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RESEARCH PAPER

Numerical simulation of chemical potential dominated fracturing fluid flowback in hydraulically fractured shale gas reservoirs WANG Fei*, PAN Ziqing MOE Key Laboratory of Petroleum Engineering in China University of Petroleum, Beijing 102249, China

Abstract: To find out the impact of chemical potential difference between the low salinity fracturing fluid and the high sa-

linity formation water on fracturing fluid flowback, a chemical potential difference expression of fracturing fluid and formation water was deduced, on this basis, a mathematical model which considers viscous force, capillary force and osmosis pressure driven gas-water flow in matrix-fracture system was built, the flow back performance of fracturing fluid driven by chemical potential difference was simulated, and the formation water saturation and salt concentration profile with flow back time were analyzed. The results show that in the process of flow back, the water molecules in the matrix driven by the chemical potential difference continually migrated to the deeper reservoirs, while salt ions in the matrix constantly spread to the fractures. After 168 h of fracturing-fluid flow back, the migration distance of water was up to 40 cm, and the salt concentration near the fracture surface increased by 0.841%, and the cumulative flowback ratio of the gas well was only 22.1%. The cumulative flowback ratio would be 23.5%, 32.4% and 41.1% respectively, without taking into account the effect of gas absorption, chemical osmosis or capillary imbibition. The capillary imbibition and chemical osmosis seriously hindered the fracturing-fluid flow back, therefore, the two factors should be fully considered in the post-fracturing evaluation of shale gas wells. Key words: shale; flow back; chemical potential; capillary force; desorption effect; osmosis pressure

Introduction As an important part of unconventional resource, shale gas has become the focus of the world, and has been developed successfully in a number of basins in the United States and Canada. Slick water fracturing is one of the key shale gas reservoir stimulation technologies. In order to achieve large stimulated reservoir volume (SRV) and necessary proppant carrying capacity, large fracturing fluid volume and high pumping rate are required[1]. Fracturing practices of shale gas reservoir in China and abroad show that the fracturing fluid has low flowback ratio in general[23]. The flowback ratio of fracturing fluid in the United States is about 20%40%, and only 5%10%[4] for some shale gas wells in Fuling, China. Most researches[58] attribute this phenomenon to fluid spontaneous imbibition caused by capillary pressure or closure of natural fractures. Shale is composed of sediments with high heterogeneity. It has higher clay content than conventional oil and gas reser-

voirs, of up to 80%[9]. Clay can work as semi-permeable membrane under subsurface condition[10], which enables osmotic migration of water molecules, namely, migration of water molecules from the low salinity side of the membrane to high salinity side. There is certain amount of formation water in shale reservoir, and due to water consumption in diagenesis and hydrocarbon generation process, the salinity of initial formation water is very high[11]. Research by Haluszczak et al[12] indicates that the salinity of formation water in shale reservoir can be as high as 28%, in contrast, the general salinity of slick water is about 0.1%, thus, the huge salinity difference between fracturing fluid and formation water surely will create huge osmotic pressure, which drives fracturing fluid to move from hydraulic fractures to matrix. Flowback of fracturing fluids is commonly considered as an immiscible displacement process. Single-porosity or dual-porosity gas-water two-phase flow models[1318] have been developed, and on this basis, numerical simulation of fractur-

Received date: 15 Apr. 2016; Revised date: 15 Sep. 2016. * Corresponding author. E-mail: [email protected] Foundation item: Supported by the China National Natural Science Foundation (No.51504266); Beijing Natural Science Foundation (No.3154038); Science Foundation of China University of Petroleum, Beijing (No.2462015YQ0212). Copyright © 2016, Research Institute of Petroleum Exploration and Development, PetroChina. Published by Elsevier BV. All rights reserved.

