capillary 1.doc

8/20/2019 CAPILLARY 1.doc

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3 CAPILLARY PRESSURE

Reservoir rock typically contains the immiscible phases: oil, water, and gas. The

forces that hold these fluids in equilibrium with each other and with the rock are

expressions of capillary forces. During waterflooding, these forces may act

together with frictional forces to resist the flow of oil. It is therefore advantageous

to gain an understanding of the nature of these capillary forces.

Definition : Capillary pressure is the pressure difference existing across the

interface separating two immiscible fluids.

If the wettability of the system is known, then the capillary pressure will always

be positive if it is defined as the difference between the pressures in the non-wetting and wetting phases. That is:

P c = P nw − P w

Thus for an oil-water system (water wet)

P c = P o − P w

For a gas-oil system (oil-wet)

P c = P g − P w

What Causes Capillary Pressure?

Capillary pressure is as a result of the interfacial tension existing at the interface

separating two immiscible fluids. The interfacial tension itself is caused by the

imbalance in the molecular forces of attraction experienced by the molecules at

the surface as shown below.


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For molecules in the interior:Net forces = 0 since there are enough molecules

around to balance out.

For molecules on the surface:

Net result of forces is a pull toward the interior

causing a tangential tension on the surface.

The net effect of the interfacial tension is to try to minimize the interfacial area in

a manner analogous to the tension in a stretched membrane. To balance these

forces and to keep the interface in equilibrium, the pressure inside the interface

needs to be higher than that on the outside.

Forces reducing interface are due to: a) Interfacial tensionb) External pressure

The effect of interfacial tension is to compress the non-wetting phase relative to

the wetting phase. The force created by the internal pressure is balancing it.

3.1 EXPRESSIONS FOR CAPILLARY PRESSURE UNER S!A!ICCONI!IONS

3.1.1 Pc In terms of radius of capillary tube

Since the interface is in equilibrium, force can be balanced on any segment. The

interfacial forces are eliminated by taking as a free body, that part of the interface

not in direct contact with the solid. A force balance would give:

(Internal pressure - External pressure) * Cross-sectional area = Interfacial tension

* Circumference

σ aw P nw

P w

air

water

θ

σ w


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Thus, P nw π r

2( )= σ Cosθ 2π r ( )+ P w π r 2( )

Therefore, P nw − P w( )π r 2

= σ Cosθ 2π r ( )

And since by definition, P c = P nw − P w , we have: P c =

2σ Cosθ

r

For an air-water system, the air is the non-wetting phase and P c =

2σ awCosθ aw

r

This equation is referred to as Laplace's Equation in some texts.

3.1. Pc In terms of !ei"!t of fluid column.

Air - water system

air

air air

P a1

P w1

P w2 P w2

P a2

w

water

(1)

(2)

P a2 = P w 2 because there is no capillary pressure across a horizontal interface. P a1 = P a2 = P w 2 but P w2 = P w1 + h ρ w g

therefore, P a1 − P w1 = h ρ w g

Since P c = P a1 − P w1 then P c = h ρ w g

Oil-water system


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oil

oil

oil

water

P o

P w1

P w2 P w2

P o

hw

(1)

(2)

P a2 = P w 2 because no capillary exists across any interface that is horizontal. P w2 = P w1 + h ρ w g P o2 = P o1 + h ρ o g

Since P w2 = P o2 , then, P w1 + h ρ w g = P o1 + h ρ o g

Therefore, P o1 − P w1 = ρ w − ρ o( )hg

That is, P c = ρ w − ρ o( )hg

Cgs Units: Field Units:

P c = dynes cm2

ρ = gm cc g = cm sec2

h = cm

P c = h ρ w

144orP c =

h ρ w − ρ o( )144

psi(or Ib sqin)

ρ w , ρ o = Ib ft 3

h = ft

The two expressions for capillary pressure in a tube, one in terms of height of a

fluid column and the other in terms of the radius of the capillary tube can be

combined to give an expression for the height of a fluid column in terms of the

radius of the tube as follows:

h ρ g =2σ cosθ

r

Therefore for an air-water system,

hw =2σ awCosθ

r ρ w g

Similarly, for an oil-water system,


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hw

=2σ owCosθ

r ρ w − ρ o( ) g

These two equations show an inverse relationship between fluid height and

capillary radius. The smaller the radius is, the higher the height of the fluid

column will be.

