Research Journal of Environmental and Earth Sciences 4(11): 962-981, 2012
ISSN: 2041-0492
© Maxwell Scientific Organization, 2012
Submitted: August 15, 2012
Accepted: September 13, 2012
Published: November 20, 2012
Well Test Analysis in Dual-Porosity Aquifers with Stress-Dependent Conductivity
1
H. Jabbari, 1Z. Zeng, 1S.F. Korom and 2M. Khavanin
Department of Geology and Geological Engineering,
2
Department of Mathematics, University of North Dakota, Grand Forks, North Dakota 58202, USA
1
Abstract: A new model for analyzing the hydraulic head in the vicinity of a vertical well in fractured, confined
aquifers is presented. This study shows that flow dynamics within the fractured aquifers are more complex than
previously believed and the fluid flow behavior can be related to rock deformation through hydraulic conductivity
change with fluid withdrawal/injection. This fluid-solid interaction is particularly significant in stress-sensitive,
fissured rocks where the rate of withdrawal/injection is high. The model is derived for cubic geometry under
hydrostatic confining pressure. The solution, however, can be extended to handle other geometries. Considering the
conductivity changes during the life of an aquifer in this study, several findings are drawn: (1) a fully coupled geo
mechanics and fluid-flow model is developed to interpret well test data from confined aquifers with linear elastic
behavior, (2) for characterizing a fissured, confined aquifer knowing three major parameters may be sufficient,
which can be obtained by proper analysis of recovery data, (3) the concept of stress-dependent skin around the wells
is discussed and (4) even though the coupled effect may not significantly influence the drawdown/recovery data
from a normal (stress-insensitive) aquifer, such a coupled model provides an iterative algorithm to estimate the main
fracture parameters, namely fracture porosity, fracture transmissivity, fracture storage capacity ratio and the
elasticity parameter. In general, the fluid-solid coupling effects cannot be ignored when analyzing stress-sensitive
aquifers unless otherwise it is shown to be negligible.
Keywords: Confined aquifer, dual porosity, recovery data, stress-dependent conductivity
•
INTRODUCTION
Groundwater flow in fractured aquifers behaves
differently from that in homogeneous ones. In
homogeneous, single porosity aquifers there is one
single flow regime, whereas in fractured aquifers there
are two: fractures and matrix. Characterizing such a
heterogeneous formation with two distinct media,
namely matrix and fractures, requires more caution and
a more sophisticated model, such as a dual-porosity
solution. As a matter of fact, quantifying the fluid flow
in fissured rock masses requires the combination of
theoretical, laboratory and field studies. In the
theoretical part for fissured aquifers, which are
heterogeneous and anisotropic formations, cogent
mathematical models are imperative (Lods and Gouze,
2008; Murdoch and Germanovich, 2006; Neuman,
2005). These solutions can be obtained based on certain
idealizations (Warren and Root, 1963; Snow, 1969;
Kazemi, 1969; Moench, 1984; Hsieh, 1983).
In fact, three different approaches may be
employed for the analysis of groundwater flow in a
fractured rock mass:
•
•
The continuum approach that treats the porous
medium and the fractures as two separate but
overlapping continua (Peters and Klavetter, 1988;
Rodriguez et al., 2006; Bear and Cheng, 2010)
The Discrete Fracture Network (DFN) approach in
which the unknown heads at the intersections of
fracture network are calculated based on the laws
for flow through individual fractures (Bacca et al.,
1984; Karim-Fard and Firoozabadi, 2003)
The Dual-Porosity (DP) model which assumes that
the medium consists of two continua, one
associated with the fracture system and the other
with the matrix (Barenblatt et al., 1960; Warren
and Root, 1963; Kazemi, 1969)
There have been several studies to implement the
dual-porosity models for characterizing fractured
systems. The main goal in dual-porosity modeling is to
simulate the effects of a fractured system on drawdown
or buildup (recovery) data. This concept was first
introduced by Barenblatt et al. (1960). It was later
applied by Warren and Root (1963) for the case of
pseudo-steady state interporosity flow model and by
Corresponding Author: H. Jabbari, Department of Geology and Geological Engineering, University of North Dakota, Grand
Forks, North Dakota 58202, USA
962
Res. J. Environ. Earth Sci., 4(11): 962-981, 2012
Kazemi (1969) for a transient flow model in naturally
fractured oil reservoirs. It took the groundwater
industry some time to adopt the proposed methods and
develop new models and primitive research on
fractured aquifers appears in the 1970s and 80s
(Streltsova, 1976; Boulton, 1977; Neuzil, 1981;
Moench, 1984). Of course, there had been some
previous research on homogeneous (single-porosity)
aquifers in the decades prior (Theis, 1935, 1938; Jacob,
1940; Cooper and Jacob, 1946; Domenico and Mifflin,
1965; Bear, 1972).
The dual-porosity concept is widely used to
analyze hydraulic heads at any point around the vertical
wells in the fractured media (Gerke and Genuchten,
1993). However, the limitation of the dual-porosity
models is that they are mainly applicable when the
fractured media is represented by regularly-shaped
objects, such as sugar cubes, slabs, or spheres. The
sugar-cube representation is an efficient configuration
method, but it cannot be applied when the fractures are
not connected (Karim-Frad and Firoozabadi, 2003;
Farayola et al., 2011).
Several approaches have been proposed to
implement the geomechanical effects into both oil
reservoir characterization (Min et al., 2004; Dautriat
et al., 2007; Tao et al., 2009) and groundwater flow
modeling (Cey, 2006; Cappa et al., 2008). However,
most proposed models are based on numerical analysis
and exact analytical solutions for nonlinear
geomechanics, fluid-flow problems are limited. It is the
objective of this study to develop a three-parameter
dual-porosity model for investigating the unsteady-state
groundwater flow in naturally fractured aquifers, with
the inclusion of the changes in effective-stress of
formations. This study is founded mainly on the model
of Warren and Root (1963) for dual-porosity systems
which was extended by Jabbari and Zeng (2011) to
incorporate the effect of rock deformation into fractured
media modeling. It, in fact, couples the geomechanical
and fluid-flow aspects for characterizing a stresssensitive fractured formation, where the elasticity
parameter (ε) is relatively significant (ε>0.01).
Before presenting the proposed model, some points
are clarified so that the mathematical model is more
easily understood. First, by “fluid-solid interaction” we
imply that the change in the fracture hydraulic head
(especially in the vicinity of the wellbore) can cause
change in the fracture aperture and, thus, fracture
hydraulic conductivity. This change in hydraulic
conductivity comes from two sources:
•
Fig. 1: Idealization of the heterogeneous porous aquifer
(Warren and Root, 1963)
•
Pore volume compressibility of the fracture
(Jabbari et al., 2011b)
In this study, we demonstrate the effects of change
in fracture conductivity on the overall behavior of a
fissured, confined aquifer. Another goal of this research
is to investigate whether or not the geo mechanical
behavior of the formation affects the transient test data
from a fissured aquifer in a dual-porosity framework.
MATHEMATICAL MODEL
Problem statement: The general configuration of a
dual-porosity model for a fissured aquifer is depicted in
Fig. 1. The concept of dual-porosity model was
originally developed by Warren and Root (1963) in
order to quantify the fluid flow in fractured rocks.
According to their model a fractured medium is
assumed to consist of two interacting, overlapping
media: a medium of low-permeability and high-storage
(matrix) and a medium of high-permeability and lowstorage (fracture). The origin of the coordinate system
is at the wellbore and the equations are derived on the
horizontal plane (x, y). The developed model should
reflect the response of a real fractured aquifer.
However, certain assumptions and idealizations are
inevitable (Warren and Root, 1963):
•
•
Volumetric change of matrix blocks (the pore
pressure effect in the matrix is ignored)
963
The material containing the primary porosity is
homogeneous and isotropic and is contained within
a systematic array of identical, rectangular
parallelepipeds (matrix).
All of the secondary porosity is contained within
an orthogonal system of continuous, uniform
fractures which are oriented so that each fracture is
parallel to one of the principal axes of hydraulic
conductivity (fractures).
Res. J. Environ. Earth Sci., 4(11): 962-981, 2012
•
 ∂ 2 h ∂ 2 h 1  S
∂h 
 ∂p1
 ∂x 2 + ∂=
2
 γ b − α n2  ∂t + γ n2 ( β 2 + c f ) ∂t 
y
K



