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Determination Techniques of Archie’s Parameters: a, m and n in

Heterogeneous Reservoirs

Article  in  Journal of Geophysics and Engineering · July 2017

DOI: 10.1088/1742-2140/aa805c

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 Water Saturation Exponent, Cementation exponent and tortuosity Factor
Using 3-D and CAPE Techniques in Heterogeneous Reservoirs

G.M. Hamada* and A.M. Mohamad , Petroleum Engineering Department and A.M.A. Salim,
Geoscience Department, Faculty of Geosciences and Petroleum Engineering, Universiti
Teknologi PETRONAS, Malaysia.
* Corresponding Author

ABSTRACT
Determination of water saturation in heterogeneous reservoir becomes more challenging as Archie's
equation is only suitable for the clean homogenous formation and high dependency of Archie's parameters
on rock properties. This study focuses on the measurement of Archie’s parameters in carbonate and
sandstone core samples around Malaysian heterogeneous carbonate and sandstone reservoirs. Three
techniques for determination of Archie’s parameters a, m and n will be implemented: Conventional
technique, Core Archie’s Parameter Estimation (CAPE) and Three-Dimensional Regression Technique
(3D). By using results obtained from three different techniques, water saturation graphs were produced to
observe a symbolic difference of Archie’s parameter and relevant impact on water saturation values. The
difference in water saturation values could be primarily attributed to showing the uncertainty level of
Archie's parameters mainly in carbonate and sandstone rock samples. It was obvious that the accuracy of
Archie's parameters has profound impact on the calculated water saturation values in carbonate sandstone
reservoir due to highly stress region will reduced electrical conduction due to raised electrical
heterogeneity the heterogeneous carbonate core samples. Due to unrealistic assumptions involved in the
conventional method, it is better to use either CAPE or 3-D method to accurately determine Archie’s
parameters in heterogonous as well homogenous reservoirs.

1. INTRODUCTION*
From the pertrophysical conception, reservoir can be divided into two main categories which is Archie
reservoir and non-Archie reservoir by referring to certain condition and properties of reservoir rock itself [1].
Archie’s reservoir also refers to clastic reservoir which means rocks are comprised of terrigenous material
formed due to weathering effect of pre-existing rocks and extensively matches with main requirements for the
application of Archie’s equation to produce quantitative well-log analysis [2, 3]. Non-Archie’s reservoir refers
to type of sedimentary rocks formed as result of chemical reactions and due to compaction from another layer.
This process mostly occurred in ocean. Both types of reservoir rock, Archie’s and non-Archie’s reservoir
contain fossils that was present when the rock was formed [4]. The high geological heterogeneity contributes
to growing uncertainty when estimating water saturation [5, 6]. The heterogeneous properties of carbonate
and shaly sandstone are the effect of consequence chemical and physical reorganization processes occur in the
preliminary stages of rock formation [7, 8]. The impact of heterogeneity properties such as variation of pore
size distribution and different wettability may cause difficulty in predicting the hydrocarbon potential of the
pay formation [9-12].
This study will discuss Archie’s parameters determination technique to determine exact water saturation value.
Based on [13] the first application of Archie’s equation is used to determine formation factors F, where
formation factor of rock is defined as the ratio of rock resistivity when 100% saturated by brine water Ro to
the brine water resistivity Rw [14,15]. According to [16], the value of Formation resistivity factor (F) for any
given rock sample will remain essentially constant for wide range of formation water resistivity (Rw) values
measured in reservoir rocks. The next application of Archie’s equation is to calculate the resistivity index Ir,
resistivity index can be defined as resistivity partially saturated with oil (Rt) divided by resistivity of rock
when 100% saturated by brine water (Ro) [17]. Then, determine Archie’s parameters by implementing
Conventional method, Cape method and 3-D method. The main objective of this paper is to propose an
accurate and reliable approach to determine Archie’s parameters in carbonate and sandstone core samples.

