GEOL 411 Final Report
Holly Burwinkle
December 14, 2007
Introduction
The majority of the work I did this semester consisted of library research, in which I
looked for methods of modeling turbulent flow in fractured media that could be applied to
the DFrx code. While conducting this research I found several papers which applied
various numerical methods for modeling such situations, most used the Forchheimer
equation or a power law function. The method presented by Louis seems to be the most
applicable to the DFrx code.
Research
In Louis’s paper,”A study of groundwater flow in jointed rock and its influence on the
stability of rock masses” he reports his findings from analytical and experimental tests for
different types of flow in jointed media. For his experimental work, Louis designed a
complex system to simulate flow in fractured rock. During the tests, water was pumped
through two slabs of concrete while changes in pressure and flow rate were measured.
To reproduce the effects of roughness in a fracture, the concrete slabs were given a
coarser texture. From these experiments Louis calculated values for the resistance
coefficient, λ, Reynolds number, Re, and relative roughness, k/Dh, where k is the
absolute roughness of the fracture and Dh is the hydraulic diameter of the fracture for
different types of flow.
First, Louis considers the case of parallel flow in a joint with constant width opening.
Although this situation does not occur often in nature, the results can be useful
nonetheless. According to Louis, flow can be divided into three flow domains
hydraulically smooth, transition zone, and completely rough, where the boundaries
between different flow domains depend on the relative roughness of the fracture. In the
case of hydraulically smooth flow λ is a function of the Reynolds number. For the
transition zone, λ is a function of Re and k/Dh, and for completely rough flow λ is a
function of k/Dh only.
Louis applies flow laws for flow in open pipes to characterize laminar and turbulent flows.
Resistance for laminar flows is dependent on the shape of the cross section of the
fracture and is given by 64/ Re. For turbulent flows, Louis adapted the results from
previous work by Schiller and Nikuradse for turbulent flows in different types of noncircular pipes by using the hydraulic diameter of the joint in the equations. From the
results, Louis assumes that the transition between laminar and turbulent flow occurs
when Re = 2300.
Louis reports that the results obtained for turbulent flows are more accurate than those
of laminar flows due to the set up of his experimental model. Furthermore, his
experiments show that the critical Reynolds number, where the transition between
laminar turbulent flow occurs, decreases as k/Dh increases. These results are different
from those reported by Lomize from his similar experiment. Below, results from the two
experiments are compared.
Louis
Laminar
Turbulent

Lomize
96 
 k 
1  8.8

Re 
 Dh 
1.5 
k

 2 log Dh 
 1.9 



1



96 
 k 
1  17

Re 
 Dh 
1.5 


k

 2.55 log Dh 
 1.24 



1
The compiled results from the experiments show that parallel flow occurs when
k/Dh<0.033 and can be divided into four regions hydraulically smooth, laminar flow,
hydraulically smooth, turbulent flow, transition zone, and completely rough flow. The
boundaries for these four flow domains are determined by the Reynolds number and the
relative roughness of the fracture. The transition between laminar and turbulent flow,
Re=2300, was taken from the Lomize results and non-parallel flow occurs when
k/Dh>0.033. Below is a graphical representation of the four flow regions and a table of
the flow laws applied in these results.
Applications
Using the flow laws presented Louis’s work, modifications can be made to the original
DFrx flux function to include turbulence. The DFrx flux function currently models
hydraulically smooth laminar flows seen in 1. This function can be adjusted to model
parallel, turbulent flows, 2,3 by , and similarly, the non-parallel flows, 4a-5b.
In order to incorporate these flow laws into the DFrx model there must be smooth
transitions from one flow domain to the next. This can be done using a sinusoidal
function or, perhaps, the smooth Heaviside function used in COMSOL.
Work in Progress
Currently, we are working on setting up the model to work for the simplest case, parallel
flows that transition from laminar to turbulent. The flux function is being modified to
include both types of flow. For this to work correctly, we must find a transition function
that will connect the two flow laws smoothly.
Once the DFrx model works for laminar and turbulent parallel flows, we will implement
the rest of the flow laws into the model and compare the results to data collected in the
field.