Research ArticleProduction Rate Analysis of Fractured Horizontal Wellconsidering Multitransport Mechanisms in Shale Gas Reservoir

Qi-guo Liu ,1,2 Wei-hong Wang,1 Hua Liu,1 Guangdong Zhang,2 Long-xin Li,3

and Yu-long Zhao 2

1State Key Laboratory of Shale Oil and Gas Enrichment Mechanisms and Effective Development, SINOPEC, Beijing, China2State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan, China3Research Institute of Petroleum Exploration and Development, PetroChina Southwest Oil and Gas Field Company,Chengdu Sichuan 610051, China

Correspondence should be addressed to Yu-long Zhao; swpuzhao@swpu.edu.cn

Received 10 August 2017; Revised 15 December 2017; Accepted 30 August 2018; Published 18 October 2018

Academic Editor: Timothy S. Collett

Copyright © 2018 Qi-guo Liu 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.

Shale gas reservoir has been aggressively exploited around the world, which has complex pore structure with multiple transportmechanisms according to the reservoir characteristics. In this paper, a new comprehensive mathematical model is established toanalyze the production performance of multiple fractured horizontal well (MFHW) in box-shaped shale gas reservoirconsidering multiscaled flow mechanisms (ad/desorption and Fick diffusion). In the model, the adsorbed gas is assumed notdirectly diffused into the natural macrofractures but into the macropores of matrix first and then flows into the naturalfractures. The ad/desorption phenomenon of shale gas on the matrix particles is described by a combination of the Langmuir’sisothermal adsorption equation, continuity equation, gas state equation, and the motion equation in matrix system. On the basisof the Green’s function theory, the point source solution is derived under the assumption that gas flow from macropores intonatural fractures follows transient interporosity and absorbed gas diffused into macropores from nanopores follows unsteady-state diffusion. The production rate expression of a MFHW producing at constant bottomhole pressure is obtained by usingDuhamel’s principle. Moreover, the curves of well production rate and cumulative production vs. time are plotted by Stehfestnumerical inversion algorithm and also the effects of influential factors on well production performance are analyzed. Theresults derived in this paper have significance to the guidance of shale gas reservoir development.

1. Introduction

Shale gas is an unconventional natural gas with the maincomponent of methane existing as adsorbed or free gas inthe rich organic shale or its interlayers, the reserve of whichis tremendous around the world. With the innovative tech-niques of horizontal well drilling as well as hydraulic fractur-ing in North American or other places of the world, theannual production of shale gas in the USA has increasedfrom 564× 108m3 in 2007 to 4382× 108m3 in 2015. More-over, well testing and production performance analysis inshale gas reservoir, especially for multiple fractured hori-zontal well (MFHW), is a hot research topic in recent years[1]. Compared to conventional gas reservoirs, shale gas reser-voirs have special features on the accumulation conditions,

occurrence manner, percolation mechanism, etc. Forinstance, shale is a self-sourcing reservoir within which thegas is stored either by compression in the pores and naturalfracture as free gas or by adsorption on the surface of thesolid material (either organic material or minerals) asadsorbed gas. According to Hill and Nelson [2], the adsorbedgas accounts for up to 85% of total gas, which is dependenton a variety of geologic and geochemical properties [3]. Thepore network of shale gas reservoir is complex, which canbe further divided into 4 categories: nanoinorganic pore,nanoorganic intragranular pore, macronatural fracture, andlarge-scaled hydraulic fractures. The first two pore typesbelong to matrix pores and the latter two belong to fractures[4–7], so the mechanisms of gas flow in shale reservoirs isvery complicated. Shale gas reservoirs are generally described

HindawiGeofluidsVolume 2018, Article ID 3148298, 17 pageshttps://doi.org/10.1155/2018/3148298

by the dual porosity model like Warren-Root model [8], inwhich the matrix blocks serve as the main storage space ofgas and are separated by fractures.

Kucuk and Sawyer [9] first analyzed the transient pres-sure response of wells in shale gas reservoirs by using the dualporosity model (assuming cylindrical and spherical matrixgeometries), and the corresponding analytical and numericalresults were obtained. However, the analytical model pre-sented in the paper did not take the effects of gas desorptionand diffusion into account and the numerical model ignoredthe effects of gas diffusion.

On the basis of isothermal adsorption equation, Bumband McKee [10] analyzed the transient pressure of a verticalwell in coal bed methane (CBM) reservoir by introducing anadditional compressibility coefficient in their model. Theirmethod has been popularly used for production performanceanalysis of unconventional gas. Many researchers followedBumb andMcKee’s [10] approach to consider the ad/desorp-tion effects in their studies; the results of which were provedto be much more realistic than previous studies [11–14].

Ertekin et al. [15] utilized a dual-mechanism model fortight gas reservoirs, which considered Darcy flow (drivenby pressure gradient) and Fick diffusion (driven by concen-tration difference). Subsequently, Carlson and Mercer [16]investigated the gas flow behavior in shale gas reservoir bycoupling the gas diffusion and desorption with Fick’s lawand Langmuir isothermal adsorption equation. And then,using the same method, Chawathe et al. [17] analyzed thepressure and saturation distribution in the reservoir bynumerical solution.

Recently, many studies related to well pressure and ratetransient analysis have been presented [5, 18–33]. El-Banbi[34] utilized a linear flow model to analyze the productionperformance of a MFHW by ignoring the gas desorptionand diffusion. In this model, the reservoir was simplified bycontinuous medium with slab dual porosity and Warren-Root model [19, 20, 35]. However, most of these studies didnot consider desorption or diffusion effects, such as “trilin-ear” model [24] and other models [36, 37].

