3.3 Capillary Pressure

3 3 Capillary pressure of reservoir

3.3 Capillary pressure of reservoir

rocks

1 Rise of fluid in Capillaries

2 Additional pressure of arbitrary curvature (Laplace equation )and its application

(Laplace equation )and its application

3 Capillary Pressure effect in Porous Mediap y 4 Capillary Pressure Hysteresis

5 measurement of Capillary Pressure 6 Application of Capillary Pressure

1 Rise of fluid in Capillaries

1) Capillary pressure of gas-liquid system

If a Capillary tube is placed in a large open vessel continuing liquid , liquid will rise in the

tube above the height of the liquid in the large vessel

liquid in the large vessel.

This rise in height is duo to the g attractive force (adhesion

tension) between the tube and th li id d th ll i ht

Fig 3-20 Pressure relations in Capillary tube

the liquid and the small weight

of the column of liquid in the tube.

Capillary pressure:

Capillary pressure:

p

_c

=

2

σ

cos

θ

r

p

_c

Definition:

The pressure difference between p

non-wet phase and wet phase is called capillary pressure.

p

expressed by P_c.

the direction was wet phase to non wet phase the direction was wet phase to non-wet phase.

2) Capillary pressure in oil-water system) p y p y

θ

σ

cos

2

r

p

_c

=

2

σ

cos

θ

r

2

_wo

cos

_wo

h

σ

θ

0

(

)

wo wo w

h

rg

ρ ρ

=

−

Fig.3-21 Pressure relations in Capillary tube

0

(

_w

)

3)Property of capillary pressure

A.

The capillary pressure exists in capillary t b l id i di ti

tube laid up in any direction

3)Property of capillary pressure

B.

P_C is proportional to cosθ, and inverse proportional to r;

Figure 7.8 Capillary rise experiments for two porous

3)Property of capillary pressure 3)Property of capillary pressure

3)Property of capillary pressure

C

if rock surface is water-wet capillary pressure 3)Property of capillary pressure

C.

if rock surface is water-wet , capillary pressure is driving force of displacing oil by water;

D.

If rock surface is oil-wet , capillary pressure is resistance of displacing oil by water;

is resistance of displacing oil by water;

F.

If rock surface is water-wet, water can ,

automatically enter rock ; but if rock surface is oil wet water can not automatically enter is oil-wet , water can not automatically enter rock .

4) Some definitions 4) Some definitions

Drainage (驱替过程）:

A process displacing the wetting phase from a porous medium with a

non-wetting phase from a porous medium with a non wetting phase is known as drainage.

Imbibition（吸吮过程）：

A process displacing

Imbibition（吸吮过程）：

A process displacing the non-wetting phase from a porous medium with a wetting phase is known as imbibition

Question: Why does the oil-water contact is a transitional zone?

2.

Additional pressure of arbitrary curvature

1) Derived equation (推导公式）

● Laplace equation can be derived by considering

the mechanical equilibrium of the interface.

●The work done in expanding the surface, by

increasing the pressure on the convex side is the increasing the pressure on the convex side, is the work against the surface tension.

⎟ ⎞ ⎜ ⎛ ⎟ ⎞ ⎜ ⎛ AB BC ' ' ' ' ' D 2 p ⎟ ⎞ ⎜ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + × ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + = dR dR BC AB dR R BC BC dR R AB AB D C B fA 2 1 ' ' ' ' 1 1 1 A ' A C' D dR ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + + × = dR R dR R BC AB 2 1 1 A ' B C 1 p R 2 R 1 R AB = θ B 1 θ θ₂ 1 R 1 R 2 BC = θ ΔW = Δp∗ fABCD ∗dR = AB∗ BC ∗Δp ∗dR 1 θ θ₂ 2 2 R 2 1 p p p_c = − ΔZ = ΔW BC AB fABCD = × _{p = Δp = ⎜⎜}⎛ 1 ₊ 1 _⎟⎟⎞ C σ ⎟⎟ ⎠ ⎜⎜ ⎝ R1 R2 p p_C

2) Equation simplification in Several i l

special cases

A the interface lies on a sphere A. the interface lies on a sphere

1 2

R

₁

=

R

₂

=

R

r

R

cos

R

θ

=

2 cos

σ

θ

Fig 24 the relationship between the capillary

2 cos

c

P

r

σ

θ

=

Fig.24 the relationship between the capillary

B.

