**4. Conclusion**

The fundamental solution for flow through packed columns (proppant packs) can be charac‐ terized by the sum of viscous and inertial forces (e.g., Ergun's equation). Coupling Ergun's equation with the Forchheimer equation results in a deterministic equation for the fracture permeability, *kf* , and inertial factor, *β* , as functions of the proppant diameter, sphericity, pack porosity and width. Plotting the dimensionless permeability *k* / *dp* 2 versus *Ω* can be a very useful diagnostic tool that can indicate: 1) sphericity, 2) channeling, 3) crushing, 4) non-uniform sphere size distribution, 5) embedment and 6) deviation of the friction multiplier *λ<sup>m</sup>* from Ergun's equation. Large deviations can also signify poor or inconsistent experimental results. This diagnostic plot can also quantify the behavior of proppant mono-layers.

*p* = Pressure

*Pf* = Perimeter

*q* = Flow rate

*w* = Width

*μ* = Viscosity

*ϕ* = Porosity *Φ* = Sphericity

*ρ* = Density

**Subscripts**

*f* = Fracture *h* = Hydraulic *p* = Proppant

= Sphere

*w* = Wall or width

**Greek**

Re = Reynolds number *υ* = Superficial velocity

*Vp* = Particle volume

*β* = Inertial or beta factor

*τw* = Shear stress - wall

*λm* = Friction factor multiplier

*Ω* = Characteristic proppant porosity parameter

**Appendix A: Flow through packed columns**

The solution methodology for flow through a proppant pack can be developed from flow through packed columns as presented by Bird, Stewart, and Lightfoot (1960). Although a

Modeling of Proppant Permeability and Inertial Factor for Fluid Flow Through Packed Columns

http://dx.doi.org/10.5772/56299

559

*Ψ* = Sphericity-specific area parameter

*υ*¯ = Cross-sectional velocity

#### **Nomenclature**


*p* = Pressure

**4. Conclusion**

558 Effective and Sustainable Hydraulic Fracturing

**Nomenclature**

The fundamental solution for flow through packed columns (proppant packs) can be charac‐ terized by the sum of viscous and inertial forces (e.g., Ergun's equation). Coupling Ergun's equation with the Forchheimer equation results in a deterministic equation for the fracture permeability, *kf* , and inertial factor, *β* , as functions of the proppant diameter, sphericity, pack

useful diagnostic tool that can indicate: 1) sphericity, 2) channeling, 3) crushing, 4) non-uniform sphere size distribution, 5) embedment and 6) deviation of the friction multiplier *λ<sup>m</sup>* from Ergun's equation. Large deviations can also signify poor or inconsistent experimental results.

2

versus *Ω* can be a very

porosity and width. Plotting the dimensionless permeability *k* / *dp*

*aD* = Dimensionless specific surface area, *aD* =*aw* / *as*

*as* = Specific surface area - sphere

*aw* = Specific surface area - wall

*A* = Cross-sectional area

*Ap* = Particle surface area

*Ca* = Concentration/area

*dh* = Hydraulic diameter

*dp* = Proppant diameter

*f* = Darcy friction factor

*g* = Gravitational constant

*L <sup>τ</sup>* = Tortuous path length

*k* = Permeability

*L* = Column length

= Equivalent proppant diameter, *dp*

*f* <sup>0</sup> = Burke-Plummer friction factor

*K* = Loss coefficient - inlet and exit

= Number of minor losses

*dp* '

*Nml*

This diagnostic plot can also quantify the behavior of proppant mono-layers.

'

=*Φdp*

*Pf* = Perimeter

*q* = Flow rate


*w* = Width

#### **Greek**

*β* = Inertial or beta factor *λm* = Friction factor multiplier *μ* = Viscosity *τw* = Shear stress - wall *ϕ* = Porosity *Φ* = Sphericity *Ω* = Characteristic proppant porosity parameter *Ψ* = Sphericity-specific area parameter *ρ* = Density

#### **Subscripts**


### **Appendix A: Flow through packed columns**

The solution methodology for flow through a proppant pack can be developed from flow through packed columns as presented by Bird, Stewart, and Lightfoot (1960). Although a detailed derivation of the equations for determining proppant permeability and inertial effects is not within the scope of this paper, the fundamentals are provided to give the reader an appreciation of the dominant parameters that affect the proppant pack permeability.

As discussed by Bird *et al.*, "the packing material may be spheres, cylinders, or various other kinds of packing shapes. It is also assumed that the packing is everywhere uniform and that there is no channeling of fluid (in actual practice, channeling frequently occurs and the formulas provided are not valid). It is further assumed that the diameter of the packing is small in comparison with the diameter of the column in which the packing is contained and that the column diameter is constant." The impact of these last two assumptions will be addressed later in this section.

#### **Governing equations**

The governing equations for flow through packed columns are formulated in this section. Friction factors for packed columns, frictional pressure loss for laminar flow, and inertial flow (non-Darcy) are presented. Derivation for the fracture permeability and inertial coefficient ( *β* ) are also presented.

