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Effects of petrophysical rock properties on tortuosity factor

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Journal of Petroleum Science and Engineering 48 (2005) 185 – 198
www.elsevier.com/locate/petrol
Effects of petrophysical rock properties on tortuosity factor
Attia M. Attia 1
Faculty of Petroleum Engineering-Suez Canal University-Suez, 43721, Egypt
Received 30 April 2004; received in revised form 26 April 2005; accepted 25 June 2005
Abstract
Tortuosity factor is an important parameter of formation resistivity factor calculations in the Archie formula, which is used to
predict water saturation. The objective of this work was to study the effects of petrophysical rock properties, i.e., amount of finegrains, porosity, cementation factor, formation resistivity factor,electrolyte concentrations and degree of brine saturation, on the
tortuosity factor using Berea and synthetic sandstones cores. This study also trying to formulate empirical correlations between
tortuosity factor and these studied petrophysical rock properties. The obtained results showed that the tortuosity factor is not a
constant value, but it varies largely according to many parameters such as were studied in the present article. It was found that it
increases as a result of decreasing the amount of fine grains, increasing formation resistivity factor, and cementation factor, and
decreasing both porosity and degree of brine saturation. Tortuosity obtained from electrical resistivity measurements is very close
to the tortuosity obtained from capillary pressure data. The analysis demonstrated that the correlations between the tortuosity
factor and the petrophysical rock properties would yield a strong relationship with most accurate coefficients.
D 2005 Elsevier B.V. All rights reserved.
Keywords: Tortuosity; Fine grains; Formation resistivity factor; Degree of brine saturation
1. Introduction
Carman (1939) as referenced by Adisoemarta et al.
(2000) shows that the microscopic flow path through
a porous media is approximately at 458 with respect to
the direction of the bulk ionic current through saturated unconsolidated material as:
Xa ¼ X =cos 45 ¼ 1:414 X :
ð1Þ
E-mail address: aattia2@Lsu.edu.
Academic Visitor at Petroleum Engineering Department, Louisiana State University, USA.
1
0920-4105/$ - see front matter D 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.petrol.2005.06.012
Where X a is the actual flow path length and X is
the thickness of porous media.
The higher value reflects a microscopic flow path
with a larger component perpendicular to bulk ionic
flow compared to the component parallel to flow. They
concluded that the minimum value of tortuosity factor
equal to one and the maximum value is 1.4, leasing the
tortuosity factor equal to one can be unfavorable to the
oil in place calculation. Also they found that for recalculation of water saturation by setting tortuosity factor
to a unity is between 5% less to 30% more than the real
assumed water saturation. In addition, Carman defined
the tortuosity factor is equal to the effective length of
186
A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198
the fluid flow path within the porous media X a over the
apparent length X. Tortuosity is the ratio between the
actual length of true flow paths to the theoretical length
of the sample or porous material.
Adisoemarta et al. (2000) reported that the tortuosity factor, which is equal to the square root of tortuosity, is a function of the average angle of electrical
movement with respect to the bulk fluid flow and
cementation factor (m) and is related to the flow area
difference between pore throat and pore body.
Hypothetically it is unfeasible to have tortuosity factor
less than one and must only advance to one for a nearly
linear ionic flow path through the porous media. Then
the minimum values of the tortuosity factor are equivalent to the theoretical length of porous media.
Koponen et al. (1996) defined tortuosity as a specific transportation mechanism, and it is evident that it
is a physical quantity not uniquely defined. They
constructed an empirical equation between porosity
and tortuosity as:
s ¼ 0:8ð1 UÞ þ 1:
ð2Þ
They reported that the idea of tortuosity is often
introduced in the context of solving the closure problem for moving in porous media by driving the
macroscopic transport equations in terms of average
quantities alone. A usual method of deducing the
appropriate form of the drag force between fluid and
the solid matrix is to use some simplified models of
the porous material, such as the capillary model and to
generalize the results for more realistic material. This
generalization may be attempted by introducing an
additional parameter that is supposed to take care of
the more complicated transportation path neglected in
the model.
Fellah et al. (2003) studied tortuosity and porosity
measurements in the pore space for packing of glass
beads. This technique for measuring porosity and
tortuosity is based on a time model of the direct and
inverse scattering problem for the propagation of
transient ultrasonic waves in a homogeneous isotropic
slab of porous material with a rigid frame.
Olny et al. (2001) reported an analytical solution for
describing tortuosity and typical length of permeable
material by acoustical measurements under the homogenization theory; wave propagation in permeable
materials is dependant upon the numerical geometrical
parameters. Normally, five parameters are used: por-
osity, static airflow resistance, tortuosity, and viscous
and thermal characteristic lengths. The three previous
parameters are normally complex to be measured with
present direct methods, for a wide range of materials.
The estimated process is based on the measurement of
dynamic density and compressibility, in order to separate viscous and thermal property.