In fact, a large number of core electron macroscope scan-ning images show that there are many organic pores existingin the shale matrix, the scale of which ranges from nanometerto macrometer. Previous studies considered the matrix poresas a whole without separating nanometer pores and macro-pores. Song [38, 39] proposed a “triple porosity” conceptualmodel to describe the gas storage mechanism, which wasequivalent to a dual porosity model combined with gasadsorption. Thereafter, Zhao et al. [40] adopted the modelto analyze the pressure transient and rate decline behaviorof MFHW in shale gas reservoirs. Recently, a comprehensivemathematical model of a MFHW in a box-shaped reservoir ispresented by Zhao et al. [32]. In this model, the reservoir wasdivided into a dual porosity system, which included naturalfractures and nanopores. The gas flow in natural fracture sys-tem obeyed the Darcy flow theory, while the Knudsen diffu-sion and slippage effect were considered for the matrix lowpermeability system.

According to Dehghanpour and Shirdel [41], the dualporosity models have been widely used in naturally fractured

shale gas reservoirs with multiple hydraulic fractures. Inthese models, only nanopores existed in the matrix blockswhich were surrounded by a fracture network. Gas flow inthe nanopores obeys the Fickian diffusion law and the storagecapability of the fracture system is very low, the theoreticalproductivity of the well from such conceptual models is quitelow, and the decline rate of well productivity is very large,which is not consistent with some practical data ([5, 24, 27]).

Most physical models proposed by previous investigatorsdid not take into account the nanopores and the macroporesin the matrix system [32, 40, 42]. According to the NMRscanning results of the core from YY32 well from the Yan-Chang formation in P.R. China (Figure 1), two peaks arethe most important characteristics for many shale gas reser-voirs, which means that there are mainly two types of poresin the matrix system: the nanopores and macropores.

In this paper, a new conceptual model is developed todescribe the gas flow in shale gas reservoir. The flow path ispresented by into three pore types: natural fractures, macro-pores, and nanopores. It is assumed that the adsorbed gas firstdesorbs from the organic particle surface into the nanoporesystem. It then diffuses into the macropore system. Thereafter,the gas in the macropores is transported into the natural frac-tures with transient interporosity flow. Based on this physicalmodel, the production performance of a MFHW in rectangu-lar outer-boundary shale gas reservoirs is analyzed.

2. Multiple Transport Mechanisms of ShaleGas Reservoirs

2.1. Gas Ad/Desorption Mechanism. A large amount of shalegas is adsorbed on the surface of rock particles. As the reser-voir pressure depletes, adsorbed gas desorbs from the matrixsurface and becomes free gas. This feature differentiates shalegas reservoirs different from conventional reservoirs. Theisothermal adsorption curves have the same shape as Lang-muir isothermal curve [10], which is frequently used todescribe ad/desorption process of shale gas.

The occurrence time of gas desorption behavior alsodepends on the amount of adsorbed gas at initial conditions.As shown in Figure 2, if the adsorbed gas is statured under

02468

10121416

0.01 0.1 1 10 100 1000 10000

Perc

enta

ge, %

Radius, nm

YY32-4YY32-14YY32-16

Figure 1: The pore size distribution map for shale gas reservoir inOrdos Basin (P.R. China).

2 Geofluids

initial conditions, the amount of adsorbed gas is at point Band the gas quickly desorbs from organic surfaces as pressuredecreases. However, if the adsorbed gas is unsaturated, theamount of adsorbed gas would be at point C (if reservoir con-ditions are at point C and there is no free gas in matrix pores)and the gas would not desorb from organic surfaces until res-ervoir pressure decreases to the pressure at point A, which iscalled critical desorption pressure. The difference betweenthe critical desorption pressure and the initial reservoir pres-sure determines the starting time of gas desorption [1].

Langmuir isothermal adsorption equation is expressed as

Cm = CLpm

pL + pm, 1

where Gm is the gas volumetric concentration, m3/m3; GL isthe Langmuir adsorption volume of shale gas, representingthe limit adsorbence of unit reservoir, m3/m3; pL is the Lang-muir pressure, which is the pressure when gas adsorbencereaches 50% of limit adsorbence, Pa; and pm is the pressurein matrix system, Pa.

The mass flow rate of desorbed gas from shale reservoirof volume Vb in unit time is

qdes = ρgscVb∂Cm∂t

, 2

where qdes is the mass of desorption gas in unit time, kg/s; t isthe time, s; Vb is the shale matrix volume, m3; and ρgsc is theshale gas density at standard conditions, kg/m3.

If gas desorption happens instantaneously and all des-orbed gas transforms into free gas, then (2) can be introduceddirectly into the continuity equation of the matrix or fracturesystem, resulting in the steady-state adsorption-desorptionmodel ([10–13]; Civan, 2010; [37]; Wang et al., 2013).

2.2. Gas Flow in Natural Fractures and Macropore System.Gas flow in natural and hydraulic fractures and macroporesis continuous flow. According to previous analysis (Civan,2010; [37]), Darcy’s flow equation used for conventionalgas reservoir can be adopted to analyze gas flow behavior infracture and macropore systems in shale reservoirs, that is,

J l = −plMg

ZRTklμg

∇pl, l = f , mc , 3

where f presents the natural fracture system and themc pre-sents the macropore system; Z is the gas deviation factor(dimensionless), Z=1 for ideal gas; R is the gas constant(8.314 J/(mol ⋅K)); T is the reservoir temperature, K; kl isthe permeability in l system, m2; Mg is the gas molar mass,kg/mol; pl is the pressure in l system, Pa.

