calculation of effective permeability for the bmp model in fractal porous media
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Author’s Accepted Manuscript
Calculation of Effective Permeability For The BMPModel In Fractal Porous Media
M. Turcio, J.M. Reyes, R. Camacho, C. Lira-Galeana,R.O. Vargas, O. Manero
PII: S0920-4105(13)00042-9DOI: http://dx.doi.org/10.1016/j.petrol.2013.02.010Reference: PETROL2363
To appear in: Journal of Petroleum Science and Engineering
Received date: 14 February 2012Accepted date: 19 February 2013
Cite this article as: M. Turcio, J.M. Reyes, R. Camacho, C. Lira-Galeana, R.O. Vargasand O. Manero, Calculation of Effective Permeability For The BMP Model In FractalPorous Media, Journal of Petroleum Science and Engineering, http://dx.doi.org/10.1016/j.petrol.2013.02.010
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CALCULATION OF EFFECTIVE PERMEABILITY FOR THE BMP MODEL IN FRACTAL
POROUS MEDIA
M. Turcio1, J. M. Reyes3, R. Camacho3, C. Lira-Galeana2,*, R. O. Vargas4, & O. Manero1
1Instituto de Investigaciones en Materiales
Universidad Nacional Autónoma de México. A.P. 70-360, México, D.F., 04510.
2Instituto Mexicano del Petróleo
Eje Lázaro Cárdenas 152, San Bartola Atepehuacán, México, D.F., 07730.
3Petróleos Mexicanos
México D.F.
4Instituto Politécnico Nacional, SEPI-ESIME-Azcapotzalco
Avenida de las Granjas 682, Sta. Catarina Azcapotzalco, C.P. 02550 México, D.F.
* Phone number: 52 04455 54558911 and 52 5591756507; e-mail: [email protected]
ABSTRACT
In this work, a fractal model is developed for the effective permeability using the BMP (Bautista-
Manero-Puig) model in porous media. The main assumptions of this analysis involve a bundle of
tortuous capillaries whose size distribution and tortuosity follow the fractal scaling laws. The
average flow velocity and the effective permeability for the BMP fluid flow in porous media are
derived. The BMP model does not contain fitting parameters and every material constant in the
model can be estimated from independent rheological measurements. This model consists in the
upper-convected Maxwell equation coupled to a kinetic equation representing the structure
modification that the complex liquid undergoes as it flows in the fractal porous media. Under simple
shear, a first Newtonian viscosity plateau at low shear rates, a second plateau at high shear rates, a
power-law region between both asymptotic regions, a real yield stress, thixotropy and elasticity are
described by the model. Predictions of the proposed fractal model are compared with those of other
models and with available experimental data.
Keywords BMP Model; Fractal medium; Permeability; Porous media; Yield stress; Petroleum
Production
1. INTRODUCTION
Many of the fluids used in the oil industry to stimulate reservoirs are significantly non-Newtonian:
they display shear-dependence of viscosity, thixotropy (or rheopexy) and elasticity. In addition, they
may be chemically complex and they can interact with the porous formations (clays, sands, rocks)
through which they are pumped and with the other fluids which they come into contact with the
porous structure.
The flow of non-Newtonian fluids through porous media is the subject of ample attention, in areas
such as soil mechanics, filtration of polymer solutions or slurries and the use of these systems in
reservoirs in secondary oil operations. In practice, the permeability of porous media is obtained by
direct measurement of pressure-drop and flow rate using cylindrical cores from blocks of coherent
porous media or granular media. It has been verified that the permeability obtained by using a
Newtonian fluid with one viscosity is the same to that obtained using another Newtonian fluid of
different viscosity, and that pressure-drop and flow rate are linearly related, thus confirming that
Stokes law applies at the pore scale. The problem arises when the fluid flowing through the medium
of known permeability is now a non-Newtonian fluid (of known bulk rheology) and an equivalent of
Darcy´s law is sought. The use of non-linear constitutive equations aims to provide extensions of
the Darcy law to cover fluids with complex rheology. The first extensions to describe the non-
Newtonian flow behavior are the generalized Newtonian fluids, in which the viscosity is not a
constant but a function of the rate of strain tensor, like power-law fluids, Carreau models, and the
Ellis model. The thixotropic fluids describe an extension of generalized Newtonian fluids in which
the viscosity is determined by the history of the rate of strain (many real fluids display such
character when subjected to non-constant shear rate). Finally, the elastic fluids cannot be
represented as in the previous fluids, but are written as a functional relation representing the past
history of the rate of strain of a fluid element that is in a certain position at a given time, for a
general memory-fluid. Typical examples are the evolutionary models of Oldroyd based on the
upper-convected Maxwell or Oldroyd derivatives. At the pore scale, flow fields are strongly affected
by elastic forces which cause the principal directions of stress and rate of strain not to be parallel,
as they have for generalized Newtonian or thixotropic fluids. However, at the Darcy scale, the
functional over past history is relevant because all complexities at the pore scale are considered in
the functional.