Perkins et al. (1956) concluded that for a predetermination of the tortuosity, of fully brine saturation
sandstones, has shown that the formation resistivity
factor, F r, the tortuosity, t, and porosity are related by
Fr ¼ s2 =U:
ð3Þ
For partial saturation in the presence of more than
one phase in the sample, the resistivity factor is related
to the porosity, tortuosity and saturation as:
Fr ðSr Þ ¼ s2 =UTSr :
ð4Þ
In addition, concluded saturation exponent dnT may
be articulated by the tortuosity and apparent cross
section area of the electrolyte during electric current
stream, the Brine saturation exponent is a function of
resistivity ratio as follows:
Fr =Fr ðSr Þ ¼ Srn ¼ Ro =Rt :
ð5Þ
Wildenschild and Jensen (1999), Knight (1991)
concluded the relationships between tortuosity and
saturation of formation. Five homogeneous and three
heterogeneous sands are used in this study. Although
the tortuosity flow patterns, they found that the effective unsaturated hydraulic conductivity as well as the
retention curves for the three heterogeneous sands
were quite related, thus signifying that this type of
heterogeneous flow system can be treated as a corresponding homogeneous medium characterized by
effective parameters.
Revil et al. (1998) built up a new electrical conductivity equation based on Bussian’s model and
accounting for the special performance of ions in the
pore space. The tortuosity of the transport of anions is
self-sufficient of the salinity and corresponds to the
bulk tortuosity of the pore space, which is obtained by
the result of the formation resistivity factor and the
porosity. For the cations, the situation is dissimilar. At
elevated salinities, the main paths for the electro
passage of the cations are situated in the interconnected pore space, and the tortuosity for the transport
of cations is therefore the bulk tortuosity. As the
A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198
Salem (1993) concluded that tortuosity is a key
parameter controlling the modification in jointly
cementation factor and Kozeny–Carman constant
designated the difficult electric and hydraulic tortuous
passageway in the sediments. In addition, tortuosity
shows specific control on the variation of formation
resistivity factor ( F r), where porous media show additional resistance to electric current. The correlation
between cementation factor and tortuosity maybe suggested that the existence of very small pores force the
fluid to stream through a longer path (higher tortuosity) than when the fluid flows in larger pores or
fractures. The empirical equation of tortuosity with
cementation factor is:
m ¼ 0:91365 ln s þ 1:55514:
ð6Þ
One potential challenge is the difficulty associated
with attempting to extract a relationship among porosity, permeability, and particles diameter in this equation. Using a simple plot, Carman–Kozeny, estimated
permeability over different data ranges to study the
sensitivities associated with tortuosity, particle size,
and porosity. Their study indicates that the most significant parameter is porosity. Tortuosity is found to
have little effect. Permeability increases significantly
at the same time that porosity and particle size
increases. Porosity is the most sensitive to permeability; the large uncertainty in particle diameter creates
the parameter that controls the accuracy of permeability estimates from the Carman–Kozeny relationships.
Particle size also controls the permeability, while
porosity exhibiting slighter significance.
Kewen and Roland (2002) obtained the pore size
distribution index (k) where Purcell (1949) intro-
10
100% fines
Tortuosity Factor, t
salinity decreases, the main paths for transport of the
cations transfer from the pore space to the mineral
water boundary and subsequently are focus to different tortuosities. This transfer is function of salinity,
ratio between the surface conductivity of the grains
and the electrolyte conductivity. The electrical conductivity of grainy porous media is resolute as a
function of pore fluid salinity, temperature, water
and gas saturations, shale content, and porosity.
They found that the model provides an excellent detail
for the variation of electrical conductivity with last
parameters. Surface conduction at the mineral water
interface is described with the Stern theory of the
electrical double layer and is shown to be independent
of the salinity in shaly sands above 10 3 mol L 1.
Stephanie and Richard (1998) concluded that the
electrical resistivity of siliciclastic rock depends on
the pore fluid resistivity and formation resistivity
factor. They studied for low and high porosity. They
found that for low porosity sediments the formation
resistivity factor depends on clay conduction and
porosity. For high-porosity sediments (0.3–0.6), the
clay conduction is a minor effect, but controlled primarily by porosity and pore geometry. Porosity versus
formation resistivity factors for Amazon Fan well logs
show two separate trends that function on the amount
of shale present. Muds, with more than about 0.4
shale fractions, follow a trend that increases too
much higher formation resistivity factor while porosity decreases, compared to sands and a little muddy
sands. They concluded that in high-porosity muds,
existence of clays decreases the formation conductivity by increasing the tortuosity of pores.
Christian David (1993) concluded that from
numerical simulations of two-dimensional networks
used as analogs to pore space in porous rocks are
presented to emphasize the existence of preferential
paths for transport processes in heterogeneous media.
Results showed that hydraulic flow and electrical
current are generally determined in the so-called
bcritical pathsQ when the pore size distribution has a
declining exponential-like shape in opposition to
nearly homogeneous or uniform like distributions.