If curved surface is cylindrical surface

1

,

2

R

= ∞

R

=

r

1

1

(

)

c

P

=

σ

+

=

σ

=

σ

1 2 2

(

)

c

R

R

R

r

c.

Capillary pressure in conical capillary

p

y p

p

y

R₁=r₁/COS(θ+β) R₂=r₂/COS(θ-β) so:

(

)

(

)

ci

r

p

=

2

σ

cos

θ

±

β

i

r

D. capillary pressure in cracks

p

y p

R = ∞ R₂ = ∞

2

/

cos

R

W

=

θ

1

R

p

=

σ

=

2

σ

cos

θ

W

R

p

_c 1

F.

The capillary pressure in packing of uniform h spheres

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

=

1

1

R

R

p

_C

σ

_⎟

⎠

⎜

⎝

R

1

R

2 C 2 1

1

1

1

R

R

R

=

R

₁

+

R

₂

R

_m c

R

p

=

σ

m

R

3 capillary effect in capillary tube

1) When the droplet (or bubble) is at the static state

The capillary pressure of spherical interface

θ σ

σ 2 cos 2

'

The capillary pressure of spherical interface

r R

p_c = σ = σ cosθ

The capillary pressure of

σ

The capillary pressure of cylindrical interface

r

p

_cz''

=

σ

Th ill ff t

(

cos

0

5

)

2

cos

2

σ

θ

σ

σ

_θ

p

The capillary pressure effect :

(

cos

−

0

.

5

)

=

−

=

Ι

θ

r

r

r

p

2) when the droplet moves in a capillary

2) when the droplet moves in a capillary

tube of constant diameter

2

σ

2

σ

' '

2

R

p

=

σ

'' ''

2

R

p

=

σ

R

R

(

'' '

)

' ''

cos

cos

2

1

1

2

σ

⎜

⎛ +

⎟

⎞

σ

θ

θ

p

p

p

2

σ

_{'' '}

⎟

=

(

cos

θ

−

cos

θ

)

⎠

⎞

⎜

⎝

⎛ +

=

−

=

r

R

R

p

p

p

_C

3) When the droplet passes through

)

p

p

g

a throat of a pore

⎞

⎛

1

1

⎟

⎠

⎞

⎜

⎝

⎛

₋

=

Ι _{'' '}

1

1

2

R

R

p

_C

σ

⎠

⎝

R

R

J

i Eff t

When the droplet flow in a nonuniform pore

Jamin Effect:

When the droplet flow in a nonuniform pore to a capillary tube of decreasing size ,a higher pressure drop is required to move the drop into the constriction the constriction.

⎞

⎛

1

1

⎟

⎠

⎞

⎜

⎝

⎛

₋

=

Ι _{'' '}

1

1

2

R

R

p

_C

σ

⎠

⎝

R

R

4

Capillary Pressure Hysteresis

1) _{Capillary Pressure Hysteresis by Contact} angle hysteresis

angle hysteresis

2)

)

Capillary Pressure Hysteresis by the

p

y

y

y

sudden change in diameter

3)

)

Capillary Pressure Hysteresis by the

p

y

y

y

gradual change in diameter

(

)

P

p

=

2

σ

cos

θ

+

β

p P

r

p

(

)

t

p

=

2

σ

cos

θ

−

β

t

r

4)

capillary hysteresis in actual rock

Visual advancing

θ

=

θ

+

β

Visual advancing angle:

θ

A

=

θ

1

+

β

β

θ

θ

Visual receding

θ

=

θ

−

β

2 R Visual receding angle: displacement:

(

)

R dr

p

=

2

σ

cos

θ

=

2

σ

cos

θ

2

−

β

displacement: t t dr

r

r

p

imbibition :

(

)

A i

p

=

2

σ

cos

θ

=

2

σ

cos

θ

1

+

β

imbibition : P P im

r

r

p

5 Measurement of Capillary Pressure Curve

5 Measurement of Capillary Pressure Curve

毛管压力曲线的测定

Semi-permeable Disk Method Mercury injection method

Centrifuge method

The dynamic capillary-pressure method The evaporation method

Definition :

C

Capillary Pressure curve:

The relationship curve between capillary pressure of reservoir p y p rock and the wetting- phase saturation is called Capillary Pressure curve

Capillary Pressure curve.

