**Abstract**

Standard industry testing procedures provide proppant quality control and methods to determine long term reference conductivity for proppants under laboratory conditions. However, test methods often lack repeatable results. Additionally, the testing procedures are not designed to account for fundamental parameters (e.g., proppant diameter, porosity, wall effects, multi-phase/non-Darcy effects, proppant and gel damage) that greatly reduce absolute proppant bed conductivity under realistic flowing conditions.

A constitutive model for permeability and inertial factor for flow through packed columns has been formulated from fundamental principles. This work provides a detailed deterministic proppant permeability correlation and defines a methodology to help explain why different proppant types behave differently under stress. The theory also characterizes the origin of inertial, or non-Darcy flow, based on a unique approach formulated from the extended Bernoulli equation based on minor losses. The physical model provides insight into the dominant parameters affecting the pressure drop in a proppant pack and improves our understanding of fluid flow and transport phenomena in porous media.

The fundamental solution for flow through packed columns can be characterized by the sum of viscous (Blake-Kozeny) and inertial forces (Burke-Plummer) in Ergun's equation. Coupling Ergun's equation with the Forchheimer equation results in a deterministic set of equations that describe the fracture permeability and inertial factor as functions of the proppant diameter, pack porosity, sphericity, and fracture width. Plotting the dimensionless permeability, (k/dp 2 ), versus the characteristic proppant porosity parameter, Ω, is a very useful diagnostic tool that can indicate: 1) sphericity, 2) channeling, 3) crushing, 4) non-uniform sphere size distri‐ bution, 5) embedment and 6) deviation of the friction multiplier *λ<sup>m</sup>* from Ergun's equation.

The dimensionless experimental proppant permeability data can be plotted as a linear function of dimensionless porosity with large deviations from these equations signifying poor or inconsistent experimental results or inadequate proppant characterization. The formulated permeability and non-Darcy equations provide the foundation for a quantitative (including quality control of the test) and qualitative analyses for determining fracture permeability and the inertial factor based on the physical properties of the proppant pack.

where the first term on the right hand side of this equation represents the viscous effects and the second term the inertial or minor loss effects. Multiphase fluid interaction (gascondensate, oil-water, etc.) causes pressure losses as multiple viscosities move through the proppant pack at different velocities (fluid mobility). The non-Darcy beta factor, *β* , is a material property of proppant that quantifies the inertial or minor losses as a result of fluid contraction and expansion. The greater the inertial losses, the greater the beta coefficient which increases the total pressure loss in the proppant pack. The effects of the beta coefficient can be reduced by increasing the porosity and permeability of the proppant pack, reducing the mesh distribution, and by using more spherical proppant with lower surface friction. Proppant crush tests are one method to determine some of these physi‐

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

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551

Standardized crush test procedures are outlined in API RP-56, RP-58 and RP-60 and are summarized in ISO 13503-2. The intent of these tests is to provide a comparison of the physical characteristics of various proppants including crush test results. Again, there are limitations of the testing methodology that do not simulate actual conditions within a producing fracture. However, the actual testing methods, specifically the loading of the cell, can be even more immediately problematic to results. Results from eleven different companies testing a common sample of 16/30 Brown Sand indicate varying test results between companies as high as 25%

This work provides a detailed deterministic proppant permeability correlation and presents a methodology to help explain why different proppant types behave differently under stress. The governing equations for flow through pack columns are formulated in Appendix A. Derivation of the theoretical fracture permeability and inertial coefficient, *β* , are also given in

This section summarizes the equations for viscous and inertial flow in packed columns and presents a correlation model for fracture permeability. The flow through packed columns may be characterized as the sum of frictional (viscous) and inertial (minor losses) forces. The

3 2 0 3

where from experimental data *λ<sup>m</sup>* =25 / 12 and *f* <sup>0</sup> =7 / 3 . Viscous forces dominate laminar flow regimes while kinetic forces dominate inertial flow. Ergun developed his famous equation for the total pressure loss in packed columns for all flow regimes by simply adding the Blake-Kozeny equation for viscous dissipation and the Burke-Plummer equation for inertial losses.