3." APPLICA!IONS OF CAPILLARY PRESSURE EXPRESSIONS INPOROUS #EIA

3..1 Application to obtain static fluid distribution in porous media

3.. Porous media modelled as a bundle of capillaries

One of the earliest and simplest depiction of porous media was as a bundle ofcapillary tubes of arbitrarily varying diameters. By applying the applicable one of

the equations:

hw =2σ awCosθ aw

r ρ w g or

hw =2σ owCosθ ow

r ρ w − ρ o( ) g

The different water heights in such a system is illustrated in the figure below

where if the number of tubes were numerous, a smooth curve will result as shown

in the lower figure. That figure is for a three-phase gas-oil-water system. The

figures also show the difference between the water-oil contact (WOC) and the


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free-water table. The WOC is the depth at which S w = 100% begins (moving

downward) while the free-water table is the depth at which

P c = 0.


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GOC

WOC

FWT Free water table

OilOil

GasGas

Water Water

Water

Dept

Water sat!ratio", %

WOC

GOC

#at!ratio", % $rossectio" o reser&oir

Free - water 'able

WO$

GO$

Gas cap

Oil o"e

Water

Dept

i*!i+, %

Water, %

S g

S o

S wi

S o + S w

S w

00 100

3..3 Porous media modelled as a pac#a"in" of uniform sp!eres

An even more realistic model is the depiction of porous media as a packaging of

uniform spheres. Applying the two expressions for capillary pressure in terms of

the radii of the interface and in terms of the height of fluid column, we have for

this system:


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R

R2

#a"+ rai"#a"+ rai"

.o" - wetti" l!i+

Wetti" l!i+

P c = σ 1

R1−

1

R2

=

h ρ w − ρ o( )144

From which,

h =144σ

1

R1−

1

R2

ρ w − ρ o

In field units, h = ft ,σ = Ib in , ρ = Ib ft 3, R1 = ft , R2 = ft


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Unfortunately, R1 and are impossible to measure in porous media and so are

usually determined empirically from other measurements in the porous media. For

this reason, it is more convenient to explicitly measure capillary pressure and use

the equation below to calculate the height of the fluid column.

h =144 P c

ρ w − ρ o


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Example 3.3

Using the drainage capillary pressure curve of the Venango Core (shown below).

How many feet above the free water table is the water/oil contact? (1 ft = 30.48

cm)

ρ w = 1 gm cm3, ρ o = 0/ gm cm

3,1 cm o% merc!ry 13,3222 dynes cm

Solution

From the figure, the capillary pressure at the water-oil contact can be read as 4cm.

Since P c = hw ρ w − ρ o( ) g = 4 13,322 +y"es cm

Then,

hw = P c

ρ w − ρ o( ) g =

4 13,2222( )1− 0/( )50

= 21/ cm

=

t 1/

4530

21/=

3.3 La$%rat%ry &eth%'s %( &easuri)* +apillary pressure

Three generally accepted methods of measuring capillary pressure in the

laboratory are:

a) The Porous Diaphragm (or restored state) Method

b) The Centrifugal Method

c) The Mercury Injection Method


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All three tests are conducted on core plugs cut from reservoir whole core samples.

Drilling fluids, coring fluids, coring procedure, core handling and transportation,

storage and experimental processes can alter the natural state of the core.Therefore, special precautions are necessary to avoid altering the natural state of

the core. If the natural state of saturation of the core had been altered, then it must

be restored to its natural state before conducting any capillary pressure tests.

Fresh Core :

Samples from core taken with either water or oil-base muds that are preserved

(with invaded fluids) and subsequently tested without cleaning and drying are

referred to as fresh cores.

Native State Cores :

Samples from core recovered with lease crude or special oil base fluids known to

have minimal influence on core wettability, and that are tested as fresh samples,

are referred to as Native State. These cores are in their native state (i.e. without

invaded fluids). Such cores coming from above the transition zone should have

the same quantity and distribution of water as in the reservoir. These samples are

preferred for water displacement tests.

Restored Cores :

$ore samples clea"e+ a"+ +rie+ prior to testi" are reerre+ to as restore+ cores

A" a+&a"tae is tat air permeability a"+ porosity are a&ailable to assist i" sample

selectio" A +isa+&a"tae is tat core wettability a"+ spatial +istrib!tio" o pore

water may "ot matc tat i" te reser&oir

The following precautions can be helpful in obtaining representative cores if the

drilling conditions permit.