)

(

f

P

f

(s

)

1) Semipermeable Disk Method

A. Normal pressure Semipermeable Disk Method

non-wetting phase is air non-wetting phase is air

wetting phase is water

the maximum capillary pressure is about 1atm

1) Semipermeable Disk Method

The bottom of the vessel consists of a

semi-permeable plate, which allows the wetting phase allows the wetting phase displaced from the sample to pass through while

blocking the passage of the non-wetting phase the non wetting phase

1) Semipermeable Disk Method P W w

V

V

S

V

−

−

=

最初岩样饱和水体积 累计排出水体积

=

∑

最初岩样饱和水体积

_V

_P

最初岩样饱和水体积

Principle and step of measurement:

● _{placing the sample, initially saturated with a}

wetting fluid, in a vessel filled with the non-wetting fluid.

● _{With the sample on the porous plate, the}

pressure of the non-wetting fluid is increased in steps and the system is allowed to achieve

equilibrium after each pressure change.

● _{The volume of wetting phase displaced at each g p p}

pressure is measured.

● _{The wetting phase saturation of the sample is} ● _{The wetting phase saturation of the sample is}

determined from the volume of wetting phase

displaced at each pressure to obtain the capillary displaced at each pressure to obtain the capillary pressure versus saturation relationship.

★

Determination data record

★

Determination data record

★

typical curve

c P 布 ，% 16 20 孔 隙大小分 布 8 12 16 孔 4 8 (%) w S 孔隙半径，r, μ_m 4.2 9 18 27 54

Fig.-1 capillary pressure curve

Fig. -2 pore size distribution curve

小

累积，%

孔隙大

小

r

1) Semipermeable Disk Method

Principle and step of measurement:

● _{l i th l i iti ll t t d ith} ● _{placing the sample, initially saturated with a}

wetting fluid, in a vessel filled with the non-wetting fl id

fluid.

● _{With the sample on the porous plate, the}

pressure of the non-wetting fluid is increased in steps and the system is allowed to achieve

equilibrium after each pressure change.

● _{The volume of wetting phase displaced at each g p p}

pressure is measured.

● _{The wetting phase saturation of the sample is} ● _{The wetting phase saturation of the sample is}

determined from the volume of wetting phase

displaced at each pressure to obtain the capillary displaced at each pressure to obtain the capillary pressure versus saturation relationship.

Advantage of the porous plate method: Advantage of the porous plate method:

● _{This method use oil and water ,therefore more ,}

nearly approaching actual wetting conditions.

● _{The method gives a reliable estimate of the}

irreducible wetting phase saturation. Disadvantage:

● _{The porous plate limits the maximum capillary}

pressure to about 200 psi. pressure to about 200 psi.

● _{It takes too long to obtain the entire capillary}

b thi th d pressure curve by this method.

2)

Mercury injection method

2)

Mercury injection method

Principle and step of measurement:

Principle and step of measurement:

non-wetting phase — mercury; wetting phase — air

● _{The core is placed in file sample chamber of the} ● _{The core is placed in file sample chamber of the}

mercury injection equipment

● _{The sample chamber is evacuated, and}

incremental quantities of mercury are injected while incremental quantities of mercury are injected while the pressure required for injection of each increment is recorded

Principle of measurement:

1

p

1 2 5 4 3

1 — Nitrogen pressure; 2 — pressure gauge;

3 i j ti 4 l ll

3 — mercury injection pump; 4 — sample cell; 5 — vacuum system

Figure 7 43 Mercury air capillary pressure curves Figure 7.43. Mercury-air capillary pressure curves

Advantage of mercury injection method :

● _{The mercury injection method is very fast} ● _{The mercury injection method is very fast.} ● _{The range of pressure is large.}

Disadvantage of mercury injection method :

● _{Core can no longer be used for other tests after}

mercury injection.

● _{The method also cannot be used to determine}

the irreducible wetting phase saturation. the irreducible wetting phase saturation.

