09.pdf

Proc. Tokai Univ., Ser. E Proc.Schl. Schl.Eng. Eng. Tokai Univ., Ser. E 41 (2016) 53-58 41（2016）■-■

Effect of Fractional Flow Curves on the Recovery of Different Types of Oil in Petroleum Reservoirs by Abdul Jamil NAZARI*1 , Ahmad Fahim NASIRY*1 and Shigeo HONMA*2 (Received on Mar. 31, 2016 and accepted on May 12, 2016)

Abstract This paper considers the effect of fractional flow curves on different types of oil recovery when injecting water into petroleum reservoirs. In the computation of oil recovery, the Buckley–Leverett frontal displacement theory has been widely used to calculate the saturation profile of two immiscible fluids, wherein saturation is largely affected by the fractional flow curve of the displacing fluid. This paper reviews a fractional flow equation and a frontal advance equation and evaluates fractional flow curves of light and heavy oil by using relative permeability curves obtained from laboratory experiments. Results indicate that the fractional flow curve of light oil exhibits a regular S-shape, and application of this curve to the waterflooding method shows that a large amount of mobile oil in the reservoir is displaced by water injection. In contrast, the fractional flow curve of heavy oil does not display an S-shape because of its high viscosity. Although the advance of the injected water front is faster than that in light oil reservoirs, a significant amount of mobile oil remains behind the water front. Keywords: Fractional flow, Relative permeability, Petroleum reservoir, Oil recovery, Waterflooding technique, Buckley–Leverett analysis

1. Introduction

the effects of capillary pressure between the two fluids and gravity are neglected. According to this theory, the advance of a saturation front by the displacing fluid is largely affected by the

In petroleum reservoir engineering, the technique of injecting

permeability of oil and water relative to reservoir rock.

water into oil reservoirs is used to maintain oil production rates during pumping operations. This so-called waterflooding technique provides high oil production rates with a high degree of

Natural water table

petroleum recovery when applied as oil production rates begin to drop1). The technique has been widely employed in oil fields around the world and in shale oil exploitation technologies

Water injection

initiated in the United States. After long-term extraction of crude

well

oil, a mixture of oil and water is pumped up in the production

Oil production

qw

well

wells. Water and oil are separated by a separator installed on-site, and the separated water is recycled for injection2). When water is injected into a reservoir, oil is displaced toward the production well in the two-phase flow situation depicted in Fig. 1. Oil and water are mutually immiscible, so this phenomenon is andFig. to the viscosity ratiotechnique between the fluids. reservoir3). 1 Waterflooding in atwo petroleum

referred to as immiscible displacement in porous media. The mechanism of immiscible displacement of two-phase

Figure 2 illustrates typical relative permeability curves from

fluids has been studied extensively in the discipline of fluid flow frontal

the petroleum engineering literature4). The relative permeability

displacement theory describes a method for calculating saturation

of oil kro and water krw are generally given as a wetting fluid

through

porous

media.

The

Buckley–Leverett

saturation Sw, usually that of water. As the water saturation

profiles on the basis of the relative permeability, assuming that

increases, kro gradually decreases and becomes zero at residual oil saturation Sor. The residual oil saturation is immobile in rock

*1 Graduate Student, Course of Civil Engineering *2 Professor, Department of Civil Engineering XLI,2016 2016 Vol. XLI,

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Abdul Jamil NAZARI, Fahim NASIRY and Shigeo HONMA Abdul Jamil NAZARI, Ahmad Fahim NASIRY and Shigeo HONMA pores. Also, krw increases as the water saturation in a medium

We start with Darcy’s equations

increases, and reaches the endpoint of the relative permeability. There exists some amount Swi of immobile water in pores, called the irreducible water saturation. The limits of the fractional flow

qo  

kkro A  po   o g sin  ,  o  x 

(1)

qw  

k krw A  pw   w g sin  w  x 

(2)

are 0 % and 100 % for Swi and Sor. At the irreducible water saturtion, the water flow rate fw is zero and, therefore, the frational flow is 0 %. At the residual oil saturation point Sor, the

and replace the water pressure by pw  po  pcow , so that

oil flow rate is zero and the fractional flow reaches its upper limit of 100 %. The shape of the fractional flow versus the water

qw  

saturation curve is characteristically an S-shape. 1.0 1.0

0.8

0.6

So

0.4

c

0.8

0.2

fw

d

After rearranging, the equations may be written as

 qo kro

’ b at f wmax max

0.4

krw

Swi

(3)

Here, pcow is the capillary pressure between oil and water.

