Journal of Water Resource and Protection, 2012, 4, 657-662
http://dx.doi.org/10.4236/jwarp.2012.48076 Published Online August 2012 (http://www.SciRP.org/journal/jwarp)
Analytical and Numerical Modeling of Flow in a
Fractured Gneiss Aquifer
Ramadan Abdelaziz, Broder J. Merkel
Department for Geology, Technische Universität Bergakademie Freiberg, Freiberg, Germany
Received May 16, 2012; revised June 22, 2012; accepted July 2, 2012
ABSTRACT
Investigating and modeling fluid flow in fractured aquifers is a challenge. This study presents the results of a series of
packer tests conducted in a fractured aquifer in Freiberg, Germany, where gneiss is the dominant rock type. Two meth-
ods were applied to acquire hydraulic properties from the packer tests: analytical and numerical modeling. MLU
(Multi-Layer Unsteady state) for Windows is the analytical model that was applied. ANSYS-FLOTRAN was used to
build a two-dimensional numerical model of the geometry of the layered aquifer. A reasonable match between experi-
mental data and simulated data was achieved with the 2D numerical model while the solution from the analytical model
revealed significant deviations with respect to direction.
Keywords: Ansys/Flotran; MLU for Windows; Gneiss; Packer Test; Fractured Aquifer
1. Introduction
Fractured aquifers are very important for groundwater
supply because about 75% of the earth’s surface consists
of fractured aquifers [1] and 25% of the global popula-
tion is supplied by karst waters [2]. Flow velocity in
fractured gneiss is known to be highly variable over a
range of scales and uncertainties which arises from het-
erogeneous flow pattern in fissures and fractures. This
has significant implications on water resource manage-
ment from borehole to catchment scales. In addition,
understanding flow heterogeneity in the aquifer is of
great importance for groundwater protection and for pre-
dicting contaminant transport.
Theis [3] was the first scientist to conduct a transient
analysis of the groundwater flow. After Theis, many re-
searchers like Warren and Root [4], Kazemi [5], Odeh
[6], Hantusch and Thomas [7], and Streltsova [8] studied
the flow through fractured rocks in the context of petro-
leum and groundwater engineering. Jenkins and Prentice
[9] described groundwater flow in a single fracture with a
very large permeability. Sen [10] used an analytical solu-
tion to analyze fractured gneiss with a linear flow pattern.
Cohen [11] used a two-dimensional numerical model to
analyze an open-well test in fractured crystalline rock.
Gernand and Heidtman [12] used the analytical model by
Jenkins and Prentice to analyze a pumping test in a frac-
tured gneiss aquifer. Schweisinger et al. [13] analyzed
transient changes in a fracture aperture during hydraulic
well tests in fractured gneiss. Wang and Cui [14] ana-
lyzed fluid flow and heat transfer by using the distributed
resistance application in ANSYS FLOTRAN. Their analy-
sis was done without comparing the modeled results with
those from experiments (Gu et al. [15] and Cen and Chi
[16]). Slack [17] proposed a theoretical analysis for the
slug test which couples elastic deformation with fluid
flow within a fracture. Molina-Aiz et al. [18] used ANYS
FLOTRAN to simulate the velocity and temperature in a
ventilated greenhouse. Crandall et al. [19] used ANSYS
FLUENT to obtain the flow solution in a fractured aqui-
fer.
Several analytical solutions are implemented in soft-
ware packages like AQTESOLV, Aquifer Win32, Aqui-
ferTest Pro, StepMaster, and MODPUMP to determine
the hydraulic parameters of aquifers. Some of these
packages offer analytical solutions for fractured aquifers.
However, these software packages have certain limita-
tions due to the more or less arbitrary selection of ana-
lytical solutions that are implemented. MLU for Win-
dows [20] is based on a completely different concept: It
is a multi-layer analytical model for confined and uncon-
fined aquifers and can thus be used for any kind of
groundwater testing scenario.
Several numerical models have been developed to si-
mulate the flow and transport in fractured aquifers. Ex-
amples are GeoSys/Rockflow and TOUGH2. Walsh et al.
[21] modeled flow and mechanical deformation in frac-
tured rock using Rockflow/GeoSys. McDermott et al.
[22], Myrttinen et al. [23] and others used the numerical
simulator GeoSys/Rockflow to simulate the flow and
C
opyright © 2012 SciRes. JWARP