2 

∂
p
p
S





− α n2  1 =Csh K1  h − 1 
  γ b
γ 
 ∂t

(3)
The complex of primary and secondary porosities
(matrix plus fractures) is homogeneous. Flow to
the wellbore can only occur from the fracture
network; hence, flow through the primary-porosity
(from matrix to wellbore) cannot occur.
Note, also, that the effect of poroelasticity in the
matrix has been ignored in this study since we have
assumed that the matrix conductivity is negligible
compared to the fracture conductivity. Additional
assumptions will be made at appropriate points in the
mathematical treatment.
The governing equation of hydraulic head for a
single phase flow of a slightly compressible liquid in
the vicinity of a point sink/source within a uniform
aquifer that is horizontal, homogeneous and anisotropic,
is partially described by applying Green’s theorem to
the Volume (V) to obtain the applicable form of the
continuity equation (Appendix A):

1 
∂h
 S
 ∂p
∇.( ρ K 2 ∇h)= 
− α n2  1 + γ n2 ( β 2 + c f )
∂
∂t
b
t
ρ
γ


p1

h =
hi
τ 0)
=
 I .C. : (=
γ

 ∂h 

Q = 2π T 

 B.C. : (τ ⟩ 0)
 ∂ ln r r = rw

Q ( re , t ) = 0

All the variables are defined in the section of
notations. Here, the x-axis and the y-axis coincide with
the principle axes of the conductivity field. In fact,
three different types of flow mechanics can be
distinguished:
•
•
•
(1)
Because water in the matrix cannot flow directly to
wellbore (dual-porosity assumption), the hydraulic head
notation does not apply to the matrix, hence, we used
pressure notation for the primary porosity.
In addition to Eq. (1), continuity on a local basis is
necessary. The matrix-fracture interaction is described
by a pseudo-steady state pressure relation
(Appendix A):
p1 
 S
 ∂ p1

 γ b − α n2  ∂ t =C sh K1  h − γ 




(4)
Transient behavior in a bounded system
Steady-state with outer boundary head
Pseudosteady-state denoting a no-flow boundary
condition. Henceforth, the transient behavior of a
finite aquifer (bounded system) is employed in this
study Eq. (4)
On the other hand, if we also consider the changes
in fracture conductivity due to water withdrawal or
fluid injection, the preceding differential system is
changed such that it reflects the effect of head and, thus,
conductivity changes over the aquifer. Because in
fissured formations it is the fracture network that
constitute the main conduits for the flow and the
fracture aperture variations are stress-dependent, the
study of the coupled effective-stress and fracture
conductivity has attracted significant attention in the
recent years (Bai and Elsworth, 1994; Zhu et al., 1995;
Takashi et al., 1995; Suri et al., 1997; Chin et al., 2000;
Gutierrez et al., 2001; Dautriat et al., 2007; Meza et al.,
2010). Notice that in Eq. (6) we defined a new term
with ε that represents this phenomenon. Therefore, the
principle differential system Eq. (3) turns out to be as
follows (Appendix B):
(2)
where, Csh , shape factor, reflects the geometry of the
matrix elements and controls the flow between matrix
and fractures.
In the existing models for fractured formations,
from both petroleum and hydrology points of view, it is
assumed that the hydraulic conductivity of a fracture
network is always constant and it does not vary with
hydraulic head change (Warren and Root, 1963;
Kazemi, 1969; Streltsova, 1976; Boulton, 1977; Neuzil,
1981; Moench, 1984). For such an aquifer that is
confined top and bottom and is bounded at the outer
boundary with a uniform initial head that is to be
produced at a constant discharge rate Eq. (4), the
differential system which represents both the flow from
fractures to wellbore and the flow from matrix to
fracture network can be written as follows:
2
2
 ∂2h ∂2h 
3(1 − 2ν )    ∂h   ∂h  
 n 
 2 + 2 + γ  β 2 + 3  1 + 2  c f +
    +    =
En2    ∂x   ∂y  
 9 
 ∂x ∂y


1  S
∂h 
 ∂p1

=
 − α n2  ∂t + ( 2n2 β 2γ ) ∂t 
K 2  γ b




  S − α n2  ∂p1 =Csh K1  h − p1 
 ∂t


  γ b
γ 


964
(5)
Res. J. Environ. Earth Sci., 4(11): 962-981, 2012
 −λ x 
 −λ x 

 − Ei 
1−ω 
 ω (1 − ω ) 
The transformed equations in polar coordinates (ξ,
θ), the initial condition and boundary conditions can be
rewritten as follows (Jabbari and Zeng, 2011):
2
 ∂ 2 h  1  ∂h
 ∂h 
∂hD
∂ψ
+ (1 − ω )
 2D +   D − ε  D =
 ω
 ∂ξ
ξ  ∂ξ
∂ξ 
∂τ
∂τ



∂ψ

 (1 − ω ) ∂τ =λ ( hD −ψ )

 I .C : (=
τ 0)


 B.Cs : (τ ⟩ 0)




Φ ( x=
) ln(η x ) + Ei 
2 3
 −λ x    λ η x 
 λx 
 −λ x  
  ln 
 − E1  1 − ω  − Ei  ω  




 1 − ω    256(1 − ω ) 
 −λ x    (1 − ω )η x 
 λx 
 λ x 
 + E1 
 ln 
 − E1   
 ω 
 ω (1 − ω )    ω 
 ω (1 − ω ) 
+ exp 
Ei And E1 are “Exponential Integral” functions,
defined as:
(6)
∞
0
hDi
= ψ=
i
∂hD
ξ =1
∂ξ
∂ξ
Ei ( − x ) =
−∫
x
=
−1
∂hD
•
(7)
 ∂hD 