As far permeability is concerned, a series of models for porous media aim to reproduce the relevant
features of real porous media. In the case of homogeneous or isotropic media, simple one-
dimensional tubular models are suggested, such as bundles of cylindrical tubes. Bundles of
identical tubes with their diameters varying along their length have also been considered. More
elaborate network models have been used on square or cubical grids, with nodes connected by
circular pipes; others have most of the porosity contained in pores connected by throats which
account for the total pressure drop. The more realistic model geometries are described topologically
rather than metrical; pores are connected to one another by a variable number of links that can
have different lengths.
The fractal geometry theory characterizes irregular or disordered objects, like sandstone pores and
granular materials (Katz and Thompson, 1985; Yu, 2005; Wu and Yu, 2007) and represents a
useful tool for analysis of porous media. A fractal geometry model for permeability of porous media
with Newtonian fluids has been reported (Yu and Cheng, 2002). Extensions to non-Newtonian fluids
include models like power-law (Zhang et al., 2006), Bingham (Yun et al., 2008) and Ellis fluids (Li et
al., 2008).These models usually relate the structural parameters of the porous media, like the fractal
dimensions, tortuosity fractal dimensions, microstructural parameters and porosity, to the
rheological material functions. The models seek relationships among the average flow velocity, the
effective permeability, the effective porosity, pressure gradient and material constants.
In this work, the BMP (Bautista, Manero, Puig) model (Bautista et al., 1999) is used to calculate the
effective permeability in fractal porous media. The constitutive equations of the model are the
upper-convected Maxwell equation coupled to a kinetic equation representing the structure
modification of the complex liquid as it flows through the fractal porous media. This new fractal
model involves a bundle of tortuous capillaries whose size distribution and tortuosity follow the
fractal scaling laws. The BMP model was chosen here on the basis of its capability to predict in
simple shear the first Newtonian plateau at low shear rates, a second Newtonian plateau at high
shear rates, a power-law region for intermediate shear rates, a real yield stress as the fluidity tends
to zero and shows elastic behavior, namely, a non-zero first normal stress difference that increases
with shear rate. In unsteady stress cycles, the model also accounts for thixotropic and rheopectic
behavior.
2.1 Rheological equation of state.
The BMP model is described by the following equations:
DG ϕ
σϕ
σ 21
0
=+∇
(1)
D:)(k)(1dtd
00 σϕϕϕϕλ
ϕ−+−= ∞ (2)
σ is the stress tensor, D is the symmetric part of the rate of strain tensor, ϕ is the fluidity (inverse
of the shear viscosity η), 0ϕ (η0-1) is the zero shear-rate fluidity, ∞ϕ is the fluidity at high shear
rates, G0 is the shear modulus, λ is the structural characteristic time and k0 is a kinetic constant
related to structure modification. The upper-convected derivative of the stress tensor is:
( )TdL L
d t
∇ σσ = − ⋅ σ + σ ⋅ (3)
where L is the velocity gradient tensor. Equations 1 and 2 reduce to the upper-convected Maxwell
model when 0ϕϕ ≡ . These equations express that the non-linear viscoelastic processes contained
in the Maxwell equation are coupled with an equation written in terms of the fluidity, which is itself a
kinetic equation with a characteristic time related to structure formation λ and a destruction term
related to structure modification with kinetic constant k0 proportional to the dissipation.
Under simple shear flow, the above equations reduce to:
ϕγσ
ϕσ =+
dtd
G0
1 (4)
γσϕϕϕϕλ
ϕ )(k)(1dtd
00 −+−= ∞ (5)
where .γ is the shear rate and the non-linear terms in equation 3 are not considered, implying that
the normal stresses generated under flow are negligibly small. If the material under study has a very
small elastic response (i.e. 01
0
≈ϕG
), this means that the stress relaxation time is very small, in
addition to the time-dependent structure change being sufficiently small (i.e.
0≈dtdϕ
), both
equations 4 and 5 can be reduced to their steady state version and they may combine to give:
02 =−+− ∞ ϕσϕϕλϕϕ )(k)( 00 (6)
Equation 6 predicts shear-thinning behavior when 0ϕϕ >∞ , shear-thickening when ∞ϕ < 0ϕ and
Newtonian behavior when ∞ϕ = 0ϕ . A plateau region is predicted in the limits of very low and very
high shear rates, with a power-law behavior in the intermediate shear rates. In addition, a real yield
stress is predicted when 0ϕ = 0. The real yield stress implies a solid-like behavior in the limit when
the shear rate approaches zero, as in the Bingham viscoplastic model. Apparent yield is predicted
for very small values of 0ϕ . In this manner, with a single model the Bingham and power-law-type
behaviors can be predicted.
