Tracer dispersion in a hydraulic fracture with porous walls

Chemical Engineering Research and Design 1 5 0 ( 2 0 1 9 ) 169–178

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Tracer dispersion in a hydraulic fracture with porous walls Morteza Dejam Department of Petroleum Engineering, College of Engineering and Applied Science, University of Wyoming, 1000 E. University Avenue, Laramie, WY 82071-2000, USA

a r t i c l e

i n f o

a b s t r a c t

Article history:

This study presents an analytical expression for the dispersion of a tracer transporting

Received 27 March 2019

through a hydraulic fracture with porous walls. Three different geometrical models for

Received in revised form 12 July 2019

the hydraulic fracture, including rectangular, triangular, and elliptical models, are applied

Accepted 25 July 2019

to evaluate the role of the hydraulic fracture geometry on the tracer dispersion coeffi-

Available online 6 August 2019

cient. It is revealed that the average tracer dispersion coefficients for all hydraulic fracture

Keywords:

magnitudes of the average tracer dispersion coefficients in hydraulic fractures with both

geometries with porous walls are smaller than those with non-porous walls. However, the Tracer dispersion coefficient

non-porous and porous walls follow an order of Triangular > Elliptical > Rectangular geome-

Hydraulic fracture

tries. The analysis recognizes three distinct regimes of diffusion-dominated, transition,

Porous walls

and advection-dominated for each hydraulic fracture geometry. In the diffusion-dominated

Half-aperture

regime, the advection is not important for the tracer transport and the ratios of the aver-

Geometrical models

age dispersion coefficients in hydraulic fractures with porous walls to those in hydraulic fractures with non-porous walls are unity (R = 1). In the transition regime, the ratios depend on the Peclet number and they vary in the range of 0.3 < R < 1. The magnitudes of the ratios follow an order of Rectangular > Elliptical > Triangular hydraulic fracture geometries in the transition regime. The average tracer dispersion coefficients in the hydraulic fractures with porous walls are 0.3 times smaller than those with non-porous walls within the advectiondominated regime (R = 0.3). Therefore, it is crucial to consider the mass transfer of a tracer from the hydraulic fractures into the matrix in derivation of the dispersion coefficient within the transition and advection-dominated regimes for all hydraulic fracture geometries. © 2019 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

1.

Introduction

It is considered that the unconventional (shale and tight) reservoirs to be the future of the fossil fuel industry in North America due to the decline in conventional reserves. Due to the very small fluid mobility in these unconventional reservoirs, the advanced extraction methods (such as hydraulic fracturing) are necessary in order to increase the fluid mobility and therefore the production from the reservoirs. One of the significant research areas in both the academia and the industry is thus defined on the improved understanding of fundamentals of fluid flow and tracer transport in hydraulically fractured reservoirs to improve recovery (Dejam, 2016, 2019a). It is worth mentioning that a hydraulically fractured reservoir resembles a

double-porosity system (Zendehboudi and Chatzis, 2011; Zendehboudi et al., 2011, 2012, 2014), comprised of a hydraulic fracture and a matrix. The tracer transport in the hydraulically fractured reservoir exhibits a coupled problem because the medium surrounding the hydraulic fracture, where the matrix and the neighboring hydraulic fracture interact with each other, is naturally porous (Dejam, 2016, 2019a; Dejam et al., 2014, 2015a, 2018a; Ling et al., 2016; Mahadevan, 2018; Kou and Dejam, 2019). Also, the hydraulic fracture walls may not be parallel (Dontsov, 2016). Based on rock mechanical tests and practical data interpretation, the main factors affecting the hydraulic fracture geometry are the horizontal principal stress difference and the angle between the perforation orientation and the maximum horizontal

E-mail address: [email protected] https://doi.org/10.1016/j.cherd.2019.07.027 0263-8762/© 2019 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

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Chemical Engineering Research and Design 1 5 0 ( 2 0 1 9 ) 169–178

Nomenclature b bD

half-aperture of hydraulic fracture, m dimensionless half-aperture of hydraulic fracture Cf tracer concentration in hydraulic fracture, mol m−3 Cf cross-sectional average of tracer concentration in hydraulic fracture, mol m−3 C f fluctuation from cross-sectional average of tracer concentration in hydraulic fracture, mol m−3 Cm tracer concentration in matrix, mol m−3 Df diffusion coefficient in hydraulic fracture, m2 s−1 Dm diffusion coefficient in matrix, m2 s−1 Dporous tracer dispersion coefficient in a hydraulic fracture with porous walls, m2 s−1 Dnon-porous tracer dispersion coefficient in a hydraulic fracture with non-porous walls, m2 s−1 Dporous

average tracer dispersion coefficient in a hydraulic fracture with porous walls, m2 s−1

Dnon-porous average tracer dispersion coefficient in a hydraulic fracture with non-porous walls, m2 s−1 ˆ D dimensionless average tracer dispersion coeffiporous cient in a hydraulic fracture with porous walls ˆ D dimensionless average tracer dispersion non-porous

h Lf Pe R

t u u u u wf x xD z

coefficient in a hydraulic fracture with nonporous walls thickness of matrix, m half-length of hydraulic fracture, m Peclet number ratio of dimensionless average tracer dispersion coefficient in a hydraulic fracture with porous walls to that in a hydraulic fracture with non-porous walls time, s velocity across and along hydraulic fracture, m s−1 cross-sectional average velocity along hydraulic fracture, m s−1 average velocity in hydraulic fracture, m s−1 fluctuation from cross-sectional average velocity along hydraulic fracture, m s−1 width of hydraulic fracture, m horizontal coordinate, m dimensionless distance vertical coordinate, m

x z

derivative respect to x derivative respect to z

Superscripts – cross-sectional average along hydraulic fracture = average in hydraulic fracture  fluctuation from cross-sectional average along hydraulic fracture dimensionless ˆ