2.3. Gas Diffusion in Nanopores. Gas flow behavior in shale isdifferent from flow in fractures because the pores are ofnanoscale. In these nanopores, it is erroneous to assumemolecular continuous Darcy flow. Shale gas flow in nano-pores includes not only viscous flow but also diffusion. Insome ultratight shale reservoirs, the flow of gas moleculesoccur by only diffusion because the gas in the nanoporesexists only as absorbed gas and not free gas.

Fick’s law of diffusion is generally used to describe shalegas migration in nanopores and at the surface of kerogen[15, 40, 43–46]. Under the influence of concentration gradi-ent, desorbed gas migrates from high concentration area tolow concentration area, and the diffusion stops when gasconcentration becomes uniform (as illustrated in Figure 3).Assume that gas molecular diffusion satisfies Fick’s law, andthen gas diffusion flux could be expressed as

J F = −MgD ⋅ ∇Cm, 4

where D is the Fick diffusion coefficient, m2/s.If gas concentration in matrix changed with coordi-

nates, meaning nonequivalent concentration at any time inthe matrix, unsteady state diffusion occurs, which can bedescribed by Fick’s second law. Otherwise, the Fick’s firstlaw is used when the concentration is assumed to be the sameat all locations in a given time (pseudosteady-state diffusion).It is assumed in this paper that the desorption of theadsorbed gas desorbed from the nanopores into the macro-pores follows unsteady-state diffusion [15].

To simplify the hypothetical shale gas reservoir model, itis assumed that the matrix element is spherical with radiusRm; each matrix element is composed of organics and claygrains, and diffusion is the only gas flow mechanism in thematrix (Figure 4). Therefore, Fick’s second law can be appliedto describe gas diffusion from matrix elements into macro-fractures or macropores.

V (m

3 /m3 )

p (MPa)pL pi

A

B

C

Adsorbed gas unsaturated

Adsorbed gas saturated

Figure 2: Shale gas isothermal adsorption curve and desorptionprocess diagram.

Diffusion

Low concentrationshale gas

High concentrationshale gas

Figure 3: Diagram of Fick diffusion.

3Geofluids

According to the unsteady-state diffusion theory, gas con-centration in the matrix Cm is a function of time and space.Therefore, the following mathematical expression can beused to describe gas concentration change in the matrix,

∂Cm∂t

=1r2m

∂∂rm

Dr2m∂Cm∂rm

5

The initial condition of matrix unsteady-state diffusionshould satisfy the equation below.

Cm t = 0, rm = Ci pi , 6

where pi is the initial reservoir pressure, Pa.The inner boundary condition can be expressed as

∂Cm t, rm = 0∂rm

= 0 7

Since the outer boundary of matrix is connected with thefracture system or macropore system, pressure at the matrix

surface equals that of fracture system, then, the outer bound-ary can be represented as

Cm t, rm = Rm = Cm p , 8

where Rm is the radius of a spherical matrix element, m.Using the above model and relevant boundary condi-

tions, the distribution of gas concentration in the shalematrix can be solved. Then, the following equation can beused to determine gas diffusion flow rate within matrix ofvolume Vb,

qF =MgVb3DRm

∂Cm∂rm rm=Rm

9

3. Solutions of a MFHW in a Rectangular ShaleGas Reservoir

3.1. Physical Model. In this section, continuous point sourcesolution for any point in a rectangular shale gas reservoirwith closed upper and lower boundaries is derived. The phys-ical model is shown in Figure 5. The assumptions for this

(a)

(b)

(c)(d)

Rm

Macro-pore Representativeelement volume ofshale matrix system

Shale particleNano-pore

Figure 4: Conceptual model used in this paper.

xe

xfi

xfi

ye

xw

j=1 j=2j=3

j=m

(0, 0)

i=1

....

i=2⁎n....Discreteelement

..

L

i=1+(j-1)2n

....

....

..i=4n

Closed outer boundaryHorizontal well

Hydraulic fracture

Figure 5: Physical model of a MFHW in a rectangular closed outer-boundary shale gas reservoir.

4 Geofluids

model are (1) the gas reservoir is homogeneous and aniso-tropic with horizontal and vertical permeability kh and kz.The width, length, and height of the reservoir are xe, ye,and h; (2) a MFHW with length L is located in the middleof the reservoir and parallel to the upper and lower bound-aries. All hydraulic fractures are symmetrical and perpendic-ular to the wellbore; the half length of the jth fracture is xf i;the fractures are fully penetrating; (3) the well is perforatedat the intersections between the fractures and the wellbore,which means that fluid flows from the reservoir to the frac-tures and then from the fractures to the wellbore. Infiniteconductivity fractures are assumed with no pressure dropacross the fracture face and along the horizontal wellborelength; (4) each fracture is evenly divided into 2N units withunified flow rate [32, 47].

3.2. Continuous Point Source Solution. During the develop-ment of shale gas reservoirs, natural fractures are the mainpaths for gas flowing from the matrix to hydraulic fracturesand wellbores (Figure 6). Therefore, flow models for thesenatural fractures are the key to deriving the point source func-tion. The PDE of pseudopressure in cylindrical coordinates offracture systems for different models are derived in Laplacedomain (detailed derivation in the Appendix), which is

1rD

∂∂rD

rD∂Δmf∂rD

+∂2Δmf∂z2D

= f s Δmf , 10

where f s is the parameter group of the corresponding modelunder different percolation mechanism, dimensionless.