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fracturing fluid flowback as well as fracturing parameter analysis have been conducted in previous researches. These flowback models assume water flows in fractures or flows both in fractures and matrix; sensitivity analysis considers relative permeability, stress sensitivity, capillary pressure as well as gravity. However, previous researches only took into account driven force of water molecules in physical level, didn’t take into account the chemical osmosis process and its force characterization and osmotic pressure. In this paper, the concept of chemical potential difference is introduced into previous mathematical model of fracturing fluid flowback, and the driving effects of viscous force, capillary pressure and osmotic pressure are considered. Through numerical simulation, the influence of different driving forces and gas desorption on fracturing fluid flowback have been verified. This study aims to understand the control mechanism of slick water migration and retention in shale gas reservoirs, and improve the understanding on mechanism of shale gas recovery.

1.

Chemical potential difference

1.1. Derivation of chemical potential difference of certain component in different solutions For multi-component solution system, the differential of chemical potential of component B can be expressed as[19] dB S B, m dT 10VB, m dp (1)

Assuming that partial molar volume doesn’t change with pressure, then the chemical potential difference of component B between different solutions can be expressed as x B T , p1 , x1 ,s B T , p2 , x2 ,s 10VB, m p1 p2 RT ln 1 x2 (8) 1.2. Chemical potential difference between fracturing fluid and formation water

During the process of hydraulic fracturing treatment, nearly ten thousand cubic meters of fracturing fluid is pumped into the formation in general. As the primary formation water has high salinity, chemical potential difference will occur between the primary formation water and low-salinity fracturing fluid. Assuming that fracturing fluid flows from wellbore to matrix through hydraulic fractures, and formation water exists in matrix, then the chemical potential difference between fracturing fluid and formation water is: x w, f w, m 10Vw pw, f pw, m RT ln f (9) xm The molar fraction of water in solution can be calculated according to salt composition and their concentrations[20]. Both chemical potential difference and pressure difference are driving force of water migration, and (μw,f－μw,m)/Vw has the dimension of pressure. Therefore, eq. (9) can be simplified as

w, f w, m

Under isothermal condition, the chemical potential of component B can be obtained by integration of eq. (1). p

pΘ

p

dB 10 VB,m dp

(2)

pΘ

Eq. (2) can also be expressed as p

B T , p B T , p Θ 10 V B, m dp

(3)

pΘ

Assuming the solution containing B is ideal dilute solution (solution obeying Raoul's law, s, different from pure liquid, l). Its chemical potential can also be written as (4) B T , p , s B T , p , l RT ln xB

According to eq. (4), the chemical potential difference of component B between two different concentrations under different pressure can be derived: B T , p1 , x1 , s B T , p2 , x2 , s

B T , p1 , l RT ln x1 B T , p2 , l RT ln x2

(5)

Combining with eq. (3),

B T , p1 , x1 ,s B T , p2 , x2 ,s T , p Θ , l 10 p1 V dp pΘ B, m B

T , p Θ , l 10 p2 V dp RT ln x1 pΘ B, m B x

(6)

2

Simplifying eq. (6), p1

B T , p1 , x1 ,s B T , p2 , x2 ,s 10 VB, m dp RT ln p2

x1 x2 (7)

Vw

10 pw, f pw, m

RT xf ln Vw xm

(10)

From eq. (10), it can be concluded that the chemical potential of water is not only related to concentration, but also pressure. When ignoring salinity difference, driving force is viscous pressure difference 10(pw,f－pw,m), which is conventional viscous pressure driving; when ignoring pressure difference, and only considering the salinity difference between formation water and fracturing fluid, the driving force becomes (RT/Vw)ln(xf /xm), which is osmotic pressure.

2. 2.1.