Keywords: Heterogeneous Reservoir; Archie’s Parameters; Water Saturation Model

1
In normal routine of formation evaluation, the primary goal which to determine total hydrocarbon in place.
The Archie’s parameters a, m and n are set to default value n=2 [18]. But, for heterogenous reservoir, the
saturation exponent (n) varies from 20 in highly oil wet reservoir condition to 2 in highly water wet reservoir
condition [19, 20]. The wettability phase can be considered as important parameter in the case of partial water
saturation present inside the pore spaces of the core sample [21, 22].
Uncertainty analysis was done for each technique and analysis for relevant impact on the water saturation
values using Archie’s equation. Determination of Archie’s parameters is very important as determination of
recoverable hydrocarbon in place is main goal of formation evaluation process [23]. The common process in
formation evaluation, the Archie’s parameters in sandstones are held constant [24, 25]. In this paper author
will analyze the exact value for all Archie’s parameters; a, m, and n for carbonate rock and sandstone rock.

2. ARCHIE’S PARAMETER DETERMINATION TECHNIQUES

2.1 Core Samples Preparation

Core samples (70mm x 35mm) were selected from various locations in Malaysian fields carbonate core
samples and sandstone core samples and are characterized by different ranges of permeability and porosity.
Most of the carbonate cores have different pore space distribution known as vugs shown in Figure 1. The
heterogeneity in the petrophysical properties of the carbonate core samples mainly influenced by the vugs and
dolomitization process and sandstone core samples with certain degree of heterogeneity. Then, the selected
core samples were subject to preparation process. This started with trimming and smoothing both ends of core
to remove any defect using gypsum. The core samples then will be cleaned with toluene for 6 hours to
eliminate residual hydrocarbon and cleaned with methanol for 12 hour to dissolve and remove the salt present
inside pore space. The core samples were dried in an oven with temperature 175 F for 3 days and recorded the
weight for each day to ensure core samples 100% dried. Then the core samples were saturated with brine at
2000 psi to ensure pores inside the core samples fully saturated using desiccator and inject oil to displace
water starting form 80% of water up until 20% of water remaining inside the core using Benchtop Permeability
System (BPS 550).

2.2 Electrical Testing and Measurement

All core samples undergo the electric testing and measurement process. In this project, two poles and four-
poles resistivity will be used to measure the resistivity of rock when 100% saturated with 30000 ppm of NaCl
brine (Ro) and resistivity of core partially saturated with oil (Rt) for 80% of water remaining until 20% of
water remaining. The computer system will measure and record temperature, pore pressure, confining pressure
and brine displacement during the experiment. The electrical measurement will be recorded continuously
during this process for each core sample until resistivity and desaturation equilibrium reached. All the
resistivity reading during data processing were adjusted to the temperature of 80 ºC which is actual reservoir
condition. When temperature equilibrium has been achieved, the confining pressure will be increased from
2000 psi to 2500 psi and the volume of brine that expelled from the core samples will be recorded. After first
stages of electrical measurement completed, desaturation of core samples will be performed gradually from 0
psi until 120 psi pore pressure. Despite four-pole resistivity will be used to determine the electrical parameters,
two-pole resistivity also will be recorded to control the contact problems that might have happened during
resistivity measurement process.

2.3. Conventional Determination of Archie’s Parameters Technique

In the preliminary stages of development of formation evaluation technique, Archie [16] proposed a set of
equations establishes the quantitative relationship between porosity (  ), rock resistivity (Ro) and hydrocarbon
saturation of reservoir rocks. Based on Archie's experimental work, it shows the resistivity of clean formation
is inversely proportional to the resistivity of the brine saturating the rock. At that moment, Archie comes out

2
with one set of the promising equation to determine the water saturation that presents inside the formation [3,
16].

Sw = [a Rw / ɸm Rt]1/n (1)

Next, Archie also introduced the resistivity of rock that 100% saturated with brine (Ro) is related to the brine
resistivity equal to (Rw). The formula to represent the value of rock resistivity in term of formation factors as
follows:
Ro  F  Rw (2)

The formation factor rearranged and replaced by other term which is tortuosity factor (a), cementation factor
(m) and porosity (  ). Then, the previous formation factor formula changed into a new set of the equation as
below:

a
F (3)
m
2.3.1 Conventional Determination of a, n and m, Archie [16]

In this section, the equation (3) has been transformed into a new equation by introducing log into that formula
and is rewritten as:
log F  log a  m log  (4)

Logarithmic plot of F against  in equation (4) provides the values of the parameters a and m for each core
samples. Figure 2 illustrates a and m values for sandstone and carbonate core samples.
The conventional determination process to determine the value of saturation exponent (n) is transformed the
equation (1) to new equation rewritten as follows:
log I r  n log S w (5)

Plot Ir against Sw on log-log scale graph will give the slope value of saturation exponent (n), Figure 3 show
saturation exponent in carbonate and sandstone core samples.

