Non-Darcy behavior is important for describing fluid flow in porous media in situations where high velocity occurs. A criterion to identify the beginning of non-Darcy flow is needed. Two types of criteria, the Reynolds number and the Forchheimer number, have been used in the past for identifying the beginning of non-Darcy flow. Because each of these criteria has different versions of definitions, consistent results cannot be achieved. Based on a review of previous work, the Forchheimer number is revised and recommended here as a criterion for identifying non-Darcy flow in porous media. Physically, this revised Forchheimer number has the advantage of clear meaning and wide applicability. It equals the ratio of pressure drop caused by liquid–solid interactions to that by viscous resistance. It is directly related to the non-Darcy effect. Forchheimer numbers are experimentally determined for nitrogen flow in Dakota sandstone, Indiana limestone and Berea sandstone at flowrates varying four orders of magnitude. These results indicate that superficial velocity in the rocks increases non-linearly with the Forchheimer number. The critical Forchheimer number for non-Darcy flow is expressed in terms of the critical non-Darcy effect. Considering a 10% non-Darcy effect, the critical Forchheimer number would be 0.11.

A Criterion for Non-Darcy Flow in Porous Media

In-plane and through-plane gas permeability of carbon fiber electrode backing layers

[1] The aim of this study is to better understand the mechanics of fracture development and propagation during hydraulic fracturing. This paper presents some development and applications of discrete particle modeling of this problem. A discontinuum modeling approach idealizes the material as separate particles bonded together at their contact points and utilizes the breakage of individual structural units or bonds to represent damage. The numerical models are correlated with existing hydrofracture laboratory experiments, which are presented in other publications. A simulation of a laboratory-scale hydrofracture experiment and the acoustic emission (AE) data from the experiment is used to validate the synthetic AEs produced in the hydrofracture model. This technique has been used to examine the mechanics of fracture initiation and time and spatial distributions of AE. The modeling results demonstrate that the mechanism of hydraulically induced fracture in the Lac du Bonnet (LdB) granite core sample is predominantly tensile failure and that the shear cracks recorded in the hydrofracture experiment were due to slip on preexisting fractures. Numerical modeling of hydrofracture on homogeneous and heterogeneous synthetic samples seems to capture much of the behavior observed in the laboratory hydrofracture experiments.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2004JB003297

Distinct element modeling of hydraulically fractured Lac du Bonnet granite

Evaluation of Forchheimer equation coefficients for non-Darcy flow in deformable rough-walled fractures

This paper experimentally investigates the role of shear processes on the variation of critical Reynolds number and nonlinear flow through rough-walled rock fractures. A quantitative criterion was developed to quantify the onset of nonlinear flow by comprehensive combination of Forchheimer's law and Reynolds number. At each shear displacement, several high-precision water flow tests were carried out with different hydraulic gradients then the critical Reynolds number was determined based on the developed criterion. The results show that (i) the Forchheimer's law was fitted very well to experimental results of nonlinear fluid flow through rough-walled fractures, (ii) the coefficients of viscous and inertial pressure drops experience 4 and 7 orders of magnitude reduction during shear displacement, respectively, and (iii) the critical Reynolds number varies from 0.001 to 25 and experiences 4 orders of magnitude enlargement by increasing shear displacement from 0 to 20 mm. These findings may prove useful in proper understanding of fluid flow through rock fractures, or inclusions in computational studies of large-scale nonlinear flow in fractured rocks.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/2013WR014610

Critical Reynolds number for nonlinear flow through rough‐walled fractures: The role of shear processes

Geomechanical Modeling of an Anisotropic Formation-Bakken Case Study

Overview of Geomechanical Properties of Bakken Formation In Williston Basin, North Dakota

It has been widely accepted in hydraulic fracturing that the higher the pressurization rate, the higher the breakdown pressure. However, linear elastic fracture mechanics (LEFM) suggests the opposite. In an effort to clarify this “injection rate paradox,” three controlled laboratory hydraulic fracturing experiments were conducted using different injection rates. These tests showed that the higher the injection rate, the lower the breakdown pressure. Further investigations indicated that the elastic- and poroelastic- models were not able to properly predict the observed breakdown pressures. Using Griffith’s energy balance concept, an LEFM-based breakdown pressure model was derived which gives satisfactory predictions.

Zhengwen Zeng

Papers

A Criterion for Non-Darcy Flow in Porous Media

Geomechanical Modeling of an Anisotropic Formation-Bakken Case Study

Overview of Geomechanical Properties of Bakken Formation In Williston Basin, North Dakota

Experimental Observation of Injection Rate Influence on the Hydraulic Fracturing Behavior of a Tight Gas Sandstone

AE b-Value Dynamic Features During Rockmass Fracturing and Their Significances.