Shogenova et al. (2001) studied the electrical properties of 273 Cambrian quartz sandstones and concluded that the lithology and cementation are the main
factors controlling properties of Cambrian quartz
sandstones rock.
187
80% fines
60% fines
40% fines
1
0
0.2
0.4
0.6
0.8
Brine Saturation, Sr
1
1.2
Fig. 1. Tortuosity factor versus brine saturation with different
percentage of fine grains at consolidation pressure 20.7 Mpa.
188
A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198
2.5
10
At 5 % NaCL
Resistivity,Ro, OHm
Tortuosity Factor, t
100% fines
"-80% fines"
60% fines
40% fines
1
0
0.2
0.4
0.6
0.8
Brine Saturation, Sr
1
At 5 % NaCL(
2
)
1.5
Ro (5%) = -0.02(FG )+ 3.19
2
R = 0.96
1
0.5
1.2
0
Fig. 2. Tortuosity factor versus brine saturation with different
percentage of fine grains at consolidated pressure 27.6 MPa.
duced lithology factor index, is equivalent to inverse
of tortuositys. It is obtained from fitting the experimental data of the normalizing water saturation versus capillary pressure in order to determine relative
permeability.
El-Khatib (1995) suggested that the J-function, as
intended by Levertt, is not only one of its kinds for all
porous media. The integral of the reciprocal of the
square of the function over the saturation range is
dependent on the tortuosity that will be diverse for
the different formations. He concluded that the modified averaged petrophysical rock parameter, J-Function, depends on the tortuosity in term of porosity,
irreducible water saturation, and permeability for a
given formation. This modified relation, J-function,
can be used to estimate permeability from logs resulting in values of brine saturation and porosity.
Most of the previous works, concluded that tortuosity,is an important parameter for the recalculation of
water saturation, lithology factor index, relative permeability and J-function calculations. They also mentioned that the tortuosity is a key parameter
0
20
40
60
80
Amount of fine grains,FG, %
100
120
Fig. 4. Effect of percentage fine grains in the samples on the
resistivity measurements at fully saturation with 5% NaCL.
controlling the modification in jointly cementation
factor. However, They did not consider the effects
of amount of fine grains, pressure consolidation and
electrolytes concentrations on the tortuosity factor.
The objective of the present work is to study the
effects of the amount of fine-grains, pressure consolidation, porosity, cementation factor, formation resistivity factor,electrolyte concentrations and degree of
brine saturation on the tortuosity factor using Berea
and synthetic sandstones cores. Moreover, in this
study it was attempted to formulate empirical correlations between tortuosity factor and the petrophysical
rock properties. The output of these correlations
would help to improve well log interpretation.
2. Results and discussion
The results of the present study are demonstrated
graphically in Figs. 1–29. These figures show the
effects of the studied parameters on the tortuosity fac2
10
Tortuosity Factor, t
100% fines
Tortuosity factor, t
80% fines
60% fines
40% fines
1
0
0.2
0.4
0.6
0.8
1
1.2
Brine Saturation, Sr
Fig. 3. Tortuosity factor versus brine saturation with different
percentage of fine grains at consolidated pressure 34.5 MPa.
1.6
t = -0.01(FG) + 2.02
2
R = 0.94
1.2
0.8
t-fines
t-fines(
0.4
0
0
20
)
40
60
80
Amount of fine grains,FG,%
100
120
Fig. 5. Effect of percentage fine grains in the sample on the
tortuosity factor at 5% NaCL.
A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198
100
10
20.7 MPa
Tortuosity Factor, t
Tortuosity Factor, t
20 MPa
27.6 MPa
34.5 MPa
10
1
0.01
189
0.1
27.6 MPa
34.5 MPa
1
0.1
1
1
Brine Saturation, Sr
Brine Saturation, Sr
Fig. 6. Effect of consolidation pressure on the tortuosity factor at
40% fine grains.
Fig. 8. Effect of consolidation pressure on the tortuosity factor at
100% fine grains.
tor: these studied influencing parameters are the
amount of the fine grained sands in the core (by
%),the cementation factor,the consolidation pressure,
the degree of saturation, the formation resistivity factors and the salinity of brine water. Tables 1 and 2
shows the summarized results of the petrophysical
rock properties for Berea and synthetic sandstone
cores, respectively. Presentation and discussion of the
obtained results are given in the following separate
heading.
