Pre

ssur

e (p

sia)

Production Date

Calculated Pressure(psia) Field Pressure(psia) Field Production Rate(mscf/d)

0

50

100

150

200

250

15

30

45

60

75

2016/4/20 2016/6/9 2016/7/29 2016/9/17 2016/11/6 2016/12/26 2017/2/14

Pre

ssur

e (p

sia)

Flow

Rat

e (m

scf/d

)

Production Date

Calculated Production Rate(mscf/d) Field Production Rate(mscf/d) Field Pressure(psia)

0

2000

4000

6000

8000

10000

12000

2016/4/20 2016/6/9 2016/7/29 2016/9/17 2016/11/6 2016/12/26 2017/2/14

Cum

ulat

ive

Gas

(msc

f)

Production Date

Field Cumulative Gas(mscf) Calculated Cumulative Gas(mscf)

Fig. 10. History match result. (a) History match of pressure;(b) History match of flow rate; (c) History match of cumulativegas production.

C. Li et al.: Oil & Gas Science and Technology - Rev. IFP Energies nouvelles 73, 62 (2018) 9

gas of the calculated value and field value. There was a goodmatch result between the calculated value and field value.

4 Conclusion

In this paper, we developed a rectangular-boundary modelfor a finite-conductivity fractured well in a coal seam. Withthe mirror image theory and Laplace transformation,the solutions of this model were obtained at a constantproduction rate or bottomhole pressure. Through the aboveanalysis, the following conclusions can be summarized:

1. By comparing the solution of the simplified model inthis paper with the classical analytical model andnumerical simulation model, the results show thatthe proposed method is more accurate and efficientto solve the fractured model and has a good matchwith the numerical solution.

2. Fracture conductivity function can dramaticallyimprove computation speed without sacrificing accu-racy for a rectangular boundary model.

3. Compared with the radial boundary, the new rectangu-lar model has two new flow stages: the first and secondboundary control flow stages, which results from thecharacteristics of rectangular reservoirs. Closed andconstant pressure boundaries show the difference stabi-lization time in pressure performance: pressure reachesstabilization quickly with the constant-pressureboundary, but the closed boundary has a transitionafter reaching the first boundary, and reaches stabiliza-tion after reaching the second boundary.

4. Interporosity coefficient influences the appearancetime to desorption and diffusion, and slightly influ-ences the degree. Storage coefficient influences thedegree of desorption and diffusion, and slightly influ-ences the appearance time. Desorption coefficientmainly describes the capacity of desorption and diffu-sion. Well storage coefficient and skin factor mainlyaffect the early stage of production. Conductivity fac-tor mainly influences the early time, after which gasflow rates from the natural fracture decrease andcan flow into the wellbore quickly – the hydraulic frac-ture equivalent for the infinity-conductivity fracture.

5. The solution of this paper can solve the problem ofhistorical matching, in which the matching speedwas fast and the result was good.

Acknowledgments. This article was supported by NationalScience and Technology Major Project of China (ProjectNo. 2016ZX05037003-002).

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