Transient Pressure Responses From Finite Conductivity on

Vertical Fracture in Shale Gas ReservoirC. Y. Ma, P. Q. Lian, Quarcoo.J

(Sinopec Research Institute of Petroleum Exploration & Development, Beijing 100083, China)

Abstract: Shale gas reservoirs usually have extremely low permeability and relatively

low porosity. Horizontal wells with massive multi stage hydraulic fracturing treatment

have been performed on these reservoirs to achieve economic breakthrough. This

paper considered shale gas dispersion and desorption in the matrix and in addition,

presents a trilinear flow model for finite conductivity on vertical fracture in shale gas

reservoir. The gas first disperses from the internal structure of the matrix to the

surface, then desorbs into the fracture system. This model was applied by the Laplace

transformation and solved with the Stehfest numerical inversion method. Typical

pressure test curves were plotted, and the effects of sensitive parameters on the

dimensionless wellbore pressure were also studied. Adsorption coefficient reflects the

adsorption capacity of the matrix on shale gas, which implies the larger the adsorption

coefficient rate, the higher the adsorption capacity, with low desorption capabilities.

Fracture conductivity factor denote the conductivity of fractures, the larger fracture

conductivity factor, the stronger the fracture conductivity capacity, which means

limited flow rate within a short time. Inter-porosity transfer coefficient determines the

appearance, time and height of the transition stage. If the inter-porosity transfer

coefficient became small, the transition stage will occur early. Storativity ratio

determines the extent of crossflow time, the smaller the storativity ratio, the earlier the

appearance time, and the longer the crossflow time.

Key words: Shale gas reservoir, vertical fracture, trilinear flow wellbore storage,

adsorption coefficient

Introduction

Shale gas reservoirs usually have extremely low permeability and relatively low

porosity, and could hold certain amount of gas absorption. In order to achieve

economic productions, horizontal wells with massive multistage hydraulic fracturing

treatment need to be performed. Due to the complex of the fracture in shale gas

reservoirs, it is difficult to accurately describe the reservoir flow mechanism. Well

testing is a useful method to evaluate the fracture pressure response, and the

parameters of hydro-fracture could be determined, which makes well testing very

important and significance in shale gas reservoir development[1-5].

Currently, a lot of research has been carried out by both domestic and international

scholars. Watson et al[6] presented a method for analyzing production data from

naturally fractured gas reservoirs, and a normalized time was used to modify analytic

solutions to the gas flow model in finite dual-porosity reservoirs.

Ozkan et al[7][8] considered stress-dependent permeability in a fractured network, they

developed a dual-mechanism dual-porosity naturally fractured reservoir formulation

and derived a new transfer function for fractured shale-gas reservoirs. Brown et al[9]

derived a model for single-phase flow of a constant compressibility fluid based on

some idealizations and simplifying assumption, but this model was not consider the

adsorption of shale gas in matrix. Duan et al[10][11] adopt the Quasi-steady state Fick

diffusion according to the point source function method, studied the single-phase flow

of shale gas in the matrix and fractures. Freeman et al[12][13] constructed a fit-for-

purpose numerical simulator which accounts for a variety of production features

pertinent to these systems, and employed the numerical simulator to examine various

tight/shale gas systems, and to identify/illustrate the various flow regimes which

progressively occur over time. Cao et al[14,15] considered absorption, desorption,

diffusion, convection and solid deformation, and presented a new simulation model

for shale gas reservoir based on the dual-porosity and dual-permeability model.

Wu et al[16][17]presented a generalized mathematical model for simulating multiphase

flow of a tightgas in porous / fractured reservoirs considering Klinkenberg effect,

non-Newtonian behavior and non-Darcy flow. Moridis et al[18][19] (2010) analyzed the

mechanisms and processes of flow in shale and tight-sand systems by means of

numerical simulation, which includes Darcy’s law as the basic equation of multiphase

flow and accurately describes the thermophysical properties of the reservoir fluids.