For unsteady-state diffusion and transient interporosityflow, the expression of f s is

f s = ωf s +λmf5

g s coth g s − 1 , 11

where g s can be expressed as

g s =15ωmλmf

+β

λσ λθmf s coth λθmf s − 1 s,

β = 1 − ωf − ωm 1 − ϕf − ϕm

12

According to the mirror image and superposition princi-ples (the schematic is shown in Figure 7) and Newman [48]as well as Ozkan and Raghavan [49], the instantaneous pointsource solution at the observation point MD′ generated byunit source sink at point MD is [1]

G MD,MD′ , s = 〠+∞

k=−∞〠+∞

m=−∞〠+∞

n=−∞

S1,1,1 + S2,1,1 + S1,2,1 + S1,1,2+ S2,2,1 + S1,2,2 + S2,1,2 + S2,2,2

13

In the above equation, Si,j,k represents the pressureresponse at any point of the gas reservoir caused by eachimage well due to closed boundary.

Gas flow from natural fractures to artificialfractures to well bore (viscous flow)

Gas flow from macro-pores tomicro-fractures

Gas desorption and flowfrom matrix tomacro-pores

Natural fracturesHydraulic fractures

Horizontal wellMacro-pores

Representativeelement volume Shale particle

Figure 6: Diagram of matrix desorbed gas flow through macropores to macrofractures.

Si,j,k =exp − f s xDi − 2kxeD 2 + yDj − 2myeD

2+ zDk − 2nhD 2

xDi − 2kxeD 2 + yDj − 2myeD2+ zDk − 2nhD 2

  i, j, k = 1, 2 , 14

5Geofluids

where

xD1 = xD − xwD,

xD2 = xD + xwD,

yD1 = yD − ywD,

yD2 = yD + ywD,

zD1 = zD − zwD,

zD2 = zD + zwD

15

Introducing the following Poisson summation formula,which is [32, 49]

〠+∞

n=−∞

exp − v a2 + ξ − 2nξe 2

a2 + ξ − 2nξe 2

=1ξe

K0 a v + 2〠∞

n=1K0 a v +

n2π2ξ2e

cos nπ ξ

ξe

16

Then, the continuous point source solution for rectangulargas reservoirs can be simplified and expressed as [32, 49]

G MD,MD′ , s =2π

hDxeD

cosh uyD1 + cosh uyD2u sinh uyeD

+ 2〠+∞

k=1cos kπ xD

xeDcos kπ xwD

xeD

cosh εkyD1 + cosh εkyD2εk sinh εkyeD

+ 2〠∞

n=1cos nπ zD

hDcos nπ zwD

hD

×cosh εn yeD − yD

εn sinh εnyeD

+ 2〠+∞

k=1cos kπ xD

xeDcos kπ xwD

xeD

cosh εk,nyD1 + cosh εk,nyD2εk,n sinh εk,nyeD

,

17

k=0, m=−1, n=0

x

xe

ye(xw, yw)

(x, y)

(0, 0)

k=0, m=0, n=0

y

k=−1, m=1, n=0 k=0, m=1, n=0 k=1, m=1, n=0

k=−1, m=0, n=0

k=1, m=−1, n=0k=−1, m=−1,

k=1, m=0, n=0

x-y plane

k=0, m=0, n=−1

y

ye(xw, yw)(x, y)

(0, 0)

k=0, m=0, n=0

z

k=0, m=−1, n=1 k=0, m=0, n=1 k=0, m=1, n=1

k=0, m=−1, n=0

k=0, m=1, n=−1k=0, m=−1, n=−1

k=0, m=1, n=0

y-z plane

Figure 7: Schematic of a point source in rectangular gas reservoir with closed boundaries.

6 Geofluids

where

yD1 = yeD − yD − ywD 18

and

yD2 = yeD − yD + ywD 19

Then, the pressure drop in rectangular reservoir gener-ated by the intensity qscins of the continuous point source is

Δmf‐ps =pscTTsc

qscins2πkfLref s

G MD,MD′ , s 20

Introducing (17) into (20) yields

Δmf‐ps =pscTTsc

qscinsπkfLrefhDs

πxeD

cosh uyD1 cosh uyD2u sinh uyeD

+ 2〠+∞

k=1cos

kπxDxeD

coskπxwDxeD

×cosh εkyD1 cosh εkyD2

εk sinh εkyeD

+ 2〠∞

n=1cos

nπzDhD

cosnπzwDhD

cosh εn yeD − yDεn sinh εnyeD

+ 2〠+∞

k=1cos kπ xD

xeDcos kπ xwD

xeD

cosh εk,nyD1 + cosh εk,nyD2εk,n sinh εk,nyeD

21

The above expression is the pressure drop at any pointof the reservoir generated by constant intensity qscins in rect-angular reservoir. Noted that it can clearly see from the aboveequation that some infinite summations of eigenfunctionsare existed, which generally have slow convergence char-acteristics. Since the solutions presented in this work arein the Laplace transform domain, we utilized the sameapproach proposed by Ozkan and Raghavan [49] to improvethe computation efficient of source function given in (21).The main routine to overcome this problem is to separatein the infinite-acting and boundary-domained flow contribu-tions in the solution and then recast the slow convergingsummations and computationally difficult integrals intocomputationally more efficient forms. In the previous paper[32], the detailed expressions during different time are given,and many authors has also reported them [49, 50].