From equation 6, the yield stress can be calculated when 0ϕ = 0 in the region of very small shear
rates. This gives:
2/10 )( −
∞= λϕσ ky (7)
Equation 6 can be solved for ϕ expressing the results in terms of the yield stress, to give:
( ) ( )[ ]⎭⎬⎫
⎩⎨⎧ +−±−= ∞
∞
2/122
022222
22 )/(/41/1//2
1)(yyy
y
σσϕϕσσσσσσϕ
σϕ (8)
The three independent parameters in Equation 8 can be evaluated from the flow curve itself
in the form of viscosity versus shear-rate. The zero shear-rate fluidity (inverse viscosity) is
extracted from the first Newtonian plateau at vanishing shear rates and the infinite shear-
rate fluidity corresponds to the plateau at high shear rates. Alternatively, both fluidities may
be evaluated from the creep curve of compliance versus time. When the yield stress is
approached the viscosity tends to a slope of -1 in a log-log plot. The usual form from which
the yield stress is evaluated considers the plateau exhibited as the shear rate tends to zero in
a plot of log stress versus log shear rate. In this context, the model does not contain fitting
parameters.
According to the model presented here, the pores are considered capillaries with different
diameters. The change in the capillary radius is taken into account by modification of the
tortuosity along the flow trajectories, and several results are shown where changes in the
tortuosity or fractal dimensions are considered. Although transient flow exists in a real
porous medium (expansion-contraction complex trajectories) in this model the change in
the geometry which modifies trajectories is considered as a change in the tortuosity and a
change in the fractal dimensions. In other words, following the averaging procedures
considered in the present model, the overall flow may be considered steady, but locally it is
intrinsically unsteady; this transient state is locally taken into account by variation of the
tortuosity or fractal dimensions of the porous medium, in a time scale shorter than that of
the global averaged macroscopic flow.
2.2 Calculation of the effective permeability.
The wall shear stress in tortuous capillaries is given by:
tw dL
dpr2
−=σ (9)
Substitution of equation A5 into 9 yields
01
02
12 dL
pdLD
rTT
T
DT
D
D
w −−−=σ (10)
Defining a unit cell, the total stress on the wall can be calculated according to:
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−=−=
−−−
∫DfD
fTT
DDf
Dr
rw
TTTT
rr
DDDrLD
dLpdrdN
max
minmax1
02
0
max
min
1)(
2)(σσ (11)
The volumetric flow rate in tortuous capillaries of radius r may be expressed in the following form:
⎥⎥⎦
⎤
⎢⎢⎣
⎡−= −
−
−1
10
1
0
2)()( T
T
TD
DT
D
rLD
rdLdprq πϕ (12)
where )(rϕ is the fluidity, that may be expressed as:
'')()(0
'
0
drrdrr r
∫ ∫⎭⎬⎫
⎩⎨⎧
= ξξξϕϕ (13)
Equation 6 indicates that the volumetric flow rate decreases due to the tortuous path in the
capillaries. In the particular case of straight capillaries, DT = 1; moreover, if the fluid is Newtonian
0)( ϕξϕ = . In this case, equations 12 and 13 reduce to the Poiseuille law:
0
4
0 8ϕπ r
dLdpq −= (14)
The total volumetric flow rate through a unit cell may be calculated by performing the sum of the
individual capillaries:
∫−=max
min
)()(r
r
rdNrqQ (15)
Substituting equations A2 and 12 into equation 15 gives
drrrDL
rDdLdprdNrqQ
r
r
r
r
DfD
TD
Dff
DT
T
T
∫ ∫ −−−
−
⎥⎥⎦
⎤
⎢⎢⎣
⎡−=−=
max
min
max
min
21
0
max1
0
)(2
)()( ϕπ
(16)
in which the ratio 0)/( maxmin ≈Dfrr .