1. Use oil-base drilling mud to minimize clay swelling

2. Use non-oxidized lease crude as a coring fluid.

3. Suitable storage procedures include submersion under degassed water, andpreservation with saran foil, and wax.

Refined oil versus crude oil

Refined oils are suitable for most tests, and are preferred when tests are at ambient

conditions.

Crude oils to be used in ambient tests should be sampled from non-water

producing wells upstream from chemical or heater treaters.

Crude oils often precipitate paraffin or asphaltenes at ambient conditions,

resulting in invalid test data.


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Reservoir condition test utilizing live crude oil at reservoir pressures and

temperatures often overcome difficulties experienced with crude at ambient

conditions.Reservoir fluid samples for special core tests may be recovered using bottom-hole

sampling techniques, or recombined from separator gas and oil samples.

3.3.1 Centrifu"al $et!od

1. Rotate at a fixed constant speed. The centrifugal force displaces someliquid, which can be read at the window using a strobe light. Thus, the saturation

can be obtained.

2. The speed of rotation is converted to capillary pressure using appropriate

equations.

3. Repeat for several speeds and plot capillary pressure with saturation.

3.3. $ercury In%ection $et!od

1. Place core sample in a chamber and evacuate it.


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2. Force mercury in under pressure. The amount of mercury injected divided

by the pore volume is the non-wetting phase saturation. The capillary pressure is

the injection pressure.3. Continue for several pressures and plot the pressure against the mercury

saturation.

Advantages: 1. Fast (minutes)

2. No threshold pressure limitation

Disadvantage: 1. Can only be used for shaped cores.

3.3.3 Porous &iap!ra"m $et!od

core

water

+isplaci" %l!i+ (air)

re!lator

6

1

2

air

air s!pply

tiss!e paper, p!l&erise+ talc,ale"a, %lo!r tomai"tai" capillaryco"tact

poro!s +iapram (%iltere+ lass +isc,cellopa"e, porcelai") sat!rate+ wit+isplace+ %l!i+

1. Saturate both the core sample and the diaphragm with the fluid to be

displaced.

2. Place the core in the apparatus as shown

3. Apply a level of pressure, wait for the core to reach static equilibrium.The capillary pressure = height of liquid column + applied pressure

#at!ratio" 6ore &ol!me - 7ol!me pro+!ce+

6ore &ol!me

Production

Equilibrium


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4. Increase the pressure and repeat step (3)

5. Plot capillary pressure versus saturation

P c

S w

P c

Disadvantages: 1. Have to work within threshold pressure of the

diaphragm

2. Takes too long to reach the equilibrium, therefore a

complete curve takes from 10 - 40 days

Mercury injection technique was developed to reduce this time.

3., Other &eth%'s

Dynamic method:

$oreas

oil

1. Simultaneous steady flow of two fluids is established in the core2. Using special welted discs, the pressure of the two fluids in the core is

measured.

The difference = Capillary pressure

3. Change the rate of one fluid and the saturation changes

4. Plot P c versus saturation.

3.'.1 (ield $et!od)

Time


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A long column of porous medium put in contact with a wetting fluid at its

base and suspended in the earths gravitational field. It is left to reach equilibrium.

Samples are taken at different heights and the capillary pressure calculated using P c = h ρ g

Disadvantage: May take very long to reach equilibrium

3.'. Capillary *ysteresis


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3.'.3 E+planations for capillary !ysteresis

1. The advancing and receding contact angles are different. If the contactangle during imbibition is the advancing contact angle, it differs from the contact

angle drainage (receding). This may explain the phenomenon of hysteresis.

a+&a"ci" co"tact a"le

P w P o P o P w

θ θ R

8ece+i" co"tact a"le

Oil

2. "Ink bottle effect

For porous media modelled as a bundle of tubes with varying diameters, a given

capillary pressure exhibits a higher fluid saturation on the drainage curve than on

the imbibition curve.


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3.'.' ,!e effect of pore si-e distribution on capillary pressure cure

The more uniform the pore sizes, the flatter the transition zone of the capillary

pressure curve.