MPa. Various brine water salinity, which varies from
0.2% to 5% NaCL. Inspection of these figures shows
that the same trend is given in all these figures,i.e.,
decreasing the values of tortousity factor with increasing the percentage of fine materials in the cores. Fig. 1.
which shows the effect of the percentage of fine grains
at consolidation pressure of 20.7 MPa, and various
brine saturation values. It indicates that the amount
of fines in core has a great influence on the tortuosity
factor, however,changing the concentration of the
brine has to a little extent,some influence at the 20.7
MPa consolidation pressure. Figs. 2 and 3 shows that
increasing the pressure to 27.6 or to 34.5 MPa, the
tortuosity factor will decrease significantly by increasing the brine saturation, however, the effect of the
percent of fine grains of sands in the n core still as it
stated above, i.e., tortuosity increases with decreases of
the percent of fines in the core. It is clear that with
decreasing percentage of fines, the tortuosity factor
increases for all synthetic cores. In addition, the
increase is also owing to the surface conduction area,
2.1. Effect of amount of fine grains on tortuosity
factor
The results are shown in Figs. 1–3 show the effects
of the amount of fines in the core specimens on tortuosity factor at various consolidation pressures. The
percentage of fine sand in the core were changed to
give different cores having 40%, 60%, 80% and 100%
fines. These cores were consolidated under three different consolidation pressure: 20.7, 27.6 and 34.5
10
2.5
20.7 MPa
Tortuosity Factor, t
Tortuosity Factor, t
27.6 MPa
34.5 MPa
t = 0.36Fr0.56
R2 = 0.92
2
1.5
1
1
0.1
1
Brine Saturation, Sr
Fig. 7. Effect of consolidation pressure on the tortuosity factor at
60% fine grains.
5
10
15
20
25
30
Formation Resistivity Factor, Fr
Fig. 9. Empirical correlation between tortuosity factor and formation
resistivity factor for Berea sandstones cores at 5% NaCL.
190
A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198
2.5
Tortuosity Factor, t
Tortuosity factor, t
2
1.5
t = 0.87Fr0.16
R2 = 0.97
1
0.5
5.5
10.5
15.5
20.5
2
25.5
Formation Resistivity Factor, Fr
Fig. 10. Empirical correlation between formation resistivity factor
and tortuosity factor for synthetic cores at 5% NaCL.
because the conductivity of the matrix is equal to the
conductivity of brine plus conductivity of the surface
conduction area. At this point, the conductivity of
brine is constant; however, the percentage of fines
varies from 40% to 100 %. The conductivity is affected
by the percentage of fine grains; Fig. 4. shows that the
resistivity increases as the percentage of fines
decrease. This is due to a decrease in surface transmission; therefore, the tortuosity increases as shown in
Fig. 5. In addition, Figs. 6–8 the tortuosity increases as
compaction increases, owing to porosity decreases and
cementation factor increases. This change in the porosity and cementation factor is due to the rearrangement of grains as pressure increases. Fines in the
sample, with pressure help to increase the cementation
factor and create a longer path (higher tortuosity), with
narrow pores. This behavior is very clear at lower brine
saturation (close to irreducible water saturation zone).
A best fit is created for the results between percentage of fine grains in samples and true resistivity,
formation resistivity factor and tortuosity factor. The
14
Fr = -0.15(FG) + 19.79
2
R = 0.96
12
10
8
6
4
1.8
1.82
1.84 1.86 1.88
1.9
Cementation Factor,m
1.92
1.94
Fig. 12. Relation between tortuosity factor and cementation factor at
5% NaCl for Berea sandstone cores.
empirical equations for each one relation were
obtained as follow:
For synthetic sandstones cores:
Rt ¼ 0:02FG þ 3:19
ð7Þ
Fr ¼ 0:15FG þ 19:79
ð8Þ
s ¼ 0:01FG þ 2:02:
ð9Þ
Where F G represents the percentage of fine grains
in samples.
2.2. Tortuosity and Formation resistivity factor
relationship
The effects of formation resistivity factor on tortuosity for Berea and synthetic cores were studied. Fig.
9. for Berea sandstones and Fig. 10. for synthetic cores
represent the different percentages of fines as grain
size increases. These figures show the influence of the
formation resistivity factor in fully saturated brine
samples on tortuosity. It is clear that with an increasing
formation resistivity factor the rock becomes more
Tortuosity Factor, t
16
1.33
2
R = 0.9619
1.5
1.78
0
0.5
Formation resistivity factor,Fr
t = 0.90 m
2
1.5
0.99
t = 0.85 m
2
R = 0.91
1
2
0
0
20
40
60
80
100
120
Amount of Fines grains,FG,%
Fig. 11. Effect of percentage of fine grains on the formation resistivity factor.
0.5
0.5
1
1.5
Cementation Factor,m
2
Fig. 13. Relation between tortuosity factor and cementation factor at
5% NaCl for synthethic cores.