To calculate the average velocity of the flow in porous media (namely, the superficial velocity), the
total flow rate should be divided by the total transversal area normal to the flow direction:
00 // LVQ
LVQ
AQv
pt φ=== (17)
where Vp is the total volume of the pores, Vt = AL0 andφ (= Vp / Vt) is the porosity. The total volume
of the pores/capillaries is then:
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−=−=
−−−−∫
DfD
fT
DDD
f
r
rtp
TTTT
rr
DDL
rDdNLrV3
max
min01
3max
max
min
2 13
2ππ (18)
and hence the superficial velocity is given by
drrrrr
rLDDD
dLdpv
r
r
DfDDfD
DfDDT
fTD
T
Y
TT
T
∫ −−
−−−
−−−
−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−=
max
min
2
13
max
min3
max22
0
22
0
)(1)3(2
ϕφ
(19)
Darcy’s law may be written in terms of the non-Newtonian fluidity as follows:
0
)(dLdpkv e σϕ−= (20)
and as a result, the effective permeability is obtained:
drrrrr
rLDDD
kr
r
DfDDfD
DfDDT
fTD
eT
Y
TT
T
∫ −−
−−−
−−−
−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−=
max
min
2
13
max
min3
max22
0
22
)()(
11)3(2
ϕσϕ
φ (21)
The Newtonian permeability may be obtained as a particular case when
04
0 )(,8/)( ϕσϕϕϕ == rr , yielding:
31
8)3(2 2
max
13
max
min22
0
22
+−⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−=
−−−
−
−
fT
DDfD
DT
fTD
e DDr
rr
LDDD
kTY
T
T φ (22)
In straight capillaries, DT =1, obtaining
f
Dff
e Dr
rrD
k−⎥
⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
−−
41
8)2( 2
max
12
max
minφ (23)
This equation agrees with those of the current literature for Newtonian fluids.
2.4 Analytical Approximation.
From equation 21, it is possible to obtain an analytical result for the permeability if )(rϕ in equation
13 can be calculated. This may be achieved assuming expressions for )(ξϕ with physical content.
Several expressions may be proposed on the basis that the fluidity in the pore/capillary attains a
minimum in the center of the geometry and describes a maximum at the walls. The non-linear
analytical expression that conveys with those limits assumes that the fluidity is a quadratic function
inside the capillary. This is:
200 ))(()( ξϕσϕϕξϕ −+= (24)
According to equation 24, the fluidity attains a minimum at the capillary center with value 0ϕ .
Similarly, the fluidity attains a maximum at the walls; with value ).(σϕ The fluidity at the wall
requires the calculation of the wall stress according to equation 10. Substituting equation 24 into
equation 13, we obtain )(rϕ . Next, )(rϕ is substituted into equation 13 to obtain:
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+−−
++−⎥
⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−=
+−−−
−−−
−
)5(24)(
)3(8)(11
)3(2 22max0
2max0
13
max
min3
max22
0
22
fT
D
fT
DDfD
DfDDT
fTD
e DDr
DDr
rr
rLDDD
kTTY
TT
T ϕσϕϕσϕ
φ
(25)
When 0)( ϕσϕ → , next to the capillary center, the permeability tends to the Newtonian constant
value given by equation 22. Near the wall, )(σϕ tends to a maximum value and the permeability
diminishes asymptotically to another constant value as a function of the maximum fluidity.
Calculation procedure.
The calculation of the permeability in equation 25 requires porosity data and mean radius of the
porous media particles, these data must be obtained from geophysical measurements. It is then
possible to calculate the microstructural parameters (rmax, rmin and L0) and the fractal dimensions (Df
and DT) from the known porous media relationships (Li et al., 2008). The three constants of the
model in equation 8 are readily obtained from the experimental flow curve.
Given a pressure gradient which generates the flow, equation 11 evaluates the total stress and the
fluidity is calculated using equation 8; the permeability is then calculated through equation 25.
For small values of the stress, 0)( ϕσϕ → and equation 25 tends to equation 22, resulting in a
Newtonian permeability. Upon increasing the pressure gradient to high values, ∞→ϕτϕ )( , and
according to equation 25, an asymptotic region is reached with lower values than those of the
Newtonian permeability.
Conductivity is defined as )(σϕek given in equation 20, as the slope of the plot of Darcy’s velocity
versus pressure gradient. For low values of the applied pressure gradient, once again, 0)( ϕσϕ →
and the slope tends to zero. For high values of the pressure gradient, ∞→ϕτϕ )( , and the
conductivity tends to a limiting slope. This is in agreement with experimental data.
3. RESULTS AND DISCUSSION
In this section we compare predictions of the proposed model with experimental available data and
with predictions from other models.
The influence of the yield behavior on the permeability is illustrated in Figures 1a and 1b. In Figure
1a, experimental data (Park, 1972) for 0.5 wt. % polyacrylamide solution is plotted with predictions
of the BMP model. This solution does not present yield stress, that is, the corresponding flow curve
(triangles) shows a smooth transition from the low-stress zone (where the solution presents a first
plateau of high Newtonian viscosity) to the high-stress zone (where the solution presents a second
plateau of low Newtonian viscosity), both characterized by constant slopes. These data have been
predicted by an Ellis model adequately (Li et al., 2008).