In the above citations, the commonly applied non-equilibrium adsorption model in

shale gas reservoir assumes that adsorbed gas in the matrix system disperses into

fracture system directly after desorption, and omit the gas diffusion in the matrix.

There is free gas in matrix, and the gas firstly disperses from the internal structure of

the matrix to the surface, then access to the fracture system. Considering the matrix

dispersion, this paper attempts to establish trilinear flow model for finite conductivity

on the vertical fracture in shale gas reservoir. The numerical solution is achieved by

Laplace transformation and Stehfest numerical inversion, with their corresponding

curves. This study provides theoretic analysis for the development of shale gas

reservoir.

1 Flow model for fractured horizontal well in shale gas reservoir

1.1 Assumed conditions

Because of the complex interplay among matrix, natural fractures, and hydraulic

fractures in shale gas reservoir, a relatively simple, trilinear flow model may represent

the key characteristic of flow toward a multiple fractured horizontal well. The trilinear

flow model couples linear flow in three contiguous flow regions as sketched in figure

1: The outer reservoir goes beyond the boundaries of the hydraulic fractures (Region

II), the inner reservoir between hydraulic fractures (Region I), and the hydraulic

fracture(F).

Fx

Fb

/ 2Fd

INNER RESERVOIR(Region I)

OUTER RESERVOIR（Region II）

Hydraulic fracture

Figure 1 Flow phase division of fractured horizontal well[9]

The contribution of Fractured horizontal wells in shale gas reservoir with permeability

in the range of micro-Darcy or even below the stimulated volume is usually

negligible, and the interference of fractures is very small, so we use one fracture as a

case study, to analyze the performance of fractured horizontal well in shale gas

reservoir. The assumed conditions are shown as follows:

(1) Shale gas reservoir is tabular, homogeneous, isotropic, and infinite in horizontal

direction.

(2) Shale gas reservoir has double porosity medium i.e matrix and fracture. The

Shale gas percolates from matrix to fracture and then flow into hydraulic

fractures, and finally into the wellbore.

(3) Vertical fracture penetrates into the reservoir completely in z-direction, and

equally spaced by a distance dF, half the length and width of fracture xF and bF

respectively. The liquid does not flow at the end of fracture.

(4) Pressure in natural fractures and hydraulic fractures are independent. The shale

gas diffusion obeys Fick's first law, ignoring the influence of gravity and

capillary pressure.

(5) Temperature is constant during the process of desorption, diffusion and

percolation of shale gas.

1.2 Mathematical model

In this section, we derive the solutions for the reservoir and the hydraulic fracture,

and are coupled using the flow rate and pressure continuity equations.

1.2.1 Equation of state

In perfect condition, the equation of state of gas is

(1)

Where, p is the absolute pressure, MPa; V is the volume of gas, m3; T is absolute

temperature, K; n is molar number, kmol; R is universal gas constant,

R=0.008314MPa·m3/(kmol·K);

It needs to consider the temperature and pressure for real gas state equation.

Therefore, a compression correction factor Z is introduced for the ideal gas equation

of state. Real gas equation of state can be written as:

， (2)

Where, denotes hydraulic fracture, region I and region II respectively.

Defining the pseudo-pressure function as the following equation:

, (3)

Where, subscript 0 denotes initial condition; subscript w denotes wellbore; μ is

viscosity of shale gas, mPa·s.

1.2.4 Fluid transfer in region II

The continuity equation of region II system is defined as:

(4)

Where, pII is pressure of region II, MPa; vII,y is the velocity in x-direction, cm/s; qII,v is

crossflow volume from matrix to natural fracture in region II.

(5)

Where, kII is permeability of region II, μm2; μ is viscosity of shale gas, mPa·s.

The crossflow volume from matrix to natural fracture in region II is defined as:

(6)

Where, FG is geometry factor of matrix; VII,m is concentration of shale gas in matrix of

region II system, m3/m3.

The diffusion equation of shale gas in Region II obeys to Fick's first law, and can

be described as follow:

(7)

Where, VE is concentration of shale gas on natural fracture interface, m3/m3; τ is

adsorption duration, day.