R. ABDELAZIZ, B. J. MERKEL 658
transport in fractured rocks. Also, Pruess et al. [24],
Pruess and García [25], and others present results for
multiphase flow in porous and fractured aquifers using
TOUGH2. A detailed review on characterizing flow and
transport in fractured geological media is presented by
Berkowitz [26]. However, models such as Rockflow,
Rockflow/Geosys, and TOUGH2 do not apply the well-
known Navier-Stokes equation. They use the fact that the
Navier-Stokes equation can be linearized as long as the
Reynolds number is less than 10 and thus can be replaced
by the much simpler Reynolds lubrication equation.
However, there are some doubts that the local cubic law
is valid in some cases and only a few publications used
models based on Navier-Stokes Equation.
The present study was motivated by the need of im-
proving conceptualization of fractured gneiss through a
combination of fieldwork and modeling. Studies of flow
properties using packer techniques and geophysical read-
ings were used to investigate groundwater flow in frac-
tured gneiss. The major task of the paper was to model
flow in fractures by means of Navier-Stokes equation on
the one hand, and to assume fractured zones as a contin-
uum and thus to apply Darcy’s Law on the other hand.
MLU for Windows was chosen as analytical model be-
cause it is an integrated tool to evaluate pumping tests for
multi-layer confined aquifers. Finally, the analytical so-
lution was compared to the numerical solution with re-
spect to evaluating and estimating permeability of frac-
tured aquifers.
2. Site Description
Gneiss is the dominant crystalline rock at the test site in
Freiberg. All six wells at the test site of the TU Berga-
kademie Freiberg with a total depth of 50 m and 100 m,
respectively, are lacking any kind of casing or screen
(except for a upper protection casing (at 3 to 5 m)) and
have diameters of 4 to 6 inch. The gneiss at the site is a
medium- to coarse-grained Inner Gneiss. Major fractures
were identified by geophysical borehole logging in six
boreholes at depths of 11 to 11.5 m and minor fractures
at 14 to 16, 22 to 23.6, 31.3 to 31.9, 37.5 to 38 and 47 to
47.6 m [27]. Caliper, Single-Point Resistance (SPR),
High Resolution Detector (HRD), Gamma-gamma sound-
ings, and neutron-neutron soundings were used to iden-
tify fracture zones [27]. Consequently, higher values of
hydraulic conductivity are associated with the horizontal
fracture zones.
The six wells serve as direct, vertical connection be-
tween the zones of higher permeability. Thus, the pres-
ence of wells intersecting with the six zones of higher
hydraulic conductivity can significantly perturb fluid
flow. The flow in the fractured gneiss aquifer is assumed
to be extremely heterogeneous with high flow velocities
in the fracture zones and very low velocities in the block
matrix. Non-fractured gneiss itself is nearly imperme-
able.
3. Methods
Hydraulic properties of rock materials can be estimated
in both: laboratory and field. However, hydraulic proper-
ties obtained in the laboratory are not a true representa-
tion of the aquifer. Therefore, especially in fractured
rocks, packer tests and tracer tests are indispensable.
Packer tests are a well-known method to determine aqui-
fer properties in open boreholes [28-30]. They can be
used in uncased boreholes to determine the hydraulic
conductivity of the individual horizon by isolating a zone
between two packers or isolating a certain part of the
borehole with a single packer. The equipment needed for
packer tests includes an air compressor, a submersible
pump, inflatable packers, and pressure transducer probes
(Diver, type CTD, Schlumberger).
Two general methods of hydraulic testing have been
used: Double packer tests, and single packer tests. Single
packer test provide hydraulic data either for the borehole
above or below the packer. The double packer test was
performed for the most dominant fault in the range 11 to
12 m. The pumping rate was kept constant at about 10
L/min until a steady drawdown was achieved. For the
test, the submersible pump is mounted between the two
packers and the resulting drawdown. The duration of the
pumping test was 6 to 7 hours.
For the single packer test the packer was mounted 13
m below ground surface. The pumping rate in the single
packer was held constant at about 16 L/min and was
maintained for 6 hours until a steady state drawdown was
achieved. In both cases recovery was monitored, too.
Two approaches are addressed in this paper to calcu-
late and evaluate the hydraulic properties for a fractured
gneiss aquifer:
 An analytical solution using MLU for Windows (ba-
sed on Darcy’s Law)
 A numerical solution using ANSYS-FLOTRAN (ba-
sed on Navier-Stokes Equation)
3.1. Analytical Solution
MLU for Windows is a tool for single- and multi-layer
aquifers (both, confined and unconfined), which com-
bines Stehfest’s method, the superposition principle, and
the Levenberg-Marquardt algorithm. Stefest’s method is
performed in the numerical solution to convert the
Laplace domain to the real domain. Parameter estimation
is performed by applying the Levenberg-Marquardt algo-
rithm [20]. MLU assumes a homogenous, isotropic, and
uniform aquifer.
A one layer model was used to calculate the hydraulic