3.'./ Conersion of Laboratory Capillary &ata to Reseroir Capillary&ata

Water (brine) - oil capillary pressure data are difficult to measure in the

laboratory. Generally, air - brine or air - mercury data are measured instead and it

becomes necessary to convert these data to equivalent oil - water data

representative of reservoir fluids. If we denote P c( )aw or P c( )a, g as P c( )!ab , and

P c( )ow as P c( )res the conversion equation can be derived as follows:

From P c( )!ab =

2σ awCosθ aw

r , we obtain

r =2σ awCosθ aw

P c( )!ab

From P c( )res =

2σ owCosθ ow

r , we obtain

r =2σ owCosθ ow

P c( )res

Assuming that the same porous medium applies in both laboratory and field, we

equate the r 9 s to obtain,

P c( )res =2σ owCosθ ow

2σ awCosθ aw P c( )!ab

ignoring the contact angles,

P c( )res = σ res

σ !ab P c( )!ab

An identical equation would be obtained by starting from the two equations:


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1

R1+

1

R2

!ab

= P c( )!ab1

R1+

1

R2

res

= P c( )resAssuming the radii of curvature in the laboratory is the same as that in the

reservoir, the RHS's can be equated and P c( )res :

P c( )res = σ res

σ !ab P c( )!ab

3.'.0 Calculatin" Aera"e ater saturation

: a reser&oir a&erae capillary press!re c!r&e (or e&e" a laboratory c!r&e) is

a&ailable, it ca" be co"&erte+ to a eit &ers!s water sat!ratio" c!r&e a"+ !se+ to

calc!late te a&erae water sat!ratio" or a"y +esire+ i"ter&al O"e simply "ee+s

to p!t a "ew scale or te eit o" te y-a;is o te 6 c rap 'e a&erae water

sat!ratio" betwee" a"y two eit i"ter&als ca" be e&al!ate+ as te area e"close+

betwee" tem +i&i+e+ by te +ista"ce betwee" te eit i"ter&als A" e;ample

ill!strates te proce+!re

Example 3.4

For a pay zone whose top and bottom are 45 ft and 25 ft from the free water table,

use the laboratory Pc graph below to calculate the average water saturation for this

pay interval.

tlbc!45t,lbc!4


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Solution to Example 3.4

First, co"&ert te 6c lab to 6c res=

labclabc

lab

labcres

resc60

)/0(

6)3(66 ==

σσ

=

.e;t, co"&ert 6c res to eit abo&e te ree water table a"+ plot o" te rit a;is by

p!tti"

a "ew scale o" te 8># or ?@ :ts scale is 0 × te scale or 6c lab 'is issow" below


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labclabc

ow

resc 6)454


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3.- Aera*i)* Capillary Pressure Cures

Consider a reservoir cross-section from which four core samples are taken at

different depth as shown below. Each core will generate its own complete

capillary pressure curve in the laboratory which can be converted to a reservoir

capillary pressure curve. Thus four different laboratory capillary pressure curves

are obtained as shown below. The question then arises:

How do we get a single P c or height &s S w curve to represent the reservoir?

The answer is to use the Leverett J-function

(1)

(2)

(3)

(4)


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3./.1 Leerett 2function

The Leverett J-function is defined as:

" = P c

σ Cosθ

#

φ

1

2

where, # = permeabilityσ = interfacial tensionθ = contact angleφ = porosity

The J-function has the effect of normalizing all curves to approach a single curve

and is based on the assumption that the porous medium can be modelled as a

bundle of non-connecting capillaries (Slider pp 279-280). Obviously the more

capillary bundle assumption deviates from reality, the less effective the J-functioncorrelation becomes. This correlation is not unique, but seems to work better

when the rocks are classed as to rock types, eg; limestone, dolomite, etc.

Given several capillary pressure curves, with corresponding values of

permeability # ( ) and porosity φ ( ), the procedure for obtaining J-function curve isas follows:

a) Pick several values of S w from 0 to 1 and read the corresponding values of P c . There will be as many P c values as there are curves.

b) For each P c value, calculate J and plot versus S w . Repeat for all S w values.

c) Put your best correlation curve through the data.


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This J-Curve is now a master curve that can be used to represent that reservoir and

in the absence of other data can be used for other reservoirs of similar rock type.

The graphs below, taken from (Amyx et.al.) shows the J-function curve for theEdwards formation showing classification as to rock types.

Fig. J-function correlation of capillary pressure data in the Edwards formation,Jourdanton Field. J-curve for (a) all cores; (b) limestone cores; (c) dolomite cores;

(d) microgranular limestone cores; (e) coarse-grained limestone cores. (Source

Amyx et.al.)