Flowback mathematical model and solution Assumptions and physical model

Assumptions: (1) shale gas reservoir is composed of matrix and hydraulic fractures; (2) matrix is considered as a homogeneous system with permeability anisotropy; (3) hydraulic fractures are vertical cracks symmetric in two wings, with height equal to the reservoir thickness; (4) isothermal flow and ignoring the influence of gravity; (5) the stress sensitivity of permeability is considered; (6) the effect of capillary pressure is considered; (7) the effect of osmotic pressure is considered; (8) the effect of gas desorption is considered; (9) in the process of fracturing fluid pumping, water enters matrix through hydraulic fractures, and in the process of flowback, gas and water flow into the horizontal wellbore through hydraulic fractures. Based on the above assumptions, shale gas reservoir can be simplified as the combination of hydraulic fracture system and

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matrix system. These two systems are coupled by the continuity of pressure, flow rate and salinity across the contact surface. Taking a horizontal well with three transverse fractures as an example, the grid representation of matrix and fractures are shown in Fig. 1. x, y and z represent the three directions of reservoir and the size of each matrix grid can be expressed by Δx, Δy and Δz. Ignoring the flow along fracture width, fracture grid has two dimensions, x and z. 2.2. 2.2.1.

Considering the gas-water two-phase flow in matrix, auxiliary equations need to be supplemented. (19) Sw, m Sg, m 1

pg, m pw, m pcgw 2.2.2.

(22) qg, mf qg, fw The salinity balance equation in fracture is: q C K K Cf f Sw, f f f rw pw, f Ci w, fw DCf (23) w t w

(11)

where qw,mf is the function of chemical potential difference of aqueous phase, and is expressed as K K (12) qw, mf w m rw w, m w, f 10w Vw The driving force in eq. (12) can be written as the combination of viscous force and osmotic pressure according to eq. (10).

w K m K rw 10 w

RT xm ln 10 pw, m pw, f Vw xf The governing equation for gas flow in matrix is qw, mf

K K m Sg, m g mg m rg g pg, m qg, mf t g

(13)

Considering the water gas two-phase flow in fracture, auxiliary equation is supplemented: (24) Sw, f Sg, f 1 In addition, because of the higher permeability in fractures, the capillary pressure in fractures approximately equals to zero. Thus, (25) pw, f pg, f 2.3.

(14)

Only when gas pressure in matrix is lower than the initial formation pressure, gas desorption term mg exists[21]. pg, m (15) mg s gsc SsVL pg, m pL

pg, m x, y, z , t

Grid division of matrix and fractures.

p0

(26)

S w, m x, y, z , t t 0 S w0, m

(27)

Cm x, y , z , t t 0 C0, m

(28)

pg, f x, z , t

(17) 2.4. 2.4.1.

t 0

p0

(29)

S w, f x, z , t t 0 S w0, f

(30)

Cf x, z , t t 0 C0, f

(31)

Boundary conditions Outer boundary condition

The calculation cell selected satisfies closed boundary condition: p 0 (32) n Γ 2.4.2.

Fig. 1.

t 0

Initial conditions for fracture

The salinity balance equation in matrix is:

Matrix permeability changes with pressure in the form of exponential function during hydraulic fracturing process[22]. (18) K m / K m0 1010 mpn 1010 m ( pc pi )

Initial conditions

Assuming matrix and fractures are both under the undeveloped reservoir condition and have the same initial pressure and initial water saturation. Initial conditions for matrix are:

qg,mf in eq. (14) denotes gas phase crossflow rate between matrix cell and fracture cell, and it only exists in the matrix cell adjacent to fracture. g K m K rg (16) qg, mf pg, m pg, f g C K K Cmm Sw, m m m rw pw, m DCm t w

(21)

K K f Sg, f g f rg g pg, f t g

Governing equations of flow in matrix

The governing equation for water flow in matrix is K K mSw, m w m rw w pw, m qw, mf t w

Governing equations of flow in fracture

The governing equation for water flow in fracture is: K K f Sw, f w f rw w pw, f qw, mf qw,fw t w

Mathematical model

(20)

Inner boundary condition

The inner boundary is shown as the fluid exchange between fracture and wellbore, i.e. qw,fw and qg,fw in eq. (21) and eq. (22). If it is a horizontal well, fluid in fractures enters the wellbore in the form of radial flow. Therefore, qw,fw and qg,fw can be calculated as 1062

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qw, fW

qg, fW

2πww Kf Krw pw, f pwf Vfw ln re / rw

2πwg Kf Krg

Vfg ln re / rw

2 2 where re 0.14 x z

1/ 2

p

pwf

g, f

(33)

3.1.