A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198
6
Tortuosity , T
Tortuosity Factor, t
5
5
T = 1.94Sr -0.43
R2 = 0.92
4
3
2
T-Sr
T-Sr(
1
0.3 %
1%NaCL
5%NaCL
4
3
2
1
0
)
0
0
0
0.2
0.4
0.6
0.8
Fig. 14. Effect of degree of brine saturation on the tortuosity for
Berea sandstone specimens at 5% NaCl concentration.
resistance to electric current. Tortuosity therefore
increases with more tortuous passages. For synthetic
cores, Fig. 11 shows the correlation between formation
resistivity factors and percentage of fines in the samples. It is clear from this figure that the formation
resistivity factor increases when the percentage of
fines is reduced; therefore, irregularity strongly affects
the formation resistivity factor and tortuosity (Fricke,
1931). In addition, a best fit is created between tortuosity and formation resistivity factor. Thereby, one can
find the empirical equations for relationships such as:
For Berea sandstones cores:
s ¼ 0:36Fr0:56 :
ð10Þ
For synthetic sandstones cores
s ¼ 0:87Fr0:16 :
ð11Þ
2.3. Tortuosity and porosity exponent (cementation
factor) relationship
T Vs Sr
10
)
-0.55
T = 1.54 Sr
2
R = 0.80
1
1.2
Figs. 12 and 13 show that the tortuosity factor
increases with increasing cementation factor (porosity
exponent) for Berea and synthetic cores, respectively.
Cementation may have an influence; therefore, the
electric flow expands and the effectiveness of electric
current path decreases. Tortuosity, therefore, increases
as cementation increases.
For synthetic cores the cementation factor can
indicate the type of rock according to grain size. For
studying each group, experiments show that the porosity and cementation factor increase as grain size
decreases, as in Table 3. As the percentage of fine
grains decrease, from 100% to 40% the degree of
sorting decrease close to poor sorting and very close
to irregularity grains which means the grains become
close to each other and consequently the pores
become slighter or even bunged. This improves the
direct interaction between cementation factor and tortuosity. The cementation factor increases as pores
decrease. Subsequently, the forces push the fluid,
thus increasing flow. Consequently, tortuosity is
5
Tortuosity Factor, t
20
0.4
0.6
0.8
Brine Saturation, Sr
Fig. 16. Effect of salinity on the tortuosity–saturation curve at 100%
fine grains and consolidation pressure 20.7 Mpa.
The relation between tortuosity factor and porosity
exponent for Berea and synthetic cores was studied.
T Vs Sr(
0.2
1
Brine Saturation, Sr
Tortuosity , T
191
0.30%
4
1%NaCL
3
5%NaCL
2
1
0
0
0
0.1
0.2
0.3 0.4 0.5 0.6 0.7
Brine Saturation , Sr
0.8
0.9
1
Fig. 15. Effect of degree of saturation on the tortuosity using
synthetic cores.
0
0.2
0.4
0.6
0.8
1
1.2
Brine Saturation, Sr
Fig. 17. Effect salinity on the tortuosity–saturation curve at 100%
fine grains and consolidation pressure 27.6 MPa.
192
A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198
8
10
8
1%NaCL
6
5%NaCL
Tortuosity Factor, t
Tortuosity Factor, t
0.30%
4
2
0
0
0.2
0.4
0.6
0.8
Brine Saturation, Sr
1
0.3 % NaCL
1%NaCL
5%NaCL
6
4
2
0
1.2
0
0.2
0.4
0.6
0.8
1
1.2
Brine Saturation, Sr
Fig. 18. Effect salinity on the tortuosity–saturation curve at 100%
fine grains and consolidation pressure 34.5 MPa.
enhancing, as fluid flows through a longer tortuous
path, compared to the path that fluid follows in a more
porous rock. Because of the increased force, the fluid
flows through a longer path, smaller pores (higher
tortuosity) than when the fluid flows in larger pores.
This phenomenon is related to pore size and pore area
in porous media. In addition best fit is created for the
results between the tortuosity and porosity exponent
or shape factor (cementation factor). The best empirical equations obtained are as follows:
For Berea sandstones cores:
s ¼ 0:90m1:33 :
ð12Þ
For synthetic sandstones cores:
s ¼ 0:93m 0:13:
ð13Þ
Moreover, the relation between tortuosity factor
and porosity for twelve synthetic cores is:
s ¼ 2:03U þ 1:78:
ð14Þ
Fig. 20. Effect of salinity on the tortuosity–saturation curve at 80%
fine grains and consolidation pressure 27.6 Mpa.
2.4. Degree of saturation and Tortuosity relationship
It is necessary to study the relation between tortousity factors and fluid saturation. With reduced fluid
saturation, the saturation of both wetting and nonwetting phase’s changes, as does the water saturation
exponent. There is an independent relation between
tortuosity and the degree of fluid saturation. Nevertheless, tortuosity is very important for explaining the
change in wettability using a water saturation exponent
and is useful for log interpretation. Figs. 14 and 15 for
Berea and synthetic cores respectively show that as
soon as the degree of brine saturation reduces, there
is an increase in tortuosity which means an increase in
electrical resistance of the core samples. Due to a
decrease in cross-sectional area of the electrolyte,
there is an increase in electrical path length. Therefore,
at surface water saturation (irreducible water saturation) the cross-sectional area is reduced to the value of a
fully saturated zone, although the tortuosity increases.