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

*p p*

2 2

 f f ru


**2. Pressure loss equations for flow through packed columns**

( )

 f mu

*m*

Placing Eq. 2 in terms of dimensionless groups we have (see Eq. A.19)

f

l

72 1 3 1

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

cal proppant parameters under in-situ conditions.

governing pressure loss equation from Eq. A.18

(Palisch *et al*., 2009).

Appendix A.

#### **1. Introduction**

Hydraulic fracturing has been the major and relatively inexpensive stimulation method used for enhanced oil and gas recovery in the petroleum industry since 1949. The primary goal of a hydraulic fracture treatment is to create a highly conductive flowpath for hydrocarbon production. Fracture conductivity is defined as the product of the packed bed width and permeability. An ideal fracture would possess infinite conductivity. However, producing proppant packs have finite permeability and conductivity. Proppant beds are also subjected to damage and conductivity degradation over time including proppant embedment, formation spalling, temperature degradation, non-Darcy flow, multiphase flow, non-uniform proppant distribution, cyclic stress, gel damage, fines migration, and other effects (Palisch *et al*., 2007).

TheAmericanPetroleumInstitute (API)developedconductivity testingprocedures outlinedin API RP-61 to provide a methodology for consistent and repeatable results. The testing condi‐ tions include using the Cooke Conductivity Cell with steel pistons loaded at 2 lb/ft2 at ambient temperature.Thestressmeasurementsaremaintainedfor15minuteswith2%KClfluidpumped at a rate of 2ml/min.Anindustry consortiumproposedchanges toAPIRP-61 to replace the steel pistons with Ohio Sandstone, increase the testing temperature to 150 oF or 250 oF and maintain the stress for 50 hours. The modified API RP-61 is referred to as "long-term" conductivity, is accepted as the standard testing procedure for proppant, and has been adopted by the Interna‐ tional Organization for Standardization (ISO) as ISO 13503-5. The original API RP-61 method is referred to as "short-term" conductivity testing. These testing procedures provide proppant conductivityunderlaminar(baselineorreference) conditionsbutfailtopredictrealistic fracture conductivity under flowing conditions because the tests do not account for the permeability reduction because of proppant pack damage mechanisms. There is tremendous superficial velocityinsideaproducinghydraulicfractureresultinginsignificantenergylossfromthekinetic and viscous energy losses and hydrocarbon inertial effects. The constitutive parameters determining the pressure losses are the rate of fluid flow, viscosity and density of the fluid, size, shape, packing orientation and surface of the proppant. In petroleum engineering for a single phase fluid, the energy loss is typically described by a form of the Forchheimer equation (Eq. A. 20) as a sum of the Darcy and non-Darcy pressure drops

$$-\frac{dp}{d\mathbf{x}} = \frac{\mu}{k}\boldsymbol{\nu} + \beta\rho\boldsymbol{\nu}\boldsymbol{\nu}\tag{1}$$

where the first term on the right hand side of this equation represents the viscous effects and the second term the inertial or minor loss effects. Multiphase fluid interaction (gascondensate, oil-water, etc.) causes pressure losses as multiple viscosities move through the proppant pack at different velocities (fluid mobility). The non-Darcy beta factor, *β* , is a material property of proppant that quantifies the inertial or minor losses as a result of fluid contraction and expansion. The greater the inertial losses, the greater the beta coefficient which increases the total pressure loss in the proppant pack. The effects of the beta coefficient can be reduced by increasing the porosity and permeability of the proppant pack, reducing the mesh distribution, and by using more spherical proppant with lower surface friction. Proppant crush tests are one method to determine some of these physi‐ cal proppant parameters under in-situ conditions.

The dimensionless experimental proppant permeability data can be plotted as a linear function of dimensionless porosity with large deviations from these equations signifying poor or inconsistent experimental results or inadequate proppant characterization. The formulated permeability and non-Darcy equations provide the foundation for a quantitative (including quality control of the test) and qualitative analyses for determining fracture permeability and

Hydraulic fracturing has been the major and relatively inexpensive stimulation method used for enhanced oil and gas recovery in the petroleum industry since 1949. The primary goal of a hydraulic fracture treatment is to create a highly conductive flowpath for hydrocarbon production. Fracture conductivity is defined as the product of the packed bed width and permeability. An ideal fracture would possess infinite conductivity. However, producing proppant packs have finite permeability and conductivity. Proppant beds are also subjected to damage and conductivity degradation over time including proppant embedment, formation spalling, temperature degradation, non-Darcy flow, multiphase flow, non-uniform proppant distribution, cyclic stress, gel damage, fines migration, and other effects (Palisch *et al*., 2007).