3.3. Solution of the MFHW Producing with ConstantBottomhole Pressure.As is shown in Figure 5, a horizontal wellwith M fully penetrated fractures is drilled in a rectangular

outer-boundary reservoir. According to the source functiontheory, the pressure drop at arbitrary point caused by eachdiscrete element can be obtained by integrating the continu-ous point source function along the fracture element, whichis [40, 51]

Δmf i xD, yD =ΓΔmf‐ps xD, yD, xwDi, ywDi, s dΓ 22

Due to the fractures are parallel to the x-axis and all ofthem are fully penetrated, the above equation can be con-verted into the following formulation:

Δmf i xD, yD =xmi+ΔL f i/2

xmi−ΔL f i/2

h

0Δmf‐ps

xD, yD, xwDi, ywDi, s dzwdxw

23

Taking (21) into (23) and arranging it yield

Δmf i =pscTTsc

qscLiπkfLrefhDs

πxeD

ΔLf icosh uyD1 + cosh uyD2

u sinh uyeD

+ 2Lref+ΔL fDi/2

−ΔL fDi/2〠+∞

k=1cos kπ xD

xeDcos kπ xmDi + α

xeD

cosh εkyD1 + cosh εkyD2εk sinh εkyeD

dα,

24

where qscLi is the continuous line source strength, whichhas the following relationship with the continuous pointsource qscins:

qscLi = qscinsh 25

Defining the following dimensionless production rateand pseudopressure, we have

qDi =qscLiΔLf i

qsc26

and

mfD =πkfhhTscpscTqsc

Δmf 27

Introducing (26) and (27) into (24), the dimensionlesspseudopressure drop generated by the ith unit at any reser-voir location can be expressed as

7Geofluids

mfDi = qDiAji,

Aji =π

sxeD

cosh uyD1 + cosh uyD2u sinh uyeD

+2

ΔLfDi

ΔL fDi/2

−ΔL fDi/2〠+∞

k=1cos kπ xD

xeDcos

kπ xmDi + α

xeD

cosh εkyD1 + cosh εkyD2εk sinh εkyeD

dα

28

Due to the horizontal well is composed of many dis-crete fracture units, the pressure at any point in the reser-voir should be the superposition of the pressure dropcaused by all these units at the same point. For the obser-vation point xDj, yDj , according to the superpositionprinciple, there is [32, 40]

mD xDj, yDj = 〠M∗2N

i=1qDiAji xDi, yDi, xDj, yDj, s 29

If setting the observation point on each fracture element,then we can obtain 2N∗M linear equations and anotherauxiliary equation of mass conservation for flux can be alsoobtained; the detailed expressions can be referred to thepaper of Zhao et al. [32].

When well producing at a constant bottomhole pressure,the dimensionless rate can be defined as follows:

qD =qscpscT

πkfhTsc mf i −mwf30

According to previous studies [52], there is the follow-ing relation formula between dimensionless pseudopressureand rate,

qD =1

mwD × s2, 31

where the definition of dimensionless pseudopressure in(31) is

mwD =πkfhTscpscTqsc

Δmf 32

The above equation is the production rate solutionwhen well producing at a constant pressure.

4. Results Analysis

4.1. Model Validation. In this section, we use the solution ofconventional dual-porosity reservoirs to prove the correct-ness of our model. Comparing to the conventional model,the difference between them is that our mode considersthe gas desorption and diffusion from the nanopores into

macropores. The model presented in this paper can simplifyinto the conventional dual-porosity model by neglecting theshale gas desorption and diffusion processes. For conven-tional reservoirs, approximation expressions for f s in (11)can be found in the literatures [8, 9, 53–55]. The expressionof f s introduced by Warren and Root for blocked matrixwith pseudosteady-state interporosity flow is

f s =ωf 1 − ωf s + λmf1 − ωf s + λmf

s 33

Using de Swaan’s approach [53], the solutions forWarren and Root’s model can be extended to the solutionsof transient interporosity flow, which are

f s = ωf s + sλmf 1 − ωf

3stanh

3 1 − ωf sλmf

34

for horizontal slab blocks and

f s = ωf s +λmf5

15 1 − ωf sλmf

coth15 1 − ωf s

λmf− 1

35

for spherical blocks.The above three types of matrix block are the most pop-

ular idealization model for naturally fractured reservoirs;they have been widely used in the petroleum industry sincethey are proposed. So, the correctness of our model andmethod can be validated by comparing the result to theconventional model with spherical blocks.

In our model, if the gas diffusion and desorption areneglected, the following relationship must be established,which is

ωf + ωm = 1 36

Substituting the above equation into the expression ofg s in (11), which becomes

g s =15 1‐ωf

λmfs 37

It can be clearly seen that (11) can be simplified intothe same solution for spherical blocks described by (35)for naturally fractured reservoirs by taking the above equa-tion into it, which prove the correctness of our solution.

4.2. Well Production Performance Analysis. On the basis ofthe above solution, the semianalytical solution in Laplacedomain can be transformed into real space using Stehfestnumerical inversion algorithm [56]. Therefore, when the wellis produced at a constant bottomhole pressure, the ratedecline semilog type curves of production rate and cumula-tive production vs time can be obtained. The input parame-ters for simulation sensitivities for the following models areshown in Table 1.

8 Geofluids

For a well producing at constant bottomhole pressure in arectangular gas reservoir, the production rate and cumulativeproduction vs. time curves in semilog coordinates with differ-ent fracture numbers are shown in Figure 8. From the plot, itcan be seen that the early time production rate is increasingwith the increasing of fracture number, which is mainlybecause the larger the M is, the bigger the drainage area ofthe well will be during the early flow period. And when thepressure wave transports to the outer region formation, thiseffect will be gradually weaken. However, the cumulative pro-duction with different M under the same production time inlater production period will be the same due to the reserve ofthem are equivalent.