3./. *o to use t!e Leerett 2function to calculate Aera"e 4aterSaturation

Values for average initial or connate water saturation are required in many

petroleum engineering calculations. Examples are: (a) average water saturation in


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a section of reservoir in order to fix effective fluid permeabilities,$ g ,$ o,$ w, and

(b) average water saturation in the whole reservoir in order to fix the initial

hydrocarbon volume in place,

% =//5 Ahφ 1− S wi( )

&oi

Under capillary equilibrium conditions, the water saturation of a particular piece,

or sample of rock not depends on several factors. It has been shown with certain

limitations, that a properly determined Leverett J-function versus water saturation

curve can be used to obtain an average water saturation from a number of

capillary pressure curves. It is assumed that a Leverett J-function curve is

available and applies to the reservoir. The objective here is to show how to use the

J-function to obtain the best possible estimate of average saturation. Recall that

the J-function is defined as:

" = P c

$

φ

σ Cosθ

By expressing the P c term in terms of height and fluid densities the equivalent

equation is:

" =h ρ w − ρ o( )

$

φ

144σ Cosφ

It is important to note while applying this equation that its units are not important.

Mixed units can be used without appropriate conversion factors. It is only

important to be sure to use the same units that went into determining the values of

J making up the original plot. In other words, find out what units were used to

calculate the J-function curve and stay consistent with those units whether they

are mixed or not.

Note also that J=constant*h. Therefore, the shape of a J-function versus S w curve

would be similar to that of a height versus S w curve. The difference is a

displacement by a factor equal to the constant. Thus, a P c curve can be converted

to a height curve simply by adding a new y-axis having its abscissa equal to the

constant* P c .


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3./..1 Case 1) Permeability5 Porosity5 and Eleation are #non foreac! sample

12

3

4

# ,φ

# ,φ

# 3,φ 3

# 4,φ 4

P c = 0

hh

2

h3h

Dat!m

This figure illustrates four reservoir samples having different values of

permeability and porosity and located at different heights above a P c = 0 datum.Assuming fluid properties are the same in all pieces, the J-function equation can

be simplified to:

" = ch $

φ were c is te co"sta"t

ρ w − ρ o144Cosθ

3./.. ,!e correct met!od

The correct method of obtaining the average saturation, S wi for the four pieces is

to calculate J for each piece, determine the corresponding water saturation, S wi of

each piece by using the J-curve and then taking the arithmetic average of the

saturations with the equation:

S wi = 1 % S wi( ) '=1

'=

∑ 'Note that this procedure correctly takes into account the vertical position of the

pieces and their corresponding permeability and porosity.

ess correct meto+s

These methods first calculate average values of $ ,φ , and h , substitute them into

the J equation to get an average J value, and then read the average water saturation

S wi from the J-function versus S w graph. The only advantage of these methods is


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that the amount of calculations is reduced. The resulting S wi will always have

error in it. How much error depends on the specific condition being calculated.

The figure below illustrates the concept behind using average values in order to

obtain an average J value.

3

4hh

h

h P c = 0

h

$

φ 1

2

There are two ways:

Method (a): Calculate

#

φ for each sample and obtain the arithmetic average for

all four. Also, obtain the arithmetic average h . It is assumed that the average

#

φ

is located at the average height h . The average J-function equation then becomes:

" = ch $

φ were c is a co"sta"t

where,

h =1

%

h '

' =1

%

∑

$

φ =

1

%

$ '

φ '

'=1

%

∑

This is the easiest of the averaging methods to do.

Method (b): The geometric average permeability and porosity are used to get the

average J-function:

" = ch # G

φ G


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# G = geometric mean permeability =

A"tilo

1

%

lo # ' '=1

%

∑ = # 1 # 2 # %

%

φ G = geometric mean porosity =

A"tilo1

%

loφ '

' =1

%

∑ = φ 1φ 2φ % %

Zero values of # ' and

φ ' are not permitted when evaluating the geometric

averages. Because porosity values usually show very limited range, the geometric

average porosity, φ G , can be replaced by the easier to calculate arithmetic

average, φ A, with little loss of accuracy. Therefore, the form used by mostengineers is

" = ch # G

φ A

where,

φ A =1

%

φ ' '=1

%

∑is the arithmetic average

3./..3 Errors due to using average values of # and φ

Standing(4) discusses the amount of error in S w introduced by using average

values of $ ,φ , and h and states that the error depends on several factors.