TheAmericanPetroleumInstitute (API)developedconductivity testingprocedures outlinedin API RP-61 to provide a methodology for consistent and repeatable results. The testing condi‐

temperature.Thestressmeasurementsaremaintainedfor15minuteswith2%KClfluidpumped at a rate of 2ml/min.Anindustry consortiumproposedchanges toAPIRP-61 to replace the steel pistons with Ohio Sandstone, increase the testing temperature to 150 oF or 250 oF and maintain the stress for 50 hours. The modified API RP-61 is referred to as "long-term" conductivity, is accepted as the standard testing procedure for proppant, and has been adopted by the Interna‐ tional Organization for Standardization (ISO) as ISO 13503-5. The original API RP-61 method is referred to as "short-term" conductivity testing. These testing procedures provide proppant conductivityunderlaminar(baselineorreference) conditionsbutfailtopredictrealistic fracture conductivity under flowing conditions because the tests do not account for the permeability reduction because of proppant pack damage mechanisms. There is tremendous superficial velocityinsideaproducinghydraulicfractureresultinginsignificantenergylossfromthekinetic and viscous energy losses and hydrocarbon inertial effects. The constitutive parameters determining the pressure losses are the rate of fluid flow, viscosity and density of the fluid, size, shape, packing orientation and surface of the proppant. In petroleum engineering for a single phase fluid, the energy loss is typically described by a form of the Forchheimer equation (Eq. A.

*dp* <sup>2</sup>

(1)

*dx k* m -= + u bru at ambient

tions include using the Cooke Conductivity Cell with steel pistons loaded at 2 lb/ft2

20) as a sum of the Darcy and non-Darcy pressure drops

the inertial factor based on the physical properties of the proppant pack.

**1. Introduction**

550 Effective and Sustainable Hydraulic Fracturing

Standardized crush test procedures are outlined in API RP-56, RP-58 and RP-60 and are summarized in ISO 13503-2. The intent of these tests is to provide a comparison of the physical characteristics of various proppants including crush test results. Again, there are limitations of the testing methodology that do not simulate actual conditions within a producing fracture. However, the actual testing methods, specifically the loading of the cell, can be even more immediately problematic to results. Results from eleven different companies testing a common sample of 16/30 Brown Sand indicate varying test results between companies as high as 25% (Palisch *et al*., 2009).

This work provides a detailed deterministic proppant permeability correlation and presents a methodology to help explain why different proppant types behave differently under stress. The governing equations for flow through pack columns are formulated in Appendix A. Derivation of the theoretical fracture permeability and inertial coefficient, *β* , are also given in Appendix A.

### **2. Pressure loss equations for flow through packed columns**

This section summarizes the equations for viscous and inertial flow in packed columns and presents a correlation model for fracture permeability. The flow through packed columns may be characterized as the sum of frictional (viscous) and inertial (minor losses) forces. The governing pressure loss equation from Eq. A.18

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

where from experimental data *λ<sup>m</sup>* =25 / 12 and *f* <sup>0</sup> =7 / 3 . Viscous forces dominate laminar flow regimes while kinetic forces dominate inertial flow. Ergun developed his famous equation for the total pressure loss in packed columns for all flow regimes by simply adding the Blake-Kozeny equation for viscous dissipation and the Burke-Plummer equation for inertial losses. Placing Eq. 2 in terms of dimensionless groups we have (see Eq. A.19)

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

**3. Proppant permeability formulation**

where

permeability ( *k* / *dp*

2

dimensionless permeability *k* / *dp*

The formulation of the proppant permeability (and inertial factor) is presented in Appendix A. It can be shown (see Eq. A.23 through Eq. A.35) that the dimensionless proppant permea‐

