## 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/d_{p}
^{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 distribution, 5) embedment and 6) deviation of the friction multiplier

## 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).

The American Petroleum Institute (API) developed conductivity testing procedures outlined in API RP-61 to provide a methodology for consistent and repeatable results. The testing conditions include using the Cooke Conductivity Cell with steel pistons loaded at 2 lb/ft^{2} at ambient temperature. The stress measurements are maintained for 15 minutes with 2% KCl fluid pumped at a rate of 2 ml/min. An industry consortium proposed changes to API RP-61 to replace the steel pistons with Ohio Sandstone, increase the testing temperature to 150^{ o}F or 250^{ o}F 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 International 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 conductivity under laminar (baseline or reference) conditions but fail to predict realistic 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 velocity inside a producing hydraulic fracture resulting in significant energy loss from the kinetic 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

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 (gas-condensate, oil-water, etc.) causes pressure losses as multiple viscosities move through the proppant pack at different velocities (fluid mobility). The non-Darcy beta factor,

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,

## 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

where from experimental data

This is the Ergun equation (see Bird 1960) where

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

where

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

where

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

The above equation works well for determining the proppant sphericity provided that the friction multiplier is a constant for all bed packing (i.e.,
*et al.* (2001) proposed a four parameter model to correlate permeability with porosity and sphere size distribution for random sphere packing. However, plotting dimensionless permeability

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

Figures 6 and 7 show a comparison of dimensionless permeability (

## 4. Conclusion

The fundamental solution for flow through packed columns (proppant packs) can be characterized 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,

## Nomenclature

### Greek

### 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 (

**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

where

**Hydraulic diameter**

The hydraulic diameter is defined as

where

**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

where the average flow rate in the cross section available for flow is given by the intrinsic velocity

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

where the Darcy friction factor is given by

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

where the cross-sectional average velocity

**Laminar flow in packed columns**

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

by replacing the cross-sectional average velocity

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

Substituting the hydraulic diameter and the relationship

Experimental measurements (Bird *et al.* 1960) indicate that if a frictional multiplier of

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

where

where

**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

where the experimental data indicated that

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

or

where

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

or

where from experimental data

**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

This is the Ergun equation (Bird 1960) where

**Darcy and non-darcy flow**

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

where

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

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

Rewriting Eq. A.17 in terms of the fracture permeability

where

Placing

**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

where

**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 (

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

The permeability for flow through an open slot or channel (i.e., no proppant as

where

The dimensionless form of Eq. A.29 is

or

where

Eq. A.32 illustrates that the dimensionless permeability ratio (

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