On the other hand, the BMP model presents a flow curve (continuous line) with a drastic transition
between both zones of Newtonian behavior, this transition takes place at a specific value of stress
(the yield stress); however, the inset shows that the BMP model predicts creeping flow with
constant viscosity (constant slope) for stresses lower than the yield value. In addition, the transition
to the second Newtonian plateau is smooth but, in contrast to the Ellis model, occurs in a small
stress interval (apparent yield stress). This comparison is intended to exhibit the difference between
a material with yield and that with no yield. As seen, the asymptotes at low and high pressure
gradients match, not in the region between both asymptotes.
In Figure 1b, a comparison with predictions from the model by Orgeas et al. (2007) for a
generalized Newtonian fluid in an anisotropic medium is presented. Permeability in this model is a
tensorial quantity and a Carreau-Yasuda viscosity function through a tri-dimensional array with
elliptic transversal section is used. The anisotropy of the macroscopic flow depends in a
complicated form on the coupling between the porous structure and fluid rheology. The inset in
figure 1b shows that the flow curve of the Orgeas et al model (triangles) presents, for small stress
values (smaller than the yield value), creeping flow with constant viscosity and, for intermediate
stress values (main figure), a drastic transition from the first Newtonian plateau to the second one.
This behavior is similar to that of a yield-stress material as the continuous line (BMP) shows.
Once more, the differences present in both predictions arise because the Carreau-Yasuda model
does not present yield stress as opposed to the BMP model.
In Figure 2, BMP model predictions of the superficial velocity as a function of the pressure gradient
for various yield stresses are shown. The superficial velocity is very small in the region of
viscoplastic behavior and as the pressure gradient identified with the yield stress is surpassed, the
velocity increases almost linearly in some cases. Expectedly, this increase is larger the lower is the
yield stress.
As pointed out, the BMP model predicts a real yield stress when 0ϕ = 0 and an apparent yield for
very small values of 0ϕ . The real yield stress implies a solid-like behavior in the limit when the shear
rate approaches zero, as in the Bingham viscoplastic model. In Figure 3, the Darcy plot is shown for
various values of 0ϕ . As the zero-shear rate fluidity approaches zero, the model predicts an
apparent yield reflected in the smooth transition from negligible velocities into a region of larger
slopes. This corresponds to the region of very small velocities in the model by Orgeas et al. In fact,
the yield behavior disappears as 0ϕ increases. On the other hand, a real yield region appears near
zero 0ϕ , where the onset for the flow region is depicted, a behavior alike to a critical percolation.
Darcy’s law (equation 20) shows a relation between superficial velocity and permeability; since
permeability depends on porosity (equation 21) the superficial velocity also depends on porosity;
this relation is presented in figure 4. A similar effect to that predicted in Figure 2 is found for various
values of the porosity shown in Figure 4. As porosity decreases, the behavior of the fluid is
equivalent to having larger yield stresses, inasmuch as the available space for flow decreases as
porosity diminishes. Notice, however, that the behavior exhibited in this Figure includes the
presence of yield stress (74.5 Pa).
The influence of the microstructural parameters and fractal dimensions on the superficial velocity is
illustrated in Figure 5a, 5b and 5c. An interesting behavior is depicted in the Darcy plot as the pore
radius ratio is varied in Figure 5a. For low pressure gradients, as the breath of radius distribution
increases, the velocity also increases. Past a critical pressure gradient, the behavior reverses: the
largest velocity is predicted for narrow size distributions. In Figures 5b and 5c the influence of the
tortuosity and that associated to the fractal dimensions of the pore volume are shown, respectively.
The limit DT =1 corresponds to straight capillaries or when the porosity tends to one. As the
tortuosity achieves lower values and tends to unity, the velocity increases and the critical yield point
diminishes, exhibiting the predicted relationship between tortuosity and yield point. The predicted
velocities as the pore volume fractal dimension increases are complicated, since the relationship is
non-linear and also that with the yield point, as observed in Figure 5c.
The predictions of the pressure gradient variation as a function of porosity and yield stress are
shown in Figures 6 and 7. In Figure 6, the pressure gradient diminishes asymptotically with
increasing porosity for various values of the tortuosity fractal dimension. Same results are obtained
for the pore volume fractal dimensions, so they are not reproduced here. As DT tends to one, the
pressure gradient is very small approaching an asymptotic value as the porosity tends to unity. In
the region of small porosity, the pressure gradient augments steeply for high tortuosity values.
These results are similar to those predicted for a Bingham fluid (Yun et al.,2008. see their Figure 4).