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

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

*a*

2

r

) should be a linear function of the characteristic proppant pack parameter

*D D*

*a a* æ ö + Y=F ç ÷

> 3 1( ) 3 *w p p*

Thus if the experimental proppant permeability data is fitted with Eq. 5, the dimensionless

( *Ω* ) with the slope represented by the proppant sphericity-specific surface area parameter

1 1 *aD* ( )

The above equation works well for determining the proppant sphericity provided that the friction multiplier is a constant for all bed packing (i.e., *λ<sup>m</sup>* =25 / 12 ), the proppant sphere size is uniform, and that the sphericity ( *Φ* ) is a constant. But in reality, *Φ* is generally a function of *Ω* , (i.e., *Φ* = *f* (*Ω*) ). Pan *et al.* (2001) proposed a four parameter model to correlate permea‐ bility with porosity and sphere size distribution for random sphere packing. However, plotting

can signify poor or inconsistent experimental results, inaccurate calculation/measurement of the mean proppant diameter (especially for slopes greater than unity), or proppant porosity

*s a*

f

*a C w*

*a d d*

= WY (5)

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553

è ø + F (7)

== = - (8)

+ -Y (9)

versus *Ω* is a very useful diagnostic tool. Large deviations


bility in terms of the proppant diameter, porosity, slot width, and sphericity is

2 *p k d*

*m*

l

( *Ψ* ). The proppant sphericity can then be found from the slope using Eq. 7

2

<sup>Y</sup> F =

*D*

*a*

f

 f

<sup>2</sup> 1 1

This is the Ergun equation (see Bird 1960) where Re=*ρυdp* / *μ* , *λ<sup>m</sup>* =25 / 12 and *f* <sup>0</sup> =7 / 3 have been substituted. To account for proppant sphericity, the particle diameter in the above equations can be replaced by ( *Φdp* ). Figure 1 shows the general behavior of the Ergun equation on a log-log plot with the Blake-Kozeny and Burke-Plummer equations for reference.

**Figure 1.** The Ergun equation with the Blake-Kozeny and Burke-Plummer equations.

Rearranging the Forchheimer equation (Eq. 1) into dimensionless groups (see Eq. A.21) we find

$$\left(-dp/d\mathbf{x}\right)\Big/\beta\rho\upsilon^2 = \mathbf{1}\Big/\mathrm{Re}\_{\beta\mathbf{k}} + \mathbf{1} \tag{4}$$

where Re*β<sup>k</sup>* =*ρυβk* / *μ* . Multiplying Eq. 4 by 7 / 4 , replacing Re*βk* with Re , and *β* and *k* in terms of the proppant diameter and porosity (see Eq. A.24) one can show that it is identical to Ergun's equation (Eq. 3).

#### **3. Proppant permeability formulation**

The formulation of the proppant permeability (and inertial factor) is presented in Appendix A. It can be shown (see Eq. A.23 through Eq. A.35) that the dimensionless proppant permea‐ bility in terms of the proppant diameter, porosity, slot width, and sphericity is

$$\frac{k}{d\_p^2} = \Omega \Psi^\mu \tag{5}$$

where

( ) <sup>3</sup>

1 Re 4

This is the Ergun equation (see Bird 1960) where Re=*ρυdp* / *μ* , *λ<sup>m</sup>* =25 / 12 and *f* <sup>0</sup> =7 / 3 have been substituted. To account for proppant sphericity, the particle diameter in the above equations can be replaced by ( *Φdp* ). Figure 1 shows the general behavior of the Ergun equation

150 1 7

f

(3)

2

ru

**Figure 1.** The Ergun equation with the Blake-Kozeny and Burke-Plummer equations.

find

equation (Eq. 3).

Rearranging the Forchheimer equation (Eq. 1) into dimensionless groups (see Eq. A.21) we

where Re*β<sup>k</sup>* =*ρυβk* / *μ* . Multiplying Eq. 4 by 7 / 4 , replacing Re*βk* with Re , and *β* and *k* in terms of the proppant diameter and porosity (see Eq. A.24) one can show that it is identical to Ergun's

b

(4)

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

 - =+ bru

f

f

on a log-log plot with the Blake-Kozeny and Burke-Plummer equations for reference.