stress (Zhang and Chen, 2009). Due to these factors, the hydraulic fracture can find different geometries such as rectangular, triangular, and elliptical geometries. However, to the best of authors’ knowledge the influences of the porous walls and the hydraulic fracture geometry on the dispersion coefficient during the transport of a tracer have not been addressed. The previous analytical studies on determination of the tracer dispersion coefficient in a hydraulic fracture have been traditionally based on the assumption of no interaction between the hydraulic fracture and the matrix, where a no-flux boundary condition is considered at the walls (Taylor, 1953; Fischer et al., 1979; Berkowitz and Zhou, 1996; Wang et al., 2012). Although the tracer dispersion in a hydraulic fracture with nonporous walls has been investigated extensively, studies that address the tracer dispersion in a hydraulic fracture with porous walls are limited. In addition, the derivation of an analytical expression for the tracer dispersion coefficient in a hydraulic fracture-matrix system has not been addressed. The current study has the following features, which make it distinct from other previous theoretical studies. In this work, first, a two-dimensional model is used where the interaction between the matrix and the hydraulic fracture is considered by applying the continuity of the tracer concentration and the mass flux at the hydraulic fracture walls. Then, for reducing the two-dimensional model, the Reynolds decomposition technique is utilized and as a result a reducedorder one-dimensional model is obtained for the advective-dispersive transport in a hydraulic fracture with porous walls, including the tracer dispersion and the advection coefficients. The objective of this study is to derive a more representative analytical expression for the tracer dispersion coefficient, which can be implemented to adequately model the tracer transport in the hydraulically fractured reservoirs. In other words, an analytical expression as a function of the Peclet number is presented for the tracer dispersion coefficient without need for simplifying assumptions used in previous works, numerical approximation, and model reduction. The developed model does not only provide the tracer dispersion coefficient for a hydraulic fracture with porous walls, but it is also capable of giving the tracer dispersion coefficient for a hydraulic fracture with non-porous walls (or a hydraulic fracture with a no-flux boundary condition at the walls), which is in agreement with the existing model in literature. Finally, different models for the geometries of the hydraulic fractures are used to obtain the average tracer dispersion coefficients. These models include the rectangular, triangu-

Greek letters hydraulic fracture porosity f matrix porosity m Subscripts 0 inlet of hydraulic fracture D dimensionless f hydraulic fracture m matrix porous porous walls non-porous non-porous walls derivative respect to t t

lar, and elliptical geometries of the hydraulic fractures. This work was performed in order to study the tracer dispersion in a single hydraulic fracture with porous walls. The proposed model can pave the way for studying tracer dispersion in more complex systems such as multiple hydraulic fractures. In other words, the results from this study and their extensions for other geometries find applications in nanofluidic and microfluidic designs, which can be utilized for a wide range of medical, biological, pharmaceutical, defense, environmental, chemical, hydrological, and petroleum applications. Examples are fluid flow and tracer transport in hydraulically fractured reservoirs (Zhang and Chen, 2009; Dontsov, 2016; Dejam, 2016, 2019a; Dejam et al., 2014, 2015a, 2018a; Ling et al., 2016; Mahadevan, 2018; Kou and Dejam, 2019), drug delivery (Beard and Wu, 2009), DNA analysis/sequencing systems (Vafai, 2011), separation in membranes and analytical chem-

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istry (Zholkovskij and Masliyah, 2004), and biological/chemical agent detection sensors on nanochips and microchips (Datta, 1990).

2.

Theoretical formulation

2.1.

Physical model and assumptions

A hydraulically-fractured reservoir rock, comprised of a matrix with a thickness of h and a hydraulic fracture with a halfaperture of b(x), a width of wf , and a half-length of Lf , is considered where the following assumptions are taken into account:

i The physical properties of the matrix and the hydraulic fracture are constant. ii The fluid flowing in the hydraulic fracture is single phase and incompressible. iii The physical properties of the fluid are constant. iv The hydraulic fracture is semi-infinite in the horizontal direction.

Different geometrical models (including rectangular, triangular, and elliptical models) are used for hydraulic fractures as shown in Fig. 1. The hydraulic fractures are two-dimensional and symmetric about a central axis at z = 0. The inlet and center of the hydraulic fracture are the origins of the x- and z-coordinates. At x = 0, a tracer with a constant concentration, Cf 0 , is introduced. It is noted that the mathematical approach used in this work was recently proposed by Dejam et al. (2018a) in order to study the shear dispersion in a rough-walled fracture.

2.2.

Governing equations for tracer transport

The mass balance results in the two-dimensional unsteady advective-diffusive equation (ADE) for governing the tracer transport in the hydraulic fracture as follows:

Cft + u(x, z)Cfx = Df (Cfxx + Cfzz )

(1)

where Cf and Df are the tracer concentration and the diffusion coefficient in the hydraulic fracture, respectively, (x,z) are the horizontal and the vertical coordinates, respectively, t is the time, and u is the velocity for laminar flow which has the following parabolic profile across and along the hydraulic fracture (Bird et al., 1960; Tamayol and Bahrami, 2009):

u(x, z) =

3 2 u(x)[1 − (z/b(x)) ] 2

(2)

where b(x) is the half-aperture of the hydraulic fracture and u(x) is the cross-sectional average velocity along the hydraulic fracture defined as (Bird et al., 1960):



b(x)

u(x) =

1 b(x)

⎛ ⎜1

u(x, z)dz = u⎝ 0

Lf

⎞−1 ⎟

b2 (x)dx⎠

Lf 0

b2 (x)

(3)

Fig. 1 – Hydraulic fractures with different geometries including (a) rectangular, (b) triangular, and (c) elliptical. where Lf is the half-length of the hydraulic fracture and u is the average velocity in the hydraulic fracture:

1 u= Lf

Lf u(x)dx

(4)

0

For the tracer transport in the hydraulic fracture, Eq. (1), the diffusion is considered in both longitudinal and transverse directions, as presented by the first and second terms on the right-hand side of Eq. (1), respectively, while the advection is only considered in longitudinal direction, as presented by the second term on the left-hand side of Eq. (1). The no-slip boundary condition is used at the hydraulic fracture walls (due to the very small velocity in the matrix) to obtain the velocity in the hydraulic fracture, Eqs. (2–4). Therefore, the velocity can be assumed two-dimensional. Form the mathematical point of view, the inertial terms in the x-momentum equation for the flow through the hydraulic fracture are ignored, which leads to Stokes formulation expressed by Eqs. (2–4). The detailed derivations of Eqs. (2–4) can be found somewhere else (Bird et al., 1960; Tamayol and Bahrami, 2009; Dejam et al., 2018a).