For Region II, the Langmuir adsorption equation for shale gas diffusion in pseudo-

steady condition is defined as:

(8)

Where, pL is Langmuir pressure, MPa; VL is Langmuir volume, m3/ m3.

Take equations (2), (5) and (6) into equation (4), percolation equation for region I

system is obtained by:

(9)

Where, subscript sc denotes standard condition;φII is porosity of region II, fraction.

Eq. (9) can be simplified by pseudo-pressure function:

(10)

The initial and boundary conditions of region II are:

(11)

(12)

(13)

1.2.3 Fluid transfer in region I

The continuity equation of region I system is defined as:

(14)

Where, pI is pressure of region I, MPa; vI,x is the velocity in x-direction, cm/s; qI,v is

crossflow volume from matrix to natural fracture in region I.

(15)

Where, kI is permeability of region I, μm2;The crossflow volume from matrix to natural fracture in region I is defined as:

(16)

Where, VI,m is concentration of shale gas in matrix of region I system, m3/m3.

The diffusion equation of shale gas in Region I obeys to Fick's first law, and can be

described as follow:

(17)

For Region I, the Langmuir adsorption equation for shale gas diffusion in pseudo-

steady condition is defined as:

(18)

Take equations (2), (15) and (16) into equation (14), percolation equation for region

I system is obtained:

(19)

Where, φI is porosity of region I, fraction.

Eq. (19) can be simplified by pseudo-pressure function:

(20)

Where, CI,t is total compressibility coefficient of region I.

The initial and boundary conditions of region I are:

(21)

(22)

(23)

Where, S is skin factor of fracture.

For region I and region II, they all belong to reservoir system, and the porosity,

permeability and compressibility coefficient are assumed to be same:

(24)

1.2.2 Fluid transfer in hydraulic fracture

The continuity equation of hydraulic fracture system is defined as:

(25)

Where, pF is fracture pressure, MPa; vF,x is the velocity in x-direction, cm/s.

(26)

Where, kF is permeability of hydraulic fracture, μm2.

Equations (2) and (26) are substituted into continuity equation (25), and the basic

differential equation of hydraulic fracture system is obtained:

(27)

Where, φF is porosity of fracture, fraction.

Eq. (27) can be simplified by pseudo-pressure function:

(28)

Where, CF,t is total compressibility coefficient of hydraulic fracture, 1/MPa.

The initial and boundary conditions of Hydraulic fracture are:

(29)

(30)

(31)

1.2.5 Parameter definition

In dual-porosity reservoir, characteristic of the matrix and natural fracture system

are incorporated into the pseudo-steady by storativity ratio and inter-porosity flow

coefficient flow capacity ratios defined respectively by:

(32)

(33)

Where, ; B is volume coefficient, , m3/m3; q is flow

rate of well, cm3/s; h is the depth of reservoir.

Dimensionless adsorption coefficient is defined as:

(34)

Where, ΨL is Langmuir pressure after pseudo-pressure conversion; h is the depth of

reservoir.

Dimensionless hydraulic fracture conductivity factor is defined as:

(35)

In order to simplify the continuity equation, a series of dimensionless variable is

also defined as bellow:

; ; ; ; ;

;

; , ; ; ; .

Where, ΨFD, ΨID, ΨIID and ΨwD are dimensionless pseudo-pressure at hydraulic

fracture, region I, region II and wellbore respectively; CD is dimensionless wellbore

storage coefficient; S is fracture skin factor; α is adsorption coefficient; ηD is the

transmissibility factor of fracture.

2 Solution for trilinear model for fractured horizontal well

2.1 Dimensionless models

The dimensionless model for region II can be obtained as follow:

(36)

The dimensionless model for region I can be obtained as follow:

(37)

The dimensionless model for hydraulic fracture can be obtained as follow:

(38)

Percolation equation of region II system is transformed by Laplace transformation:

(39)

(40)

(41)

(42)

Where, is the image function of in Laplace space; s is Laplace operator.