0

krw,kro

fw 0.6

kkrw A  ( po  pcow)   w g sin .  w  x 



po  o g sin  , x

( 4)

w



po pcow   w g sin . x x

(5)

kkro A

 qw

Sor

o

kkrw A

0.2 0

Subtracting Eq. (4) from Eq. (5), we get

a 0

0.2

0.4 SBL 0.6

Sw

0.8

1.0



Fig. 2 Relative permeability and fractional flow rate curves4). )



p 1  w    qw  qo o    cow   g sin  . kA  krw kro  x

Substituting

There are two general methods to determine relative

qT  qw  qo ,

permeability, the steady-state (SS) method and the unsteady-state (USS) method. The SS method aims to achieve steady-state flow

fw 

qw qT

(7)(8)

and solving for the fractional flow of water, we obtain the

at different fractional flow ratios, yielding unique core saturation

following expression for the fraction of flowing water:

at each ratio. The results are easy to interpret, but it takes a long time to achieve steady-state conditions. In the USS method, the

1

core saturated with oil is flooded by water at a constant total rate

fw 

until no more oil is produced. Flooding experiments record the fractional flow ratio, the pressure at both ends, and the breakthrough time of the injected fluid. From fractional

(6)

flow

k kro A  pcow    g sin   qT o  x  . kro  w 1 o krw

(9)

For the simplest case of horizontal flow with negligible capillary

theory, the two-phase relative permeability can then be

pressure, the expression reduces to5)

determined as a function of saturation at the effluent core end. For this reason, fundamental equations related to these

fw 

phenomena are presented below.

1

1

2. Buckley-Leverett Analysis

k ro  w  o k rw

.

(10 )

2.2 Buckley–Leverett equation For a displacement process where water displaces oil, the

2.1 Fractional flow equation

mass balance of water around a control volume of length ⊿x over

Derivation of the fractional flow equation for an oil–water

a time period of ⊿t is considered.

system with one-dimensional flow is as follows: Consider displacement of oil by water in a system with dip angle α.

qw

x

⊿x

α

Fig. 4 Mass balance in a flow system.

Fig. 3 One dimensional oil-water flow system.

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Proceedings of the School of Engineering, Tokai University, Series E

Effect of fractional flow curves on the recovery of different types of oil in petroleum reservoirs Effect of Fractional Flow Curves on the Recovery of Different Types of Oil in Petroleum Reservoirs heavy oil (type A), are used. Table 1 shows the properties of these

The mass balance may be written as

 q     q   w

w x

w

w x  Δx

Δt  A Δx  (S

oils and water.

 )  ( S w w )  (11)

w w

t Δt

t

Table 1 Physical property of oils and water.

which reduces to the continuity equation when Δt  0 and

Properties

Δx  0



 qw  w   A  Sw  w  x t

Light oil

(12)

Heavy oil

Water

Density ρ (g/cm )

0.795

0.837

1.00

Viscosity μ (Pa・s)

0.00242

0.0167

0.001

3

of fluid fractional flow curvesmay on the of that different oil inexperimental petroleum reservoirs Assume Effect that the compressibility be recovery neglected; is, types ofThe apparatus depicted in Fig. 5 is used. In the ρw is a constant. Also, we have that first, the sand samples are saturated by oil, and then oil and water are simultaneously pumped at different pumping ratios. The (13 ) heavy oil (type A), are used. Table 1 shows the properties of these experiment starts from high ratio of oil and low ratio of water (the oils and water. iｍbibition process). The experiment measures the pressure and t f wEffect Δt A ΔxAφof(fractional SSwww ) t Δtflow (Sw  curves on) the recovery of different types of oil in petroleum reservoirs w )  (11   . (14) amount of discharge of of oiloils andand water. Table 1 Physical property water.