R. ABDELAZIZ, B. J. MERKEL 659
conductivity for the first horizontal fault zone. There, the
fractured layer is embedded in impermeable gneiss. The
numerical values for the hydraulic conductivity are pre-
sented in Table 1. The values for the permeability were
obtained from forward modeling using the Theis method
assuming a thickness of 0.5 m and evaluating each single
observation well individually.
Conductivity increases in direction of well 2 in com-
parison to wells 3, and 4 (for spatial distribution see Fig-
ure 1).
A two layer model was used to calculate the hydraulic
conductivity from the single packer test for the fracture
zones below the packer. Only two fracture zones were
modeled because they are assumed to be the most sig-
nificant ones regarding depth and permeability. The val-
ues determined for the hydraulic conductivity of the dif-
ferent monitoring wells are presented in Table 2.
The hydraulic conductivity of well 2 and well 4 is
higher than that of well 3 and well 6. Concerning techni-
Table 1. Hydraulic conduc tivity (double packer test) for the
upper fracture zone.
Well No. Hydraulic Conductivity (m/sec)
Well 2 5.83 × 10–4
Well 3 1.17 × 10–4
Well 4 1.17 × 10–4
Well 6 1.25 × 10–4
Table 2. Hydraulic conduc tivity (single packer) for the frac-
ture zones below the packe r at a depth of 14 m.
Well No. Hydraulic Conductivity (m/s)
Well 2 1.25 × 10–4
Well 3 8.33 × 10–5
Well 4 1.25 × 10–4
Well 6 2.5 × 10–5
0 5 10 15 20 25 30
15
10
5
0
well2
well3
well6
well4
well5
Figure 1. Drawdown at the Freiberg test site at the end of
the double packer test.
cal issues, it was difficult to measure the drawdown in
the pumping well (well 5). So, no parameter was calcu-
lated for it.
3.2. Numerical Model
The ANSYS/FLOTRAN CFD (Computational Fluid Dy-
namics) software package offers comprehensive tools for
analyzing two-dimensional and three-dimensional fluid
flow fields. FLOTRAN is a finite element analysis pro-
gram for solving fluid flow and conjugate heat transfer
problems. The governing equations solved by FLOT-
RAN are the Navier-Stokes equations combined with the
continuity equation, and the thermal transport equation.
The general purpose of the CFD modul of ANSYS FEA
systems is to solve a large variety of fluid flow problems.
ANSYS/FLOTRAN simulates laminar and turbulent com-
pressible and incompressible flows, single or multiple
fluids, and thermal/fluid coupling [31]. FLUID141 and
FLUID142 are two element models in ANSYS/FLOT-
RAN [32,33]. In this paper, FLUID141 was utilized for
the 2D model.
In FLOTRAN CFD elements, the velocities are ob-
tained from the conservation of momentum principle,
and the pressure is obtained from the conservation of
mass principle [34]. The matrix system derived from the
finite element discretization of the governing equation is
solved separately for each degree of freedom. The flow
problem is nonlinear and the governing equations are
coupled. The number of global iterations requires achiev-
ing a converged solution that may vary considerably,
depending on the size and stability of the problem. The
degrees of freedom are velocity, pressure, and tempera-
ture [35].
Figure 2 shows the geometry, node locations, and the
coordinate system for a typical quadrilateral and triangu-
lar element. The element is defined by three nodes (tri-
angle) or four nodes (quadrilateral) and by isotropic ma-
terial properties.
The fluid properties density and viscosity were speci-
fied for the element and were then meshed automatically
in ANSYS. The smaller the size of elements is defined,
the more accurate is the result that can be achieved by the
model. However, very fine element sizes result in high
CPU time. In the current study, models with different
meshing sizes were applied and evaluated.
ANSYS-FLOTRAN can account for fractures influ-
enced by the concept of distributed resistanceand was
applied in the element 141 through the use of real con-
stants. The distributed resistance refers to the macro-
scopic representation of geometric features that are not
directly concerned with the region of interest. This con-
cept is a convenient way to approximate the effect of
porous media (such as a filter) or other flow domain fea-
tures without actually modeling the geometry of those

