- Modeling of Proppant Permeability and Inertial Factor for Fluid Flow Through Packed Columns
- B.R. Meyer ; L.W. Bazan ; D. Walls
- Book Title / Journal: Effective and Sustainable Hydraulic Fracturing
- Year: 2013 , Series: Chapter 26
- Numerical & Constitutive Modeling ; Groundwater & Seepage
- Keywords: permeability ; fluid flow ; 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.
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.