The conventional method has three serious limitations;1) Calculation of a and m is based on the assumption
of fully water saturated core samples, 2) Calculation of n is based on the assumption of core sample having
same porosity and 3) Conventional method separate Archie’ formula into two separate part (Eqs. 3 and 5). In
fact there are no core samples with the same porosity and Archie’ formula deals with integrated reservoir not
separated reservoir. For this reason, Three-Dimensional Technique (3-D) and Core Archie Parameter
Estimation (CAPE) are presented in the following section.

2.3.2 Three-Dimensional Technique (3-D)

Hamada et al., [15] proposed a three-dimensional method to determine Archie's Parameters which is tortuosity
factor (a), cementation exponent (m) and saturation exponent (n) via standard resistivity measurement on core
samples to reduce the error percentage in the water saturation value.
This method is the basis is to investigate S w in general Archie’s Equation as the variable in three-dimensional
regression plot of S w , Rw Rt and  . By solving the three equations simultaneously to determine Archie's
parameters. The equation then will be reordered after multiplying both side of equation (1) with logarithmic
explain as follows:
RW
log   log a  m log   n log S w (7)
RT
3
Equation (7) can be extracted to form three new equations which can be as the equation of a plane in three-
dimensional space of x, y and z coordinate.
 
x  log  , y  log Sw ,  z  log Rw 
 Rt 

Intersection of the three-dimensional when x = 0.0 will give cementation factor value (m) from the slope of
straight line, when y = 0.0 will give saturation exponent value (n) from the slope of the straight line and when
z = 0.0 will give tortuosity exponent (a). Equation (7) can be modified by replacing the variable with x, y and
z for each i measurement points:
Zi   A  mX i  nYi (8)

Equation (8) will form three set of the simultaneous equation after normalizing for N reading:
 Zi   NA  m X i  nYi (9)

 X Z   NA X  m X  n X Y
i i i i
2
i i
(10)

Y Z   NAY  m X Y  nY
i i i i i i
2
(11)

The Archie’s parameters a, m and n for each core sample can be obtained by solving the equations (9-11).
Solution of these simultaneous equation gives the Archie’s parameters a, m, and n for any set of experiments
for given core samples.

2.3.3 Core Archie’s Parameters Estimation (CAPE)

Based on the previous study by Maute et al [26] it stated that a data analysis approach was presented to
determine Archie’s parameter and ‘n’. The saturation exponent (n) value also can be obtained from standard
resistivity measurement of the core sample. The investigation method, Core Archie-Parameter Estimation
(CAPE) will determine the value of (m) and (n) spontaneously (a) by minimizing the error between computed
water saturation and measured water saturation. The mean square of water saturation error (ε) is stated as
follows:

2
 1

 a R 
    s wij   m wij 
n
 (6)
 
j i
  j  Rtij  

Equation (6) determines the minimum error between measured core water saturation and computed water
saturation by using Archie’s equation, by modifying m, n and optionally a in the equation. Where water
saturation minimum error (ɛ) set by researcher between measured water saturation (Swij) and Archie’s
equation calculated water saturation for assumed values of a, m and n parameters for set of experiments (i) on
number of core samples (j). In this method, generally we start by the conventional values of a, m and n as 1,
1, 2 for carbonate and 0.62, 2.15 and 2 sandstone core samples.

3. RESULTS AND DISCUSSION

Archie’s parameters must be determined for each reservoir, it depends on rock heterogeneity and rock
wettability conditions. In carbonate reservoir as well as in shaly sandstone reservoir. Determination of
Archie’s parameters is tedious in case of carbonate rocks attributed to complicated grain pattern distribution
and dolomitization process in case of dolomitic limestone reservoirs. Dolomitization of a limestone with inter-
particle or vuggy porosity without cementation of the earlier void space permits preservation of that void
space, such conditions will create electrical heterogeneity, and so, Archie’s parameters will be affected and
becoming different from conventional values and laboratory determination setting of Archie’s parameters will
4
face failure by water leakage from core and need to repeat the test. In case of shaly sandstone, The existence
and distribution of clay minerals within sand grains create certain degree of electrical heterogeneity and then
affect Archie’ model and introduce complication on the determination of Archie’ s parameters.