Figure 9 shows the influence of Langmuir volume CL onthe well production performance. Due to the mechanism,physical model in this paper has divided the pores in matrixsystem into the following two subspace: macropores andnanopores. The adsorbed gas will be desorbed from the par-ticle surface in the matrix and then flow into the macroporesby Fick diffusion. When the pressure in the macropore sys-tem is lower than the Langmuir pressure, desorption phe-nomenon will happen. So, when the well producing with

the constant bottomhole pressure of 4MPa, the pressure inthe vicinity of the well will be lower than the Langmuir pres-sure, which will cause the adsorbed gas to desorb frommatrixsurface. So, we can find from Figure 9 that the productionrate is larger with the increasing of Langmuir volume. Butthis effect is very small. When well is producing for a longtime, the cumulative production of well with bigger CL willbe larger than the others.

Figure 10 shows the effects of permeability in naturalfracture system on the well production performance.Because the natural fracture system is the main flow pathof gas flow from the formation into hydraulic fractures, ithas a significant influence on well production performance.According to the productivity equation of wells, the wellproduction rate has a proportional relationship with thenatural fracture permeability. And due to the constant con-trol volume of our model, the bigger production rate atearly flow period will lead to a bigger rate decline in themiddle flow period. According to the cumulative produc-tion curves on Figure 10, we can clearly see that the largerthe kf is, the larger the cumulative production will be inour given time period.

Table 1: Synthetic data used in the examples.

Parameters Value Parameters Value

Initial reservoir pressure, pi, MPa 25 Reservoir temperature, T , K 320

Formation thickness, h, m 60 Gas specific gravity, rg, fraction 0.65

Well length, L, m 1200 Constant bottomhole pressure, pwf , MPa 4

Natural fracture system permeability, kf , mD 0.005 Natural fracture system porosity, ∅f , fraction 0.01

Macropore system permeability, km, mD 10−6 Macropore system porosity, ∅m, fraction 0.05

Langmuir pressure, PL, MPa 4 Langmuir volume, CL (m3/m3) 10

Number of fractures, M (integer) 10 Half fracture length, Lf , m 100

Outer boundary size, xe, m 800 Outer boundary size, ye, m 3000

0

0.1

0.2

0.3

0.4

0.5

0

5

10

15

20

25

30

1.E+0 1.E+1 1.E+2 1.E+3 1.E+4

Prod

uctio

n ra

te (1

04 m3 /d

)

Cum

ulat

ion

rate

(108 m

3 )

t (day)

M=6M=8M=10

Figure 8: Influence of fracture number on well production performance.

9Geofluids

Figure 11 shows the effects of permeability in macro-pores on well production rate and cumulative productionrate curves. The bigger km represents a stronger supple-ment of fluid from matrix system into natural fractures.Because the fluid flow from matrix system into naturalfractures follows the transient interporosity flow, this sup-plement will happen immediately after the well opens. Thebigger the km is, the larger the well production rate willbe, and this trend will be lasting for a long time. So, forshale gas reservoir development, the permeability of the

matrix system is a very important parameter to evaluatethe well production performance.

Figure 12 shows the effects of macropores porosity (∅m)on well production performance. The value of ∅m presentsthe amount of free gas in macropore system of shale matrixsystem. With the increase of the ∅m, the quantity of freegas flows from macropores into natural fracture system byinterporosity will increase. So, when well produces at a con-stant bottomhole pressure, the higher the porosity is, thelarger the well production rate will be. Although the

0

0.2

0.4

0

5

10

15

20

25

30

35

40

1.E+0 1.E+1 1.E+2 1.E+3 1.E+4t (day)

CL=10CL=0CL=20

Prod

uctio

n ra

te (1

04 m3 /d

)

Cum

ulat

ion

rate

(108 m

3 )

Figure 9: Influence of Langmuir volume on well production performance.

0

0.1

0.2

0.3

0.4

0.5

0

5

10

15

20

25

30

1.E+0 1.E+1 1.E+2 1.E+3 1.E+4t (day)

kf=0.005mD

kf=0.001mDkf=0.01mD

Prod

uctio

n ra

te (1

04 m3 /d

)

Cum

ulat

ion

rate

(108 m

3 )

Figure 10: Influence of natural fracture permeability on well production performance.

10 Geofluids

increment of production rate is not quite obvious, for the sit-uation listed in this paper, the increment is still nearly5000m3/d. If this increment for conventional gas reservoircould be negligible, it is very considerable for shale gas reser-voir due to the long production period.

Figure 13 shows the effects of gas desorption time (τ) onthe well production performance. As we know, the gasdesorption time presents the gas desorption rate of adsorbedgas desorbed into free gas. The smaller the τ is, the faster thedesorption process happens. It can be clearly seen fromFigure 12 that the smaller the gas desorption time is, thelarger the well production time will be, and the correspond-ing cumulative production will be larger too.