At high brine saturation, the water in the pore space
becomes highly conductive and the surface conduction
8
8
1%NaCL
6
Tortuosity Factor, t
Tortuosity Factor, t
0.3 NaCL
5%NaCL
4
2
0
0.3 % NaCL
1%NaCL
5%NaCL
6
4
2
0
0
0.2
0.4
0.6
0.8
1
1.2
Brine Saturation, Sr
Fig. 19. Effect of salinity on the tortuosity–saturation at 80% fine
grains and consolidation pressure 20.7 MPa.
0
0.2
0.4
0.6
0.8
Brine Saturation, Sr
1
1.2
Fig. 21. Effect of salinity on the tortuosity–saturation curve at 80%
fine grains and consolidation pressure at 34.5 Mpa.
A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198
100
0.3 % NaCL
1%NaCL
5%NaCL
4
Tortuosity Factor, t
Tortuosity Factor, t
5
193
3
2
1
0.3 % NaCL
1%NaCL
5%NaCL
10
1
0
0
0.2
0.4
0.6
0.8
1
0.1
1.2
0
Brine Saturation, Sr
Fig. 22. Effect of salinity on the tortuosity–saturation curve at 60%
fine grains and consolidation pressure at 20.7 Mpa.
of brine/air interface is very small, relative to the total
conductivity of the rock. Hence, tortuosity is small
when brine saturation is high, due to the higher efficiency of the current flow. This is confirmed by Kozeny
–Carman equation with decreasing the specific surface
area led to increasing the tortuosity. With decreasing
the brine saturation the surface area of the air/water
interface increases; however, this increases the amount
of insulator in the porous media, consequently decreasing the conductivity of the rock and also the efficiency
of electric current flow through the porous media;
hence, the tortuosity increases. These processes continue with the decrease of saturation (gas extended),
thus, increasing air/brine interface and decreasing conductivity; hence, tortuosity increases as the efficiency
of electric flow path decreases, Knight (1991). This
process continue until no decrease in brine saturation
while the cores having a small amount of brine saturation, this brine exists in the pore space as thin film of
brine covered the surface of the rock (surface water)
defined as connate water saturation. At this stage, the
nonwetting phase saturation (insulator) is very high
0.2
0.4
0.6
0.8
Brine Saturation, Sr
consequently smallest conductivity; hence, the maximum value of tortuosity factor at the smallest conductivity (highest resistivity value) is obtained.
In addition, a best fit is created for the results
between tortuosity and degree of saturation. Moreover, empirical equation is given as follows:
For Berea sandstones cores:
s ¼ 1:94Sr0:43 :
ð15Þ
For synthetic sandstones cores:
s ¼ 1:54Sr0:55 :
ð16Þ
2.5. Tortuosity and electrolyte concentrations
relationship
The relation between tortuosity factor and electrolyte concentration is an autonomous relationship as
shown in Figs. 16–27. It’s clear that tortuosity factor
increases with salinity increase especially for all cores
having amount of fines less than 100%. Notice that
8
Tortuosity Factor, t
Tortuosity Factor, t
10
0.3 % NaCL
1%NaCL
5%NaCL
6
4
2
0
1
0
0.2
0.4
0.6
0.8
Brine Saturation, Sr
1
1.2
Fig. 23. Effect of salinity on the tortuosity–saturation curve at 60%
fine grains and consolidation pressure at 27.6 Mpa.
1.2
Fig. 24. Effect of salinity on the tortuosity–saturation curve at 60%
fine grains and consolidation pressure at 34.5 Mpa.
100
0.3 % NaCL
1%NaCL
5%NaCL
1
0
0.2
0.4
0.6
0.8
1
1.2
Brine Saturation, Sr
Fig. 25. Effect of salinity on the tortuosity–saturation curve at 40%
fine grains and consolidation pressure at 20.7 Mpa.
A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198
16
0.3 % NaCL
1%NaCL
5%NaCL
6
4
2
0
0
0.2
0.4
0.6
0.8
Brine Saturation, Sr
1
1.2
Formation resistivity Factor,Fr
Tortuosity Factor, t
8
10
12
8
10
8
6
2
6 Ro = -10.39Rw +2 20.986Rw - 0.8425
R =1
Salinity-resistivity
4
F-salinity
Salinity-resistivity. (
2
F-salinity. (
)
4
)
0
0
Fig. 26. Effect of salinity on the tortuosity–saturation curve at 40%
fine grains and consolidation pressure at 27.6 Mpa.