3.1 Conventional Technique

Referring to Figure 2 illustrates a , m and n parameters and Figure 3 gives value of saturation exponent (n).
Table 1 shows a, m and n for sandstone cores and table 2 illustrates a, m and n for carbonates core samples.

3.2 Core Archie’s Parameters Estimation Technique (CAPE)

Tables 3 and 4 show the Archie’s parameters obtaind by implementing CAPE technique in sandstone and
carbonate core samples.

3.3 Three-Dimensional Regression Technique (3-D)

Table 5 and table 6 shows the values of Archie’s parameters a, m and n from the implementation the three-
dimensional technique (3-D) on sandstone and carbonate core samples.

3.4. Comparison between Three Determination Methods of Archie’s Parameters

From the table 7 and table 9, the values of water saturation calculated by three different method and compared
with measured value obtained during experimental work. Based on the result in table 8 and table 10, the 3-D
method show the lowest delta compared to CAPE and conventional technique. Figure 4 show the plot of delta
water saturation for three techniques against measured water saturation. The blue line (3D technique), red line
(CAPE technique) and green line (conventional technique). Referring to that plot, 3D technique generates the
lowest delta compared to conventional and CAPE technique for sandstone core samples.

Figure 4, shows the comparison of delta water saturation model by three technique.Based on Figure 5, the
graph shows that the water saturation value calculated using 3D technique (blue) is identical and almost same
with the measured value compare with two other CAPE (red) and conventional (black) which have more
difference that measured value.

Referring to table 1, table 3 and table 5, the value of tortuosity factor (a) for sandstone ranging from 1.00 until
1.78, cementation exponent ranging from 1.4 until 2.4 and the saturation exponent for this project is lower
compare to other literature and pass study which is around 0.6 to 2.1.

Figure 6 shows the difference of water saturation model based on three technique in carbonate core samples.
The graph clearly shown that the 3-D technique produce the lowest delta water saturation model compared to
other method. This indicates that 3-D technique is reliable to replace conventional technique to determine
Archie’s parameters in carbonate.

Based on table 2 (Conventional technique), the Archie’s parameters which is tortuosity factor (a) and
cementation exponent (m) is constant at 1.196 and 1.929. But the saturation exponent (n) is vary from 1.772
to 3.403. From table 4 (CAPE technique) the tortuosity factor is ranged from 1.608 to 1.780, cementation
exponent from 1.564 to 2.197 and saturation exponent from 2.035 to 2.360. Then, table 6 (3D technique) show
that the tortuosity factor is constant at 1.00, cementation exponent and saturation exponent is varied from
1.760 to 3.011 and 1.638 to 3.403.

From table 9 and 10, the water saturation value has been calculated by implementing three different technique
which is same as sandstone core. Conventional technique, CAPE technique and 3D technique. So, based on
result obtain in table 9 and 10 it shows that the 3D technique provides the less difference (delta) by comparing
with measured water saturation obtain in the laboratory with compute water saturation value from three
different technique. Besides, Figure 6 demonstrates the result in table 10 and from the plot, 3D technique

5
(blue) shows the lowest different compared to CAPE (red) and conventional (green). The plot based on delta
water saturation for each technique against measured water saturation.

Figure 7, indicates the value of water saturation calculated based on three different technique compared to
measured water saturation obtained directly during experimental work. The 3D technique represented by blue
line, CAPE technique represents by red line, conventional technique represents by black line and the measured
water saturation represent by green line. The green cannot be detected it falls exactly below the blue line. This
plot shows how accurate the 3D technique to determine water saturation value in carbonate compared to other
method. Consequently, it is recommended to apply 3-D method as a promising technique to get an accurate
values of Archie’s parameter and thereby good estimation of water saturation.

CONCLUSION
1. Archie’s parameters must be determined for each reservoir, it depends on rock heterogeneity and
rock wettability conditions. In carbonate reservoir as well as in shaly sandstone reservoir,
determination of Archie’s parameters is tedious work due to electrical heterogeneity in case of
carbonate rocks attributed to complicated grain pattern distribution and in case of shaly sandstone, it
is due to mainly local conductivity created by clay minerals.
2. Conventional determination technique of Archie’s parameters can give acceptable values, however,
the results are subjected to assumptions of full water saturation core samples in order to get a and m
parameters and homogenous sample of the same porosity to get saturation exponent n.
3. CAPE method and 3-D method solved the constraints of the conventional method and 3-D method is
easier and more accurate than CAPE and conventional method.