#### **Friction factor**

The friction factor is normally defined as the ratio of friction forces to inertial forces. This factor is commonly used to determine the frictional dissipation in closed conduits and is defined as

$$f = \frac{4\pi\_w}{1/2\rho\upsilon^2} = \frac{2d\_h(-dp/dx)}{\rho\upsilon^2} \tag{13}$$

2 32 *<sup>m</sup>*


http://dx.doi.org/10.5772/56299

561

Modeling of Proppant Permeability and Inertial Factor for Fluid Flow Through Packed Columns


= = (17)

<sup>=</sup> (18)


l mu

*dp*

*λ<sup>m</sup>* =3 / 2 , respectively.

velocity from Eq. A.3 is

where the Darcy friction factor is given by

conduit porosity (i.e., *υ*¯ =*υ* /*ϕ* ).

**Laminar flow in packed columns**

diameter and porosity.

A.3

*dx d*

*h*

where the average flow rate in the cross section available for flow is given by the intrinsic velocity *υ*¯ , *λ<sup>m</sup>* is a friction factor multiplier that is a function of the closed conduit geometry, and *dh* is the hydraulic diameter. Theoretically, the friction multiplier for flow of a Newtonian fluid in a pipe, narrow elliptical slot, and rectangular slot are *λ<sup>m</sup>* =1 , *λ<sup>m</sup>* <sup>=</sup>*<sup>π</sup>* <sup>2</sup> / 8 , and

The pressure loss in terms of the Darcy friction factor based on the cross-sectional average flow

2 *<sup>h</sup>*

64 64

Re *<sup>h</sup>* r u *d* m

where the cross-sectional average velocity *υ*¯ is related to the superficial velocity *υ* by the

The frictional pressure loss through a proppant pack (or packed bed) can be derived from Eq.

2 32 *<sup>m</sup>*

l mu

*dp*

*dx d*

*h*

by replacing the cross-sectional average velocity *υ*¯ by the superficial velocity *υ* (i.e., *υ*¯ =*υ* /*ϕ* ) and the equivalent hydraulic diameter of the proppant pack in terms of the particle

*<sup>f</sup> <sup>d</sup>* lm

The Reynolds number for flow of a Newtonian fluid in a conduit is defined as

r u

*dp f dx d* r u 2

Re *m m h*

 l

where *f* is the Darcy friction factor, *τw* is the wall shear stress, *υ* is the superficial velocity ( *υ* =*q* / *A* ), and *dh* is the hydraulic diameter. The pressure gradient in the conduit is −*dp* / *dx* . The hydraulic diameter in packed columns is sometimes replaced with the equivalent particle diameter or other characteristic dimension.

#### **Hydraulic diameter**

The hydraulic diameter is defined as

$$d\_h = \frac{4A}{P\_f} \tag{14}$$

where *Pf* is the conduit wetted perimeter and *A* is the flow cross-sectional area.

#### **Laminar flow**

The equation of motion for laminar flow in closed conduits (e.g., pipes, slots, annuli and other non-circular conduits) can be represented by

Modeling of Proppant Permeability and Inertial Factor for Fluid Flow Through Packed Columns http://dx.doi.org/10.5772/56299 561

$$-\frac{dp}{d\mathbf{x}} = \frac{32\lambda\_m \mu \left< \overline{\boldsymbol{\nu}} \right>}{d\_h^2} \tag{15}$$

where the average flow rate in the cross section available for flow is given by the intrinsic velocity *υ*¯ , *λ<sup>m</sup>* is a friction factor multiplier that is a function of the closed conduit geometry, and *dh* is the hydraulic diameter. Theoretically, the friction multiplier for flow of a Newtonian fluid in a pipe, narrow elliptical slot, and rectangular slot are *λ<sup>m</sup>* =1 , *λ<sup>m</sup>* <sup>=</sup>*<sup>π</sup>* <sup>2</sup> / 8 , and *λ<sup>m</sup>* =3 / 2 , respectively.

The pressure loss in terms of the Darcy friction factor based on the cross-sectional average flow velocity from Eq. A.3 is

$$-\frac{dp}{d\mathbf{x}} = \frac{f}{2} \frac{\rho \left< \overline{\upsilon} \right>^2}{d\_h} \tag{16}$$

where the Darcy friction factor is given by

detailed derivation of the equations for determining proppant permeability and inertial effects is not within the scope of this paper, the fundamentals are provided to give the reader an

As discussed by Bird *et al.*, "the packing material may be spheres, cylinders, or various other kinds of packing shapes. It is also assumed that the packing is everywhere uniform and that there is no channeling of fluid (in actual practice, channeling frequently occurs and the formulas provided are not valid). It is further assumed that the diameter of the packing is small in comparison with the diameter of the column in which the packing is contained and that the column diameter is constant." The impact of these last two assumptions will be addressed later

The governing equations for flow through packed columns are formulated in this section. Friction factors for packed columns, frictional pressure loss for laminar flow, and inertial flow (non-Darcy) are presented. Derivation for the fracture permeability and inertial coefficient

The friction factor is normally defined as the ratio of friction forces to inertial forces. This factor is commonly used to determine the frictional dissipation in closed conduits and is defined as

> 2 2 4 2( )

where *f* is the Darcy friction factor, *τw* is the wall shear stress, *υ* is the superficial velocity ( *υ* =*q* / *A* ), and *dh* is the hydraulic diameter. The pressure gradient in the conduit is −*dp* / *dx* . The hydraulic diameter in packed columns is sometimes replaced with the equivalent particle

> 4 *h*

where *Pf* is the conduit wetted perimeter and *A* is the flow cross-sectional area.