 ∂ξ ξ =1
hwD= hD − S 
γ 

2
η = 4 exp 
(9)
where S' (stress-dependent skin factor) is given by:
24
9γ
τ
2
32
+
∫
0
+
Φ ( x)
4x
dx
u
du
( x⟩ 0)
(13)
(14)
where, 𝛾𝛾� =
�0.57721 is the Euler-Mascheroni constant
(also called Euler’s constant). Among the three major
dimensionless groups (ω, λ, ε) two of them, ω and λ,
are important; they are the groups by which we can
describe the deviation of the behavior of an aquifer with
dual-porosity from that of a homogenous one (Warren
and Root, 1963). The fracture storage capacity (ω) is a
measure of the fluid stored in fractures as compared
with the total water present in the aquifer and it has a
value between zero and unity. A value close to 1
indicates that most of the water is stored in the
fractures, whereas a value of zero indicates that no
water is stored in the fractures. A value of 0.5 indicates
that the water is stored equally in matrix and fractures.
The inter-porosity flow coefficient (λ) is a measure
of the heterogeneity scale of the system and quantifies
the water transfer capacity from matrix to fracture
network. A value of unity for λ indicates the absence of
fractures or, ideally, that fractures behave like the
matrix such that there is physically no difference in
petro physical properties; in other words, the formation
is homogeneous. Low values of λ, on the other hand,
indicate slow water transfer between matrix and
fractures. However, the actual range of λ in oil
reservoirs could be 10-9, which indicates poor fluid
transfer, to 10-3, which indicates a very high fluid
transfer between fractures and matrix (Warren and
Root, 1963).
(8)
=
S (1 + 0.75γε ) S d + ε S ′
eu − 1
and the constant η is:
Therefore, the total skin is obtained as (Jabbari and
Zeng, 2011):
−
(12)
= Ei ( x ) − π i
The skin effect condition is defined by Da Prat (1981):
π2
(x⟩ 0)
du
0
The mechanical skin due to near-wellbore damage
(S d )
The stress-dependent skin which accounts for the
effective-stress change in the near wellbore region
and its impact on the hydraulic conductivity of the
fracture network (S’)
S′ =
u
E1( x ) =
γ + ln( x) + ∫
=0
where, the dimensionless groups are defined in
Table 1.
The effect of the change in fracture hydraulic
conductivity around the wellbore can be quantified by
introducing a new skin factor which is called stresssensitive skin. Therefore, the total skin factor is
composed of two elements:
•
e −u
x
ξ =ξ D
(11)
+ exp 
(10)
where,
τ
: The time during drawdown
Φ(x) : The elasticity function defined as:
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Res. J. Environ. Earth Sci., 4(11): 962-981, 2012
Table 1: Dimensionless parameters
2π T
hD=
(ξ , θ , τ )
2
i
Q
x +y
ξ=
(h
− h (ξ , θ , τ ) )
 hi −
Q 
−1  y 
θ = tan  
x
2
rw
τ =
ω=
γ


Csh K1 rw
λ=
rw ( S + ( β 2 + c f − α )γ bn2 )
2
K2
n2 ( β 2 + c f )
γb
p1 (ξ , θ , τ ) 
2
Tt
S
2π T 
ψ=
(ξ , θ , τ )
=
ε
+ ( β 2 + c f − α ) n2
γQ 
3(1 − 2ν ) 
 n2  
 β 2 + 3  1 + 9   c f + En  


2
2π T 
Moreover, the third parameter (ε) reflects the impact of effective-stress change (due to head change) on the
conductivity of fractures and on the behavior of the aquifer. This parameter depends on rate of withdrawal/injection
(Q), geo mechanical properties (E, v), water compressibility (β 2 ), fracture porosity (n 2 ) and Transmissivity (T)
(Table 1).
General solutions in the Laplace domain: The differential system in Eq. (6) is nonlinear and is solved by using
regular perturbation theory (He, 1999, 2000) and the Laplace transformation method. The solution to this nonlinear
system with the prescribed boundary conditions in Eq. (7) is obtained in the Laplace space as follows:
hD ( ξ , s ) =
)
) (
) (
s sf ( s )  I ( ξ sf ( s ) ) .K ( sf ( s ) ) − I ( sf ( s ) ) .K ( ξ sf ( s ) ) 
 c  I ( ξ sf ( s ) ) .K ( sf ( s ) ) + I ( sf ( s ) ) .K ( ξ sf ( s ) ) 
) (
(
I o ξ sf ( s ) .K1 ξ D sf ( s ) + I1 ξ D sf ( s ) .K 0 ξ sf ( s )
1

−ε 
− I o (ξ

D
1
o
1
1
o
1
ξ sf ( s )
sf ( s )
) ∫
D
1
(
χ K 0 ( χ ).ℑ( χ ) d χ + K o ξ sf ( s )
)
ξ
sf ( s )


sf ( s )

∫ χ I 0 ( χ ).ℑ( χ )d χ 

sf ( s )
(15)
where,
= 𝑙𝑙[ℎ𝐷𝐷 (𝜉𝜉, 𝜏𝜏)]
S
: The Laplace transformation variable, ℎ𝐷𝐷 (𝜉𝜉, 𝑠𝑠) �
I n (x) & K n (x) : Modified Bessel functions of the first and second kind of the nth order, respectively and the
constant c along with the function 𝑠𝑠̃ (𝑥𝑥) are defined as:
ξD
(
) ∫
I ( ξ sf ( s ) ) .K (
I1 ξ D
sf ( s )
sf ( s )
c=
1
1
D
 ∂hD 0  2 
 
 ∂ξ  ξ =
 
ℑ( χ ) =
ξD
(
) ∫
sf ( s ) ) − I ( sf ( s ) ) .K ( ξ
sf ( s )
χ K 0 ( χ ).ℑ( χ ) d χ + K1 ξ D sf ( s )
sf ( s )
χ I 0 ( χ ).ℑ( χ ) d χ
sf ( s )
1
1
D
sf ( s )
)
(16)
χ
sf ( s )
sf ( s )
(17)
where h D0 is the solution for the case without considering the effect of elasticity (the inversion of the first fraction
term on the RHS of Eq. (15). The general solution for the case of an infinite acting system can also be obtained by
allowing the aquifer size grow to infinity, ξ D →∞. This gives us the following:
966
Res. J. Environ. Earth Sci., 4(11): 962-981, 2012
Table 2: Different types of interporosity flow models
Block geometry
Sugar-cube model
(Warren and Root, 1963)
Model
f (s) = ω
Slab-shaped matrix
(Deruyck et al., 1982)
+
(1 − ω ) λ
(1 − ω ) s + λ
λ (1 − ω )
f ( s=
) ω+
3s
Spherically-shaped matrix
(Deruyck et al., 1982)
hD ( ξ , s ) =
f (s)
(
K 0 ξ sf ( s )
s sf ( s ) K1
(
)
sf ( s )
=ω+


5s 
λ
3(1 − ω ) s

λ
15(1 − ω ) s
λ




15(1 − ω ) s

λ
coth 
 
 − 1
 
)

 ∞
K0 (χ )
d χ   I o ξ sf ( s ) .K1
sf ( s ) + I1
sf ( s ) .K o ξ sf ( s )
 ∫

sf ( s ) χ
ε 
sf ( s ) K 1


− 
s
ξ sf ( s )
ξ sf ( s )

K0 (χ )
I0 (χ )
+
ξ
sf
(
s
)
d
χ
ξ
sf
(
s
)
−
I
K
(
)
(
)
∫ χ
∫ χ dχ
o
 o

sf ( s )
sf ( s )
(

tanh 
(
)
) (
) (
) (

) 
(18)





In Eq. (18) it is clear that as we move away from the wellbore, the effect of geomechanics will become
vanishingly small since the coefficient of ε (the terms in the brackets) becomes smaller. The function f(s) depends
on the assumed matrix-fracture flow model. The different inter porosity flow models, which can be used for
different characteristic shapes of matrix blocks, are shown in Table 2.
The solution at the inner boundary (wellbore), i.e., ζ = 1, with the consideration of the total skin around the
wellbore Eq. (9) and for different flow periods, namely the early-, intermediate- and late-time periods; are obtained
for both finite and infinite confined, fractured aquifers.
Solutions for closed outer boundary aquifers: The dual-porosity solution for the case of a finite, confined aquifer
with sugar-cube geometry is obtained for the three time periods. At early time t→0, so s→∞, giving f(s)→ω
(Sageev et al., 1985). The inverse transform of Eq. (15) for this case can then be obtained by using the Heaviside
expansion theorem:


2(ξ D − 1)
τ
 1 − 62
)
+
hwD (1, τ=
3
ω (ξ D − 1)
 π


∞
∑
n =1
 − n 2π 2τ  
2  
2
2
 ω (ξ D − 1)   + (1 + 0.75γε ) S + ε  π − 9γ 
d
 24 32 
n2





exp 
(19)
At intermediate time λ controls the flow and f(s) →λ/s (Sageev et al., 1985). These yields:
hwD (1, τ )
( λ ) .K (ξ λ ) + I (ξ λ ) .K ( λ ) + (1 + 0.75γε )S
λ  I (ξ λ ) .K ( λ ) − I ( λ ) .K (ξ λ ) 


Io
1
1
1
D
1
D
0
D
1
1
 π 2 9γ 2 1   λ  γ  2
 −
+  ln 
+ 
 24 32 2   2  4 
+ε 
ξ
λ

1 
 I1 ξ D λ ∫ χ K 0 ( χ ).Θ( χ )d χ + K1 ξ D λ
−
λ
 λ λ 
(
)
D
d
D
(
where,
967
)




ξ
λ

(
).
(
)
χ
I
χ
χ
d
χ
Θ

∫ 0
λ
 
D
(20)
Res. J. Environ. Earth Sci., 4(11): 962-981, 2012
Θ( χ ) =
(
)
(
)
 I 1 ξ D λ . K1 ( χ ) − I 1 ( χ ) . K1 ξ D λ 


(
) ( λ ) − I ( λ ) .K ( ξ
 I 1 ξ D λ . K1

1
1
D
2
)
λ 

3
(21)
At late time, t⟶ ∞, so s⟶ 0 (Sageev et al., 1985), then Eq. (15) for a finite naturally fractured, confined
aquifer reduces to:
 (1 − ω ) 2 

 −λτ 
τ + λ 1 − exp  ω (1 − ω )  

2 




hwD (1, τ )

 + (1 + 0.75γε ) S d
2
2
ξD − 1  ξD 
− λτ   
 −λτ 

ln τ + 0.80908 + Ei 
+

 − Ei 
 4 
 1 − ω   
 ω (1 − ω ) 
1
 π 2 9γ 2 τ Φ ( x )

×
2
 24 − 32 + ∫ 4 x dx −

 1 
0


1
−
 ξ2 



D 


ξ
sf ( s )
sf ( s )
ξ


+ε 
K0 (χ )
I0 (χ )
 I1 ξ D sf ( s )
∫ χ d χ + K1 ξ D sf ( s ) ∫ χ d χ  

sf ( s )
sf ( s )

× −1 


s sf ( s )  I1 ξ D sf ( s ) K1
sf ( s ) − I1
sf ( s ) K1 ξ D sf ( s )   




 

+
(
)
(
D
(
D
) (
) (
) (
2
)
)
(22)
2
τ ⟩100ωξ D , if λ ⟨ ⟨1 , or , τ ⟩100ξ D − 1 / λ , if ω ⟨ ⟨1
The Laplace inversion term in Eq. (22) can be executed numerically since an analytical solution is either
unavailable or complex. The algorithms for numerical Laplace inversion are available in the engineering
mathematics literature (Stehfest, 1970; Bellman et al., 1976; Crump, 1976; Talbot, 1979; Ilk et al., 2005; Iseger,
2006; Al-Ajmi et al., 2008).
Solutions for extremely large aquifers: The solutions for the case of an infinite-acting system, namely an
extremely large aquifer in the radial direction, are obtained as follows:
At early time:
π2
τ
hwD=
(1, τ ) 2
πω
+ (1 + 0.75γε ) S d + ε 
 24
−
9γ 2 
(23)

32 
At intermediate time:
( λ ) + (1 + 0.75γε )S
λK ( λ )
Ko
hwD (1, τ ) =
d
(24)
1
π2
+ε 
 24

−
9γ
2
32
+
1  λ
 ln 
2  2
2
 γ
1
+  −
3
4
λ
λ
K


1

∞
( λ)
∫ χK
0
( χ ).K12 ( χ ) d χ 
λ


And at late time:
 1 + 0.75γε   ln τ + 0.80908 + Ei  −λτ  − Ei  −λτ  + 2 S 
h=






wD (1, τ )
d 
2


1−ω 
 ω (1 − ω ) 

968
(25)
Res. J. Environ. Earth Sci., 4(11): 962-981, 2012
For high values of shut-in time Eq. (26) turns into:
hws = hi −
Q (1 + 0.75γε )
4π T
 t p + ∆t 

 ∆t 
(28)
ln 
where τ p , the dimensionless producing time, in practical
field units is:
τp =
T tp
2
w
1440r
( S + ρ bn (β
2
(29)
+ c f − α ) /144 )
2
where,
heads
: In ft
Q
: In ft3/day
T
: In ft2/day
: In ft
rw
ρ
: In lb m /ft3
b
: In ft
β 2 , c f & α: In psi-1
: In minutes
tp
Fig. 2: Interpretation of buildup data
RECOVERY DATA INTERPRETATION
Also, the fracture Transmissivity (T) can be
determined from combining the slope of the straight
lines Eq. (26) with the elasticity parameter ( ε in
Table 1). This, in field units, gives us:
In hydrology, one way to get information about the
fracture properties and the behavior of a fissured
aquifer is by interpreting recovery data. Analyzing a
drawdown test is limited by the flow rate fluctuations
n 
3(1 − 2ν )   (30)
0.0915Q 

=
T
1 + 1 + 0.01ρ m  β + 3  1 +   c f +
inherent to pumping. The zero flow
rate that
  
En
9
m 




 
corresponds to buildup (recovery) does not have this
problem. By superimposing the drawdown equation for
where m defined as:
an infinite-acting system Eq. (25) for the case of an
infinite-acting system, the behavior of a shut-in well
0.183Q (1 + 0.75γε )
following a constant pumping rate is considered. If the
(31)
m=
duration of pumping, t p , is fairly long and if certain
T
3
conditions hold τp≻ 𝑎𝑎𝑎𝑎𝑎𝑎 ∆𝜏𝜏 ≻ 100𝜔𝜔 𝑖𝑖𝑖𝑖 𝜆𝜆 ≪ 1, 𝑜𝑜𝑜𝑜,
𝜆𝜆
is the slope of the semi-log straight line, in ft/cycle, to
1
∆𝜏𝜏 ≻ 100 − 𝑖𝑖𝑖𝑖 𝜔𝜔 ≪ 1 (Warren and Root, 1963), then
be obtained from the buildup data (Fig. 2). Substituting
𝜆𝜆
the calculated transmissivity Eq. (30) along with the
the buildup head may be obtained by Eq. (26), which
given slope (m) into Eq. (31), we can obtain ε .
uses Horner method and is depicted in Fig. 2.
The fracture storage capacity, ω, can also be

obtained
by combining Eq. (27) and (28) to yield:
−λ ∆ τ  
Q  1 + 0.75γε    t + ∆ t 
Φ( x)
 −λ ∆ τ 

=
hws hi −
dx 
  ln 
 − ε ∫

 − Ei 
 + Ei 
2π T 
2
4x
   ∆t 
 1−ω 
 ω (1 − ω ) 
2
2
2
τ p +∆ τ
p


∆τ

(26)
where,
h ws = Shut-in head
∆𝑡𝑡 = The time since pumping stopped (shut-in time)
∆𝜏𝜏 = The dimensionless shut-in time
ω =10
hws
hi −