f wqwritten T  qw ,as The mass balance may be so

 qw wx   qw wx  Δx 

qT t

x

which reduces to the continuity equation when Δt  0 and Since f S w  , the equation may be rewritten as be written as Δx The 0 mass balancew may

Properties Light oil Heavy oil Water heavy oil (type3 A), are used. Table 1 shows the properties of these 0.795 0.837 1.00 Density ρ (g/cm ) oils and water. 3.0 0.00242 0.0167 0.001 Viscosity μ (Pa・s) V = 140.0 cm3

Acrylic pipe

L=20cm

V = 140.0 cm3

(18)

3.1 Relative permeability and fractional flow curve of light oil Glass beads 3.0

The permeability to either fluid is expected to be lower than Pressure transducer Fraction collector (Glass tube)

that for the single fluid, because it saturated occupiesbyonly Sand (Initially oil) part of the pore L=20cm

df w Aφ Sthe . From S w  x, t   f, w weS wcan dt write A dSww. expression for saturation (15) qT t S w x hange as T

Acrylic pipe

.3 Frontal advance equation dx q

L=20cm

 qw  w   A  Sw  w  (12)  x  qw wx   qw wx ΔxfwtΔtSw AΔxAφ (Sww.w ) tΔt (Sw w ) t  (11) (15) Fraction collector (Glass tube) Table 1 Physical property of oils and water. qT t Sand (Initially saturated by oil) S w x Assume that the fluid compressibility may be neglected; that is, 2.64 g/cm3In the 6) The experimental apparatus depicted in Fig.ρs5= is used. . Equation (15) is known as the Buckley–Leverett equation which reduces to the continuity equation when Δt  0 and Properties Light oil Heavy Water D =oil 0.105-0.425 mm is a constant. Also, we have that w first, the sand samples are saturated by oil, and then oil and water Δx  0 3 ) 0.837beads 1.00The Density ρ (g/cm Glass are simultaneously pumped at0.795 different pumping ratios.  f wqT 2.3  qw ,q (13)  w  w  advance  Frontal  A equation Sw w  (12) μ (Pa ・ s) 0.00242 0.0167 0.001 Viscosity experiment starts from high ratio of oil and lowPressure ratio oftransducer water (the From x S w  x, t  ,twe can write the expression for saturation o iｍbibition process). The experiment measures the pressure and change S w compressibility may be neglected; that is, f that Athe φ asfluid Assume Data In the experimental Tubin  w  . S ( 14 ) amount The of discharge of oilgapparatus and water.depicted in Fig. 5 is used. recorder qT Also, w that S w ρw is aconstant. t we dS pump x have  dx  dt . ( 16 ) first, the sand samples are saturated by oil, and then oil and water w P x t P are simultaneously pumped at different pumping ratios. The ince f w S w  , the may be rewritten as f wqequation (13) T  qw , In the Buckley–Leverett solution, we follow a fluid front ofexperiment starts from high ratio of oil and low ratio of water (the so constant saturation during the displacement process as Water 3.0 Oil iｍbibition process). The experiment measures the pressure and f w Sf w Aφ S w Fraction collector (Glass tube)  (15) (14)   .. amount of discharge of oil and water. q0T  St w dx  S w dt. Sand (Initiallyapparatus saturated by (17) S w x Fig. 5 Experimental foroil) measuring x t ρs =permeabilities. 2.64 g/cm3 6) relative Equation (15) is known as the Buckley–Leverett equation . D = 0.105-0.425 mm Since f w S w  , the equation may be rewritten as Substituting into the Buckley–Leverett equation, we get 