Figure 7 exhibits the linear relation existing between the pressure gradient and yield stress, for
various porosity values. The general tendency is that as porosity increases, the pressure gradient
diminishes for a given yield stress. The linear relationship is also predicted for a Bingham fluid (Yun
et al., 2008. see Figure 3) and it is in accord to the theoretical expression developed by Wang et al.
(2006), for heavy crude oils in porous media, i.e.,
ey kL
p8
20
φτ=Δ
(26)
For the region of constant ke, the slope of the pressure gradient-yield stress straight line should
increase with porosity. However, for variable ke, the slope may decrease, which is the effect we see
in Figure 7.
With the purpose to qualitatively compare the model predictions with other models, in Figures 8 we
plot the permeability as a function of porosity, for various values of the pressure gradient (a),
tortuosity fractal dimension (b) and yield stress (c); these figures underline the dependence of
permeability upon those properties. In general, the flow curve presents a region of low fluidity
( 0ϕ ) for small pressure gradients, usually named creeping flow. This region is followed by a yield
stress region (plateau in the stress), followed in turn by a region of high fluidity ( ∞ϕ ), for large
pressure gradients. In Figure 8a, for small pressure gradients, permeability increases
monotonically with porosity (pressure gradient is smaller than that corresponding to the yield stress)
in the creeping flow region (permeability has no maximum). As porosity increases, the fluid
percolates easily through the porous media. Nonetheless, for higher pressure gradients the
permeability describes a maximum due to the presence of the yield stress (see equation 25). Larger
pressure gradients lead to decreasing permeability due to the approach to the yield stress region.
This behavior of the permeability has not been predicted before. In fact, predictions for a power-law
fluid (Zhang et al., 2006) describe a monotonic growth for all power-law exponents and this growth
increases as the power-law exponent increases. Predictions by Zhang et al. (2006) do not include
variation of the permeability for various pressure gradients.
A qualitative similar behavior of the permeability with porosity is predicted for various values of the
tortuosity fractal dimension, as shown in Figure 8b. Maxima are predicted as the fractal dimension
tends to one (straight capillaries), and a monotonic increase in the permeability with porosity is seen
for larger values of the tortuosity. As the tortuosity increases, we can expect lower permeability for
larger porosity. This effect is due to the dominating influence of the tortuosity on any increase in
porosity. The maximum in permeability is again due to the approach to the yield stress region.
In this regime, the permeability decreases with increasing tortuosity.
Once again, the yield stress influences the magnitude of the permeability in the region of high
porosity, as illustrated in Figure 8c. For high values of the yield stress, permeability increases
monotonically because the material is not allowed to yield (or the creeping flow region is wide). For
small values of the yield stress, permeability shows a maximum due to the yielding of the material.
The decrease in the yield stress reduces the region of creeping flow, and therefore the influence of
the yield stress affects the permeability greater. A trend toward saturation is predicted for small yield
values.
Finally, to illustrate the variation of the permeability with the applied pressure gradient, Figures 9 (a-
d) describe the predictions as the microstructure (radius ratio, tortuosity) is varied and also the
resulting variation with yield stress and porosity change. In Figure 9a, permeability diminishes with
pressure gradient from a high value (corresponding to the first Newtonian fluidity) down to a lower
asymptotic value (corresponding to the plateau fluidity at high pressure gradients). The larger
permeability is predicted for small radius ratios, or a more narrow size distribution. This behavior is
similar to that predicted for different pore volume fractal dimensions and for this reason is not shown
here. The permeability variation with pressure gradient depends also on the tortuosity of the porous
medium. As seen in Figure 9b, for straight capillaries (DT = 1), the slope of the decreasing
permeability is steeper, diminishing for increasing tortuosity, i.e., the decrease is more gradual with
increasing tortuosity. Of course, in general, smaller permeability is predicted as the tortuosity of the
medium increases.
The microstructure of the porous medium is reflected in the macroscopic porosity, and as an
illustration, Figure 9c shows the variation of the permeability with pressure gradient as the porosity
is changed. The general behavior is similar to that exhibited in Figure 9b, this is, as the porosity
increases, curves similar to decreasing tortuosity are obtained. Two regions include a sudden
decrease for low pressure gradients followed by an asymptotic region of low permeability for larger
pressure gradients. The slope increases with increasing porosity. These results agree with the
behavior predicted for Ellis-type fluids (Li et al., 2008) for various power-law indexes. Likewise,
when the yield stress is changed, Figure 9d illustrates that the decrease in permeability with
pressure gradient is larger as the yield stress diminishes. These results are novel and to our
knowledge, they have not been discussed with the amplitude given here.