(34)

Vf xzw

2.4.3. Boundary condition of contact surface between matrix and fracture

The boundary condition of contact surface is the fluid exchange between fracture and matrix, i.e. qw,mf and qg,mf in eq. (11) and eq. (12). Taking gas phase as an example, the boundary condition of contact surface is derived. In Fig. 1, assuming the pressure of contact surface between the fracture cell and its adjacent matrix cell is p, the pressure in the center of matrix cell is pg,m, and the pressure in the center of fracture cell is pg,f, and they have the same flow rate at the contact surface. Flow rate from matrix to contact surface is: (35) qg, m jg, m pg, m p

g K m K rg ΔxΔz g Δy / 2

where jg, m

Flow rate from contact surface to fracture is:

qg, f jg, f p pg, f

where jg, f

(36)

g K f K rg ΔxΔz g w / 2

Owing to the continuity of flow, qg,m=qg,f=qg,mf. Thus, qg,mf can be expressed as (37) qg, mf jg, mf pg, m pg, f

where jg, mf

jg, m jg, f

Model solution

The implicit pressure, explicit saturation method, namely IMPES method, is used to solve this problem with finite-difference discretization using Fortran programming language. There are six main steps in the IMPES algorithm: (1) discretize the pressure, saturation and salinity equations according to semi-implicit method, thus the previous nonlinear partial differential equations are discretized into linear difference equations; (2) calculate the convergence time step Δt according to the convergence conditions of the equations; (3) introduce initial conditions; (4) compute the transmissibility term according to the value at b-th time step, in other words, obtain transmissibility term explicitly; (5) solve the discretized pressure, saturation and salinity equations with boundary conditions, initial conditions and convergence time step Δ; (6) check the total computing time, if the set time reaches, end program and output the results of each time step, otherwise, return to step (4), and compute at (b+1)-th time step.

Numerical simulation analysis Simulation model description

The simulation model is set up according to the geological and construction parameters of a typical fractured horizontal well in Marcellus shale gas basin in the United States[23]. The shale gas reservoir 1 500 m long, 600 m wide and 42 m thick, has an initial formation pressure of 26 MPa, formation temperature of 324 K, initial water saturation of 20%, matrix porosity of 8%, matrix permeability of 3×107 μm2, and matrix compressibility of 4.4×104 MPa1, respectively. The horizontal well was fractured 20 stages with 4 clusters per stage. The length of horizontal section is 1 000 m; the fractures have a half length of 180m, conductivity of 6.5 µm2·cm. the viscosity of natural gas and injected water are 0.022 mPa·s and 1 mPa·s, respectively. The salinity of fracturing fluid and formation water are 0.1% and 28%, respectively. The partial molar volume of water is 18.02×106 m3/mol, and the diffusion coefficient of salt is 109 m2/d[24]. The density of oil source rock is 3.56×103 kg/m3. The density of shale gas under standard condition is 0.77 kg/m3. The ratio of the source rock volume to the total reservoir volume is 0.1. Langmuir volume and Langmuir pressure are 3.32×103 m3/kg and 5.8 MPa, respectively. In this model, the gas-water relative permeability and capillary pressure are obtained through core flooding experiments. The process of pumping fracturing fluid is simulated as injection of water. Injection rate and injection volume are kept the same by adjusting permeability variation coefficient. The adjusted stress sensitivity of permeability is shown in Fig. 2. Simulation analysis is implemented using the above fracturing fluid flowback model of shale gas reservoir. 3.2.

jg, m jg, f

The crossflow of water phase can be derived in the same way. 2.5.