*f <sup>A</sup> <sup>d</sup>*

The equation of motion for laminar flow in closed conduits (e.g., pipes, slots, annuli and other

 ru


*<sup>P</sup>* <sup>=</sup> (14)

*w h d dp dx <sup>f</sup>*

1 2

diameter or other characteristic dimension.

non-circular conduits) can be represented by

The hydraulic diameter is defined as

t

ru

appreciation of the dominant parameters that affect the proppant pack permeability.

in this section.

**Friction factor**

**Governing equations**

560 Effective and Sustainable Hydraulic Fracturing

( *β* ) are also presented.

**Hydraulic diameter**

**Laminar flow**

$$f = \frac{64\lambda\_m \mu}{\rho \left\{ \overline{\upsilon} \right\} d\_h} = \frac{64\lambda\_m}{\text{Re}}\tag{17}$$

The Reynolds number for flow of a Newtonian fluid in a conduit is defined as

$$\text{Re} = \frac{\rho \{\overline{\upsilon}\} d\_h}{\mu} \tag{18}$$

where the cross-sectional average velocity *υ*¯ is related to the superficial velocity *υ* by the conduit porosity (i.e., *υ*¯ =*υ* /*ϕ* ).

#### **Laminar flow in packed columns**

The frictional pressure loss through a proppant pack (or packed bed) can be derived from Eq. A.3

$$-\frac{dp}{d\mathbf{x}} = \frac{3\mathbf{2}\lambda\_m \mu \{\overline{\boldsymbol{\nu}}\}}{d\_h^2} \tag{19}$$

by replacing the cross-sectional average velocity *υ*¯ by the superficial velocity *υ* (i.e., *υ*¯ =*υ* /*ϕ* ) and the equivalent hydraulic diameter of the proppant pack in terms of the particle diameter and porosity.

The equivalent hydraulic diameter from Eq. A.2 for spherical particles with a diameter of *dp* and a packed porosity of *ϕ* is

$$d\_h = \frac{2}{3} \frac{\phi}{1 - \phi} d\_p \tag{20}$$

2 2( )

*h*


*d dp dx <sup>f</sup>* ru

*h*

*d*

where

**Inertial flow in packed columns**

pressure loss equation for inertial losses is

*<sup>μ</sup>*(1−*ϕ*) >1000 .

**Viscous and inertial flow in packed columns**

on the extended Bernoulli equation with minor losses is

1 2

r

*p p L*

which is valid for (*ρυ*)*dp*

below.

64 400 3 Re Re

*m*

l

= =

Re *<sup>h</sup> d h* <sup>=</sup> ru

f

*h h*

Modeling of Proppant Permeability and Inertial Factor for Fluid Flow Through Packed Columns

 m

The pressure loss in packed columns as a result of inertial forces (minor losses) was originally derived by Burke and Plummer assuming turbulent flow in packed columns (see Bird 1960). Burke and Plummer assumed that for highly turbulent flow that the friction factor was only a function of roughness and that the roughness characteristics were similar for all packed columns. Based on these assumptions Burke and Plummer could then justify a constant friction factor *f* <sup>0</sup> that would be used to characterize turbulent flow. Then from Eq. A.4 the resulting

2

2 0 3

 f

f

*h*

*p*

where the experimental data indicated that *f* <sup>0</sup>≅7 / 3 . This is the Burke-Plummer equation

The form of the Burke-Plummer equation can also be derived assuming inertial forces (minor flow loss) through the proppant pack using the extended Bernoulli equation as presented

The flow through packed columns may also be characterized as a sum of frictional (viscous losses) and inertial (minor losses) forces. The general pressure loss equation formulation based

2 2

 u


2 2 *<sup>h</sup>*

*f k g dg g* t u

3 1


ru

*f d*

0

r u

*dp f dx d*


2

4

*d* (25)

(24)

563

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(26)

*d d*

 f

Substituting the hydraulic diameter and the relationship *υ*¯ =*υ* /*ϕ* into Eq. A.7, we find

$$\begin{split}-\frac{dp}{dx} &= \frac{32\lambda\_m \mu \left< \overline{\psi} \right>}{d\_h^2} = \frac{32\lambda\_m \mu \left(\nu/\phi\right)}{\left(\frac{2}{3}\frac{\phi}{1-\phi}d\_p\right)^2} \\ &= \frac{72\lambda\_m \left(1-\phi\right)^2 \mu \nu}{\phi^3 d\_p^2} \end{split} \tag{21}$$

