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

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 λm from Ergun's equation. © 2013 Meyer 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. 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.

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.

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 2 dp dx k m u bru 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 physical proppant parameters under in-situ conditions. 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.

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 ( ) This is the Ergun equation (see Bird 1960) where Re = ρυd p / μ , λ m = 25 / 12 and f 0 = 7 / 3 have been substituted. To account for proppant sphericity, the particle diameter in the above equations can be replaced by ( Φd p ). 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.

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 Thus if the experimental proppant permeability data is fitted with Eq. 5, the dimensionless permeability ( k / d p 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 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 m = 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 permeability with porosity and sphere size distribution for random sphere packing. However, plotting dimensionless permeability k / d p 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 / d p 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 / d p 2 = a 0 + a 1 Ω + a 2 Ω 2 or k / d p 2 = aΩ α ) 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 ϕ → 1 , Ω → w 2 / (d p     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 = 2e − 05 is suspect (see Figure 7).  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 where f is the Darcy friction factor, τ w is the wall shear stress, υ is the superficial velocity ( υ = q / A ), and d h 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 where P f 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 where the cross-sectional average velocity ῡ is related to the superficial velocity υ by the conduit porosity (i.e., ῡ = υ / ϕ ).

Laminar flow in packed columns
The frictional pressure loss through a proppant pack (or packed bed) can be derived from Eq. by replacing the cross-sectional average velocity ῡ by the superficial velocity υ (i.e., ῡ = υ / ϕ ) and the equivalent hydraulic diameter of the proppant pack in terms of the particle diameter and porosity. where L is the length of the column and L τ is the tortuous path length the fluid takes. Further assume that the number of minor losses N ml in a column of length L can be approximated by 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 2 . Substituting the bed friction factor and superficial velocity into Eq. A.16, we find