● _{mercury vapor is toxic, so strict safety precautions}

must be followed when using mercury must be followed when using mercury.

3) Centrifuge method

Measuring Principle and step:

● _{th l t t d ith tti fl id i l d} ● _{the sample saturated with a wetting fluid is placed}

in a centrifuge cup containing the non-wetting fluid

● _{The sample is rotated at a series of constant}

angular velocities and the amount of wetting fluid angular velocities and the amount of wetting fluid

displaced at equilibrium at each velocity is measured

● _{This process is continued until no more fluid} ● _{This process is continued until no more fluid}

2

F

=

mw r

1

F

mw r

2

2

2

2

1

1

(

)

c

P

= Δ

ρ

w r

(

₂

−

r

₁

)

2

c

ρ

Advantage of Centrifuge method :

Advantage of Centrifuge method :

● _{The centrifuge method is fast}

● _{The method is good for determining the}

irreducible water saturation.

● _{It can simulate the process of water or gas}

displace oil it is a promising method

Disadvantage:

.

displace oil .it is a promising method.

● _{inability to obtain spontaneous imbibition}

capillary pressure curve.p y p

● _{the calculated water saturation at the core inlet is}

4) Converting the laboratory data

i

di i

to reservoir conditions

θ

2

2

θ

r

p

L L cL

θ

σ

cos

2

=

cL L L

p

r

=

2

σ

cos

θ

→

r

p

cL R R

θ

σ

cos

2

₂

_σ

_cos

_θ

r

p

R R cR

θ

σ

cos

2

=

cR R R

p

r

=

2

σ

cos

θ

→

R R

θ

σ

cos

cL L L R R cR

p

p

θ

σ

θ

σ

cos

cos

=

L L

The conversion between semi-permeable disk method and oil-water capillary pressure under reservoir

conditions.

1

0

25

°

θ

wg wg wg wg wg ow ow ow

p

p

p

p

3

1

0

cos

72

0

cos

25

cos

cos

₌

×

×

=

=

_°

θ

σ

θ

σ

5) The characteristics of capillary pressure curve 5) The characteristics of capillary pressure curve

Swi

—irreducible saturation of wetting fluid

Pt

— threshold displacement pressure corresponds to the pressure, corresponds to the onset of invasion of the

medium medium

P

_c50 — median pressure,

Fig. Qualitative characteristic

of

capillarity pressure curve

c50 p

corresponds to the nonwetting phase saturation of 50% .

★

Mercury injection efficiency:

★

Mercury injection efficiency:

W

(S

S

) / S

W

_E

= (S

_Hgmax

– S

_Hgmin

) / S

_Hgmax W_E: Mercury injection efficiency;

S i t ti

S_Hgmax: maximum mercury saturation; S_{H i} : minimum mercury saturation S_Hgmin: minimum mercury saturation

The mercury injection efficiency can be regarded as The mercury injection efficiency can be regarded as the oil recovery in a strongly water-wet oil reservoir

typical capillary pressure curve

(a) Well sorted sample, with medium-size pores;

(b) Nonsorted sample;

(c) Well sorted sample, with large pores; (d) Well sorted sample with fine pores; (d) Well sorted sample, with fine pores;

(e) Poorly sorted sample, with more fine pores; (f) Poorly sorted sample, with more large pores.

6 Application of Capillary Pressure curves

1) Determining rock wettability

A. Determining by Wettability number

cos

_wo

P

_{Two og}

W

=

θ

=

σ

cos

_{og Tog wo}

W

P

θ

σ

W=1 →complete wetting by water; W=0 →complete wetting by oil;

B. Determining by apparent contact angle

Two og Two og

P

σ

P

σ

cos

_wo Two og _wo

arccos

Two og

Tog wo Tog wo

P

P

θ

θ

σ

σ

=

⇒

=

Tog wo Tog wo

=00 _{→ complete wetting by water;}

wo

θ

wo

C D t i i b D ld ’ th d C. Determining by Donaldson’s method

Principle of Determination

Determining by comparing the area unclosed Determining by comparing the area unclosed by the curve of water displacing oil with the area unclosed the curve of oil displacing water.