Experimental measurements (Bird *et al.* 1960) indicate that if a frictional multiplier of *λ<sup>m</sup>* =25 / 12 is used, the above theoretical equation matches extremely well with the experimental data. Insertion of this friction multiplier value into Eq. A.9 then gives

$$1 - \frac{dp}{d\mathbf{x}} = \frac{150\left(1 - \phi\right)^2 \mu \nu}{\phi^3 d\_p^2} \tag{22}$$

which is the Blake-Kozeny equation. This equation is generally good for void fractions less than 0.5 and is valid in the laminar flow regime given by (*ρv*)*dp <sup>μ</sup>*(1−*ϕ*) <10 (Bird 1960). The bed friction multiplier based on the proppant diameter is

$$\begin{split} f\_{d\_p} &= \frac{2d\_p(-dp/dx)}{\rho \upsilon^2} \\ &= \frac{144\lambda\_m}{\text{Re}\_{d\_p}} \frac{\left(1 - \phi\right)^2}{\phi^3} = \frac{300}{\text{Re}\_{d\_p}} \frac{\left(1 - \phi\right)^2}{\phi^3} \end{split} \tag{23}$$

where Re*dp* =*ρvdp* / *μ* . The bed friction factor based on the hydraulic diameter is

#### Modeling of Proppant Permeability and Inertial Factor for Fluid Flow Through Packed Columns http://dx.doi.org/10.5772/56299 563

$$\begin{split} f\_{d\_k} &= \frac{2d\_h(-dp/dx)}{\rho \upsilon^2} \\ &= \frac{64\lambda\_m}{\phi \text{Re}\_{d\_k}} = \frac{400/3}{\phi \text{Re}\_{d\_k}} \end{split} \tag{24}$$

where

The equivalent hydraulic diameter from Eq. A.2 for spherical particles with a diameter of *dp*

f

2 2

l muf

f

æ ö ç ÷ è ø -

2 3 1 ( )

*p*


*<sup>μ</sup>*(1−*ϕ*) <10 (Bird 1960). The bed

*d*

f

Substituting the hydraulic diameter and the relationship *υ*¯ =*υ* /*ϕ* into Eq. A.7, we find

*m m*

( )

72 1


l

l mu


*dp*

*dx d*

Insertion of this friction multiplier value into Eq. A.9 then gives

*dp*

than 0.5 and is valid in the laminar flow regime given by (*ρv*)*dp*

friction multiplier based on the proppant diameter is

*p*

*d*

where Re*dp*

*m*

*h*

32 32

3 2

*d*

f

*p*

 f mu

2

Experimental measurements (Bird *et al.* 1960) indicate that if a frictional multiplier of *λ<sup>m</sup>* =25 / 12 is used, the above theoretical equation matches extremely well with the experimental data.

> ( ) 2

> > f

150 1

*dx d*

2

144 1 1 300 Re Re


f

*p p*

=*ρvdp* / *μ* . The bed friction factor based on the hydraulic diameter is

*d d*

f

2( )

*m*

l

*p*


*d dp dx <sup>f</sup>* ru

3 2

which is the Blake-Kozeny equation. This equation is generally good for void fractions less

( ) ( )

3 3

2 2

 f

 f

*p*

f mu

<sup>=</sup> - (20)

(21)

(23)

2 3 1 *h p d d* f

and a packed porosity of *ϕ* is

562 Effective and Sustainable Hydraulic Fracturing

$$\text{Re}\_{d\_h} = \rho \upsilon d\_h / \mu \tag{25}$$

#### **Inertial flow in packed columns**

The pressure loss in packed columns as a result of inertial forces (minor losses) was originally derived by Burke and Plummer assuming turbulent flow in packed columns (see Bird 1960). Burke and Plummer assumed that for highly turbulent flow that the friction factor was only a function of roughness and that the roughness characteristics were similar for all packed columns. Based on these assumptions Burke and Plummer could then justify a constant friction factor *f* <sup>0</sup> that would be used to characterize turbulent flow. Then from Eq. A.4 the resulting pressure loss equation for inertial losses is

$$\begin{split}-\frac{dp}{d\boldsymbol{x}} &= \frac{f\_0}{2} \frac{\rho \left\{\overline{\boldsymbol{\upsilon}}\right\}^2}{d\_h} \\ &= \frac{3}{4} f\_0 \frac{\rho \boldsymbol{\upsilon}^2}{d\_p} \frac{1-\phi}{\phi^3} \end{split} \tag{26}$$

where the experimental data indicated that *f* <sup>0</sup>≅7 / 3 . This is the Burke-Plummer equation which is valid for (*ρυ*)*dp <sup>μ</sup>*(1−*ϕ*) >1000 .

The form of the Burke-Plummer equation can also be derived assuming inertial forces (minor flow loss) through the proppant pack using the extended Bernoulli equation as presented below.