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The mass balance leads to the one-dimensional unsteady diffusive equation for governing the tracer transport in the matrix as follows: Cmt = Dm Cmzz

(5)

where Cm and Dm are the tracer concentration and the diffusion coefficient in the matrix, respectively. For the tracer transport in the matrix, Eq. (5), only the diffusion is considered in transverse direction, as presented by the term on the right-hand side of Eq. (5).

2.3.

Initial and boundary conditions

The suitable initial and boundary conditions for the tracer concentrations in the hydraulic fracture, Eq. (1), and the matrix, Eq. (5), are described as follows. The initial tracer concentrations in the hydraulic fracture and the matrix are zero, Cf (x,z,t = 0)=Cm (z,t = 0) = 0. The tracer concentration is constant at the inlet of the hydraulic fracture, Cf (x = 0,z,t)=Cf 0 . The tracer concentration is zero at the far end of the hydraulic fracture, Cf (x → ∞,z,t) = 0. The tracer concentration gradient is zero at the center of the hydraulic fracture because of the symmetry, Cfz (x,z = 0,t) = 0. The continuity of the tracer concentration and the mass flux at the interface between the hydraulic fracture and the matrix results in Cf [x,z = b(x),t] = Cm [z = b(x),t] and Cfz [x,z = b(x),t] = (D)m /f Cmz [z = b(x),t], respectively, where (D)m /f = Dm m /Df f in which f and m are the hydraulic fracture and matrix porosities, respectively. There is no-flux condition at the top boundary of the matrix, Cmz (z = h/2 + wf /2,t) = 0, where h is the thickness of the matrix and wf is the width of the hydraulic fracture. Eq. (1) accompanied with Eq. (5) resemble a coupled problem because of the continuity of the tracer concentration and the mass flux at the hydraulic fracture walls.

2.4.

Reynolds decomposition technique

For reduction of the two-dimensional advective-diffusive transport in the hydraulic fracture, Eq. (1), to a onedimensional advective–dispersive transport in the hydraulic fracture, the Reynolds decomposition technique is applied. This technique presents the tracer concentration and the velocity in the hydraulic fracture as the summation of the cross-sectional averages and their fluctuations from the average values (Bird et al., 1960): {Cf (x, z, t), u(x, z)} = {Cf (x, t), u(x)} + {C f (x, z, t), u (x, z)}

(6)

where Cf is the cross-sectional average of the tracer concen-



b(x)

tration in the hydraulic fracture defined as

1 b(x)

Cf dz, and 0

C f and u are the fluctuations of the tracer concentration and the velocity in the hydraulic fracture from their corresponding average values. The cross-sectional average values of the fluctuations are zero based on the definition of the fluctuations (Bird et al., 1960):



{C f , u }dz = 0 0

Reduction of tracer transport model

Substitution of Cf and u from Eq. (6) into Eq. (1) leads to the following equation: Cft + C ft + u(x, z)Cfx + u(x, z)C fx = Df (Cfxx + C fxx + C fzz )

(8)

In addition, the initial and boundary conditions are turned to Cf (x,t = 0) + C f (x,z,t = 0) = 0 (initial condition), Cf (x = 0,t) + C f (x = 0,z,t) = Cf 0 (inner boundary condition), Cf (x → ∞,t) + C f (x → ∞,z,t) = 0 (outer boundary condition), C fz (x,z = 0,t) = 0 (central symmetry condition), Cf (x,t) + C f [x,z = b(x),t] = Cm [z = b(x),t], and C fz [x,z = b(x),t] = (D)m /f Cmz [z = b(x),t] (continuity conditions). Taking the cross-sectional average from both sides of Eq. (8) and using Eq. (7) and the symmetry condition at the center of the hydraulic fracture lead to:



Cft + u(x)Cfx + u(x, z)C fx = Df Cfxx +



1  C fz [x, z = b(x), t] b(x)

(9)

Subtraction of Eq. (9) from Eq. (8) gives: C ft + [u(x, z) − u(x)]Cfx + u(x, z)C fx − u(x, z)C fx =



Df C fxx + C fzz −



1  C fz [x, z = b(x), t] b(x)

(10)

Eq. (10) is exact. Now, three following assumptions are implemented here for derivation of a tracer dispersion coefficient in a hydraulic fracture with porous walls (Taylor, 1953; Fischer et al., 1979). i The vertical diffusion tapers off the vertical concentration fluctuations (Cf » C f ) after passing an enough time from the introduction of a tracer at the inlet of the hydraulic fracture. Therefore, a quasi-steady-state condition can be achieved implying C ft = 0 (Taylor, 1953; Fischer et al., 1979). ii The concentration fluctuations in the hydraulic fracture, C f , vary slowly. This causes the third and fourth terms on the left-hand side of Eq. (10) to nearly balance each other (Fischer et al., 1979). iii The horizontal advection is considered to be dominant compared to the horizontal diffusion, which means that [u(x,z)-u(x)]Cfx » Df C fxx . In other words, the horizontal diffusion is not effective and the tracer transport in the horizontal direction is mainly handled by the horizontal advection (Taylor, 1953; Fischer et al., 1979). By applying these three assumptions to Eq. (10) and then using Eq. (2), one can obtain: C fzz = [1 − 3(z/b(x)) ] 2

u(x) 1  C + C [x, z = b(x), t] 2Df fx b(x) fz

(11)

Integrating twice from Eq. (11) with respect to z and then implementing central symmetry condition and Eq. (7) for finding the constants of integration result in:

 C f = z2 −

b(x)

1 b(x)

2.5.

