Effects of the fracture wall-roughness on fluid flow is of immense importance in underground transport modelling. For this
computational fluid dynamics study, we developed 2 sets of three-dimensional meshes with different wall-roughness, which we
characterized using the Joint Roughness Coefficient and the fractal dimension of the fracture profiles. By determining the effective fluid
flow using Navier-Stokes numerical models we developed a relationship between the macroscopic roughness parameters and flow
parameters like transmissivity and tortuosity. It was observed considering rough-walled fractures reduces transmissivity 35 times smaller
than the smooth-walled and exhibits tortuous flow paths.
Keywords: Fracture roughness, Navier-Stokes, Rock Fractures, Macroscopic roughness Parameters
1. Introduction
“A practical method of describing flow in natural rock fractures is needed for modeling various subsurface activities,
including geothermal heat extraction, contaminant transport in aquifers, and geologic sequestration of CO2 “1,2. The
number and interconnectivity of fractures that may be present in a subsurface region of interest require that modeled flows
are approximated by simple mathematical relationships. “The behavior of fluids moving through individual fractures has
been shown to be complex; small-scale wall-roughness, and the resultant variable fracture apertures have been shown to
induce tortuous flow paths within fractures” 3–8. Defining quantitative relationship between wall roughness and its effect
on fluid flow through fractures has been undefinable. The goal of this study is obtaining such relationship by using finitevolume
Computational Fluid Dynamics (CFD) models. “Methods of describing fracture wall-roughness have been
identified by previous researchers “5, 9. For defining fracture geometries we determined two fairly common roughness
parameters, Joint roughness Coefficient (JRC) and the fractal dimension (Df) of fracture profiles. “The effectiveness of
using the Df to describe the roughness of rock fracture profiles has been demonstrated for fractures in concrete 5, 10 and
natural rocks” 11–13. In addition, a number of simulated rock fractures have been created from fractal surfaces for
numerical studies 6, 14–16. The Hurst exponent (H) is the fractal parameter often reported in the literature and is related
to the Df of 2D fracture profiles by Df=2–H. The H of a profile can be defined by using the variable-bandwidth technique
5 where the difference in height between two points distance ‘s’ apart is measured and the standard deviation (
? s
) of this
displacement is calculated as follows:
2
1
1
( ) M s
s i s i i
y y
M s
?
?
? ?
? ?
?
?
(1)
Where M is the number of data points and yi is the fracture profile height at a distance along the fracture, xi. This process is
repeated for a range of s. “If the fracture profile is well described by fractional Brownian motion, a non-Fickian diffusive
scaling will exist between
? s
and H” 5:
XXX-2
H
s ? ? s (2)
Therefore, H can be determined from the slope of a log–log plot of
? s
against s. This variable-bandwidth method has been
shown 5 to give more accurate results than spectral analysis, and to give a similar Df value to box-counting. The JRC is an
index from 0 to 20 that was initially proposed by Barton and Choubey 9 as a method of describing the shear strength of
rock fractures. Tse and Cruden 17 established an accurate empirical relationship between the root-mean-square of the
fracture profile wall slope (Z2) and the JRC. A number of detailed measurements of the fracture surface height, y, some
small distance apart, ?x, can be used to find Z2 of a fracture profile
2
2
1
1
2 ( )
( )
M
i s i
i
Z y y M x ?
?
? ?
? ?
(3)
This was determined by Tse and Cruden 17 to fit the empirical relationship
32.2 32.47log 2 JRC Z ? ? (4)
“This fit has a high degree of correlation to the original JRC descriptions given by Barton and Choubey” 9, but is least
accurate for JRC less than 8 (i.e. smooth fractures).
Two distinctive features of fluid flow in fractures are: (1) the fluid velocities are small and (2) fluid passes through narrow
openings. For characterizing the speed of fluid motion Reynolds number (Re) is often used. In this research we used Re
formulation described by Konzuk and Kueper 18,
Re D Qh
A
?
?
? (5)
Where Q is volumetric flow rate through the fracture,
? is the fluid density,
?
is the absolute fluid viscosity, A is the
cross-sectional area of the fracture perpendicular to the mean flow, Dh is the mean hydraulic diameter of the fracture (
4
h
A
D
P
?
), and p is the perimeter of A. We are primarily interested in the flow regime where Re<1 (creeping flow), as this can be applied to most geological flows 2. Fracture aperture is defined by measuring the vertical distance between the fracture walls. "When modeling flow through fractures on the reservoir-scale, the fluid transport through fracturenetworks is often of interest "19. "This fluid motion has been described by the solution to the Navier–Stokes equations through wide, closely spaced parallel plates, moving due to an imposed pressure head, ? P" 20. The flow through such a fracture can be described as 3 ( ) 12 Wb Q p ? ? ? ? (6) Which is known as the cubic-law. "This relationship is useful in predicting fluid transport through highly fractured reservoirs, but does not account for the small scale roughness of the fracture" 23. Numerous methods of incorporating roughness into cubic-law simulations have been suggested 16, 18, 22, but currently none of these procedures are universally accepted. In our research we tried to implicitly model the roughness effect on fluid flow by defining transmisiivity of the fracture (T) determined from Darcy's law 22, p Q T ? ? ? ? (7) XXX-3 The parameter T can be interpreted as macroscopically flow resistance in a fracture. In this research we tried to solve the full Navier-Stokes equations using finite volume CFD solver instead of simplifying the fluid flow equations. Equations the conservation of mass and the conservation of momentum need to be included: ? ? . 0 u (8) 2 ? ? ( . ) u u P u F ? ? ?? ? ? ? (9) Where u is the fluid velocity and F is a body force. "3D Navier–Stokes simulations in modeled fracture geometries have been used to examine the relationship between the aperture distribution and the fracture permeability" 16. For our study we used two different models, a case with no roughness and one with a geometry with rough walls. With these two cases we generate meshes and analyzed flow properties. The roughness of the fracture geometries was determined using the JRC and the Df calculations described in the literature. We solved the Navier–Stokes equations of fluid motion within these unstructured tetrahedral meshes under steady-state, low Re conditions. 2. Fracture mesh properties The fracture geometries converted into 3D meshes, appropriate for solving the Navier-Stocks equations of fluid motion. We used GAMBITTM (Ansys Inc., Canonsburg, PA) for meshing. The resulting meshes were used in the finite-volume CFD solver, FLUENTTM (Ansys Inc., Canonsburg, PA), to obtain the flow solution. The fracture meshes are all 5mm long with fracture widths of 5mm. For each fracture geometry the computational mesh was refined to capture the details of flow. Refinement was performed within FLUENT until no variation in the flow rate and the average fluid velocity was observed. A rigorous interpolation of open fracture was done in AutoCAD but meshing such geometry costs system with lots of RAM, consequently we tried to create and mesh the model in GAMB