C Determining by Donaldson’s method C. Determining by Donaldson s method

1

l

A

0

0.7 Ⅰ 1 2

log

0

A

>

water-wet; 0 Ⅲ Ⅰ A₁ 1

log

A

0

A

<

Oil-wet; 0 Ⅱ A2 2

A

A

₁ 2

log

A

0

A

=

intermediate wetting -0.7 0 100 2

2) determining the pore size distribution

)

g

p

of porous materials

(确定孔隙大小分布） 20 小 分布，% 12 16 20 孔隙大 小 8 12 4 4.2 9 18 27 54 孔隙半径，r, μm

2) determining the pore size distribution

)

g

p

of porous materials

(确定孔隙大小分布）

r

_max

= 2σcosθ

r

_max

2σcosθ

P

_T

75

0

T

p

r

_max

=

0

.

75

3)

Calculation of permeability from

3)

Calculation of permeability from

drainage capillary pressure curve

根据驱替曲线计算渗透率

A.

Calculation of absolute permeabilityp y

4) St d i

il

4) Studying oil recovery

%

100

max

−

R Hg

S

S

E

100

%

max max

×

=

Hg R Hg w

S

E

5) Studying initial static fluid distribution in

petroleum reservoir

（

研究流体在油藏中的分布

）

5) Studying initial static fluid distribution in

petroleum reservoir

（

研究流体在油藏中的分布

）

petroleum reservoir

（

研究流体在油藏中的分布

）

①

_{the oil water contact level (100% water}

①

_{t e o ate co tact e e ( 00% ate}

saturation lever)

→

P

_T

②

the free water level

→

P

_CC

=0

③

connate water saturation level

→

S

_CW

③

connate water saturation level

S

_CW

④

theoretical transition zone

→

the height

④

theoretical transition zone

→

the height between 100% water saturation lever and

t t t ti l l

⑤

→

S

⑤

level of f_w=100%

→

S

_or

⑥

actual transition zone

→

the height

between level of fw=100% and connate water between level of fw 100% and connate water saturation level

1.0 0.8 0.6 0 4 0.4 0.2 0 20 40 60 80 100 0 含水饱和度Sw（%） 纯油产区 Pc(R) H 纯油产区 （含束缚水饱和度） Swi B C 油水同产区 C 纯水产区（含 残余油） 0 20 40 60 80 100 含水饱和度（%） 自由水面

To convert capillary pressure data to

h i ht b f t f

100 P

height above free water surface

0

100

_cR w

P

h

ρ

ρ

=

−

₀ w

ρ

ρ

h:

height above free water level m

P

_cR

:

capillary pressure at some particular

h:

height above free water level ,m

cR p y p p

saturation for reservoir conditions,（MPa）；

d i f d il i

ρ

_w

,ρ

_o

:

density of water and oil at reservoir conditions of water and oil（g/cm3₎

E l 1 Th ill f h b

Example 1:The capillary force curve has been

obtained from laboratory. if the water saturation is 35% d th ill i 0 126MP

35%, and the capillary pressure is 0.126MPa, calculate the height of water saturation of 35%

l b f t l l

plane above free water level. L t t i diti

，

Let at reservoir conditions , σ_wo = 24 mN/m, 1 088 / 3 ， ρ_w = 1.088 g / cm3_, ρ_o = 0.848 g / cm3_, t t h i at atmospheric pressure σ_wg= 72 mN/m.

Example 2: The air –water capillary pressure curve is b i d b i bl di h h d i h

obtained by semi-permeable diaphragm method in the

laboratory. When the water saturation is 50%, the capillary pressure is measured to be p =0 06Mpa The surface tension pressure is measured to be p_cL=0.06Mpa.The surface tension of water is 72 mN/m in surface conditions. While in the

reservoir conditions ,the interfacial tension between water , and oil is 24 mN/m.The water density is ρ_w = 1.088 g / cm3 and oil density is ρ_o = 0.848 g / cm3 .The altitude of free

t l l i 1000 th i k i t t d th water level is -1000m.the reservoir rock is water-wet, and the contact angle between water and reservoir rock is assumed to be the same as that at surface condition

to be the same as that at surface condition.

Calculate:

(1)The distance of the water level where water saturation is (1)The distance of the water level where water saturation is

50% to the free water level.