(7) −



7 2 u(x) 1 z4 − C b (x) 2 b2 (x) 30 4Df fx

b(x) 2 [1 − 3(z/b(x)) ]C fz [x, z = b(x), t] 6

(12)

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For closing the formulation, the term u(x, z)C fx in Eq. (9) should be determined. One can derive C fx by taking the first derivative of Eq. (12) with respect to x:

 C fx = z2 −



4

7 2 1 1 − b (x) 2 b2 (x) 30 4Df

(z/b(x)) −

−



z4

7

15

b(x)





u(x)Cfxx +

1 db(x) 2 [1 + 3(z/b(x)) ]C fz [x, z = b(x), t] − 6 dx The cross-sectional average of u(x,z)C fx is obtained by using Eqs. (2) and (13) as follows:



b(x)

u(x, z)C

fx

1 = b(x)

[u(x, z)C fx ]dz 0

=−



2 b2 (x)u(x) du(x) u(x)Cfxx + C 105 Df dx fx

Cft +

2

1 du(x) b2 (x)u(x) 2 db(x) u (x) 7 u(x) − + b(x) 5 175 dx Df 175 dx Df

 Cfx =



1 u2 (x)b2 (x) Cfxx 175 Df  3Df 6 1 db(x) + +{Cm [z = b(x), t] − Cf } u(x) 5 b(x) dx b2 (x)

(18)

Df +

(13)

b(x) 2 [1 − 3(z/b(x)) ]C fzx [x, z = b(x), t] 6



du(x) C + dx fx

db(x) u(x) C dx 4Df fx

Combining Eqs. (15–17) and making some rearrangements give:

(14)

Eq. (18) exhibits a reduced-order one-dimensional model for the advective-dispersive transport in a hydraulic fracture with porous walls, including the tracer dispersion and the advection coefficients. For solving Eq. (18), the appropriate initial and boundary conditions are found by taking the cross-sectional averages of initial, inner, and outer conditions and applying Eq. (7) as Cf (x,t = 0) = 0, Cf (x = 0,t) = Cf 0 , and Cf (x → ∞,t) = 0, respectively. In order to solve Eq. (5), which refers to the main governing equation for tracer transport in the matrix (Cmt = Dm Cmzz ), the appropriate initial and boundary conditions are as Cm (z,t = 0) = 0, Cmz (z = h/2 + wf /2,t) = 0, and

2

−

db(x) u (x) 2 u(x)b(x)  b(x) Cfx − C fzx [x, z = b(x), t] 21 dx Df 15

−

4 db(x)  u(x) C fz [x, z = b(x), t] 15 dx

Cm [z = b(x), t] −

Combination of Eq.s (9) and (14) and making some rearrangements give:

2

db(x) u (x) 2 du(x) b2 (x)u(x) 2 Cft + u(x) − b(x) − 105 dx Df 21 dx Df





Df  2 u2 (x)b2 (x) Cfxx + C fz [x, z = b(x), t] 105 Df b(x)

Cfx =

(15)

u(x)b(x)  + C fzx [x, z = b(x), t] 15 +

where Eq. (19) is derived by combination of the continuity of mass flux at the hydraulic fracture walls and Eq. (16).

2.6. Average tracer dispersion coefficient in a hydraulic fracture with non-porous walls



Df +

b(x) u(x)b2 (x) Cfx (D)m/f Cmz [z = b(x), t] = Cf + 3 15Df (19)

The tracer dispersion coefficient in a hydraulic fracture with non-porous walls, Dnon-porous , as a function of x is obtained using Eq. (15): Dnon-porous (x) = Df +

db(x)  4 C fz [x, z = b(x), t] u(x) 15 dx

In order to eliminate C fz [x,z = b(x),t] in Eq. (15), Eq. (12) is applied for z = b(x) and then it is combined with the continuity of the tracer concentration at the hydraulic fracture walls, which leads to:

2 u2 (x)b2 (x) 105 Df

(20)

Combination of Eq.s (3) and (20) gives:

⎛ Dnon-porous (x) = Df +

2 2⎜ 1 u ⎝ 105 Lf

⎞−2

Lf

⎟ b6 (x)

b2 (x)dx⎠

Df

(21)

0

u(x)b(x) 3 3 C fz [x, z = b(x), t] = Cm [z = b(x), t] − Cf − Cfx 5Df b(x) b(x) (16) 

For elimination of C fzx [x,z = b(x),t] in Eq. (15), Eq. (13) is applied for z = b(x) and then it is combined with Equation (16) and the first derivative of the continuity of the tracer concentration at the hydraulic fracture walls with respect to x, which results in: C fzx [x, z = b(x), t] =



−

Dnon-porous , as follows:

Dnon-porous

1 = Lf

(17)

Dnon-porous (x)dx 0

= Df +



Lf

⎛ 2

6 db(x) {Cm [z = b(x), t] − Cf } b2 (x) dx

1 du(x) b2 (x) 4 db(x) u(x) 3 1+ + Cfx b(x) 15 dx Df 15 dx Df b(x) u(x)b(x) − Cfxx 5Df

Taking the axial average of Eq. (21) over the half-length of the hydraulic fracture leads to the average tracer dispersion coefficient in a hydraulic fracture with non-porous walls,

2 u 105 Df

⎜1 ⎝L

Lf

⎞⎛ ⎟⎜ 1

b6 (x)dx⎠ ⎝

f 0

Lf

⎞−2 ⎟

b2 (x)dx⎠

Lf

(22)

0

The interaction between the hydraulic fracture and the matrix is not considered in determination of the average

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tracer dispersion coefficient in a hydraulic fracture with nonporous walls, Eq. (22), and instead it is assumed that there is a non-porous (or no-flux) boundary condition at the hydraulic fracture walls. For a hydraulic fracture with smooth walls, Eq. (22) turns to the tracer dispersion coefficient in a hydraulic fracture with non-porous walls presented by Berkowitz and Zhou (1996) and Wang et al. (2012).