3.

Simulation and result analysis

First, fracturing fluid was pumped into the fractures from perforations of horizontal wellbore continuously for 30 min, while slick water is leaking off into matrix. This process can simulate how fracturing fluid enters matrix and imbibes deeper through capillary pressure. As soon as pumping ended, the well was opened to start flowback at once. The flowback was simulated by producing for 7 days at 5 MPa of bottomhole flowing pressure.

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Fig. 2.

Curve of stress sensitivity of permeability.

WANG Fei et al. / Petroleum Exploration and Development, 2016, 43(6): 1060–1066

Fig. 3 shows the water saturation profile at different distances from fracture along y axis from the beginning to the end of flowback. Simulation results indicate that at the end of pumping, the fracture is filled with water, and the region within 20 cm from the fracture is invaded by water due to leak-off and spontaneous imbibition. During the process of flowback, water saturation of the area near the fracture (within 13 cm) decreases continuously with flowback time, while water saturation of the area away from fracture (13 cm outside) increases with time, which suggests that fracturing fluid continues to move further into matrix even during the flowback process. From Fig. 3, it can be seen that water saturation front has moved 40 cm forward from 20 cm at the beginning of flowback to 60 cm at the end of flowback (168h). Fig. 4 shows the salt concentration profile at different positions from fracture along y axis. The simulation results indicate that high concentration ions in matrix diffuse into fracture and flow out of formation with fracturing fluid during flowback. The salinity within fracture increases by 0.841% from an initial salinity of 0.1% to 0.941% after 168h. The salinity keeps at 28% about 60 cm away from the fracture. The flowback rate and flowback ratio of the fractured horizontal well are shown in Fig. 5. It can be seen from Fig. 5 that flowback rate increases with time rapidly at first, then decreases rapidly. Flowback rate peaks at 786 m3/d and reduces to almost zero after 68 h. The cumulative flowback ratio is only 22.1% after 168 h. Fig. 6 shows the comparison of flowback ratio of four models (① basic model; ②model not considering capillary pressure, i.e. only considering viscous force and osmotic pressure; ③model not considering osmotic pressure, i.e. only considering viscous force and capillary pressure; and model not considering effect of gas desorption). It can be seen from Fig. 6 that if not taking into account the effect of gas desorption, flowback ratio after 168 h is 23.5%, which is most close to actual field condition. At the beginning of flowback, formation pressure is higher than initial formation pressure, and there is no gas desorbed. Thus, the curves considering and not considering effect of gas desorption overlap at the early period of flowback and only after formation pressure is lower than initial formation pressure, the effect of gas desorption takes

Fig. 4.

Fig. 5. Flowback rate and cumulative flowback ratio driven by chemical potential.

Fig. 6.

Water saturation profile along y axis.

Cumulative flowback ratio under different driving force.

effect. The cumulative flowback ratio not considering osmotic pressure or capillary pressure are 32.4% and 41.1%, respectively. Therefore, capillary pressure and chemical osmosis effect have obvious inhibition towards water flowback, and ignoring either of them will cause great deviation on fracturing fluid flowback simulation and post fracturing evaluation.

4.

Fig. 3.

Salt concentration profile along y axis.