5. Discussion

This paper established a new mechanism model to describethe gas flow in multiscaled shale gas reservoir, which dividedthe pores in matrix systems into two subtypes: macroporesand nanopores. The gas flow in nanopores obeys the Fick’ssecond diffusion and in macropores follows the inter-porosity flow as conventional dual-porosity model. Onthe basis of the above flow mechanism, the productionperformance of a MFHW in a rectangular outer-boundary shale gas reservoir is analyzed by semianalyticalsolution method. Comparing to the solutions of conven-tional naturally fractured reservoir, the correctness of our

0

0.2

0.4

0.6

0.8

1

0

10

20

30

40

50

60

1.E+0 1.E+1 1.E+2 1.E+3 1.E+4t (day)

km=10^-6mD

km=10^-5mDkm=10^-4mD

Prod

uctio

n ra

te (1

04 m3 /d

)

Cum

ulat

ion

rate

(108 m

3 )

Figure 11: Influence of permeability in macropore system on well production performance.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

5

10

15

20

1.E+1 1.E+2 1.E+3 1.E+4

t (day)

Ø=0.05Ø=0.1

1

2

3

4

1.E+2 1.E+3

Prod

uctio

n ra

te (1

04 m3 /d

)

Cum

ulat

ion

rate

(108 m

3 )

Figure 12: Influence of porosity in macropores on well production performance.

11Geofluids

model is certified by neglecting the gas desorption and dif-fusion properties. After that, the effects of correspondingsensitive parameters on well production rate and cumula-tive production rate curves are analyzed. According tothe results, we can conclude that the fracture number onlyaffects the well performance in early flow period; the per-meability of natural fractures mainly affects the early per-formance due to the finite drainage volume in ourphysical model. Comparing to the permeability of naturalfracture system, the effects of permeability in macroporesystem on early and middle flow periods are obvious. So,for some high rate well in field, the macropore may be animportant storage space, and which cannot be neglectedor combined with other system to analyze.

Appendix

Continuity Equation in Natural Fracture System

The continuity equation of shale gas flow in macrofracturesystem is given in the cylindrical coordinate system by [32]

1r∂∂r

r∂mf∂r

+∂2mf∂z2

=ϕfμgicfgi

kf

∂mf∂t

−1kf

2RTMg

qm A 1

For transient interporosity flow from matrix to fractures,the continuity equation for flow in matrix is

1r2m

∂∂rm

r2m∂mm∂rm

=ϕmμgicmgi

km

∂mm∂t

+1km

2RTMg

qF A 2

The interporosity flow rate qm is expressed as

qm = −3ρgRmac

kmμg

∂pm∂rm

rm=RmacA 3

Gas diffusion from matrix to fractures in unit reservoirvolume is

qF =Mg 1 − ϕf − ϕmdCmdt

A 4

For unsteady-state gas diffusion from matrix to fractures,there is

dCmdt

=3DFRm

∂Cm∂rm rm=Rm

A 5

For transient interporosity flow model, define the follow-ing dimensionless variables:

λmf = αkmkf

L2ref ,

rD =r

Lref,

rmD =rmRmac

,

tD =kf t

ΛL2ref,

ωf =ϕfμgicfgi

Λ ,

ωm =ϕmμgicmgi

Λ ,

λ =kmτ

ΛL2ref

A 6

The other variable groups used in the above equations aredefined as

0

0.2

0.4

0.6

0.8

1

0

5

10

15

20

𝜏=10^6𝜏=10^5𝜏=10^4

1.E+1 1.E+2 1.E+3 1.E+4

t (day)

Prod

uctio

n ra

te (1

04 m3 /d

)

Cum

ulat

ion

rate

(108 m

3 )

Figure 13: Influence of desorption time on well production performance.

12 Geofluids

Δmf =m pi −m pf ,

Δmm =m pi −m pm ,

Λ = ϕmμgicmgi + ϕfμgicfgi +6kfhqsc

R2macL2ref

,

τ = R2m

DF, θmf =

kfkm

,

CmD = Cm pm − Cm pi ,

CED = CE pm − Cm pi ,

A 7

where the expression of pseudopressure m p is defined as

m p =p

po

p′μZ

dp′, A 8

where λ is the dimensionless interporosity flow coefficientfrom nanopore into macropore system; λm−f is the dimen-sionless interporosity flow coefficient from macropore intonatural fracture system; τ is the sorption time constantwhich can be measured in the lab; ω is the storabilityratio, dimensionless; and Lref is the characteristic length,m; in this paper, we can choose the horizontal well lengthas the reference length.

And then the continuity equation in the natural fractureand macropore system in the matrix system can be pre-scribed as the following in Laplace domain, which are

1r2D

∂∂rD

r2D∂Δmf∂rD

= ωf sΔmf +λmf5

∂Δmm∂rmD rmD=1

, A 9

∂∂r2mD

r2mD∂Δmm∂rmD

=15ωmλmf

sΔmm

−Mg

ρsc

qscpscTkfhTsc

β

λ

∂CmD∂rmD rmD=1

,A 10

where β = 1 − ωf − ωm 1 − ϕf − ϕm .According to the mathematical model of gas diffusion in

the nanopores described in (5)–(8) and combining thedefined dimensionless variables with them, we can obtainedthe dimensionless diffusion equation in the Laplace domainis [1]

1r2mD

∂∂rmD

r2mD∂CmD∂rmD

= λθmf sCmD, A 11

∂CmD s, rmD = 0∂rmD

= 0, A 12

CmD s, rmD = 1 = CED m pf A 13

Define the following group as

WD = CmDrmD A 14

Substituting the above equation into (A.11) yields

∂2WD∂r2mD

= λθmf sWD A 15

The general solution is

WD = A sinh λθmf srmD + B cosh λθmf srmD

A 16

Substituting (A.16) into (A.14) yields

CmD =A sinh λθmf srmD + B cosh λθmf srmD

rmD

A 17

According to the inner boundary condition of matrix sys-tem, we have

limrmD→0

CmD

= limrmD→0

A sinh λθmf srmD + B cosh λθmf srmD

rmD

= finite valueA 18

According to the L’Hospital rule, the solution of Bparameter can be derived from (A.18)

