Assessment of the validity of Stokes and Reynolds equations for fluid flow through a rough‐walled fracture with flow imaging

Abstract

Understanding fluid flow through a rough‐walled fracture is important in many problems such as petroleum and geothermal reservoir exploitation, geological storage of CO2, and sitting of radioactive waste repositories. In order to advance the understanding of fracture flow, we conducted the first direct measurement of flow velocity across rough‐walled fractures at Reynolds number (Re) of 0.014 to 0.086. The results were used for an order of magnitude analysis to evaluate assumptions underlying the Stokes and the Reynolds equations, which are derived from simplifying the Navier–Stokes equations. Even at very rough subregions, viscous forces were at least 2 orders of magnitude greater than inertial forces, indicating that the Stokes equations are valid for Re < 0.1. However, the assumption made in the derivation of the Reynolds equation that ∂2ux/∂z2 is dominant over other viscous terms was not satisfied even at moderate roughness for Re < 0.1. The Reynolds equation overestimated flow rate.

A Review of Critical Conditions for the Onset of Nonlinear Fluid Flow in Rock Fractures

“…When the flow rate/velocity is very small, which corresponds to a small Reynolds number (Re), the inertial term (the left term of (5)) can be negligible, resulting in the expression of the Stokes equations [83]:…”

Roughness decomposition and nonlinear fluid flow in a single rock fracture

“…The reason for overestimation (up to as high as 47-60% at highly rough-walled regions of the tested sample) of flow rate by solving the Reynolds equation was regarded as caused by the abruptly changed geometry during this test. It was also found in [14] that a minor roughness change in the fracture made the Reynolds equation overestimate the flow through the fractures even for Reo1. Unfortunately, no roughness analysis was reported in [14], so that a quantitative correlation analysis between roughness and nonlinear flow was not available.…”

“…However, the assumption of the Cubic Law and its local validity may be acceptable only when the flow is largely laminar, and might not be able to provide adequate support for investigating the impacts of surface roughness on energy and mass transport processes in natural fractures, as reported in literature with both laboratory experiments and numerical simulations (e.g. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]). …”

“…However, no results of roughness analysis or non-laminar flow regions (Eddy) from the numerical modeling were presented or analyzed. A recent integrated laboratory testing and numerical modeling by solving Stokes equation and Reynolds equations was reported in Lee et al [14], with detailed and quantitative observations, measurements and presentations of the flow field, including Eddy flow regions. The comparison between measured and calculated results indicated important impact of nonlinearity caused by sudden change of surface geometry, when Reo0.1.…”

Migração de solutos em basalto fraturado: quantificação experimental em laboratório e validação matemática

“…A abertura mecânica, be, da amostra natural de basalto pode ser considerada pequena em relação às aberturas (medidas e não estimadas) de outros estudos da literatura, os quais os quais frequentemente realizam experimentos com traçadores dentro de fraturas milimétricas (entre 0,2 e 5,0 mm) naturais (Huber et al, 2012;Wang e Cardenas, 2014) e artificiais (Stockman et al, 2001;Boutt et al, 2006;Cai et al, 2007;Schmittbuhl et al, 2008;Bauget e Fourar, 2008;Zheng et al, 2008;Lee et al, 2014).…”