æ öæ ö - - =+ ç ÷ç ÷ ç ÷ç ÷ - è øè ø

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

552 Effective and Sustainable Hydraulic Fracturing

$$
\Omega = \frac{\phi^3}{72\lambda\_m(1-\phi)^2} \left(1 + a\_D\right)^{-2} \tag{6}
$$

$$\Psi = \Phi^2 \left( \frac{1 + a\_D}{1 + \Phi a\_D} \right)^2 \tag{7}$$

$$a\_D = \frac{a\_w}{a\_s} = \frac{d\_p}{\Im\left(1-\phi\right)w} = \frac{d\_p\rho}{\Im\mathcal{C}\_a} \tag{8}$$

Thus if the experimental proppant permeability data is fitted with Eq. 5, the dimensionless permeability ( *k* / *dp* 2 ) should be a linear function of the characteristic proppant pack parameter ( *Ω* ) with the slope represented by the proppant sphericity-specific surface area parameter ( *Ψ* ). The proppant sphericity can then be found from the slope using Eq. 7

$$\Phi = \frac{\sqrt{\Psi}}{1 + a\_D \left(1 - \sqrt{\Psi}\right)}\tag{9}$$

The above equation works well for determining the proppant sphericity provided that the friction multiplier is a constant for all bed packing (i.e., *λ<sup>m</sup>* =25 / 12 ), the proppant sphere size is uniform, and that the sphericity ( *Φ* ) is a constant. But in reality, *Φ* is generally a function of *Ω* , (i.e., *Φ* = *f* (*Ω*) ). Pan *et al.* (2001) proposed a four parameter model to correlate permea‐ bility with porosity and sphere size distribution for random sphere packing. However, plotting dimensionless permeability *k* / *dp* 2 versus *Ω* is a very useful diagnostic tool. Large deviations can signify poor or inconsistent experimental results, inaccurate calculation/measurement of the mean proppant diameter (especially for slopes greater than unity), or proppant porosity (and width) measurement errors as a function of closure. A diagnostic plot of *k* / *dp* 2 versus *Ω* will provide insight into the topics discussed above and also provide a comparison of different proppants and their relative pack permeability as closure stress increases (i.e., low values of *Ω* ). The main emphasis of this paper is not to provide a detailed deterministic proppant permeability correlation but rather to provide a methodology to help explain and understand why different proppant types behave differently under stress.

Although Eq. 5 is a very good correlation for diagnostics, other forms of this equation (e.g., *k* / *dp* <sup>2</sup> <sup>=</sup>*a*<sup>0</sup> <sup>+</sup> *<sup>a</sup>*1*<sup>Ω</sup>* <sup>+</sup> *<sup>a</sup>*2*<sup>Ω</sup>* <sup>2</sup> or *k* / *dp* <sup>2</sup> =*aΩ <sup>α</sup>* ) also fit the data very well over limited ranges for some proppants. The other major advantage of correlating the permeability data with *Ω* is that *Ω* has the correct limits for mono-layers (i.e., as *<sup>ϕ</sup>* <sup>→</sup>1 , *<sup>Ω</sup>* <sup>→</sup>*<sup>w</sup>* <sup>2</sup> /(*dp* 2 12) ). Figures 2 and 3 illustrate images for a 20/40 Northern White Sand and a 20/40 Brown Sand, respectively. The Northern White has a sphericity of about 0.73 while the Brown Sand has a much lower sphericity of about 0.5. Sintered bauxite and resin coated sands have much higher sphericity of approxi‐ mately 0.90 and 0.80-0.85 respectively as illustrated in Figures 4 and 5.