#### **Viscous and inertial flow in packed columns**

The flow through packed columns may also be characterized as a sum of frictional (viscous losses) and inertial (minor losses) forces. The general pressure loss equation formulation based on the extended Bernoulli equation with minor losses is

$$\frac{p\_1 - p\_2}{\rho g} = \sum f \frac{L\_\tau}{d\_h} \frac{\left\langle \overline{\upsilon} \right\rangle^2}{2g} + \sum k \frac{\left\langle \overline{\upsilon} \right\rangle^2}{2g} \tag{27}$$

or

$$-\frac{dp}{d\mathbf{x}} = \sum f \frac{L\_\tau}{L} \frac{\rho \left< \overline{\upsilon} \right>^2}{2d\_h} + \sum K \frac{L\_\tau}{L} \frac{\rho \left< \overline{\upsilon} \right>^2}{2d\_h} \tag{28}$$

( ) <sup>3</sup>

1 Re 4

This is the Ergun equation (Bird 1960) where Re=*ρυdp* / *μ* , *λ<sup>m</sup>* =25 / 12 and *f* <sup>0</sup> =7 / 3 have been

150 1 7

f

Modeling of Proppant Permeability and Inertial Factor for Fluid Flow Through Packed Columns

(33)

(34)

<sup>=</sup> (35)

(32)

565

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(36)

2

ru f

f

The equation to describe non-Darcy flow is a form of the Forchheimer (1901) equation

*dp* <sup>2</sup>

( ) <sup>2</sup> 1 Re 1 *<sup>k</sup> dp dx*

 - =+ bru

where the dimensionless Reynolds number for non-Darcy flow is given by

Re *<sup>k</sup>*

32


*k*

2

*h*

l

f m u bru

= +

*m*

l mu

f

*dp <sup>f</sup> dx d d*

b

b

2

ru

0

f

<sup>=</sup> <sup>=</sup> (37)

*d*

*k*

rub

m

Rewriting Eq. A.17 in terms of the fracture permeability *k* and inertial factor *β* from Eq. A.20,

2 2 0

*h h*

2

 f

2

2 and

*m h*

b

32 2

*d f <sup>k</sup>*

where *k* is the permeability of the porous media and *β* is the non-Darcy or inertial factor. Clearly, the first term in this equation accounts for viscous effects and the second term for

*dx k* m -= + u bru

Rearranging Eq. A.20 in terms of the dimensionless groups we find

æ öæ ö - - =+ ç ÷ç ÷ ç ÷ç ÷ - è øè ø

*<sup>p</sup> dp d dx*

substituted.

we have

where

**Darcy and non-darcy flow**

inertial or minor loss effects.

where *L* is the length of the column and *L <sup>τ</sup>* is the tortuous path length the fluid takes. Further assume that the number of minor losses *Nml* in a column of length *L* can be approximated by *Nml* = *L <sup>τ</sup>* / *dh* . Then the exit and entrance losses as the fluid expands and contracts through the packed column from Eq. A.15 can be written as

$$\begin{split} -\frac{dp}{dx} &= f \frac{L\_r}{L} \frac{\rho \left< \overline{\upsilon} \right>^2}{2d\_h} + \sum\_{i=1}^{N\_{\text{nl}}} \left( K\_{\text{inlet}} + K\_{\text{exit}} \right) \frac{\rho \left< \overline{\upsilon} \right>^2}{2d\_h} \\ &= f\_{d\_k} \frac{\rho \left< \overline{\upsilon} \right>^2}{2d\_h} + f\_0 \frac{\rho \left< \overline{\upsilon} \right>^2}{2d\_h} \end{split} \tag{29}$$

The inertial pressure loss is identical to the form of the Burke-Plummer equation even though one was based on inertial effects and the other on turbulence. This, however, should not be surprising since both inertial and turbulent losses are proportional to *ρυ* <sup>2</sup> . Substituting the bed friction factor and superficial velocity into Eq. A.16, we find

$$-\frac{dp}{dx} = \frac{32\lambda\_m \mu \nu}{\phi d\_h^2} + f\_0 \frac{\rho \nu^2}{2\phi^2 d\_h} \tag{30}$$

or

$$-\frac{dp}{dx} = \frac{72\lambda\_m \left(1 - \phi\right)^2 \mu \nu}{\phi^3 d\_p^2} + \frac{3}{2} f\_0 \frac{1 - \phi}{\phi^3} \frac{\rho \nu^2}{2d\_p} \tag{31}$$

where from experimental data *λ<sup>m</sup>* =25 / 12 and *f* <sup>0</sup> =7 / 3 .

#### **Ergun's equation**

The total pressure loss formulation for all flow regimes may thus be obtained by simply adding the Blake-Kozeny equation for viscous dissipation and the Burke-Plummer equation for inertial losses. The above equation can be written in terms of the dimensionless groups as follows

Modeling of Proppant Permeability and Inertial Factor for Fluid Flow Through Packed Columns http://dx.doi.org/10.5772/56299 565

$$-\left(\frac{dp}{d\propto}\frac{d\_p}{\rho\upsilon^2}\right)\left(\frac{\phi^3}{1-\phi}\right) = \frac{150\left(1-\phi\right)}{\text{Re}} + \frac{7}{4} \tag{32}$$

This is the Ergun equation (Bird 1960) where Re=*ρυdp* / *μ* , *λ<sup>m</sup>* =25 / 12 and *f* <sup>0</sup> =7 / 3 have been substituted.