2π T 
2
m

τp
ε
1.151 (1+ 0.75γε )
∫+
0

Φ( x)
dx 
4x


(32)
where δ is the vertical displacement between the early
and late semi-log straight lines for a buildup curve
(Fig. 2) Also, the mechanical skin factor due to near
wellbore damage, S d , can be obtained from the
superposition concept as follows:
For relatively small times the early straight line in
the semi-log coordinates acts as though the aquifer is
homogeneous. This behavior is described by the
following relationship:
Q  1 + 0.75γε 
− δ +
 hws (1min) − hwf (t p )
S d = 1.151 
τ
p
 (27)
Φ( x)
 t p + ∆t 
dx 
 ln 
−ε ∫
  ω∆t  0 4 x


m

− log 
(
T
 rw S + ρ bn2 ( β 2 + c f − α ) / 144
2

) 

+ 1.024 

(33)
969
Res. J. Environ. Earth Sci., 4(11): 962-981, 2012
An iterative calculation technique has been
introduced in this study for reaching satisfactory results
from pumping test analysis. The flowchart in Fig. 3
demonstrates the steps of this trial-and-error algorithm.
The method is to try out various values of the
secondary porosity until the resulting error is
sufficiently reduced or eliminated. This process can use
the value obtained from well logging, if available, as
the first guess. Using well test data through an iterative
procedure, we can achieve much more accurate fracture
porosity. This fracture porosity can be considered as an
average value for the part of the aquifer involved in the
cone of depression (radius of investigation) (Jabbari
and Zeng, 2011).
On the other hand, the procedure to answer a
nonlinear problem, such as that in Eq. (6), by means of
the iterative algorithm shown in Fig. 3 might run into a
convergence problem. The iterative solution is to make
an initial guess for the desired variable, here fracture
porosity. Then, the procedure calculates the fracture
porosity and cycle begins again. This continues
ESTIMATING THE FRACTURE PROPERTIES
Since fracture porosity is a scale-dependent
parameter, any method to estimate it is prone to error. It
can be obtained from well logging or well testing.
However, the porosity computed from well logs and
well tests may differ from one another and they are
different most of the time. Also, the radius of
investigation in well logging is limited to a few feet
around the wellbore. Therefore, the value obtained from
well logs does not represent an average value for the
reservoir sector under study, in spite of use of
sophisticated computation methods.
Note, also, that the value of the secondary porosity
has a significant impact on the properties of a fractured
aquifer Eq. (29) to (33). The value of n 2 can be any
number at any scale over the aquifer, but the average
value for the whole aquifer is generally less than 1%
(Garcia, 2005). According to Nelson (2001) fracture
porosity is always less than 2%; in most formations it is
less than 1%, with a general value of less than 0.5%.
Fig. 3: Algorithm to determine the average values of the properties of a fractured aquifer
970
Res. J. Environ. Earth Sci., 4(11): 962-981, 2012
Also, we assume that λ is not as sensitive as fracture
properties to matrix-fracture elasticity and fracture
porosity and it can be obtained from the proper analysis
of conventional type-curve matching developed by
Onur et al. (1993).
until the fracture porosity settles to a value which is
within a specific tolerance limit. This limit, however,
can be altered using various option parameters in the
algorithm, such as error in the algorithm shown in
Fig. 3. If the fracture porosity does not converge within
a certain number of iterations, the loop will not cease.
If the fracture porosity does not converge, the
values of input data, such as Young’s modulus (E),
Poisson’s ratio (v), the slope of the semi-log straight
line (m) and interporosity flow coefficient (λ) should be
reviewed to ensure their values are appropriate. If the
procedure fails to converge, the first guess for the
fracture porosity may be changed. Nevertheless,
convergence problem from the fracture porosity would
result if its first guess is too low or too high. Almost
always, 1% fracture porosity would be a reasonable
first guess for the calculations.
However, when a convergence problem is
encountered, it is better to start with relatively lower
fracture porosity, say 0.5% (Nelson, 2001) and proceed
with the subsequent suggestions until convergence is
achieved. The sequence of the suggestions is structured
so that they can be incrementally added to the program.
All in all, the proposed algorithm in Fig. 3 seems not to
have convergence problems if the required data are
input as described above.
As stated earlier, we can use the above rule of
thumb (n 2 = 0.5 and 1%) for the first guess to input to
the iterative algorithm if another estimate is not
available.
DISCUSSION
As discussed earlier, for stress-sensitive fissured
aquifers, effective-stress changes induce changes in
fracture hydraulic conductivity around the wellbore.
This, in turn, depending upon the value of elasticity
parameter (ε), would affect the buildup/drawdown
responses from a stress-sensitive aquifer. In this
section, we investigate the effect of on the response of
drawdown curves from both finite and infinite-acting
aquifers namely ω , λ , and ε.
Head variations vs. time plots (both dimensionless)
shown in Fig. 4 and 5 clearly show the effect of
elasticity parameter on the transient test curves. The
results presented in these figures indicate that
drawdown tests, through the elasticity parameter (ε),
can identify whether a fissured aquifer is stresssensitive or stress-insensitive. The elasticity parameter,
however, can vary from aquifer to aquifer which is
dependent on the solid-fluid parameters (Table 1). The
range of variation of the elasticity parameter will be
discussed later in this study.
Fig. 4: Behavior of closed-boundary, naturally fractured aquifers (ω = 0.01, λ = 5×10ￚ5)
971
Res. J. Environ. Earth Sci., 4(11): 962-981, 2012
Fig. 5: Behavior of infinite-acting, naturally fractured aquifers (ω = 0.01)
Fig. 6: Asymptotic solutions for a finite, naturally fractured aquifer (ε = 0)
interaction between ω and λ, which is the
communication between primary porosity and
secondary porosity. At late times, the curves show the
behavior of both matrix and fractures together.
Also, Fig. 6 and 7 present suites of the solutions in
real space (time domain) for the cases of both finite
(with ξ D = 500) and infinite-acting aquifers Eq. (22)
and (25) and for various values of λ and ω. The
elasticity
of
Figure 4 and 5 shows that the influence of the
formation elasticity on the drawdown curves may not
be significant, whereas one advantage of considering
such a parameter in modeling is that it provides the
estimation of the average fracture porosity via the
iterative algorithm in Fig. 3. In Fig. 4 and 5, all the
curves are nearly identical at early times, representing
well discharge from storage in the fissures, followed, at
intermediate time, by the transitional curves that
connect the two linear portions, representing the
972
Res. J. Environ. Earth Sci., 4(11): 962-981, 2012
Fig. 7: Asymptotic solutions for an infinite-acting, naturally fractured aquifer (ε = 0)
the formation is ignored in these schematics. Note, also,
that Eq. (22) (used in Fig. 6) is appropriate when certain
conditions hold (τ>100ω 𝜉𝜉𝐷𝐷2 , 𝑖𝑖𝑖𝑖 𝜆𝜆 ≪ 1, 𝑜𝑜𝑜𝑜, 𝜏𝜏 > 100𝜉𝜉𝐷𝐷2 −
1
, 𝑖𝑖𝑖𝑖 𝜔𝜔 <). Hence, for the case where, ξ D = 100, with
𝜆𝜆
ω = 0.001, 0.1, or 0.1, the asymptotic solutions are valid
when τ>25×103, τ>25×104, or τ>25×105, respectively.
Likewise, Eq. (25) (used in Fig. 7) is appropriate
for all values of ω and λ when τ>100. However, for
small values of ω and λ, τ>100ω, if λ<<1, or, τ>1001/λ, if ω<<1. Hence, for the case of an infinite-acting
aquifer with ω = 0.001, 0.01 , or 0.1 , the asymptotic
solutions are valid when τ>0.1, τ>1, or τ>10,
respectively.
The dual-porosity model of Warren and Root
(1963) for a fractured medium is valid if we assume
that there is no change in fracture conductivity with
respect to change in hydraulic head. However, changes
in hydraulic head (pressure) due to water withdrawal or
fluid injection (e.g., in disposal wells), theoretically,
would affect the aquifer effective-stress and this, in
turn, would influence the fracture hydraulic
conductivity. Hence, this change in hydraulic
conductivity may influence the behavior of an aquifer at
the near wellbore region, which can cause changes in
total skin around the wellbore Eq. (9).
Table 3: Recovery data from an oil well (Najurieta, 1980)
∆t, min
(t p +∆t)/∆t
1
516668
2
258335
4
129168
8
64584
16
32283
32
16147
64
8073
128
4037
256
2019
512
1010
1024
505.6
2048
253.3
∆h s , ft
325.1
315.6
307.3
299.5
283.3
272.1
264.9
245.8
230.3
204.4
187.5
172.1
a naturally fractured, oil reservoir, discussed in the
study by Najurieta (1980). Table 3 shows the observed
head data and Fig. 8 shows the corresponding Horner
plot. Basic data include: Q = 14, 318 ft3/day, r w = 0.375
ft, α 6.9×10-5, psi-1, n 1 = 0.2, β 1 = β 2 = 3.2×10-6 psi-1,