2.7. Average tracer dispersion coefficient in a hydraulic fracture with porous walls The tracer dispersion coefficient in a hydraulic fracture with porous walls, Dporous , as a function of x is obtained using Eq. (18): Dporous (x) = Df +

1 u2 (x)b2 (x) 175 Df

(23)

Combination of Eq.s (3) and (23) results in:

⎛ Dporous (x) = Df +

1 2⎜ 1 u ⎝ 175 Lf

⎞−2

Lf

⎟ b6 (x)

b2 (x)dx⎠

Table 1 – The dimensional and dimensionless half-apertures for different (including rectangular, triangular, and elliptical) geometries of hydraulic fractures. Geometry of hydraulic fracture

b(x), 0≤x≤Lf

bD (xD ), 0≤xD ≤1

Rectangular Triangular

wf /2 wf (Lf − x)/2Lf

1 1 − xD

Elliptical

wf



2

1 − (x/Lf ) /2



2 1 − xD

Table 2 – The dimensionless average tracer dispersion coefficients for different (including rectangular, triangular, and elliptical) geometries of hydraulic fractures with non-porous and porous walls. Geometry of hydraulic fracture

ˆ D non-porous

ˆ D porous

Rectangular Triangular Elliptical

1 + 2Pe2 /105 1 + 6Pe2 /245 1 + 24Pe2 /1225

1 + Pe2 /175 1 + 9Pe2 /1225 1 + 36Pe2 /6125

(24)

Df

0

Taking the axial average of Eq. (24) over the half-length of the hydraulic fracture gives the average tracer dispersion coefficient in a hydraulic fracture with porous walls, Dporous , as follows:

Dporous

1 = Lf

Lf Dporous (x)dx 0

⎛ 2

= Df +

1 u 175 Df

⎜1 ⎝L

Lf

⎞⎛ ⎟⎜ 1

b6 (x)dx⎠ ⎝

f 0

⎞−2

Lf

⎟

b2 (x)dx⎠

Lf

(25)

0

The interaction between the hydraulic fracture and the matrix is incorporated in derivation of the average tracer dispersion coefficient in a hydraulic fracture with porous walls, Eq. (25). For a hydraulic fracture with smooth walls, Eq. (25) turns to the tracer dispersion coefficient in a hydraulic fracture with porous walls presented by Dejam et al. (2014, 2015a, 2018a); Dejam (2018, 2019b, 2019c).

2.8.

Geometries of hydraulic fractures

Various models for geometries of hydraulic fractures are used to obtain average tracer dispersion coefficients. These models include rectangular, triangular, and elliptical geometries of hydraulic fractures. For hydraulic fractures of half-length Lf and width wf with rectangular, triangular, and elliptical geometries (see Fig. 1), the dimensional and dimensionless half-apertures are presented in Table 1, where bD = 2b/wf and xD = x/Lf , in which bD and xD are the dimensionless halfaperture and the dimensionless distance, respectively. Eq.s (22),(25), and the half-apertures for rectangular, triangular, and elliptical geometries of hydraulic fractures in Table 1 are combined to develop the average tracer dispersion coefficients in hydraulic fractures with non-porous and porous walls. In order to make the developed expressions dimenˆ is the ˆ = D/D and Pe = u w /2D are used, where D sionless, D f

f

f

Fig. 2 – Dimensionless half-apertures versus dimensionless distance for (a) rectangular, (b) triangular, and (c) elliptical hydraulic fractures. dimensionless average tracer dispersion coefficient and Pe is the Peclet number. By using the dimensionless variables, the dimensionless forms of the average tracer dispersion coefficients can be obtained. Table 2 summarizes the dimensionless average tracer dispersion coefficients for different (including rectangular, triangular, and elliptical) geometries of hydraulic fractures with non-porous and porous walls. The detailed derivations of the dimensionless average tracer dispersion coefficients for rectangular, triangular, and elliptical hydraulic fractures with non-porous and porous walls in Table 2 can be found in Appendixes A, B, and C, respectively.

3.

Results and discussion

3.1. Half-apertures for different geometries of hydraulic fractures The dimensionless half-apertures, bD , versus the dimensionless distance, xD , for rectangular, triangular, and elliptical hydraulic fractures are demonstrated in Fig. 2 using Table 1. Based on the definitions of the dimensionless half-aperture and the dimensionless distance, it is possible to con-

Chemical Engineering Research and Design 1 5 0 ( 2 0 1 9 ) 169–178

Fig. 3 – Dimensionless average tracer dispersion coefficients versus Peclet number for rectangular, triangular, and elliptical hydraulic fractures with non-porous and porous walls. clude that 0 ≤ bD ≤ 1 and 0 ≤ xD ≤ 1. The rectangular hydraulic fracture has a constant dimensionless half-aperture equal to 1, bD = 1, along xD . The dimensionless half-aperture for triangular hydraulic fracture decreases linearly from bD = 1 to bD = 0 when the dimensionless distance increases from xD = 0 to xD = 1, respectively. The dimensionless halfaperture for elliptical hydraulic fracture varies nonlinearly from (xD = 0,bD = 1) to (xD = 1,bD = 0). The dimensionless halfaperture for elliptical hydraulic fracture is between the dimensionless half-apertures for rectangular and triangular hydraulic fractures.

3.2. The role of hydraulic fracture geometry on average tracer dispersion coefficient in a hydraulic fracture with porous walls Fig. 3 shows the dimensionless average tracer dispersion coefficients versus the Peclet number for rectangular, triangular, and elliptical hydraulic fractures with non-porous and porous walls using Table 2. As expected, it is revealed that three distinct regimes can be recognized for all hydraulic fracture geometries with non-porous and porous walls: diffusiondominated, transition, and advection-dominated regimes. However, for each hydraulic fracture geometry with porous walls a larger Peclet number is required which the dimensionless average tracer dispersion coefficient can move from the diffusion-dominated regime to the transition regime compared to that with non-porous walls. The results also demonstrate that the dimensionless average tracer dispersion coefficients in rectangular, triangular, and elliptical hydraulic fractures with porous walls are smaller than those with non-porous walls. This performance is expected if one considers the retardation of the tracer due to the mass exchange between the matrix and the hydraulic fracture. As a general outcome, it is worth noting that the magnitudes of the dimensionless average tracer dispersion coefficients in hydraulic fractures with both non-porous and porous walls follow an order of Triangular > Elliptical > Rectangular geometries. In the case of rectangular hydraulic fracture with nonporous walls, the dimensionless average tracer dispersion coefficient is 1 + 2Pe2 /105, which was developed by Berkowitz and Zhou (1996) and Wang et al. (2012). The dimensionless average tracer dispersion coefficient in a rectangular hydraulic

175

Fig. 4 – Ratios of dimensionless average tracer dispersion coefficients in hydraulic fractures with porous walls to those in hydraulic fractures with non-porous walls versus Peclet number for rectangular, triangular, and elliptical geometries of hydraulic fractures.

fracture with porous walls is 1 + Pe2 /175, which was presented recently by Dejam et al. (2014, 2015a, 2018a); Dejam (2018, 2019b, 2019c).