12
Fr = -6.3024Rw2 + 2.9813Rw + 13.764
14
R2 = 1
2
Core Resistivity,Ro,OHm
194
0
1.5
0.5
1
Brine resistivity,Rw,OHm
Fig. 28. Effect of salinity on the resistivity and formation resistivity
factor.
the tortuosity factor increases in conjunction with the
electrolyte concentration, until the level of concentration becomes less effective with increasing salinity
concentrations. In fact, the conductivity of the rock
is equal to the conductivity of electrolyte concentration, plus the conductivity of the surface area in
question, which is constant while the conductivity of
brine changes; thus, the conductivity of the rock
depends on the salinity concentration. As the conductivity of electrolyte increases the formation resistivity
factor decreases; thus, resistivity decrease. Fig. 28
shows the relation between formation resistivity factors and electrolyte resistivity. It is clear that the
formation resistivity factor increases with increasing
resistivity (decreasing electrolyte concentrations).
2.6. Tortuosity from capillary pressure and from electrical resistivity
methods such as, capillary pressure, acoustic properties and electrical properties of core samples. Fig. 29.
Show the tortuosity from capillary pressure and tortuosity obtained from electrical measurements. It is
clear from this figure that similarity for results but the
accuracy is not high. The results from electrical properties are more accurate because these are measured
data. However, the tortuosity from capillary pressure
depends on the slope of the capillary pressure versus
normalized water saturation.
2.7. Comparison between the correlations in the literature and correlations obtained in this article
Figs. 30–34 show that the comparison between the
correlations which were mentioned in the introduction
in the present study and those obtained from resistiv-
Tortuosity is very important in fluid flow through
porous media. Tortuosity can be obtains by different
6
5
0.3 % NaCL
1%NaCL
5%NaCL
Tortuosity
Tortuosity Factor, t
100
10
4
3
2
electrical
1
Capillary pressure
0
1
0
0.2
0.4
0.6
0.8
Brine Saturation, Sr
1
1.2
Fig. 27. Effect of salinity on the tortuosity–saturation curve at 40%
fine grains and consolidation pressure at 34.5 Mpa.
21 23 24 30 31 32 33 34
3.01 4.35 4.26 3.78 4.7 4.04 3.82 4.48
electrical
Capillary pressure 3.84 4.75 3.87 3.61 3.6 4.02 4.95 4.33
Core #
Fig. 29. Tortuosity obtained from capillary pressure and obtained
from electrical properties.
A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198
195
Table 1
Summarized results of porosity, formation resistivity factor, tortuosity, brine saturation exponent for Berea sandstone cores
Core
no.
21
23
24
30
31
32
33
34
Porosity
Tortuosity, T
0.19
0.19
0.186
0.187
0.181
0.182
0.192
3.84
4.35
4.26
3.78
4.71
4.04
3.82
4.48
Tortuosity
factor, t
Cementation
factor, m
Formation resistivity
factor, F r
Water saturation
exponent, n
2.09
2.06
1.94
2.17
2.01
1.95
2.12
1.89
1.88
1.79
1.92
1.82
1.79
1.91
22.78
22.21
20.34
25.2
22.34
21
23.34
2.12
1.8
2.42
1.93
1.98
1.62
2.02
1.96
Table 2
Physical properties of the synthetic cores
Specimen
name
Porosity, /
2
6
14
18
22
28
32
38
42
48
54
58
0.26
0.29
0.29
0.22
0.27
0.25
0.19
0.24
0.22
0.20
0.20
0.19
Permeability,
K air (mD)
Consolidation
pressure (MPa)
Grain Size (%)
Quartz
flour
From 0.126
to 0.149 mm
From 0.149
to 0.177 mm
From 0.177
to 0.210 mm
48
37
32
68
49
40
70
55
38
333
132
118
20.7
27.6
34.5
20.7
27.6
34.5
20.7
27.6
34.5
20.7
27.6
34.5
100
100
100
80
80
80
60
60
60
40
40
40
0
0
0
6
6
6
12
12
12
18
18
18
0
0
0
6
6
6
12
12
12
18
18
18
0
0
0
8
8
8
16
16
16
24
24
24
Table 3
Petrophysical rock properties for synthetic cores
Core
no.
Fine grains,
(%)
Porosity
Formation resistivity
factor at 5% NaCl, F r
Water saturation exponent
at 5% NaCl, n
Cementation factor
at 5% NaCl, m
Tortuosity factor
at 5% NaCl
2
6
14
18
22
28
32
38
42
48
54
58
100
100
100
80
80
80
60
60
60
40
40
40
0.26
0.29
0.29
0.22
0.27
0.25
0.19
0.24
0.22
0.20
0.20
0.19
5.6
7.1
7.5
6.8
9.8
10.2
11.1
13.8
15.2
14.1
21.1
18.6
1.7
1.8
2.2
1.7
2
1.7
1.7
2
1.6
2.1
2.2
1.8
1.3
1.5
1.6
1.3
1.6
1.6
1.5
1.6
1.5
1.6
1.7
1.7
1.31
1.44
1.48
1.38
1.61
1.61
1.64
1.81
1.83
1.71
2.06
1.86
A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
2.2
Tortuosity factor
Tortuosity predicted
196
tortuosity presented in this work
Tortuosity by Koppen (1996)
Maximum value by Carman
1.4
Minimum value by Carman
1
0.6
0.26 0.29 0.29
0.27 0.25
0.24 0.22 0.2
Porosity
0
0.19
Fig. 30. Comparison between tortuosity calculation by Koponen and
tortuosity presented in this article.
ity measurements data. Fig. 30 shows that the comparison between tortuosity factor that was predicted
by Koponen et al. (1996) and tortuosity factor that
was obtained in the present study. It is clear from this
figure that the values of tortuosity factor for both
correlations were very close to each other.