A further comparison with other models is provided in Figures 10a and 10b. Here, the permeability
is plotted with velocity for various yield stresses, for straight capillaries (Figure 10a) and when the
tortuosity dimension is 1.15 (Figure 10b). In Morais et al. (2009), a theoretical description given by
simulation of non-Newtonian flow through three dimensional disordered porous media is made,
wherein a Hershel-Bulkley model approximates the rheology. This model combines the effects of
Bingham and power-law behavior for a fluid. Results in this reference show maxima in the
permeability as a function of the Reynolds number for various yield stresses, and the maxima shift
to higher Reynolds numbers as the yield stress is increased. This is precisely what is predicted for
the BMP model (in this case the permeability is plotted with velocity) in Figures 10a and 10b. In
Figure 10a, the shifting with increasing yield stress is apparent, and for large yield stress a
monotonic increasing permeability is predicted. Similar results for increasing tortuosity is shown in
Figure 10b, with a pressure gradient of 30 kPa/m, with shifting maxima until a monotonically
increasing permeability is predicted at high yield stresses.
A useful correlation relates the change in permeability to a change in porosity in sandstones. For
high permeability formations, a good estimate relates the permeability to the porosity according to a
power law, mk φ∝ . Indeed, the correlation suggested by Labrid [10], i.e., 3)/(/ iikk φφ= is based
on the initial values of porosity and permeability ( iφ , ki) before an acid treatment of the formation. In
Figures 11a, 11b and 11c, we compare the BMP predictions (continuous lines) with the Labrid
correlation for various initial conditions. In Figure 11a, a quantitative agreement of both predictions
for high porosity and Df up to 1.5 is apparent, revealing that the cubic law agrees with the BMP
curves for several pore volume fractal dimensions. Similarly, in Figure 11b agreement is found for
increasing tortuosity of the formation. Finally, in Figure 11c, agreement is found in most porosities,
revealing the relation between initial conditions to various yield stresses.
5. CONCLUSIONS
In this work, a fractal model was developed for the effective permeability using the BMP (Bautista-
Manero-Puig) model in porous media. The BMP model accounts for the rheology and flow curve
exhibited by complex fluids, like the prediction of two Newtonian asymptotic regions at low and high
shear rates, a power-law region (shear-thinning and shear-thickening) and yield stresses (apparent
and real) under steady-state flows. Transient behavior of these fluids can also be accounted for.
Predictions of the permeability as a function of the microstructure parameters (radius distribution,
tortuosity fractal dimension and pore volume fractal dimension) and macroscopic quantities
(porosity, yield stress, applied pressure gradient) agree with previous descriptions based on the
power-law model, Carreau viscosity function, Bingham behavior and other models, like the Ellis
equation, and with available experimental data. Results show clearly the influence of yield stresses
on the permeability behavior. The expression obtained for the permeability can be used in
calculations related with fracturing fluids and with analysis of data from pressure decline curves,
since the input for the permeability calculation requires only the value of the applied pressure
gradient in addition to the porous medium characteristics.
6. ACKNOWLEDGEMENTS.
The authors thank the Mexican Council of Science and Technology (CONACYT), PEMEX and
Mexican Institute of Petroleum (IMP) for their permission to publish this work as a part of the project
of investigation Y.00106
Appendix A. Fractal scaling laws in porous media.
Here it is assumed that the porous medium consists in a set of capillaries, whose size and tortuosity
distributions follow the fractal scaling laws. The scaling relationship of the cumulative pore/capillary
number (N) each one with radius r is given by:
Df
rr
rLN ⎟⎠⎞
⎜⎝⎛=≥ max)( (A1)
where L is a characteristic length scale, rmax is the maximum pore/capillary radius and Df is the
fractal dimension. The number of capillaries with sizes within the interval r, r + dr is then
drrrDrdN DfDff
)1(max)( +−=− (A2)
The negative sign implies that the capillary number decreases with the increase in the capillary size,
such as –dN > 0; a property of fractal objects is that the number of capillaries tend to infinity as r
tends to zero. The total number of capillaries, from the smallest rmin to the largest rmax may be
obtained from equation A1 as:
Df
t rr
N ⎟⎟⎠
⎞⎜⎜⎝
⎛=
min
max (A3)
where Df (1 < Df <3) is the pore fractal dimension, and in two dimensions 1 < Df < 2.