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6
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7
 Table 1: Archie's Parameters of sandstone by Conventional Technique

Core a m n
1 1.147 2.135 0.848
2 1.147 2.135 1.109
3 1.147 2.135 0.726
4 1.147 2.135 0.665
5 1.147 2.135 0.944

Table 2: Archie's Parameters of carbonate by Conventional Technique

Core a m n
1 1.196 1.929 1.772
2 1.196 1.929 2.086
3 1.196 1.929 3.403
4 1.196 1.929 2.511
5 1.196 1.929 1.919

Table 3: Archie's Parameters of sandstone by CAPE Technique

Core a m n
1 1.742 1.496 2.180
2 1.772 1.560 2.296
3 1.734 1.510 2.042
4 1.716 1.504 2.140
5 1.774 1.546 2.190

Table 4: Archie’s Parameters of carbonate by CAPE Technique

Core a m n
1 1.780 1.800 2.151
2 1.721 1.564 2.177
3 1.780 2.197 2.360
4 1.764 1.603 2.053
5 1.608 1.579 2.035

Table 5: Archie’s Parameters of Sandstone by 3D Regression Technique

Core a m n
1 1.000 2.135 0.578
2 1.000 2.253 0.671
3 1.000 2.135 0.578
4 1.000 2.212 0.614
5 1.000 2.241 0.680

Table 6: Archie’s Parameters of carbonate by 3D Regression Technique

Core a m n
1 1.000 2.245 1.638
2 1.000 1.760 2.086
3 1.000 1.933 3.403
4 1.000 2.358 1.988
5 1.000 3.011 1.828
8
 Table 7: Comparison of measured water saturation with three method in sandstone core

Sw Sw Sw Sw
(Measure) (Conventional) (CAPE) (3D)
1.00 0.986 0.692 0.996
0.80 0.828 0.656 0.824
0.60 0.628 0.602 0.611
0.40 0.395 0.521 0.369
0.20 0.231 0.441 0.207

Table 8: Difference of measured water saturation with three method in sandstone core

ΔSw ΔSw ΔSw

(Conventional) (CAPE) (3D)
0.014 0.308 0.004
-0.028 0.144 -0.024
-0.028 -0.002 -0.011
0.005 -0.121 0.031
-0.031 -0.241 -0.007

Table 9: Comparison of measured water saturation with three method in carbonate core

Sw Sw Sw Sw
(Measure) (Conventional) (CAPE) (3D)
1.000 1.000 1.000 1.000
0.958 1.000 0.965 0.909
0.931 1.000 0.831 0.778
0.915 1.000 0.839 0.785
0.758 0.993 0.818 0.765
0.536 0.496 0.421 0.382
0.233 0.355 0.306 0.274

Table 10: Difference of measured water saturation with three method in carbonate core

ΔSw ΔSw ΔSw

(Conventional) (CAPE) (3D)
0.000 0.000 0.000
-0.042 -0.006 0.050
-0.069 0.100 0.154
-0.085 0.077 0.130
-0.234 -0.060 -0.007
0.040 0.115 0.154
-0.122 -0.073 -0.041

9
 Figure 1. Studied cores A) Sandstone B) Carbonate

Figure 2: 'a' and 'm' value on studied cores A) sandstone B) carbonate

10
 Figure 3:'n' value on studied coresA) sandstone B) carbonate

0.4

0.3

0.2

0.1
ΔSw

0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

-0.1

-0.2
Conventional CAPE 3D

-0.3
Water Saturation
Figure 4: Water saturation difference between three techniques for sandstone cores

11
 1.2

1.0

0.8
Water Saturation

0.6

0.4

0.2
Measured Conventional CAPE 3D

0.0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Number of Measurement
Figure 5: Water saturation value for three technique compare with measure water saturation in
sandstone cores.

0.2

0.2

0.1

0.1

0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
-0.1
ΔSw

-0.1

-0.2

-0.2

-0.3
Conventional CAPE 3D
-0.3

Water Saturation

Figure 6: Water saturation difference between three techniques for carbonate cores

12
 1.20

1.00

0.80
Water saturation

0.60

0.40

0.20

Measured Conventional CAPE 3D

0.00
0 1 2 3 4 5 6 7
Number of measurement

Figure 7: Water saturation value for three techniques compare with measure water saturation

13

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