Percolation equation of region I system is transformed by Laplace transformation:

(43)

(44)

(45)

(46)

Percolation model for hydraulic fracture system is transformed by Laplace

transformation:

(47)

(48)

(49)

2.2 Solution of equations

2.2.1 Region II system

Substitute eq. (40) into eq. (39), and the following equation can be obtained:

(50)

The solution of eq.(50) is:

(51)

Where, .

According to the internal and external boundary conditions (eq.(41) and eq.(42)),

the coefficients of eq.(51) can be derived:

,

The expression of region II system can be concluded:

(52)

2.2.2 Region I system

According to eq. (44) and eq. (52), the following equation can be concluded:

, (53)

Substitute eq. (53) into eq. (42), and the following equation can be obtained:

(54)

The solution of eq. (54) is:

(55)

Where, .

According to the internal and external boundary conditions (eq.(45) and eq.(46)),

the coefficients of eq.(55) can be derived:

,

The solution of region I in Laplace space is:

(56)

2.2.2 Hydraulic fracture system

According to eq. (56), eq. (57) can be derived:

(57)

Substitute eq. (57) into eq. (47), we can obtain the following equation:

(58)

The solution of formula (58) is:

(59)

Where, .

According to the internal and external boundary conditions(eq.(48) and eq.(49)),

the coefficients of eq.(59) can be derived:

, .

The pressure expression of fracture system can be concluded as follow:

(60)

When , , the bottom hole pressure is obtained as follow:

(61)

Since eq. (61) is the semi-analytical solution for bottom hole pressure in Laplace

space, the numerical solution can be obtained by Stehfest numerical inversion [20][21].

3 The sensitivity analysis for trilinear flow model

The solution of the trilinear flow model is expected to capture the characteristics of

the early-time regimes in hydraulic fractures. However, the accuracy of the

computations is preliminary constrained by the assumption of the boundaries. In order

to enhance the accuracy of the model, Skin factor and wellbore storage were taken

into account for practical purposes.

Figure 2 is the effect of dimensionless wellbore storage CD on the dimensionless

wellbore pressure. It has significant impact on the liquid flow, especially at the early

transition and stage. If the wellbore storage remains constant for a long time, it

implies the transition stage would occurred later. The influence on dimensionless

pressure by wellbore storage coefficient decreases with an increase in time with

pressures curves intersecting at later stage.

Figure 3 shows the effects of skin factor S on the dimensionless wellbore pressure.

S affects the transition stage primarily whilst pressure curve at early stage appears to

be smooth and steady, while low skin factor at middle stage results in premature

occurrence of transition stage. Pressures curves tend to coincided at later stage.

Higher skin factor means higher dimensionless bottom-hole pressure, because the

additional pressure draw-down near wellbore was caused by the skin factor.

tD

ΨwD CD=0.01

CD=0.1

CD=1

tD

ΨwD

S=10

S=1

S=0

Figure 2 Effects of dimensionless wellbore

storage CD on the dimensionless wellbore

pressure

Figure 3 Effects of skin factor S on the

dimensionless wellbore pressure

Figure 4 and figure 5 show the effects of adsorption coefficient α1 and α2 on the

dimensionless wellbore pressure. Adsorption coefficient predominantly reflects the

matrix ability of adsorbing methane. Large adsorption coefficient results in strong

adsorption capacity, which makes desorption of adsorbed gas difficult, resulting in

lower pressure and premature occurrence of later transient stage. Because region I is

transition region of fluid flow, adsorption coefficient α1 mainly influences transition

section. The bigger α1, the easier it will be for the fluid to flows into the fracture and

the smaller the dimensionless pressure will be. Region II is main drainage area of the

shale gas. It mainly provide shale gas to region I by desorption.The adsorption

coefficient α2 mainly influences the middle and late period of flow state. Therefore,

the bigger α2, the more desorption gas and the deeper the "concave".