The fractal scaling law for the tortuosity is:
TTT DDDt LrLL 0
110 )2( −−= (A4)
where L0 is the representative length and Lt is the tortuous length along the flow direction. DT is the
fractal tortuosity dimension (1< DT <2) in two dimensions, representing the convolutedness of
capillaries for fluid flow through porous media. Differentiating equation 12 with respect to L0 gives:
011
0 )2( dLDrLLd TDD
tTT −−= (A5)
The microstructural parameters of the porous media are may be evaluated according to (Li et al.,
2008):
( )φπ−
=132
0 RL (A6)
φφ−
=12
2maxRr (A7)
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−
+=+
φ
φ
121ln
ln2d
Df (A8)
0
2dRd =+
(A9)
In these relations φ is the effective porosity of the medium, R is the average radius of the cluster
and 0d is the minimum particle diameter within a cluster; this last parameter was not reported by Li
et al., hence, in this study fD was considered as a first fitting parameter.
Finally, tortuosity and tortuosity fractal dimension are given by (Li et al., 2008):
av
avT L
D
λ
τ0ln
ln1+= (A10)
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
+⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−+−+=
φ
φφφτ
11
411
11
11211
21
2
av (A11)
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
−1
max
minmax 1
12 fD
f
fav r
rrD
Dλ (A12)
( )φ−=
1224
min
max
rr
(A13)
Where λav and τav are the average pore size and the average tortuosity, respectively, and DT is the
tortuosity fractal dimension. As previously stated, Df is been used as a fitting first parameter
because the absence of a particular parameter needed in its evaluation; since there is a non lineal
relation between both fractal dimensions, in the present work DT is also used as a fitting parameter.
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Highlights • A fractal model is developed for the effective permeability using the BMP model.
• Predictions of permeability for the BMP model agree with those of other models.
• Results show clearly the influence of yield stresses on the permeability behavior.
• This expression for permeability can be used in calculations for fracturing fluids.
Figure captions
Figure 1.- (a) Superficial velocity as a function of pressure gradient. Experimental data of
polyacrylamide by Park (1972). Continues Line: BMP model predictions (σy=74.5 Pa, Df=1.79 and
DT=1.42). Data is predicted by the Ellis model (Li et al. (2008)), which has no yield stress. (b)
Predictions by the model of Orgeas et al. (2007) using a Carreau-Yasuda viscosity function and
those of the BMP model (σy=89.1 Pa, Df=1.30 and DT=1.10).
Figure 2.- Predictions of the BMP model. Darcy velocity as a function of the applied pressure
gradient for various yield stresses (�0=0.278 Pa-1 s-1, Df=1.79 and DT=1.42). Porosity data and
porous radius were taken from Li et al. (2008).
Figure 3.- Predictions of the Darcy velocity as a function of the pressure gradient for various values
of the zero shear-rate fluidity (σy=74.5 Pa, Df=1.79 and DT=1.42). Porosity data and porous radius
were taken from Li et al. (2008).
Figure 4.- Predictions of the superficial velocity as a function of the applied pressure gradient for
various values of the porosity (σy=74.5 Pa, Df=1.79 and DT=1.42). Porous radius were taken from Li
et al. (2008).
Figure 5.- Darcy plot of the superficial velocity as a function of the applied pressure gradient for
various values of the microstructure parameters (a) maximum/minimum radius ratio, (b) tortuosity
fractal dimension, (c) pore volume fractal dimension (parameters used σy=74.5 Pa, Df=1.79 and
DT=1.42).
Figure 6.- The starting pressure gradient is plotted as a function of porosity for various values of the
tortuosity fractal dimension. Predictions of the BMP model (σy=74.5 Pa and Df=1.79).
Figure 7.- Predictions of the starting pressure gradient versus yield stress, for various porosity
values (Df=1.79 and DT=1.42).
Figure 8.- Predictions of the permeability as a function of porosity, for various values of the (a)
applied pressure gradient (σy=74.5 Pa, Df=1.79 and DT=1.42), (b) toruosity fractal dimension
(σy=74.5 Pa and Df=1.79), and (c) yield stress (Df=1.79 and DT=1.15).
Figure 9.- Permeability versus applied pressure gradient for various values of the microstructure
parameters: (a) radius ratio, (b) tortuosity fractal dimension, and for various values of the porosity
(c) and yield stress (d) (σy=74.5 Pa, Df=1.79 and DT=1.42). As before, porosity data and porous
radius were taken from Li et al. (2008).
Figure 10.- Permeability as a function of velocity for various values of the yield stress. (a) Straight
capillaries (DT = 1), (b) DT = 1.15, (Pressure gradient = 10 kPa/m, Df=1.79 and DT=1.42).
Figure 11.- Comparison of BMP model predictions with the Labrid correlation with several initial
conditions. (a).- various pore volume fractal dimensions, (b).- various tortuosity dimensions and (c).-
various yield stresses (σy=74.5 Pa, Df=1.79 and DT=1.42).