Figs. 31 and 32. show that, the difference between
tortuosity factor according to Carman ( which ranges
from 1 and 1.4 ) and tortuosity factor obtained in the
present study. It is clear that from this comparison
that, for Berea sandstones cores, these values of tortuosity factor were more than the maximum value
(1.4) that predicted by Carman. Fig. 32. shows that
some of these values of tortuosity factor for synthetic
cores (compacted cores) are in the interval of Carman
range and others values found were above the maximum level (1.4) of Carman. These results showed
that tortuosity factor is not a constant value, but it
varies largely according to type of rocks and petrophysical rock properties.
2.6
20
40
Cores #
60
80
Fig. 32. Comparison between tortuosity obtained from Carman and
obtained from presented in this article using synthetic sandstones
cores.
Fig. 33 Shows a good agreement between tortuosity factor obtained according to Eq. (11) and tortuosity
factor obtained from resistivity measurements. Fig. 34
Represent the relations, between tortuosity factor
obtained by Salem (1993) and those obtained in the
present article, using Berea and synthetic cores. It is
clear from the figure, that a variation between the
results from these correlations. This may be due to;
the difference of core preparations, amount of fine and
consolidation pressure in synthetic cores.
3. Conclusions
1. For all cores tested, the results showed that tortuosity factor is not a constant value, but it varies
largely according to many parameters such as studied in the present article.
2. Tortuosity factor increases as a result of decreasing
the amount of fine grains, increasing formation
resistivity factor, consolidation pressure and cementation factor.
2.2
5
Tortuosity factor
predicted,tp
Tortuosity factor
Tortuosity factor
of Synthetic cores
1.8
1.8
Maximum value by Carman
1.4
Minimum value by Carman
1
4
1:1
tp vs. tm
3
2
1
0
0.6
0
10
20
30
40
0
1
2
4
3
5
Tortuosity Factor from electrical measurements ,tm
Cores #
Fig. 31. Comparison between tortuosity obtained from Carman and
obtained from presented in this article using Berea sandstones cores.
Fig. 33. Validation of tortuosity factor from formation resistivity
factor according to the empirical equation (t = 0.87F0.2
r ) for synthetic
cores at 5% NaCL.
A.M. Attia / Journal of Petroleum Science and Engineering 48 (2005) 185–198
Tortuosity factor
2.5
k
U
2
197
Lithology factor index
Porosity
1.5
1
Acknowledgments
" Berea"
" synthetic cores"
0.5
0
"Tortuosity predicted by
Salem(1993)
1.21 1.41 1.43 1.21 1.54 1.45 1.35 1.46 1.42 1.49 1.55 1.50
Cementation factor
Fig. 34. Comparison between tortuosity factor obtained by Salem
(1993) and presented in this article as a function of cementation
factor.
3. Tortuosity factor increases with decreasing both
porosity and degree of brine saturation.
4. Tortuosity factor increases with electrolyte concentration increase for all cores having different amount
of fine-grains.
5. Tortuosity factor obtained from electrical resistivity
measurements very close to the tortuosity factor
obtained from capillary pressure data.
6. New correlations for the tortuosity factor and the
petrophysical rock properties such as porosity,
cementation factor and formation resistivity factor
have been developed to cover a wide range of the
amount of fine grains in the samples.
7. The correlations between the tortuosity factor and
petrophysical rock properties yielded the strongest
relationship with most accurate coefficients.
Nomenclature
FG
Percentage of fine grains in samples
Fr
Formation resistivity factor
J
Average rock properties ( J-Function)
m
Cementation factor
n
Brine saturation exponent
Rw
Water resistivity, V.m
Rt
True rock resistivity, V.m
Ro
Formation resistivity when 100%
saturated with brine, V.m
Sr
Brine saturation
T
Tortuosity
s
Tortuosity factor
Xa
Actual flow path length
X
Thickness of porous media
Sincere appreciation to Dr. Zaki Bassiuoni for his
valuable advice and helpful discussions. I would like
to thanks Dr. Hamid Khattab and Dr. Amen AbdElraheem for useful discussions and review this
paper. A special thanks to Dan Lawrence for his
helping me during experimental work. The author
acknowledges the support of the Department of Petroleum Engineering at the Louisiana State University
and A and M College.
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