**Figure 3.** Brown Sand, sphericity ~ 0.50 – Photo courtesy: U.S. Silica Company.

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

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555

**Figure 4.** Bauxite, sphericity ~ 0.90 – Photo courtesy: Oxane.

**Figure 2.** Northern White Sand, sphericity ~ 0.73 – Photo courtesy: Santrol.

Modeling of Proppant Permeability and Inertial Factor for Fluid Flow Through Packed Columns http://dx.doi.org/10.5772/56299 555

**Figure 3.** Brown Sand, sphericity ~ 0.50 – Photo courtesy: U.S. Silica Company.

(and width) measurement errors as a function of closure. A diagnostic plot of *k* / *dp*

why different proppant types behave differently under stress.

has the correct limits for mono-layers (i.e., as *<sup>ϕ</sup>* <sup>→</sup>1 , *<sup>Ω</sup>* <sup>→</sup>*<sup>w</sup>* <sup>2</sup> /(*dp*

**Figure 2.** Northern White Sand, sphericity ~ 0.73 – Photo courtesy: Santrol.

mately 0.90 and 0.80-0.85 respectively as illustrated in Figures 4 and 5.

or *k* / *dp*

*k* / *dp*

<sup>2</sup> <sup>=</sup>*a*<sup>0</sup> <sup>+</sup> *<sup>a</sup>*1*<sup>Ω</sup>* <sup>+</sup> *<sup>a</sup>*2*<sup>Ω</sup>* <sup>2</sup>

554 Effective and Sustainable Hydraulic Fracturing

will provide insight into the topics discussed above and also provide a comparison of different proppants and their relative pack permeability as closure stress increases (i.e., low values of *Ω* ). The main emphasis of this paper is not to provide a detailed deterministic proppant permeability correlation but rather to provide a methodology to help explain and understand

Although Eq. 5 is a very good correlation for diagnostics, other forms of this equation (e.g.,

proppants. The other major advantage of correlating the permeability data with *Ω* is that *Ω*

images for a 20/40 Northern White Sand and a 20/40 Brown Sand, respectively. The Northern White has a sphericity of about 0.73 while the Brown Sand has a much lower sphericity of about 0.5. Sintered bauxite and resin coated sands have much higher sphericity of approxi‐

<sup>2</sup> =*aΩ <sup>α</sup>* ) also fit the data very well over limited ranges for some

2

2

12) ). Figures 2 and 3 illustrate

versus *Ω*

**Figure 4.** Bauxite, sphericity ~ 0.90 – Photo courtesy: Oxane.

**Figure 6.** Correlation of Dimensionless Permeability for various types of 20/40 Proppants - Linear Plot.

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

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557

**Figure 7.** Correlation of Dimensionless Permeability for various types of 20/40 Proppants - Log-Log Plot.

**Figure 5.** Resin Coated Sand, sphericity ~ 0.80 – Photo courtesy: Santrol.

Figures 6 and 7 show a comparison of dimensionless permeability ( *k* / *dp* 2 ) versus the charac‐ teristic proppant pack parameter, *Ω* , for selected 20/40 Brown Sand (BS), 20/40 White Sand (WS), 20/40 resin coated sand, and 20/40 bauxite proppants at a concentration of *Ca* =2 lb*<sup>m</sup>* / *<sup>f</sup> <sup>t</sup>* <sup>2</sup> . As illustrated, these proppants generally follow the correlation of Eq. 5. However, the substantial permeability reduction as a result of the low sphericity is evident for the BS and to a lesser extent in the resin coated. The WS high permeability at about *Ω* =2*e* −05 is suspect (see Figure 7).

**Figure 6.** Correlation of Dimensionless Permeability for various types of 20/40 Proppants - Linear Plot.

**Figure 7.** Correlation of Dimensionless Permeability for various types of 20/40 Proppants - Log-Log Plot.

**Figure 5.** Resin Coated Sand, sphericity ~ 0.80 – Photo courtesy: Santrol.

*Ca* =2 lb*<sup>m</sup>* / *<sup>f</sup> <sup>t</sup>* <sup>2</sup>

is suspect (see Figure 7).

556 Effective and Sustainable Hydraulic Fracturing

Figures 6 and 7 show a comparison of dimensionless permeability ( *k* / *dp*

teristic proppant pack parameter, *Ω* , for selected 20/40 Brown Sand (BS), 20/40 White Sand (WS), 20/40 resin coated sand, and 20/40 bauxite proppants at a concentration of

However, the substantial permeability reduction as a result of the low sphericity is evident for the BS and to a lesser extent in the resin coated. The WS high permeability at about *Ω* =2*e* −05

. As illustrated, these proppants generally follow the correlation of Eq. 5.

2

) versus the charac‐