#### **Darcy and non-darcy flow**

or

564 Effective and Sustainable Hydraulic Fracturing

or

**Ergun's equation**

follows

2 2

 t  ru


r u

(29)

2 2 *h h*

( ) 2 2

*inlet exit h h i*

2


ru

2 2

The inertial pressure loss is identical to the form of the Burke-Plummer equation even though one was based on inertial effects and the other on turbulence. This, however, should not be surprising since both inertial and turbulent losses are proportional to *ρυ* <sup>2</sup> . Substituting the

2 2 0

3 2 0 3

The total pressure loss formulation for all flow regimes may thus be obtained by simply adding the Blake-Kozeny equation for viscous dissipation and the Burke-Plummer equation for inertial losses. The above equation can be written in terms of the dimensionless groups as

*h h*

2

 f

<sup>2</sup> <sup>2</sup>

*p p*

2 2

 f f ru


where *L* is the length of the column and *L <sup>τ</sup>* is the tortuous path length the fluid takes. Further assume that the number of minor losses *Nml* in a column of length *L* can be approximated by *Nml* = *L <sup>τ</sup>* / *dh* . Then the exit and entrance losses as the fluid expands and contracts through the

> 1 2 2 0

*ml*

*N*

*f KK dx L d <sup>d</sup>*

å

=

 ru

2 2

32

*m*

f

( )

 f mu

*m*

f

l

where from experimental data *λ<sup>m</sup>* =25 / 12 and *f* <sup>0</sup> =7 / 3 .

72 1 3 1

*dp <sup>f</sup> dx <sup>d</sup> <sup>d</sup>*

*dp <sup>f</sup> dx d d* l mu

*f f d d*


*h h*

*h*

t r u

ru

= +

bed friction factor and superficial velocity into Eq. A.16, we find

*d*

*dp L*

*dp L L f K dx L d L d*

ru

t

packed column from Eq. A.15 can be written as

The equation to describe non-Darcy flow is a form of the Forchheimer (1901) equation

$$-\frac{dp}{d\mathbf{x}} = \frac{\mu}{k}\nu + \beta\rho\nu^2\tag{33}$$

where *k* is the permeability of the porous media and *β* is the non-Darcy or inertial factor. Clearly, the first term in this equation accounts for viscous effects and the second term for inertial or minor loss effects.

Rearranging Eq. A.20 in terms of the dimensionless groups we find

$$\left(-dp/d\mathbf{x}\right)\Big/\beta\rho\upsilon^2 = 1/\text{Re}\_{\beta\mathbf{k}} + 1\tag{34}$$

where the dimensionless Reynolds number for non-Darcy flow is given by

$$\text{Re}\_{\beta k} = \frac{\rho \upsilon \beta k}{\mu} \tag{35}$$

Rewriting Eq. A.17 in terms of the fracture permeability *k* and inertial factor *β* from Eq. A.20, we have

$$\begin{split}-\frac{dp}{d\mathbf{x}} &= \frac{\Im 2\lambda\_m \mu \nu}{\phi d\_h^2} + f\_0 \frac{\rho \nu^2}{2\phi^2 d\_h} \\ &= \frac{\mu}{k} \nu + \beta \rho \nu^2 \end{split} \tag{36}$$

where

$$k = \frac{\phi d\_h^2}{32\lambda\_m} \text{ and } \beta = \frac{f\_0}{2d\_h\phi^2} \tag{37}$$

Placing *β* in terms of the fracture permeability we find

$$\beta = \frac{3/4 \, f\_0}{\sqrt{72 \, \lambda\_m}} \frac{1}{\sqrt{k\phi^3}} \tag{38}$$

The permeability for flow through an open slot or channel (i.e., no proppant as *ϕ* →1 ) with a

2 2 9 72 12 *<sup>m</sup> w w <sup>k</sup>* l

( )

( ) <sup>3</sup> <sup>2</sup> <sup>2</sup> 1 72 (1 ) *<sup>D</sup>*

*a*

2

*D D*

f

characteristic proppant pack parameter ( *Ω* ), and the proppant sphericity-specific surface area parameter ( *Ψ* ) for all proppants. The dimensionless specific area ratio of the fracture (slot

Pan et al. (2001) provides a good review of permeability versus porosity correlation for random sphere packing. Pan also proposed a modification to Ergun's equation for low Reynolds

2

*a a* æ ö + Y=F ç ÷ *p*


Modeling of Proppant Permeability and Inertial Factor for Fluid Flow Through Packed Columns

f

<sup>2</sup> 3 2

F æ ö F

2 2 1

f

l

*p m*

72 (1 ) 3 1

2 *p k d*

*m*

l

f

 f

<sup>2</sup> 1 1

3 1( ) *<sup>D</sup> w p*

*a d*

*s*

*a*

*a*

Eq. A.32 illustrates that the dimensionless permeability ratio ( *k* / *dp*

number with a four parameter fit model to correlate *k* / *dp*

wall) and proppant is represented by *aD* .

size distribution.