cm3

3

V = 140.0



Acrylic pipe

ρs = 2.64Data g/cm by interaction with other phases. Tubing space and may also be affected with to time as EquationIntegration (15) the Buckley–Leverett equation6). recorder mm Swis known S wasrespect D = 0.105-0.425 pump dSw  dx  dt. (16) The relative permeability of oil and water, kro and krw, are P x Effectdx t P in petroleum reservoirs qTflow df wcurves on the recovery of different types of oil of fractional Glassasbeads dt  dt (19) calculated by Darcy’s law 2.3 Frontal advance dt equation A follow dS w a fluid front of n the Buckley–Leverettt solution,t we Pressure transducer From S w  x, t  , we can write the expression for saturation onstant saturation during the displacement processofasthe water front as Water qw  w p qo oOil p yields an expression for the position , kthe change as may be written as / of these (21)( 22) rw properties ro  Table /1 shows heavy oil (type A), arekused. e mass balance Data L k A L kA Tubing S w SS w S recorder  w dt 0  dS dx (17) (16) oils and water. Fig. pump 5 Experimental apparatus for measuring q. T t wdtdf . w  , (20) x w  xxtdx f  t t Δt P t  p relative permeabilities. P   where is the injection pressure, and L is the length of the sand   Aφ (SdS w  w x   qw  w x  Δx Δt  A Δx w w ) f  ( S w w )  (11) Tablesample. 1 Physical of water oils and water. is calculated based on the ubstituting into the Buckley–Leverett equation, we get Theproperty degree of saturation In the Buckley–Leverett solution, we follow a fluid front of which is often called the frontal advance equation2). 3.1 Relative permeability andconcept fractional fractional flow as flow curve of light oil ich reduces the when Δt  constant the displacement process WaterLight 0 and as Oil Properties oil Heavy oil Water dx tosaturation qT continuity df w duringequation The permeability to either fluid is expected to be lower than  . (18) 0 3 V p dt A dS s ρ (g/cm S Sw ) S 0.795 0.837 only 1.00of the pore Density Experiments (17) that for the single because ( fod itapparatus foccupies 1 measuring Spart (23)  0  w dx  w3.dtLaboratory . Fig. 5fluid, Experimental for o, w o ) dVp  0  qw  w  A  S wt w  (12) x μ (Pa ・ s) 0.00242 0.0167 0.001 Viscosity space and may also be affected by interaction with other phases. relative permeabilities. ntegration with respect to time as x t





fod is theoffractional displaced oil, and fo is the experiments forequation, measuring relative permeability krw, are The relative where permeability oil and discharge water, kroofand SubstitutingLaboratory into the Buckley–Leverett we get dx fluid compressibility qT df w sume that the may be neglected; that is, 3.1 Relative permeability and fractional flow curve of light The experimental apparatus used. In pumping the oil rate. (For dt fractional flow dt are performed based(19 fractional ratioinofFig. oil 5onisthe total and on) the steady-state calculated by Darcy’s lawpumping as depicted dx we qt7,8) df wdSthat dtAlso, TA w t is a constant. have The samples permeability tocalculation, eitherbyfluid is expected lower than  . different types of oils, light oil(18 ) first, and the sand saturated oil,see and then andbewater ) to details are of the Ref. 7) oil . Two (kerosene) method dt A dSw q q   p p that for the single fluid, because it occupies only part of w w pumping ratios. Thethe pore ields an are simultaneously at different f wexpression qT  qw , for the position of the water front as(13) , k rw (21)( 22 ) k ro  o o /pumped / k A L k Aand L ratio ofwith spacestarts and from may also affected by interaction other Integration with respect to time as experiment high be ratio of oil low water (thephases. qT t  df w  ―3relative ― k and krw, are The permeability of oil and water,  ,  2016 ro (20) x  iｍwhere bibition The experiment measures pressure p isprocess). Vol. XLI, the injection pressure, and L is the the length of theand sand 2016 q df w SdS f wf Vol. dt φXLI, Adx ww  f T − 55 − dt calculated by Darcy’s as   . (14) (19) amount of The discharge andlaw water. sample. degreeofofoilwater saturation is calculated based on the t t A dS w x t qdt T 2) which is often called the frontal advance equation .









Abdul Jamil NAZARI, Fahim NASIRY and Shigeo HONMA Abdul Jamil NAZARI, Ahmad Fahim NASIRY and Shigeo HONMA Table 2 Relative permeabilities and fractional flow data for light oil.