3.3. Comparison between average tracer dispersion coefficients in hydraulic fractures with non-porous and porous walls Fig. 4 illustrates the ratios of the dimensionless average tracer dispersion coefficients in hydraulic fractures with porous walls to those in hydraulic fractures with non-porous walls ˆ ˆ /D ) versus Peclet number for (defined as R = D porous

non-porous

rectangular, triangular, and elliptical geometries of hydraulic fractures using Table 2. The results shown in Fig. 4 reveal that the dimensionless average tracer dispersion coefficients in hydraulic fractures with porous walls are smaller than those with non-porous walls. The reduced dimensionless average tracer dispersion coefficients in the case of porous walls can be justified as follows. First, the tracer dispersion increases as the velocity gradient increases. Second, the porous walls discharge the tracer in the area close to the walls where velocity gradients are larger than those in the central area of the hydraulic fracture. The consequence is a smaller velocity gradient and therefore, dispersion decreases if the retardation of the tracer due to the mass exchange between the matrix and the hydraulic fracture is considered (Sankarasubramanian and Gill, 1973; Dejam et al., 2015b, 2016, 2018b). Fig. 4 recognizes three distinct regimes of diffusiondominated, transition, and advection-dominated more clearly for each hydraulic fracture geometry. In the case of small Peclet numbers within the diffusion-dominated regime, the advection is not important for the tracer transport through the rectangular, triangular, and elliptical hydraulic fractures where the ratios are unity (R = 1). In the transition regime, the ratios depend on the Peclet number and they vary in the range of 0.3 < R < 1. The magnitudes of the ratios follow an order of Rectangular > Elliptical > Triangular hydraulic fracture geometries in the transition regime. The dimensionless average tracer dispersion coefficients in the hydraulic fractures with porous walls are 0.3 times smaller than those in the hydraulic fractures with non-porous walls within

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the advection-dominated regime (R = 0.3). The presented outcomes exhibit the significance of the porous nature of the walls resulting in the communication between the matrix and hydraulic fractures, which was recently studied in details by Dejam et al. (2014, 2015a, 2018a); Dejam (2018, 2019b, 2019c) for the case of the rectangular hydraulic fractures. The findings reveal that it is crucial to consider the mass transfer of a tracer from the hydraulic fractures into the matrix in derivation of the dimensionless average tracer dispersion coefficient within the transition and advection-dominated regimes for all hydraulic fracture geometries. The findings of this study can help for better understanding of the tracer dispersion in a hydraulic fracture with porous walls.

4.

Summary and conclusions

The tracer transport in the hydraulically fractured reservoir exhibits a coupled problem because the medium surrounding the hydraulic fracture, where the matrix and the neighboring hydraulic fracture interact with each other, is naturally porous. Also, the hydraulic fracture walls may not be parallel. In this study, first, a two-dimensional model is used where the interaction between the matrix and the hydraulic fracture is considered by applying the continuity of the tracer concentration and the mass flux at the hydraulic fracture walls. Then, for reducing the two-dimensional model, the Reynolds decomposition technique is utilized and as a result a reduced-order one-dimensional model is obtained for the advective-dispersive transport in a hydraulic fracture with porous walls, including the tracer dispersion and the advection coefficients. The objective of this study is to derive a more representative analytical expression for the tracer dispersion coefficient, which can be implemented to adequately model the tracer transport in the hydraulically fractured reservoirs. The developed model is also capable of giving the tracer dispersion coefficient for a hydraulic fracture with non-porous walls (or a hydraulic fracture with a no-flux boundary condition at the walls). Finally, different models for the geometries of the hydraulic fractures are used to obtain the average tracer dispersion coefficients. These models include the rectangular, triangular, and elliptical geometries of the hydraulic fractures. The following results and remarks are obtained from this study:

i The average tracer dispersion coefficients for all hydraulic fracture geometries with porous walls are smaller than those with non-porous walls. ii The magnitudes of the average tracer dispersion coefficients in hydraulic fractures with both non-porous and porous walls follow an order of Triangular > Elliptical > Rectangular geometries. iii The analysis recognizes three distinct regimes of diffusion-dominated, transition, and advectiondominated for each hydraulic fracture geometry. iv In the diffusion-dominated regime, the advection is not important for the tracer transport and the ratios of the average tracer dispersion coefficients in hydraulic fractures with porous walls to those in hydraulic fractures with non-porous walls are unity (R = 1). v In the transition regime, the ratios depend on the Peclet number and they vary in the range of 0.3 < R < 1. The magnitudes of the ratios follow an order of Rectangu-

lar > Elliptical > Triangular hydraulic fracture geometries in the transition regime. vi The average tracer dispersion coefficients in the hydraulic fractures with porous walls are 0.3 times smaller than those with non-porous walls within the advectiondominated regime (R = 0.3). vii It is crucial to consider the mass transfer of a tracer from the hydraulic fractures into the matrix in derivation of the tracer dispersion coefficient within the transition and advection-dominated regimes for all hydraulic fracture geometries. This work was performed in order to study the tracer dispersion in a single hydraulic fracture with porous walls. The proposed model can pave the way for studying tracer dispersion in more complex systems such as multiple hydraulic fractures.

Conflicts of interest The author declares no conflict of interest.

Acknowledgments The author is gratefully appreciative of the financial support from the Department of Petroleum Engineering in the College of Engineering and Applied Science at the University of Wyoming.