= + ç ÷ - - è ø

*k d d w*

 f

= = (43)

http://dx.doi.org/10.5772/56299

= WY (45)

è ø + F (47)

<sup>2</sup> ) is a linear function of the

as a function of porosity and sphere

*<sup>w</sup>* = = - (48)


(44)

567

slot width of *w* from Eq. A.29 is

where *λ<sup>m</sup>* =3 / 2 for slot flow.

or

where

The dimensionless form of Eq. A.29 is

#### **Sphericity**

Sphericity is a measure of how closely a grain approaches the shape of a perfect sphere compared to roundness which is a measure of the sharpness of grain corners. The sphericity of a particle is the ratio of the surface area of a sphere (with the same volume as the given particle) to the surface area of the particle

$$\Phi = \frac{\pi^{1/3} \left(6V\_p\right)^{2/3}}{A\_p} \tag{39}$$

where *Vp* is the volume and *Ap* is the surface area of the particle. For non-spherical particles the characteristic particle diameter in the above equations must be replaced by

$$
\dot{d\_p} = \Phi d\_p \tag{40}
$$

#### **Proppant permeability**

The proppant permeability can be theoretically calculated from Eq. A.24 provided that the hydraulic diameter and porosity are known. The hydraulic diameter for flow in a slot of width ( *w* ), packed with a proppant of uniform porosity ( *ϕ* ), and diameter ( *dp* ) from Eq. A.2 is

$$d\_h = \frac{4\phi}{\left(a\_s + a\_w\right)}\tag{41}$$

where the specific surface areas for the proppant spheres and fracture wall are *as* =6(1−*ϕ*) / *dp* and *aw* =2 / *w* , respectively.

The proppant permeability in terms of the proppant diameter, porosity, slot width, and sphericity is found by substituting Eq. A.26 through A.28 into A.24

$$k = \frac{\phi^3 \Phi^2 d\_p^2}{72\lambda\_m (1-\phi)^2} \left( 1 + \frac{\Phi d\_p}{3\left(1-\phi\right)w} \right)^{-2} \tag{42}$$

The permeability for flow through an open slot or channel (i.e., no proppant as *ϕ* →1 ) with a slot width of *w* from Eq. A.29 is

$$k = \frac{9\pi v^2}{72\lambda\_m} = \frac{\pi v^2}{12} \tag{43}$$

where *λ<sup>m</sup>* =3 / 2 for slot flow.

The dimensionless form of Eq. A.29 is

$$\frac{k}{d\_p^2} = \frac{\phi^3 \Phi^2}{72\lambda\_m (1-\phi)^2} \left( 1 + \frac{\Phi d\_p}{3\left(1-\phi\right)w} \right)^{-2} \tag{44}$$

or

Placing *β* in terms of the fracture permeability we find

566 Effective and Sustainable Hydraulic Fracturing

particle) to the surface area of the particle

*as* =6(1−*ϕ*) / *dp* and *aw* =2 / *w* , respectively.

**Proppant permeability**

**Sphericity**

0

l

b

3 4 1 72 *<sup>m</sup> f*

3

= (38)

F = (39)

*p p d d* = F (40)

<sup>=</sup> <sup>+</sup> (41)

(42)

*k*

Sphericity is a measure of how closely a grain approaches the shape of a perfect sphere compared to roundness which is a measure of the sharpness of grain corners. The sphericity of a particle is the ratio of the surface area of a sphere (with the same volume as the given

> ( ) 2 3 1 3 <sup>6</sup> *<sup>p</sup>*

p

the characteristic particle diameter in the above equations must be replaced by

'

*V A*

*p*

where *Vp* is the volume and *Ap* is the surface area of the particle. For non-spherical particles

The proppant permeability can be theoretically calculated from Eq. A.24 provided that the hydraulic diameter and porosity are known. The hydraulic diameter for flow in a slot of width ( *w* ), packed with a proppant of uniform porosity ( *ϕ* ), and diameter ( *dp* ) from Eq. A.2 is

> ( ) 4

*a a* f

*s w*

where the specific surface areas for the proppant spheres and fracture wall are

The proppant permeability in terms of the proppant diameter, porosity, slot width, and

2 3 22


*p p*

*d d*

<sup>2</sup> 1 72 (1 ) 3 1

= + ç ÷ - - è ø

 f ( )

f

*w*

*h*

*d*

sphericity is found by substituting Eq. A.26 through A.28 into A.24

f

l

*m*

*k*

f

$$\frac{k}{d\_p^2} = \Omega \Psi^\mu \tag{45}$$

where

$$
\Omega = \frac{\phi^3}{T 2\lambda\_m (1 - \phi)^2} \left( 1 + a\_D \right)^{-2} \tag{46}
$$

$$\Psi = \Phi^2 \left( \frac{1 + a\_D}{1 + \Phi a\_D} \right)^2 \tag{47}$$

$$a\_{\phantom{w}} = \frac{a\_w}{a\_s} = \frac{d\_p}{3\left(1-\phi\right)w} \tag{48}$$

Eq. A.32 illustrates that the dimensionless permeability ratio ( *k* / *dp* <sup>2</sup> ) is a linear function of the characteristic proppant pack parameter ( *Ω* ), and the proppant sphericity-specific surface area parameter ( *Ψ* ) for all proppants. The dimensionless specific area ratio of the fracture (slot wall) and proppant is represented by *aD* .

Pan et al. (2001) provides a good review of permeability versus porosity correlation for random sphere packing. Pan also proposed a modification to Ergun's equation for low Reynolds number with a four parameter fit model to correlate *k* / *dp* 2 as a function of porosity and sphere size distribution.

### **Acknowledgements**

The authors wish to thank the management of Baker Hughes Incorporated for the opportunity to perform this research and publish the findings. We would also like to thank Bill Brinzer and Jennifer Pusch for their grammatical review and proofing of the manuscript.