b = 18 ft, from which we can obtain S = 4.2×10-3. The
drawdown at the end of the flowing test, h vf (t p ), is 960
ft and the producing time is t p = 516, 667 min. Also, we
assume n 2 = 1% as the first guess.
As is clear in Fig. 8, the test has not distinctly
shown the early straight line. Hence, this line is
obtained from an extrapolation.
From the buildup plot the slope of the semi-log
straight line, m and δ are obtained as 71.2 ft/cycle and
250 ft, respectively. Using the conventional type-curve
match presented in the literature (Onur et al., 1993), the
following results are obtained: ω = 0.0004, λ = 2×10-6
and S d = 4.8. Since there is no information available
regarding the geomechanical parameters of the
CASE STUDY
The mathematical model presented in this study is
tested by analyzing a recovery (buildup) test taken from
973
Res. J. Environ. Earth Sci., 4(11): 962-981, 2012
Fig. 8: Recovery data interpretation
Table 4: Final results from recovery data interpretation using the iterative algorithm in Fig. 3
Geomechanical
2
Runs
parameters
n 2 , (%)
T, (ft /d)
ω
ε
♠E 1 = 4.5E6
1
ν 1 = 0.26
36.88
0.38
0.00035
0.0068
2
ν 2 = 0.30
36.86
0.39
0.00036
0.0055
3
ν 3 = 0.35
36.84
0.41
0.00038
0.0041
E 2 = 7.5E6
4
ν 1 = 0.26
36.84
0.41
0.00038
0.0039
5
ν 2 = 0.30
36.83
0.42
0.00039
0.0033
6
ν 3 = 0.35
36.82
0.43
0.00040
0.0025
E 3 = 9.5E6
7
ν 1 = 0.26
36.82
0.42
0.00039
0.0031
8
ν 2 = 0.30
36.82
0.43
0.00040
0.0026
9
ν 3 = 0.35
36.81
0.43
0.00040
0.0020
♠ (psi)
𝑠𝑠̅
5.63
5.61
5.58
12
10
9
5.58
5.57
5.56
10
8
7
5.57
5.56
5.55
6
6
6
Table 5: Sensitivity analyses on the different parameters affecting ε
Base case: n 2 = 0.3%, Q = 33690 (ft3/day), E = 7.5×106 (psi), v = 0.26, c f = 10-5 (psi-1), and T = 36.8 (ft2/day)
n 2 (%)
Q (ft3/day)
0.3
0.6
1
5615
33690
0.0142
0.0082
0.0057
0.0024
0.0142
ε
E (psi)
v (dimensionless)
0.26
0.3
4.5×106
7.5×106
9.5×106
0.0223
0.0142
0.0117
0.0142
0.0122
ε
2
C f (psi-1)
T (ft /day)
30
36.8
5×10-6
1×10-6
5×10-5
0.0133
0.0142
0.0218
0.0174
0.0142
ε
formation from which the test is taken, namely E and v,
we executed the iterative algorithm (Fig. 3) for 9
different cases with different combinations of E and v.
This helps us determine the effect of these parameters
on the resulting fracture parameters (Table 4). The
results obtained from the iterative algorithm, with a
relative error of 10-8, are shown in Table 4.
The effect of the change in fracture hydraulic
conductivity on the recovery data is studied using 9
simulation runs (runs 1 through 9 in Table 4). The
Iteration No. Until
convergence
56150
0.0237
0.35
0.0097
60
0.0087
simulation conditions for all 9 cases are identical except
for the matrix geomechanical parameters, namely
Young’s modulus (E) and Poisson’s ratio (v). As is
clear in Table 4, the matrix-fracture elasticity
interaction (different geo mechanical parameters) may
affect the behavior of buildup curves from fissured
aquifers. Although the elasticity parameter seems not to
affect the buildup curve in this specific case since the
transmissivity shows little change, using the introduced
iterative algorithm in this study (Fig. 3), provides a
974
Res. J. Environ. Earth Sci., 4(11): 962-981, 2012
practical method to estimate the values of average
secondary porosity (n 2 ) and fracture storage capacity
(ω), which both of the results are in accordance with
what was claimed earlier in the context. In terms of the
estimated fracture porosity, generally speaking, the
iterative algorithm works well in practice since the
estimated secondary porosities is near the value stated
by Nelson (2001), namely 0.5%. Moreover, the
approximated values of the fracture storage capacity are
well in line with the result obtained by Najurieta (1980)
which is 0.0004.
The estimated elasticity parameter (ε) values for
different combinations of geo mechanical parameters
show that the fractured medium in this example is
stress-insensitive. This is because the estimated
elasticity parameter values are very small (ε<<0.01)).
Also, relatively higher values of skin factor in Table 4,
compared with that obtained from conventional typecurves, which is 4.8, are due to the inclusion of
elasticity concept in the calculations, namely the stressdependent skin defined in Eq. (9).
To better understand the concept of elasticity
parameter and determine its contribution to the head
traces in drawdown and buildup tests, arbitrary ranges
of variation of each parameter were chosen and their
corresponding elasticity parameters were calculated.
The results are shown in Table 5. The base-case data
are identical for all 12 cases except for the parameter
for which the sensitivity analysis was run (Table 5).
The results in Table 5 indicate that the effect of
matrix block elasticity becomes insignificant as the
secondary porosity (n 2 ) increases. This seems logical
since in such a case where fracture compressibility is
the dominating parameter as for the elasticity of the
combined matrix and fracture system. The rate of
withdrawal/injection (Q) is the main controlling
parameter that is linearly related to ε. The larger the Q,
the larger the elasticity parameter (ε) and the more
significant the effect of elasticity on the aquifer
characterization calculations. As shown in Table 5, for
the case with a higher pumping rate (56150 ft3/day or
equivalently 10000 BBL/day), a higher elasticity
parameter of 0.0237 is obtained which seems to be
fairly influential in the calculations (values of ε in
Table 5). As for the effect of geo mechanical
parameters, one would deduce that the elasticity
parameter is lower in cases of higher values of E and v.
This, in fact, makes sense because for a stiffer matrix
block with a higher bulk modulus (K) (due to high E
and v), the external head (pressure) change needed to
change the unit block volume is higher. Hence, for a
specific state of stress, the impact of elasticity
parameter becomes weaker with higher module. Also,
the higher fracture Compressibility (C f ) leads to a
higher elasticity parameter and, therefore, a higher
relative volume change when associated with a change
in fracture hydraulic head. Last but not the least, when
the fracture transmissivity is high, the elasticity
parameter becomes vanishingly small since the relative
change in a high fracture hydraulic conductivity would
be insignificant when hydraulic head changes.
Conversely, it would be significant when the fracture
transmissivity is small.
CONCLUSION
In this study a mathematical model, founded on the
original model of Warren and Root (1963), is
developed for coupling the fluid-flow and rock
deformation interactions in aquifers with dual-porosity
behavior. The model is to study the effect of stressdependent fracture hydraulic conductivity on the
hydraulic head transient data from a fissured, confined
aquifer. The following conclusions are drawn from this
study:
•
•
•
•
•
975
A general mathematical model for hydraulic head
trends in finite and infinite-acting, confined,
fissured aquifers with the consideration of fracture
hydraulic conductivity changes is developed. The
model is successfully employed as a tool to analyze
the well test problems in heterogeneous aquifers.
Effective-stress change in the system usually does
not affect the shape of the head trends, since the
elasticity parameter (ε) is in the order of 10-2
(Table 5). Therefore, this parameter will not
significantly influence the fracture transmissivity
and Storativity (S). However, it provides us a
promising method, through the iterative algorithm
(Fig. 3), to quantify the fracture parameters, such
as fracture porosity, fracture storage capacity,
elasticity parameter and stress-dependent skin
around the wellbore.
All other parameters being constant, flow rate (Q)
and fracture Compressibility (C f ) variations have
the largest impact on the elasticity parameters-due
to the linear relationship among them (Table 5).
The other parameters, affect the elasticity
parameter but to a less extent.
For a weakly fissured aquifer with a stressdependent, fracture conductivity, the recovery trace
may show differences from that with constant
conductivity. Specific combinations of the main
parameters, namely fracture conductivity level,
flow rate, fracture compressibility and matrix geo
mechanical parameters, may reduce the effect of
elasticity on the resulting fracture parameters from
the well test interpretations (small ε , Table 5).
Long producing times intensify the severity of the
aquifer stress sensitivity which is evident on the
resulting total skin factors, through the stressdependent skin Eq. (9).
Res. J. Environ. Earth Sci., 4(11): 962-981, 2012
If we define the Storativity (S) of a confined aquifer as the product of
the Specific Storage (S s ) and the aquifer thickness (b) (Fetter, 2001),
we have:
APPENDIX A
Considering the continuity equation with isotropy in fracture
conductivity (Warren and Root, 1963), namely K 2x = K 2y = K 2z , we
have:


∂ ( ρ n )1 ∂ ( ρ n ) 2
∇.( ρ K 2 ∇=
h)
+
∂t
∂t
S = γ b( n1 β 1 + α ) ⇒ n1 β1 + α (1 − n2 ) =


⇒
∂Vs
∂t

= Vb  −
∂ ( ρ n )1
 S
 ∂p
= ρ
− α n2  1
∂t
 γb
 ∂t
1
.
∂n1
−
n1
(1 − n2 ) 2
.
Vs
Vt
⇒
∂n2
∂t
∂n2  
n1  ∂Vb
= 0
 + 1 −

∂t   1 − n2  ∂t
∂ ( ρ n) 2
∂t
∂Vb  Vb  ∂n1
∂n
=
− 1 ⇒
=


∂t
∂t
 1 − n2  ∂t
∂t
 1 − n2  ∂Vb ∂σ e
∂p1
 ∂σ t ∂p1 
×
=−(1 − n2 )α 
−
 =(1 − n2 )α

V
∂
σ
∂
t
∂
t
∂
t
∂t


 b  e
(A4)
Assuming that the fractures are fully filled with water, one would say
β2 =
� C f . This gives us:
where,
V s = Volume of solid phase
V b = Bulk volume of a matrix block
n m = Matrix porosity
σ e = Effective stress
α = The compressibility of matrix skeleton which is defined as:
− 1 ∂ Vb
Vb ∂ σ e
∂ ( ρ n) 2
∂t
(A5)
∂h
∂t
(A13)
p1 
 S
 ∂ p1

 γ b − α n2  ∂ t =C sh K1  h − γ 




(A14)
Finally, the differential system for a naturally fractured aquifer results
in the following:
(A6)

1 
 ρ ∇.( ρ K 2 ∇ h)=


  S − α n  ∂ p1
2 


 ∂t
 γb
This yield:
∂ ( ρ n )1
∂p ∂ρ
∂n
∂p
= n1 1 .
+ ρ 1 = ρ [ n1 β1 + α (1 − n2 ) ] 1
∂t
∂ t ∂ p1
∂t
∂t
= 2 ργβ 2 n2
If we also consider pseudo-steady state condition, the following
equation holds for the interaction between matrix and fractures
(Warren and Root (1963)):
and from the definition of water compressibility, we have:
∂ p1
∂ρ
∂h
ργβ 2
= ρβ
=
1
∂t
∂t
∂t
∂ p2 ∂ ρ
∂n
∂p
.
+ ρ 2 = ρ n2 ( β 2 + c f ) 2 (A11)
∂ t ∂ p2
∂t
∂t


∂h 
 S
 ∂p
∇.( ρ K 2∇
=
h) ρ  − α n2  1 + γ n2 ( β 2 + c f )  (A12)
∂
∂t 
b
t
γ


=
α=
= n2
(A10)
Hence, A1 becomes:
(A3)
∂n1
(A9)
1 ∂ n2
1 ∂ n2
1 ∂n
cf =
−
=
=2
n2 ∂ σ e n2 ∂ p 2 γ n2 ∂ h
(A2)
n1 + n2 =
1−
(A8)
To determine the rate of change of mass in fracture network, we have:
1 − n2 
 1 − n2 ∂t
− α n2
Therefore,

 = const
n1
γb
(A1)
where, the subscripts 1 and 2 denote primary and secondary porosity,
respectively. To determine the rate of change of mass in matrix we
start with (Bear, 1972):
Vs = Vb (1 − nm ) = Vb  1 −
S
(A7)
∂h
 S
 ∂ p1
 γ b − α n2  ∂ t + ( 2 n2 β 2γ ) ∂ t



=C sh K1  h −

(A15)
p1 
γ 
APPENDIX B
Equation (A1) gives the following form, upon taking the derivatives:



 


∂ ( ρ n )1 ∂ ( ρ n ) 2
∇.( ρ K 2 ∇ h=
+
) ρ K 2 ∇ 2 h + K 2 ∇ ρ .∇ h + ρ ∇ K 2 .∇ h=
∂t
∂t
(B1)
To determine the gradient terms we consider planar variation of functions for which we obtain:
 ∂ρ   ∂h 
 ∂ρ ∂h   ∂h 
2
2
 ∂x   ∂x 
 ∂h × ∂x   ∂x 
 
∂ρ   ∂h   ∂h  
 .  K 2 
 .  K 2
K 2 ∇ ρ .∇ h K 2 =
=
=
   +   
∂h   ∂x   ∂y  
 ∂ρ   ∂h 
 ∂ρ ∂h   ∂h 
 ∂y   ∂y 
 ∂h × ∂y   ∂y 
  

 
976
(B2)
Res. J. Environ. Earth Sci., 4(11): 962-981, 2012
And
 ∂K 2   ∂h 
 ∂K 2 ∂h   ∂h 
2
2
 ∂x   ∂x 
 ∂h × ∂x   ∂x 


∂K  ∂h
 ∂h  
 .  ρ 
 .   ρ 2    +   
ρ ∇ K 2 .∇ h ρ  =
=
=
∂ h   ∂ x   ∂ y  
 ∂K 2   ∂h 
 ∂K 2 × ∂h   ∂h 
 ∂y   ∂y 
 ∂h ∂y   ∂y 