Appendix A. Derivations of Dimensionless Average Tracer Dispersion Coefficients for Rectangular Hydraulic Fractures with Non-Porous and Porous Walls Replacement of the dimensional half-aperture for rectangular hydraulic fracture in Table 1, b = wf /2, in the average tracer dispersion coefficient in a hydraulic fracture with non-porous walls, Eq. (22), leads to:

⎛ 2

Dnon-porous = Df +

2 u 105 Df

⎜1 ⎝L

⎞

Lf 6

⎟

(wf /2) dx⎠

f 0

⎛ ⎜1 ⎝L f

⎞−2

Lf 2

⎟

(wf /2) dx⎠

(A1)

0

After integration and rearrangement, Eq. (A1) turns to: 2

Dnon-porous = Df +

2 u (wf /2) Df 105

2

(A2)

By using the definitions for the dimensionless average tracer dispersion coefficient in a hydraulic fracture with ˆ =D /D , and the Peclet non-porous walls, D non-porous

non-porous

f

number, Pe = u wf /2Df , Eq. (A2) can be written as follows: 2 ˆ Pe2 D non-porous = 1 + 105

(A3)

Eq. (A3) is the dimensionless average tracer dispersion coefficient for rectangular hydraulic fracture with non-porous walls.

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Replacement of the dimensional half-aperture for rectangular hydraulic fracture in Table 1, b = wf /2, in the average tracer dispersion coefficient in a hydraulic fracture with porous walls, Eq. (25), leads to:

⎛ 2

⎜1 ⎝L

1 u Dporous = Df + 175 Df



Lf

⎞⎛ ⎟⎜ 1 (wf /2) dx⎠ ⎝ L



6

f

⎞−2

Lf 2

⎟

(wf /2) dx⎠

f

0

0

(A4)

dispersion coefficient in a hydraulic fracture with porous walls, Eq. (25), leads to:

⎛ 2

Dporous = Df +

⎜1 ⎝L

1 u 175 Df

⎞

Lf 6

⎟

[wf (Lf − x)/2Lf ] dx⎠

f 0

⎛ ⎜1 ⎝L

⎞−2

Lf

⎟

2

[wf (Lf − x)/2Lf ] dx⎠

f

(B4)

0

After integration and rearrangement, Eq. (A4) turns to: After integration and rearrangement, Eq. (B4) turns to: 2

Dporous = Df +

1 u (wf /2) 175 Df

2

Dporous = Df +

By using the definitions for the dimensionless average tracer dispersion coefficient in a hydraulic fracture with ˆ =D /D , and the Peclet number, porous walls, D porous

2

(A5)

porous

f

Pe = u wf /2Df , Eq. (A5) can be written as follows:

9 u (wf /2) 1225 Df

2

(B5)

By using the definitions for the dimensionless average tracer dispersion coefficient in a hydraulic fracture with ˆ =D /D , and the Peclet number, porous walls, D porous

porous

f

Pe = u wf /2Df , Eq. (B5) can be written as follows: 1 ˆ D Pe2 porous = 1 + 175

(A6) ˆ D porous = 1 +

Eq. (A6) is the dimensionless average tracer dispersion coefficient for rectangular hydraulic fracture with porous walls.

Appendix B. Derivations of Dimensionless Average Tracer Dispersion Coefficients for Triangular Hydraulic Fractures with Non-Porous and Porous Walls

2

Dnon-porous

2 u = Df + 105 Df

⎜1 ⎝L

⎞



Lf 6

⎜1 ⎝L

Lf

hydraulic fracture in Table 1, b(x) = wf

⎛ 2

⎞−2

2

1 − (x/Lf ) /2, in the

average tracer dispersion coefficient in a hydraulic fracture with non-porous walls, Eq. (22), leads to:

Dnon-porous = Df +

⎜1 ⎝L

2 u 105 Df

Lf [wf

f

⎞



6 2

⎟

1 − (x/Lf ) /2] dx⎠

0

⎟ [wf (Lf − x)/2Lf ] dx⎠ 2

f

Replacement of the dimensional half-aperture for elliptical 

⎟

0

⎛

Eq. (B6) is the dimensionless average tracer dispersion coefficient for triangular hydraulic fracture with porous walls.

[wf (Lf − x)/2Lf ] dx⎠

f

(B6)

Appendix C. Derivations of Dimensionless Average Tracer Dispersion Coefficients for Elliptical Hydraulic Fractures with Non-Porous and Porous Walls

Replacement of the dimensional half-aperture for triangular hydraulic fracture in Table 1, b(x) = wf (Lf − x)/2Lf , in the average tracer dispersion coefficient in a hydraulic fracture with non-porous walls, Eq. (22), leads to:

⎛

9 Pe2 1225

(B1)

⎛ ⎜1 ⎝L

0

Lf [wf

f

⎞−2



2 2

⎟

1 − (x/Lf ) /2] dx⎠

(C1)

0

After integration and rearrangement, Eq. (B1) turns to: 2

Dnon-porous = Df +

6 u (wf /2) 245 Df

After integration and rearrangement, Eq. (C1) turns to:

2

(B2) 2

By using the definitions for the dimensionless average tracer dispersion coefficient in a hydraulic fracture with ˆ =D /D , and the Peclet non-porous walls, D non-porous

non-porous

f

number, Pe = uwf /2Df , Eq. (B2) can be written as follows:

Dnon-porous = Df +

24 u (wf /2) 1225 Df

(B3)

Eq. (B3) is the dimensionless average tracer dispersion coefficient for triangular hydraulic fracture with non-porous walls. Replacement of the dimensional half-aperture for triangular hydraulic fracture in Table 1, b = wf /2, in the average tracer

(C2)

By using the definitions for the dimensionless average tracer dispersion coefficient in a hydraulic fracture with ˆ =D /D , and the Peclet non-porous walls, D non-porous

6 ˆ D Pe2 non-porous = 1 + 245

2

non-porous

f

number, Pe = u wf /2Df , Eq. (C2) can be written as follows: 24 ˆ D Pe2 non-porous = 1 + 1225

(C3)

Eq. (C3) is the dimensionless average tracer dispersion coefficient for elliptical hydraulic fracture with non-porous walls.