Conclusions

The gas-water two-phase flow mathematical model driven by chemical potential difference has been set up based on the semi-permeability of shale and high salinity of formation water. The method using fracture-matrix hybrid model to simulate flowback of fractured shale gas well has been proposed. The simulation results of formation water saturation profile indicate fluid leaks off and imbibes spontaneously into the formation during pumping and creates a high-water-saturation 1064

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zone near the fracture; driven by chemical potential difference, water molecules in matrix still move further into formation during flowback, and water molecules migrated 40 cm 168 h into flowback. The salt concentration profile indicates that ions in matrix diffuse into formation continuously during flowback and after 168 h, the salinity of fracture surface increases by 0.841%. The flowback rate and flowback ratio of fractured shale gas well are mainly influenced by viscous force, capillary pressure as well as chemosmosis. Gas desorption has little impact on flowback behavior. Osmosis and capillarity can hinder flowback significantly, thus, they are two key factors need to be considered in post fracturing evaluation of shale gas wells.

which only exists in the fracture cells adjacent to wellbore, g/(cm3·s); qw,mf —water crossflow rate between fracture and matrix which only exists in the matrix cells adjacent to fracture, g/(cm3·s); re—equivalent radius of cell where the wellbore is, cm; rw—wellbore radius, cm; R—ideal gas constant, 0.008314 kJ/(mol·K); s—water solution obeying Raoul's law; SB,m—partial molar entropy of component, B, kJ/(mol·K); Sg,f —gas saturation at fracture, %; Sg,m—gas saturation at matrix, %; Ss—volume proportion of source rock; Sw,f —water saturation at fracture, %; Sw,m—water saturation at matrix, %; Sw0,f—initial water saturation at fracture, %;

Nomenclature

Sw0,m—initial water saturation at matrix, %; t—computing time, s;

B—time step sequence number;

Δt—time step, s;

Cf—salinity in fracture, %;

T—temperature, K;

Ci—salinity of injecting fluid, %;

VB,m—partial molar volume of component B, m3/kmol;

Cm— salinity of fluid in matrix, %;

Vf —fracture grid volume, cm3;

C0,f—initial salinity of formation water in fracture, %;

VL—Langmuir volume, cm3/g;

C0,m—initial salinity of formation water in matrix, %;

Vw—water partial molar volume, m3/kmol;

2

D—ion diffusion coefficient, cm /s;

w—width of fracture, cm;

Kf —fracture permeability, μm2;

xB—molar fraction of component B in solution, dimensionless;

2

Km—matrix permeability, μm ;

xf, xm—water molar fraction in fracturing fluid and formation wa-

Km0—initial matrix permeability, μm2;

ter, dimensionless; x1, x2—water molar fracture in two different concentration solu-

Krg—gas relative permeability, dimensionless; Krw—water relative permeability, dimensionless;

tions, dimensionless;

l—pure liquid;

α—shape factor between matrix and fracture, cm2;

m—permeability variation coefficient, 1/MPa;

Γ—outer boundary of shale reservoir;

mg—mass of adsorbed gas per formation volume, g/cm3;

x, y, z—Cartesian coordinate system;

n—normal direction of outer boundary;

Δx, Δy, Δz—grid size along x, y and z direction, cm;

1

ηg—viscosity of gas, mPa·s;

p—pressure vector, 10 MPa; p—pressure, 101 MPa;

ηw—viscosity of water, mPa·s;

pc—cell pressure, 101 MPa;

μB—chemical potential of component B in solution, kJ/mol;

pcgw—capillary pressure, 101 MPa;

μw,f —water chemical potential in fracturing fluid, kJ/mol;

1

pg,f —gas pressure at fracture, 10 MPa;

μw,m—water chemical potential in formation water, kJ/mol;

pg,m—gas pressure at matrix, 101 MPa;

ρg—density of gas, g/cm3;

1

ρgsc—gas density under standard condition, g/cm3;

pi—initial formation pressure, 10 MPa; 1

ρs—density of source rock, g/cm3;

pL—Langmuir pressure, 10 MPa; pn—net pressure, 101 MPa, pn=pcpi;

ρw—density of water, g/cm3; 1

pwf —bottom hole flowing pressure, 10 MPa;

φm—matrix porosity, %;

pw,f —water pressure at fracture, 101 MPa;

φf —fracture porosity, %.

pw,m—water pressure at matrix, 101 MPa;

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