(2)The altitude of the water level where water saturation is (2)The altitude of the water level where water saturation is

D t i ti Thi k f il t

Determinating Thickness of oil-water production P_c1 2 2 or c

S

→

P

→

h

P_c2

S

cw

→ →

P

c1

h

1 S_cw S_or ᅀH = h1 - h2

6.Averaging capillary-pressure data

The definition of J-function :

g g

p

y p

1

P

K

₂

(

)

(

)

cos

c W

P

K

J S

σ

θ φ

=

★

_{Leverett J-function suggests that porous}

★

_{Leverett J function suggests that porous}

media that have the same pore structure but ff

different permeability and porosity will have the same Leverett J-function.

Formula Derivation:

2 1

8

K

⎛ K

⎞

2

2

σ

cos

θ

8

⎟

⎠

⎞

⎜

⎝

⎛

=

=

φ

φ

K

c

K

r

c

p

r

=

2

σ

cos

θ

⎠

⎝

φ

φ

p

_c 1

⎞

⎛

⎛

⎞

1

p

K

c

σ

θ

φ

cos

2

2

=

⎟

⎠

⎞

⎜

⎝

⎛

2

cos

2

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=

φ

θ

σ

K

p

c

c c

p

φ

_⎠

⎝

c

σ

cos

θ

⎝

φ

⎠

1

( )

2 1 cos ⎟⎟⎠ ⎞ ⎜⎜ ⎝ ⎛ = φ θ σ K p s J _w c cosθ ⎝ φ ⎠ σ

The characteristic of J-function

:

（a）全部岩样；（b）石灰岩岩样；（c）白云岩岩样 （d）细晶灰岩岩样；（e）粗晶灰岩岩样

Water-gas system in laboratory: 70 N/ θ 0° σ＝70mN/m,θ=0° core P K 10-3 _{Φ J(50)} c p

_J

core number P_C (50) K,10 3 μm2 Φ J(50) 1 3 66 11 2 0 147 0 46 1 3.66 11.2 0.147 0.46 2 2.5 34.0 0.174 0.50 3 1 38 157 0 208 0 54 1 3 1.38 157 0.208 0.54 4 0.85 569 0.275 0.55 1 2 3 4 J 4 J 5 . 0

⎟

⎠

⎞

⎜

⎝

⎛

φ

K

p

0

1

70

×

⎟

⎠

⎜

⎝

=

p

φ

J

c % w S

0

.

1

70

×

At reservoir conditions:

σ_R＝_{28mN/m,θ=0°every permeability is166μm}2_，

σ_R 28mN/m,θ 0 every permeability is166μm ， average porosity is 0.208 S_w% J(s_w) P_c(s_w) 100 0.35 0.35

p

70 0.40 0.40 54 0.45 0.45 c

p

J

44 0.60 0.59 30 1.45 1.43 20 3.15 3.12 % w S

J

J

J

99

0

10

28

cos

θ

×

×

σ

(

J

)

J

K

J

p

_c

0

.

99

208

.

0

166

10

28

cos

5 . 0 5 . 0

=

=

⎟

⎠

⎞

⎜

⎝

⎛

=

φ

θ

σ

208

.

0

⎟

⎠

⎜

⎝

φ

7 Capillary Pressure Hysteresis

7 Capillary Pressure Hysteresis

200 g /cm²) 100 sure （ kg 10 1 ry Pres s 10 I（displacement ） ① R ③（ ） 0.1 Capilla r R ③（displacement）阻滞滞 后 trap hysteresis ②（Inhaledt） 0 20 S 40 60 80 100 Swi 0.001 后 W 0 20 Sor 40 60 80 100 Swi Mercury saturation (%)

1.0 0.8 0.6 0 4 0.4 0.2 0 20 40 60 80 100 0 Water saturation Sw（%）

Pure oil producing areas(纯油产区)

Pc(R) H

Pure oil producing areas(纯油产区)

irreducible water saturation （含束缚水饱和度)

Swi B

C

Producing oil and water(油水同产区)

100％producing water surfa

(100％产水面)

C

Pure water producing areas irreducible oil （含残余油)

( )

Sw=1-Sor

0 20 40 60 80 100

Water saturation （%）