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Replacement of the dimensional half-aperture for elliptical hydraulic fracture in Table 1, b = wf /2, in the average tracer dispersion coefficient in a hydraulic fracture with porous walls, Eq. (25), leads to:

⎛ 2

Dporous = Df +

⎜1 ⎝L

1 u 175 Df

Lf



2

⎟

1 − (x/Lf ) /2] dx⎠

[wf

f

⎞ 6

0

⎛ ⎜1 ⎝L

Lf [wf

f

⎞−2



2 2

⎟

1 − (x/Lf ) /2] dx⎠

(C4)

0

After integration and rearrangement, Eq. (C4) turns to: 2

Dporous = Df +

36 u (wf /2) 6125 Df

2

(C5)

By using the definitions for the dimensionless average tracer dispersion coefficient in a hydraulic fracture with ˆ porous walls, D =D /D , and the Peclet number, Pe= porous

porous

f

u wf /2Df , Eq. (C5) can be written as follows: 36 ˆ D Pe2 porous = 1 + 6125

(C6)

Eq. (C6) is the dimensionless average tracer dispersion coefficient for elliptical hydraulic fracture with porous walls.

References Dejam, M., 2016. Shear Dispersion in Double-Porosity Systems; PhD Dissertation. University of Calgary, Calgary, Alberta, Canada. Dejam, M., 2019a. Advective-diffusive-reactive solute transport due to non-Newtonian fluid flows in a fracture surrounded by a tight porous medium. Int. J. Heat Mass Transf. 128, 1307–1321. Zendehboudi, S., Chatzis, I., 2011. Experimental study of controlled gravity drainage in fractured porous media. J. Can. Pet. Technol. 50 (2), 56–71. Zendehboudi, S., Chatzis, I., Shafiei, A., Dusseault, M.B., 2011. Empirical modeling of gravity drainage in fractured porous media. Energy Fuels 25 (3), 1229–1241. Zendehboudi, S., Elkamel, A., Chatzis, I., Ahmadi, M.A., Bahadori, A., Lohi, A., 2014. Estimation of breakthrough time for water coning in fractured systems: experimental study and connectionist modeling. Am. Inst. Chem. Eng. J. 60 (5), 1905–1919. Zendehboudi, S., Rezaei, N., Chatzis, I., 2012. Effects of fracture properties on the behavior of free-fall and controlled gravity drainage processes. J. Porous Med. 15 (4), 343–369. Dejam, M., Hassanzadeh, H., Chen, Z., 2014. Shear dispersion in a fracture with porous walls. Adv. Water Resour. 74, 14–25. Dejam, M., Hassanzadeh, H., Chen, Z., 2015a. Shear dispersion in combined pressure-driven and electro-osmotic flows in a channel with porous walls. Chem. Eng. Sci. 137, 205–215.

Dejam, M., Hassanzadeh, H., Chen, Z., 2018a. Shear dispersion in a rough-walled fracture. Soc. Pet. Eng. J. 23 (5), 1669–1688. Ling, B., Tartakovsky, A.M., Battiato, I., 2016. Dispersion controlled by permeable surfaces: surface properties and scaling. J. Fluid Mech. 801, 13–42. Mahadevan, J., 2018. A study of the effect of fracture walls on tracer transport. Soc. Pet. Eng. J. 23 (1), 157–171. Kou, Z., Dejam, M., 2019. Dispersion due to combined pressure-driven and electro-osmotic flows in a channel surrounded by a permeable porous medium. Phys. Fluids 31 (5), 056603. Dontsov, E.V., 2016. Tip region of a hydraulic fracture driven by a laminar-to-turbulent fluid flow. J. Fluid Mech. 797, R2. Zhang, G., Chen, M., 2009. Complex fracture shapes in hydraulic fracturing with orientated perforations. Petrol. Explor. Dev. 36 (1), 103–107. Taylor, G., 1953. Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186–203. Fischer, H.B., List, E.J., Koh, R.C.Y., Imberger, J., Brooks, N.H., 1979. Mixing in Inland and Coastal Waters. Academic Press, San Diego. Berkowitz, B., Zhou, J., 1996. Reactive solute transport in a single fracture. Water Resour. Res. 32 (4), 901–913. Wang, L., Cardenas, M.B., Deng, W., Bennett, P.C., 2012. Theory for dynamic longitudinal dispersion in fractures and rivers with Poiseuille flow. Geophys. Res. Lett. 39 (5), L05401. Beard, D.A., Wu, F., 2009. Apparent diffusivity and Taylor dispersion of water and solutes in capillary beds. B. Math Biol. 71 (6), 1366–1377. Vafai, K., 2011. Porous Media: Applications in Biological Systems and Biotechnology. CRC Press; Taylor & Francis Group, New York. Zholkovskij, E.K., Masliyah, J.H., 2004. Hydrodynamic dispersion due to combined pressuredriven and electroosmotic flow through microchannels with a thin double layer. Anal. Chem. 76 (10), 2708–2718. Datta, R., 1990. Theoretical evaluation of capillary electrophoresis performance. Biotechnol. Progr. 6 (6), 485–493. Bird, R.B., Stewart, W.E., Lightfoot, E.N., 1960. Transport Phenomena. John Wiley, New York. Tamayol, A., Bahrami, M., 2009. Analytical determination of viscous permeability of fibrous porous media. Int. J. Heat Mass Transfer 52, 2407–2414. Dejam, M., 2018. Dispersion in non-Newtonian fluid flows in a conduit with porous walls. Chem. Eng. Sci. 189, 296–310. Dejam, M., 2019b. Hydrodynamic dispersion due to a variety of flow velocity profiles in a porous-walled microfluidic channel. Int. J. Heat Mass Transfer 136, 87–98. Dejam, M., 2019c. Derivation of dispersion coefficient in an electro-osmotic flow of a viscoelastic fluid through a porous-walled microchannel. Chem. Eng. Sci. 204, 298–309. Sankarasubramanian, R., Gill, W.N., 1973. Unsteady convective diffusion with interphase mass transfer. Proc. R. Soc. Lond. A 333, 115–132. Dejam, M., Hassanzadeh, H., Chen, Z., 2015b. Shear dispersion in combined pressure-driven and electro-osmotic flows in a capillary tube with a porous wall. Am. Inst. Chem. Eng. J. 61 (11), 3981–3995. Dejam, M., Hassanzadeh, H., Chen, Z., 2016. Shear dispersion in a capillary tube with a porous wall. J. Contam. Hydrol. 185–186, 87–104. Dejam, M., Hassanzadeh, H., Chen, Z., 2018b. A reduced-order model for chemical species transport in a tube with a constant wall concentration. Can. J. Chem. Eng. 96 (1), 307–316.