B = 0 A 19

Then, the following equation can be obtained from theinner boundary condition:

limrmD→0

Aλθmf s cosh λθmf srmD − sinh λθmf srmD

r2mD= 0

A 20

Similarly, the above equation can be simplified as

Aλθmf s2

limrmD→0

sinh λθmf srmD = 0 A 21

Combining with the outer boundary condition and tak-ing B into the corresponding equation, the parameter A canbe obtained as

A =CED m pf

sinh λθmf sA 22

13Geofluids

Substituting (A.19) and (A.22) into (A.17), and then thesolution of the gas concentration is

CmD =CED m pf

sinh λθmf s

sinh λθmf srmD

rmDA 23

According to the Langmuir isotherm desorption equa-tion, the following equation is established:

CED m pf =ρscMg

L VLm pfm pL +m pf

−VLm pi

m pL +m piA 24

Define the following desorption coefficient σ as follows:

σ =VLm pL

m pL +m pf m pL +m pi

qscpscTkfhTsc

A 25

Then, (A.24) becomes

CED m pf = ρscMg

σ m pf −m pi = −ρscMg

kfhTscqscpscT

σΔmf

A 26

Substituting (A.26) into (A.23) yields

CmD = −ρscMg

kfhTscqscpscT

σ

sinh λθmf s

sinh λθmf srmD

rmDΔmf

A 27

Taking the derivation with rmD for the above equation,we have

∂CmD∂rmD rmD=1

= −ρscMg

kfhTscqscpscT

σ λθmf s coth λθmf s − 1 Δmf

A 28

Then, substituting (A.28) into (A.10), we have

1r2mD

∂∂rmD

r2mD∂Δmm∂rmD

= g s Δmm, A 29

where g s = 15ωm/λmf + β/λ σ λθmf s coth λθmf s− 1 s .

Then, based on the previous research [8], the relationshipof macropores in matrix system with pressure in natural frac-ture system can be solved. So, the uniform expression of thegoverning equation in natural system can be changed into

1rD

∂∂rD

rD∂Δmf∂rD

+∂2Δmf∂z2D

= f s Δmf , A 30

where f s = ωf s + λmf /5 g s coth g s − 1 .

Nomenclature

Latin

cmgi, cfgi: Gas compressibility in fracture and matrixsystem, respectively, Pa−1

Cm: Molar mass of gas in matrix, mol/m3

CE: Volume of gas adsorbed per unit volume ofthe shale gas reservoir in equilibrium atpressure pm, sm3/m3

CL: Langmuir volume, sm3/m3

D: Diffusion coefficient, m2/sf s : The parameters group of the corresponding

model under different percolation mecha-nism, dimensionless

f, mc: Fractures and macropore mediah: Formation thickness, mJ F: Fick mass diffusion flux (gas mass passing

through unit acreage in unit time), kg/(m2 ⋅ s)

J l: Mass velocity of gas in media l, kg/(m2 ⋅ s)kf : Permeability of natural fracture system,

m2

km: Permeability of macropore in matrix system,m2

kh, kz: Horizontal and vertical permeability ofreservoir, respectively, m2

L: Well length, mLref : The characteristic length, mM: Number of fracturesN : Discrete number of each half fractureMg: Molar mass of gas, kg/molp: Pore pressure, PaPL: Langmuir pressure, Papi: Initial reservoir pore pressure, Papf , pm: Pressure in fracture and matrix system,

respectively, Papsc: Pressure at standard condition, psc = 0 1MPapl: Pressure in media l, Paqm: Interporosity flow rate, kg/sqscin: The intensity of the cylinder source sinkqdes: Mass flow rate of gas desorption from

reservoir of volume Vb, kg/sqsc: Well production rate, m3/sqF: Fick mass flow(gas mass passing through

volume Vb in the unit time), kg/srm: Inner diameter of sphere matrix element, mRm: Radius of sphere matrix, ms: Laplace variables, dimensionlesst: Production time, hour, and dayTsc: Standard temperature, Tsc = 273KT: Reservoir temperature, KVb: Shale matrix volume, m3

x f i: The half length of the jth fracture, mx, y, z: Distance coordinates, mxe, ye: Effective reservoir width and length, mxD, yD, zD: Dimensionless space coordinates,

dimensionless

14 Geofluids

xwD, ywD, zwD: Dimensionless space distance between thewell with the bottom boundary,dimensionless

Z: Gas deviation factor, fraction.

Greek

λ: Interporosity flow coefficient, dimensionlessτ: Sorption time, constant, sω: Storability ratio, dimensionlessμg Gas viscosity, Pa ⋅ sρgsc: Shale gas density at standard conditions, kg/m3

∇: Gradient operator, ∇ = ∂/∂x i + ∂/∂y j + ∂/∂z k∅f ,∅m: Natural fracture and shale matrix porosity, respec-

tively, fractionΣ: Desorption coefficientπ: 3.141…..

Superscript

−: Laplace domain.

Subscripts

D: DimensionlessI: Initial conditionsc: Standard statew: Wellboref: Fracture systemm: Matrix systemwD: Dimensionless wellbore.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Basic ResearchProgram of China (Grant No. 2014CB239205) and theNational Natural Science Foundation of China (Grant Nos.51704247, 51504202).

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