tD

ΨwD

α 1=2.0α 1=0.5

α 1=0.1

tD

ΨwD

α 2=0.1

α 2=2.0

α 2=0.5

Figure 4 Effects of adsorption coefficient α1 on

the dimensionless wellbore pressure

Figure 5 Effects of adsorption coefficient α2 on

the dimensionless wellbore pressure

Figure 6 shows the effects of fracture diffusivity coefficient ηD on the

dimensionless wellbore pressure. Due to the impact of the fracture at the formation

near the wellbore, the fracture parameters have significant influence of early flow

period. Figure 3.6 shows that as the fracture diffusivity coefficient become bigger, the

flow resistance become less, and the dimensionless bottom hole pressure become

bigger.

Figure 7 shows the effects of fracture conductivity factor CFD on the dimensionless

wellbore pressure. Fracture conductivity factor denotes the conductivity of fractures,

the larger fracture conductivity factor, the stronger fracture conductivity capacity, the

smaller percolation resistance in the fracture leading to lower pressure with a

corresponding shift down the curve, and shorter duration of early flow stage. The

pressure curves tend to be parallel as the increase of dimensionless time at later stage.

tD

ΨwD

η D=1

ηD=10

ηD=100

tD

ΨwD

CfD=0.1

CfD=1

CfD=10

Figure 6 Effects of fracture diffusivity

coefficient ηD on the dimensionless wellbore

pressure

Figure 7 Effects of fracture conductivity factor

CfD on the dimensionless wellbore pressure

Figure 8 shows the effect of inter-porosity transfer coefficient on the dimensionless

wellbore pressure. This parameter mainly affect the transient flow period from matrix

to fracture, and have negligible influence at early and later stages. The inter-porosity

transfer coefficient λ determines the time and height for transition stage occurrence. if

the inter-porosity transfer coefficient becomes small, the transition stage would

appear early and placement of pressure curves would become low.

Figure 9 shows effects of elastic storativity ratio ω on the dimensionless wellbore

pressure. It can be seen from figure 7 that storativity ratio determines the extent of

crossflow time. Lower storativity ratio means higher dimensionless pressure and

earlier appearance time of transition stage. The dimensionless pressure is irrelevant to

storativity ratio at later stage. When storativity ratio values equals to 1, dimensionless

pressure curve performs the characteristic of an isotropic gas reservoir.

tD

ΨwD

λ =10

λ=103

λ=105

tD

ΨwD

ω=0.2

ω=0.05ω=0.1

Figure 8 Effects of inter-porosity transfer

coefficient λ on the dimensionless wellbore

pressure

Figure 9 Effects of storativity ratio ω on the

dimensionless wellbore pressure

4 Conclusion

(1) A trilinear flow model is established for fractured horizontal well in shale gas

reservoir respecting the influence of both wellbore storage and skin factor. In

addition, the numerical solution is obtained by Laplace transformation and

Stehfest numerical inversion.

(2) If the wellbore storage remains constant for a long time, it implies the transition

stage will occur late. Skin factor mainly affects the transition stage meaning, the

smaller the Skin factor, the earlier the appearance of the transition stage.

(3) Adsorption coefficient reflects the adsorption capacity of matrix on methane gas,

which implies large adsorption coefficient results in strong adsorption capacity,

with difficulty in the gas to desorb.

(4) The larger dimensionless fracture conductivity factor, the stronger fracture

conductivity capacity and the lower the dimensionless bottom hole pressure. This

factor tends to shortens early flow period.

(5) Inter-porosity transfer coefficient mainly affect the transient period flow from

matrix to fracture, and have negligible influence at early and later stages whereas

the cross flow coefficient determines the time and height for transition stage

occurrence.

Acknowledgments

The authors are grateful for the financial support by National Science and Technology

Key Project “Complex oil & gas geology and EOR technology” (No. 2011ZX05009)

and National Natural Science Foundation of China “Seepage mechanics and theory

study on fracture anisotropy in shale gas reservoir” (No. 51374222).

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