R. ABDELAZIZ, B. J. MERKEL 660
K
L
J I J
K,L
I
Y
X
1
2
3
4
Figure 2. FLUID141 2D fluid-FLOTRAN element (modified
after ANSYS, 2009).
features. Also, it can be used to simulate the flow in a
fractured aquifer. A distributed resistance is an artifi-
cially imposed, unrecoverable loss associated with the
geometry that is not explicitly modeled. Any fluid ele-
ment with a distributed resistance will have a real con-
stant set number greater than 1 [36].
The flow resistance, modeled as a distributed resis-
tance, is caused by one or a combination of these factors:
localized head loss (k), friction factor (f), and permeabil-
ity (C). The equation for the total pressure gradient in x
direction is shown below (1) [31]. It is a sum of the fac-
tors mentioned before.
x
x
f
uuCu   

h
P
kuu
xD

 
 (1)
where:
ρ = is the density (mass/length³)
μ = is the viscosity (mass/(length*time))
f = is a friction coefficient (dimension-less; calculated
by the program): f = aRE-b
RE = is the local value of the Reynolds Number
(calculated by the program): RE uDh

 
a, b = are the coefficient and exponent of Reynolds
number, respectively, used in fraction factor calculation
C = is the FLOTRAN permeability (2
length).
FLOTRAN permeability is the inverse of the intrinsic
or physical permeability.
The unit of the distributed resistance is 2 length.
The permeability of the cells in fracture zone 1 was
found to be in the range of 0.00003 to 3e−8 m/s. This
value has to be converted into a value that can be put into
ANSYS. The flow rate of water through fractured gneiss
is proportional to the hydrostatic pressure difference (δP).
The hydrostatic pressure is normally expressed as a
pressure potential

hpg  , where

is the liquid
density 3
ML, g is geravical accelarion 2 LT , h has
the dimension L and is equivalent to the hydrostatic head.
The calculation was carried out using ANSYS V.12.1.
The permeability value depends both on the material
and the fluid. The permeability for Newtonian liquids
during laminar flow through inert non-swelling media is
inversely proportional to the fluid viscosity η. Therefore,
the intrinsic permeability for the material is defined as
k′ = k·η. In this equation k′ is a material property inde-
pendent from the fluid and with a dimension of L2 [36].
The results from numerical modeling using ANSYS-
FLOTRAN were calibrated with data from the packer
test. The horizontal hydraulic conductivity was assumed
to be anisotropic within a model cell. Heterogeneity was
simulated by varying the horizontal hydraulic conduc-
tiveity between individual model cells or layers. The ver-
tical hydraulic conductivity is based on specified values
of the horizontal hydraulic conductivity.
Figure 1 illustrates high drawdown in observation
well 6. Moreover, the shape of the drawdown and the
high hydraulic gradients between wells 5 and 6, imply
that there is a fault between them. Pumping generates a
cone of depression in the hydraulic potential field that
both expands outward and deepens with time. Drawdown
values were obtained from packer tests using double
packers (Figure 1). A higher drawdown was observed at
monitoring well 6 with about 2.7 m and a lower one at
monitoring well 2 and well 3 with about 30 cm. Inverse
distance weight interpolation was used to generate the
contour lines. Drawdown at the pumping well (No. 5)
could not be monitored, because a double packer system
with the submersible pump between both packers was
used. Thus the drawdown shown in Figure 1 was esti-
mated to be 5 m.
For the double packer test the first fracture zones were
modeled as a confined aquifer with constant head
boundary zones on both sides and no recharge. The mesh
size used was 0.04 m around the pumping well and 0.1 m
close to the margin of the model. Due to the geometry
the mesh used comprised over 40,000 elements. The pre-
ssure equation was solved using a pre-conditioned con-
jugate gradient method for the incompressible flow. Pre-
conditioned Conjugate Gradient (PCG) is the most robust
iterative solver in ANSYS. The exact method is the
semi-direct conjugate direction method that iterates until
a specified convergence criterion is reached.
4. Results and Discussion
The uppermost fractured zone located at between 11 and
12 m depth has an average thickness of 0.5 m. The simu-
lation period of 402 minutes was chosen according to the
time of the packer test. The differences between the
measured and computed values are mainly due to the
strong dependence of the coefficients on hydraulic con-
ductivity which is not constant in the aquifer but highly
heterogeneous. The horizontal hydraulic conductivity is
assumed to be isotropic within a single model cell (Fig-
ures 3 and 4).
The simulation period was 6 hours. Concerning bound-
ary conditions it was assumed that there is no drawdown
at the left and right margin. The pressure is continuous
across the fracture from block to block. Five fracture
layers were included in this model. Moreover, hydraulic