 

 
(B3)
Substituting the change in density from Eq. (A6) into (B2) and (B3) yields:
  ∂h  2  ∂h 
 
=
K 2∇ ρ .∇ h K 2 β 2γρ    +  
  ∂x   ∂y 



∂K
=
ρ ∇ K 2 .∇ h ρ 2
∂h
2



(B4)
  ∂h  2  ∂h  2 
   +   
  ∂x   ∂y  
Hence,
2



1 
1 ∂K 2    ∂h   ∂h 
∇.( ρ K 2∇ h)= K 2 ∇ 2 h +  β 2γ +
    +  
ρ
K 2 ∂h    ∂x   ∂y 


2

 
 
(B5)
Combining Eq. (A12) and (B5), we obtain the following:


1 ∂K 2
∇ 2 h +  γβ 2 +
K 2 ∂h

2
2
   ∂h   ∂h   1   S
∂h 
 ∂ p1
    +  =
  − α n2  ∂ t + ( 2 n2 β 2γ ) ∂ t 
 


   ∂ x   ∂ y   K 2   γ b
(B6)
Hence, the following non-linear system is obtained:
2
2
 ∂2h ∂2h 
1 ∂K 2   ∂h   ∂h   1  S
∂h 
 ∂p1
 2 + 2 +  β 2γ +

   +  =

 γ b − α n2  ∂t + ( 2n2 β 2γ ) ∂t 
x
y
K
h
x
y
K
∂
∂
∂
∂
∂



  



 
2
2 

p1 
  S
 ∂p1

  γ b − α n2  ∂t =Csh K1  h − γ 



 
(B7)
Plugging Eq. (C2) (Appendix C) into Eq. (B7) gives us:
2
2
 ∂2h ∂2h
3(1 − 2ν )   ∂h 

 ∂h  
 n 
 2 + 2 + γ β2 + 3 1 + 2   c f +
  +    =


En2   ∂x 
∂y
9 

 ∂y  

 ∂x


∂h 
1  S
 ∂p1

=
 − α n2  ∂t + ( 2n2 β 2γ ) ∂t 
K 2  γ b




∂
p
p
S





− α n2  1 =Csh K1  h − 1 
  γ b
γ 
 ∂t


(B8)
To transform the equations above to act in polar coordinate we use the dimensionless groups presented in Table 1. The following partial
differential terms can be obtained by transformation of variables in polar coordinates (ξ, θ):
∂h
sin θ ∂hD 
Γ
.
=
−  cos θ . D −

ξ
∂x
∂ξ
∂θ 
rw 
∂h
(B9)
∂h
cos θ ∂hD 
Γ
.
=
−  sin θ . D +

ξ
∂y
∂ξ
∂θ 
rw 
∂h
∂ 2 hD sin 2 θ ∂ 2 hD sin 2θ ∂ 2 hD sin 2θ ∂hD sin 2 θ ∂hD 
Γ
.
.
.
.
=
− 2  cos 2 θ .
+
−
+
+

ξ 2 ∂θ 2
ξ
ξ2
ξ
∂x
∂ξ 2
∂ξ∂θ
∂θ
∂ξ 
rw 
∂2h
2
∂ 2 hD cos 2 θ ∂ 2 hD sin 2θ ∂ 2 hD sin 2θ ∂hD cos 2 θ ∂hD 
Γ
.
.
.
.
=
− 2  sin 2 θ .
+
+
−
+

ξ2
ξ
ξ2
ξ
∂y
∂ξ 2
∂θ 2
∂ξ∂θ
∂θ
∂ξ 
rw 
∂2h
2
977
(B10)
Res. J. Environ. Earth Sci., 4(11): 962-981, 2012
Γ ∂hD
.
∂t
Θ ∂τ
∂p1
γ Γ ∂ψ
= −
.
∂t
Θ ∂τ
∂h
= −
where, Г and
Γ=
=
Θ
Θ
(B11)
Q
2π T
(B12)
Substituting the terms above into Eq. (B8) and considering symmetry
in angular direction, we obtain a simpler nonlinear differential system
in polar coordinates Eq. (6):
2
 ∂2h
 1  ∂h
 ∂h 
∂h
∂ψ
D
ω D + (1 − ω )
+   D − ε  D=


2
 ∂ξ
ξ  ∂ξ
∂ξ 
∂τ
∂τ



∂ψ

(1 − ω )
=λ ( hD −ψ )

∂τ

(B13)
APPENDIX C
The change in intrinsic permeability of fractures, in petroleum
engineering notations, is obtained as (Jabbari et al., 2011):
1 dk f
k f dp2
 φf
=+
3 1
9

3 (1 − 2ν ) 


  c f +
Eφ f 

(C1)
where,
k f = Fracture intrinsic permeability
P 2 = Fracture pressure
Ф f = Secondary porosity
C f = Fracture compressibility which can be considered equal to β 2
(Appendix A)
v = Poisson’s ratio
E = Young’s modulus.
:
:
:
:
:
:
:
h ws
K1
K2
n1
n2
p1
:
:
:
:
:
:
:
:
The authors would like to thank the following
sponsors for their financial support: US DOE through
contract
of
DE-FC26-08NT0005643
(Bakken
Geomechanics), North Dakota Industry Commision
together with five industrial sponsors (Denbury
Resources Inc., Hess Corporation, Marathon Oil
Company, St. Mary Land and Exploration Company
and Whiting Petroleum Corporation) under contract
NDIC-G015-031 and North Dakota Department of
Commerce through UND’s Petroleum Research,
Education and Entrepreneurship Center of Excellence
(PREEC).
(C2)
NOTATIONS
b
cf
C sh
E
H
hD
h wD
S
Pumping rate, L3/T
Radius, L
Wellbore radius, L
Storativity, dimensionless
ACKNOWLEDGMENT
Using hydrology notations, we derived the following:
1 dK 2
3(1 − 2ν ) 
 n 
3γ 1 + 2   β 2 +
=

9 
K 2 dh
En2 

:
:
:
:
Total skin factor, dimensionless
Sd
Mechanical skin due to near wellbore damage,
dimensionless
S ′ : Stress-dependent skin, dimensionless
t
: Time, T
T
: Transmissivity, L2/T
α
: Compressibility of aquifer skeleton, LT2/M
β 1 : Water compressibility in primary porosity,
LT2/M
β 2 : Water compressibility in secondary porosity,
LT2/M
: ≅ 0.57721, Euler’s constant
γ
γ
: Specific weight, M/L2T2
λ
: Inter porosity flow coefficient, dimensionless
ω
: Fracture storage capacity ratio, dimensionless
ε
: Elasticity parameter, dimensionless
Φ(x) : Elasticity function, dimensionless
ν
: Poisson’s ratio, dimensionless
ξ
: Radial coordinate, dimensionless
ξ D : Aquifer size in the radial direction,
dimensionless
ρ
: Fluid density, M/L3
Ψ
: Pressure decline in primary porosity,
dimensionless
τ
: Time, dimensionless
are defined as:
rw2  S

 + ( β 2 + c f − α )γ n2 
K2  b

Q
r
rw
S
Aquifer thickness, L
Fracture compressibility, LT2/M
Shape factor, 1/L2
Young’s modulus of elasticity, M/LT2
Hydraulic head in the fractures, L
Hydraulic head in fracture, dimensionless
Hydraulic head in fracture (at the wellbore),
dimensionless
Hydraulic head in fracture (shut-in), L
Matrix hydraulic conductivity, L2
Fracture hydraulic conductivity, L2
Primary porosity, fraction
Secondary porosity, fraction
Matrix pressure, M/LT2
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