Figure 7 shows the fractional flow curve in the displacement of light oil by water. The curve has an elongated S-shape in the

Sw

krw

kro

fw

0.17

0.00

1.00

0

0.34

0.03

0.55

0.117

0.36

0.04

0.42

0.187

0.41

0.08

0.35

0.356

0.44

0.10

0.25

0.492

0.47

0.11

0.20

0.571

0.50

0.15

0.18

0.669

0.52

0.19

0.11

0.807

Table 3 and Fig. 8 show the change in relative permeabilities

0.55

0.21

0.07

0.879

and fractional water flow for the displacement of heavy oil by

0.63

0.34

0.00

1.00

water. The data show that the residual oil saturation Sor is smaller

range of effective saturation of the displacing fluid (Swi < Sw < 1 - Sor). The degree of saturation at the tangent point of a straight line drawn from the irreducible saturation on fractional flow curve SBL is used to determine the saturation value at the water front according to Buckley–Leverett theory. 3.2 Relative permeability and fractional flow curve of heavy oil

than light oil, but the relative permeability of water krw is very small as compared with the light oil displacement. This is attributed to the viscosity of heavy oil being about 17 times larger than that of water, and the relative permeability of water calculated by Eq. (22) becomes very small even though water flow occurs together with oil flow through sand. Figure 9 illustrates the fractional flow curve for heavy oil. The curve does not display an S-shape; it swells on the low-saturation side,

krws = 0.34

because of the very small values of krw under low water-saturation conditions. Table 3 Relative permeabilities and fractional flow data of heavy oil. Sw

krw

kro

0.12

0.00

0.95

0

0.15

0.01

0.77

0.178

Table 2 and Fig. 6 show the change in relative permeabilities

0.18

0.01

0.64

0.207

and fraction of water flow with the change in water saturation.

0.24

0.02

0.54

0.382

0.28

0.03

0.45

0.527

0.34

0.03

0.36

0.582

0.44

0.04

0.27

0.712

0.50

0.04

0.19

0.779

0.60

0.05

0.10

0.893

0.73

0.06

0.00

1.00

Fig. 6 Relative permeability curves for the displacement of light oil by water.

The fractional water flow fw was calculated from Eq. (10) using the fluid properties listed in Table 1 and the values of relative permeabilities. Relative permeability curves of light oil and water exhibit normal cross curves, and the end-point value of relative water permeability was 0.34. At that point, only water was pumped into the sand column, and a significant amount of

fw

residual oil remains in sand pores (Sor = 0.37).

krws = 0.06

Fig. 8 Relative permeability curves for the displacement of heavy oil by water.

Fig. 7 Fractional flow curve in the displacement of light oil by water.

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Proceedings of the School of Engineering, Tokai University, Series E

Effect of fractional flow curves on the recovery of different types of oil in petroleum reservoirs Effect of Fractional Flow Curves on the Recovery of Different Types of Oil in Petroleum Reservoirs progresses with a constant speed toward the production site (right side). Although there is a large amount of residual oil in the reservoir, water displaces the most of mobile oil. The oil recovery factor for this situation is computed as

RF 

S w  Swi . 1 Swi

(24) (24)

From the above, RF = 0.51, from which the total amount of oil produced by waterflooding is AφB× RF = 28,050,000 m3 ( = 175 million barrels) for the given reservoir. 1.0 Residual oil Sor = 0.27

Sw

Fig. 9 Fractional flow curve in the displacement of heavy oil by water. The difference in these fractional flow curves on the recovery of reservoir oils is investigated in the next section.

0

Displaceable mobile oil

0.5

Irreducible water Swi = 0.12 0

100

200

x (m)

300

400

800 day

700

600

Advance of the saturation front in the waterflooding method

Water

500

0

200

t=100

Waterflooding Method in Reservoirs

1.0 500

Fig. 11 Displacement of heavy oil calculated by the frontal advance equation.

may be calculated by using the frontal advance equation given by Eq. (20), which involves derivatives of the fractional rate of flow with respect to water saturation. Each saturation advances into the

Figure 11 illustrates the calculated results of the saturation

system at a rate in direct proportion to d fw /dS w. The curve

profile for a heavy oil reservoir. A significant amount of mobile

generally displays a smooth convex curve toward the flow

oil remains in the reservoir after displacement, even though the

direction, and the position of abrupt change in saturation, i.e., the

effective saturation for this displacement is larger than that for

water front, is determined from the value of SBL previously shown

light oil. The total amount of oil produced by waterflooding is

in Figs. (7) and (9).