R. ABDELAZIZ, B. J. MERKEL 661
Figure 3. Map with arbitrary numbers in meter of observed
drawdown during double packer test versus drawdown
modeled with ANSYS-FLOTRAN.
Figure 4. Map with arbitrary numbers in meter of observed
drawdown during single packer test versus drawdown
modeled with ANSYS-FLOTRAN.
conductivity was determined in the fracture zones be-
tween 3 × 10–6 to 9 × 10–6 m/s. In fact, the magnitude of
the response of groundwater levels in the gneiss aquifer
spatially varied indicating heterogeneity in the fractured
gneiss. Similar to the first model, a simplified 2D-
groundwater flow model was built to simulate the single
packer test in the fractured gneiss.
Figure 4 depicts the difference between the observed
and the calculated head for the single packer test. Good
matching was observed at well 3 and well 4. In contrast,
only sufficient matching can be seen at well 2 and well 6.
Calibration was performed by trial and error. The hy-
draulic conductivity of the fracture layer was between 3
× 10–5 to 3 × 10–8 m/s and 10–15 m/s or less for the non-
fractured gneiss. The hydraulic conductivity of the two-
dimensional model decreases with increasing depth. As
stated before, having more and better information about
the fracture’s geometry, roughness and the network
would give better matching between the observed and
simulated heads.
5. Conclusion
Navier-Stokes equation is essential to predict the ground-
water flow in the fractured gneiss aquifer at this scale
and when turbulent flow is likely to occur. The two
methods used to identify the permeability of the aquifer
were analytical and numerical modeling. They are rea-
sonable but care should be taken with respect to the in-
terpretation of the results because of the uncertainties in
the characterization of aquifer properties. In general it
can be concluded that it is recommendable to carry out
tracer tests in order to increase the accuracy of the model.
At the investigated test site, hydraulic conductivity de-
creases from the right side to the left side due to the de-
crease in fracture thickness. The hydraulic conductivity
decreases with increased distance from the ground sur-
face (depth) because the number and width of fracture
openings decrease with depth. Generally, the hydraulic
conductivity values of the analytical solution were higher
than the values obtained using the numerical approach.
Finally, both the analytical and the numerical model
proved to be useful tools for improving the knowledge of
the fractured gneiss aquifer and for identifying the vari-
ous flow components. ANSYS-FLOTRAN will give a
better prediction of aquifer response than MLU for Win-
dows because it is based on a non-linear equation.
6. Acknowledgements
The authors wish to thank Wondem Gezahegne and
Rudy Abo for their supporting during fieldwork. Also,
we want to thank anonymous reviewers for their valuable
comments and suggestions. This work has been sup-
ported by the Department of Hydrogeology at TU
Freiberg.
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