[9] Palisch, T, Duenckel, R, Bazan, L, Heidt, H, & Turk, G. Determining Realistic Frac‐ ture Conductivity and Understanding its Impact on Well Performance". SPE 106301,

Modeling of Proppant Permeability and Inertial Factor for Fluid Flow Through Packed Columns

http://dx.doi.org/10.5772/56299

569

[10] Palisch, T, Duenckel, R, Chapman, M, Woolfolk, S, & Vincent, M. C. How to Use and Misuse Proppant Crush Tests- Exposing the Top 10 Myths". SPE 119242, January

[11] Pan, C, Hilpert, M, & Miller, C. T. Pore-scale modeling of saturated permeabilities in

random sphere packings," Physical Review E, November (2001). , 066702

January (2007).

(2009).

#### **Author details**

Bruce R. Meyer1 , Lucas W. Bazan2 and Doug Walls3

\*Address all correspondence to: lucas.w.bazan@bazanconsulting.com

1 Meyer Technologies, LLC, Natrona Heights, PA, USA

2 Bazan Consulting, Inc., Houston, TX, USA

3 Baker Hughes Incorporated, Natrona Heights, PA, USA

#### **References**


[9] Palisch, T, Duenckel, R, Bazan, L, Heidt, H, & Turk, G. Determining Realistic Frac‐ ture Conductivity and Understanding its Impact on Well Performance". SPE 106301, January (2007).

**Acknowledgements**

568 Effective and Sustainable Hydraulic Fracturing

**Author details**

Bruce R. Meyer1

**References**

(1995).

York, (1960).

Ingenieuer, 45 edition, (1901).

ductivity of Proppants". ISO.org

The authors wish to thank the management of Baker Hughes Incorporated for the opportunity to perform this research and publish the findings. We would also like to thank Bill Brinzer and

Jennifer Pusch for their grammatical review and proofing of the manuscript.

\*Address all correspondence to: lucas.w.bazan@bazanconsulting.com

and Doug Walls3

[1] API, American Petroleum Institute,: "Recommended Practices for Testing Sand Used

[2] API, American Petroleum Institute,: "Recommended Practices for Testing Sand Used

[3] API, American Petroleum Institute,: "Recommended Practices for Testing High-Strength Proppants Used in Hydraulic Fracturing Operations", API RP 60, December

[4] Bird, R. B, Stewart, W. E, & Lightfoot, E. N. Transport Phenomena," Wiley, New

[5] Forchheimer, P. Wasserbewegung durch Boden. Zeitschrift des Vereines Deutscher"

[6] ISO, (2006). Amendment 13503-2: "Measurement of properties of Proppants used in

[7] ISO, (2006). Amendment 13503-5: "Procedures for Measuring the Long-Term Con‐

[8] Much, M. G, & Penny, G. S. Long-Term Performance of Proppants Under Simulated

in Hydraulic Fracturing Operations", API RP 56, March (1983).

in Gravel Packing Operations", API RP 58, December (1995).

Hydraulic Fracturing and Gravel-Packing Operations". ISO.org

Reservoir Conditions," SPE/DOE 16415, May (1987).

, Lucas W. Bazan2

2 Bazan Consulting, Inc., Houston, TX, USA

1 Meyer Technologies, LLC, Natrona Heights, PA, USA

3 Baker Hughes Incorporated, Natrona Heights, PA, USA


**Chapter 27**

**A New Approach to Hydraulic Stimulation of**

Nima Gholizadeh Doonechaly, Sheik S. Rahman and

Additional information is available at the end of the chapter

properties are considered to simulate rock deformation.

**Opening**

Andrei Kotousov

**Abstract**

http://dx.doi.org/10.5772/56447

**Geothermal Reservoirs by Roughness Induced Fracture**

Hydraulic fracturing by shear slippage mechanism (mode II) has been studied in both laboratory and field scales to enhance permeability of geothermal reservoirs by numerous authors and their success stories have been reported. Shear slippage takes place along the planes of pre-existing fractures which causes opening of the fracture planes by the fracture asperities (roughness induced opening). Simplified empirical relationships, which are derived based on simple fracture experiments or best guess, are used to calculate compressive normal surface traction, residual aperture and shear displacement. This introduces ambiguity into the

simulation results and often leads to erroneous predictions of reservoir performance.

In this study an innovative analytical approach based on the distributed dislocation technique is developed to simulate the roughness induced opening of fractures in the presence of compressive and shear stresses as well as fluid pressure inside the fracture. This provides fundamental basis for computation of aperture distribution for all parts of the fracture which can then be used in the next step of modeling fluid flow inside the fracture as a function of time. It also allows formulation of change in aperture due to thermal stresses. The stress distribution and the fluid pressure are calculated using the fluid flow modeling inside the fracture in a numerical framework in which thermo-hydro mechanical effects are also consid‐ ered using finite element methods (FEM). In this study, fractures with their characteristic

This new approach is applied to the Soultz-Sous-Forets geothermal reservoir to study changes in permeability and its impact on temperature drawdown. It has been shown that the analytical

and reproduction in any medium, provided the original work is properly cited.

© 2013 Doonechaly et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,