23,100,000 m3 (RF = 0.42) for the given reservoir.

A petroleum reservoir of extent area A = 10 km2, thickness B

The advance of the water front for the heavy oil reservoir is faster

= 25 m, length L = 1 km, and porosity φ = 0.22 is considered. The total amount of water injected is assumed to be qw = qT = 1,000

than that for the light oil reservoir. These results are reflected in the shape of the fractional flow curve of the reservoir oil. The

m3/day.

viscosity ratios with water used in the waterflooding simulation

1.0

are 2.42 for light oil and 16.7 for heavy oil. This suggests that the

0 Sor = 0.37

Residual oil

Sw SBL=0.53

viscosity ratio between the displaced liquid and displacing liquid has a significant influence on the degree of oil recovery9). The use

So

of heated water in waterflooding may be a reasonable method for improving the mobility of the reservoir oils, requiring

Displaceable mobile oil

Water

100

200

x (m)

300

thermodynamic analysis between the immiscible fluid flow and

1000 day

900

0

800

700

600

500

400

300

200

t = 100 day

0.5

Irreducible water 0

0.5

SBL=0.29

4. Application of the Fractional Flow Curves to the

0.5

So

porous media. This might produce a challenging problem for waterflooding oil recovery.

Swi = 0.17

400

5. Conclusion

1.0 500

The effect of fractional flow curves on different types of oil

Fig. 10 Displacement of light oil calculated by the frontal advance equation.

recovery in petroleum reservoirs was investigated in this paper.

Figure 10 illustrates the calculated results of a saturation

through laboratory experiments, and fractional water flow was

Relative permeabilities of light and heavy oils were measured evaluated using the fractional flow equation. The fractional flow

profile for a light oil reservoir. Here, the saturation front

XLI,2016 2016 Vol. XLI,

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Abdul Jamil NAZARI, Fahim NASIRY and Shigeo HONMA Abdul Jamil NAZARI, Ahmad Fahim NASIRY and Shigeo HONMA curve of light oil showed a regular S-shape, and application of

A

Cross-sectional area

this curve to the waterflooding method showed that a large

B

Thickness of reservoir

amount of mobile oil in the reservoir is displaced by water

RF

Recovery factor

injection. In contrast, the fractional flow curve of heavy oil did not display an S-shape because of its high viscosity, and a References

significant amount of mobile oil remains in the reservoir behind the water front. With the fractional flow data employed in this

1) R.C.Craft and M.Hawkins, Revised by R.E.Terry: Applied Petroleum Reservoir Engineering, (Prentice-Hall, 1991) pp.1-6. 2) A.Y. Dandekar : Petroleum Reservoir, Rock and Fluid Proper-

study, the oil recovery rate by waterflooding of light oil reservoirs was about 52 % and that of heavy oil reservoirs was 42 %.

ties, (CRC Press, 2013) pp.45-83. Nomenclature

3) Water Flooding Concept : youtube TekOil http://www.youtube.com/watch?v=Y4Ipuo1IBdk

k

Intrinsic permeability

4) B. Philip and M. A. Celia: Practical implementation of the

krw

Relative permeability of water

fractional flow approach to multi-phase flow simulation,

kro

Relative permeability of oil

Advance in water resource, Vol.22, No.5, (1991), pp 461-487.

krws

End-point relative permeability

μw

Viscosity of water

5) A. Arabzai and S. Honma: Numerical simulation of the Buckley-Leverett problem, Proc. School of Eng. of Tokai

μo

Viscosity of oil

ρw

Density of water

ρo

Density of oil

placement in sands, Transactions AIME, Vol.146, (1942),

pw

Water pressure

pp.107-116.

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Oil pressure

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Total pumping rate of oil and water

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Amount of water

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Water saturation

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Residual oil saturation

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Fractional water flow

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Fractional oil flow

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Porosity of reservoir

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Pore volume

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8) A.J. Nazari, A.F. Nasiry and S. Honma: Measurement of rela-

9) A.J. Nazari, A.F. Nasiry, K.N. Seddiqi and S. Honma: Influ-

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Proceedings of the School of Engineering, Tokai University, Series E