One factor is the distribution of $ 9 s in a vertical sense. If the $ 9 s are distributed

randomly, no error will be involved. On the other hand, if high permeabilities

predominate in one portion of the section and low permeabilities in another, some

error will be introduced.

A second factor is the shape of the " &s S w curve. Where log J is linear to S w , no

error will result from geometric average $ . Where J is linear withS

w , some errorwill result.

A third factor is the range of permeability values. Little error is introduced when

the range is small and more error is introduced when the range is large. The best

way to minimize errors of averaging is not to average. Use the correct method.

3./..' Case ) Permeability and porosity are un#non as functions

of eleation. &istance from P c = 0 6distance from freeatertable7 is #non


29/34

The petroleum engineer often needs to develop a value for average water

saturation but does not have detailed information on permeability and porosity as

a function of elevation. (Many wells are not core analyzed). However, he mayknow from results of pressure buildup tests that the average permeability in the

region of the wellbore is, say, 100 md. Also, he may know from well logs that an

average porosity is, say 18%. With these average permeability and porosity values

plus information on the distance to the appropriate P c = 0 datum and informationon fluid properties, he can make a reasonable calculation of the average water

saturation.

To illustrate the method of getting S wi , consider the sketch above. At the wellbore

location, the bottom and top of the formation are hbottom andhtop from the P c = 0

datum. For given values of # ,φ , ρ w , ρ o,σ Cosθ , a"+ h , calculate " top a"+ " bottom.

Shade the area enclosed by " top and " bottom on the " curve and calculate the

average initial water saturation S wi .

The simplest way of determining S wi is by graphical integration. Thus, determine

the area under the curve, divide this area by the value

" top − " bottom

( ) and the resultwill give S wi . That is:

S wi =S wi " bottom

" top

∫ d" " top − " bottom

Example 3.5

oil zone

free water

GOC

WOC

J top

J bottom

Sw 100%

gas

oil

fault


30/34

Example calculation of the use of Capillary Pressure Data to Obtain Average

Water Saturation Using J-Function

It is desired to calculate the initial oil in place for an oil reservoir having a gas cap

as illustrated below. There is no prior J-function curve available and no well logs

to give permeability, porosity, and saturation data with depth. All we have are old

cores from storage.

The bulk volume of the oil zone is 1,000 acre-ft. The thickness of the oil zone is

20 ft. Four core samples were taken from the oil zone in the middle of 5 ft.

intervals. From laboratory measurements of porosity and permeability, the data

are:

Interval depth Permeability Porosity4,000 - 4,005 11.2 0.147

4,005 - 4,010 34.0 0.174

4,010 - 4,015 157.0 0.208

4,015 - 4,020 569.0 0.275

as

a!lt

oil o"e

ree - water table

well

GOC

WOC

The free-water table is at a depth of 4030 ft. In addition to porosity and

permeability, the capillary pressure for each sample was measured using air

displacing water in a centrifuge. These laboratory derived capillary pressure

curves are shown below. The water/oil interfacial tension for this reservoir is

estimated to be 28 dynes/cm, the reservoir (water/oil) wetting angle is 0.0. The

air/water interfacial tension is 70 dynes/cm with a wetting angle of 0.0 also,

ρ w =


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Solution

a) Convert the P c!ab data to cres

P data and calculate the J-function curve using:

==

!ab

resc!abcres

resres

cres P P # P

"

σ

σ

φ θ σ

were

cos

This has been calculated and plotted below.


32/34

b) Calculate the value of " at each "h " of each core and read the corresponding

water saturation from the " curve.

" = h


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" = 00030/∗ 20 ∗/


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b) An elliptical shape pore throat of d1 = 0.0001 inches and d2 = 0.001 inches

d2d1

c) An infinite horizontal fracture of fracture width = 0.0001 inches.

Use σ = 35.2 dynes/cm, and θ = 0.0

References

1. Clark, Norman J. "Elements of Petroleum Reservoirs" Henry L. Doherty

Series, Society of Petroleum Engineers of AIME, Dallas, 1960.

2. Slider H.C., “Worldwide Practical Petroleum Reservoir Engineering Methods”,

Penwell Books, 1983

3. Wilhite, G.P. : “Waterflooding”, SPE Textbook Series, Vol. 3, 1986.

4. Standing, M.B.: Lecture notes, Stanford University, 1977

5. Amyx, J. W., Bass, Jnr. D. M., Whiting, R. L. : Petroleum Reservoir

Engineering, McGraw-Hill, 1960

capillary 1.doc

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