1

Study of Pressure Front Propagation in a Reservoir from a Producing Well

by

Hsieh, B.Z., Chilingar, G.V., Lin, Z.S.

May 4, 2007

2

Outline

• Introduction• Purpose• Basic theory and simulation tool• Results and discussions• Conclusions

3

Introduction

4

Producing rate and flowing pressure at wellbore

Pwf

(psi)

p=pi

q(stb/day)

q=constant

t (hours)t=tit=0

r

rw

t (hours)t=ti

5

Pressure distribution in reservoir at t = ti

P(psi)

r (ft)

p=pi

t=ti

q(stb/day)

q=constant

t (hours)

Radius of investigation (ri)

Pressure front

Pressure disturbance area ( Drainage area )

Non-disturbed area

t=ti

r=rw

t=0

ri

rw

t=ti

r=ri

6

Pressure distribution in reservoir at various times

P(psi)

r (ft)

p=pi

t=t3

q(stb/day)

q=constant

t=t2

t=t1

t (hours)

ri 3ri 1 ri 2

t=t1 t=t2 t=t3

pressure front s

7

t1

rw

r1 r2

r3

t2

t3

Plane view of pressure fronts at various times

8

Dimensionless Variables

Dimensionless radius

Dimensionless time

Dimensionless pressure Bq

ppkhp i

D 2.141

2

000264.0

w

Drc

ktt

wD r

rr

9

Pressure distribution in a reservoir in terms of dimensionless variables

PD

rD

tD3

tD2tD1

riD1 riD2 riD3

pressure front s

10

Radius of investigation (riD) and time (tD)

• The relationship between the dimensionless radius of investigation (riD) and the dimensionless time (tD) is (Muskat, 1934; Tek et al., 1957; Jones, 1962; Van Poolen, 1964; Lee, 1982; Chandhry, 2004, etc.)

riD2 = α tD

where the radius coefficient (α) is a constant and varied in different studies, from 3.18 to 16

11

Literature on radius of investigation equation

Author (Year)

Radius of investigation

equation

Definition of radius of investigation

or method used

Muskat (1934) riD2= 4tD Material balance method

Tek et al. (1957) riD2= 18.4tD The fluid flow rate at the radius of

investigation is 1% of that flowing into the wellbore

Hurst et al. (1961) riD2= 6.97tD Pressure build-up test

Jones (1962) riD2= 16tD The pressure drop at the radius of

investigation is 1% of pressure drop at the wellbore

Van Poolen (1964) riD2= 4tD Y-function of an infinite and a finite

reservoir

Hurst (1969) riD2= 8tD The analytical van Everdingen and

Hurst solution

12

Literature on Radius of Investigation (Cont.)

Author (Year)

Radius of investigation equation

Definition of radius of investigation

or method used

Earlougher (1977) riD2= 3.18tD After van Poolen (1964)

Lee (1982) riD2= 4tD The solution of the diffusivity

equation for an instantaneous line source in an infinite medium

Kutasov and Hejri (1984) riD2= 4.12tD Constant bottom-hole pressure test

Johnson (1986) riD2= 7.89tD The radius enclosing a volume in the

reservoir accounts for a specified fraction of the cumulative production of 96.1%

Chandhry (2004) riD2= 4tD Pressure transient analysis of pressure

build-up test

13

Purposes of the study

• To estimate the propagation of the radius of investigation from a producing well by using both analytical and numerical methods, including variable flow rates case, skin factor, and wellbore storage effect.

• To estimate the starting time of transient pressure affected by the reservoir boundary to concurrently determine the radius coefficient

14

Basic theory and simulation tool

15

Analytical Solution – Ei solution

• The analytical solution of the diffusivity equation for a well (line source) producing in an infinite cylindrical reservoir is (van Everdingen and Hurst, 1949; Earlougher, 1977):

)4

(2

1 2

D

DD t

rEip

n

k

kk

x

u

i kk

xxdu

u

exExE

1

1

1 )!(

)1(ln0.5772)()(

Bq

ppkhp i

D 2.141 where

2

000264.0

w

Drc

ktt

wD r

rr

16

Numerical Solution of Diffusivity Equation

• Numerical solutions are also used in this study for the cases that no analytical is available or the comparisons are required.

• The IMEX simulator (CMG) is used in this study to generate results in numerical simulation.

17

Basic reservoir parameters used in this study

18

The pressure behavior check-- infinite reservoir

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

tD

PD

van Everdingen & Hurst solution

CMG calculated PD

Even the specific oil reservoir is used in this study, the pressure behavior(dimensionless pressure as function of time and radius) is checked by comparison with analytical solution that exist in the literature

19

The pressure behavior check-- bounded reservoir

7

8

9

10

11

12

13

14

1.E+06 1.E+07 1.E+08

tD

PD

re/rw=2000

re/rw=2200

re/rw=2400

re/rw=2600

re/rw=2800

re/rw=3000

re/rw=infinite

CMG re/rw=2000

CMG re/rw=2200

CMG re/rw=2400

CMG re/rw=2600

CMG re/rw=2800

CMG re/rw=3000

CMG re/rw=infinite

20

The pressure behavior check-- bounded reservoir

6

7

8

9

10

11

12

13

1.E+05 1.E+06 1.E+07tD

PD

re/rw=700

re/rw=800

re/rw=900

re/rw=1000

re/rw=1200

re/rw=1400

re/rw=1600

re/rw=1800

re/rw=infinite

CMG re/rw=700

CMG re/rw=800

CMG re/rw=900

CMG re/rw=1000

CMG re/rw=1200

CMG re/rw=1400

CMG re/rw=1600

CMG re/rw=1800

CMG re/rw=infinite

21

Results and Discussions

22

Definition of pressure front

P(psi)

r (ft)

p=pi

t=ti

Δp1 Δp2 Δp3

α1α2

α3

△p= pressure drop defined at the pressure front

α= radius coefficient

● ● ●

● ● ●

23

Definition of pressure front

PD

rD

tD5

△pD= the dimensionless pressure drop defined at the pressure front α= radius coefficient

α1 α2 α3

ΔpD1 ΔpD2 ΔpD3

24

Pressure front and radius of investigation • From Ei solution such as

• By defining or giving ΔpD (or y), the following equation can be derived

.4

2

constt

r

D

iD

)4

(2

1 2

D

iDD t

rEip

DD

iD pt

rEior 2)

4(

2

DiD tror 2

Note: The radius coefficient (α) is dependent on the criteria defined at the pressure front (the value of ΔpD).

25

Radius coefficients from analytical solution with constant flow rate in an infinite reservoir

• By defining a small dimensionless pressure value (ΔpD) at the pressure front, the value of riD

2/4tD in the Ei solution can be estimated.

26

Radius coefficients from analytical solution with constant flow rate in an infinite reservoir

0.0E+00

5.0E+10

1.0E+11

1.5E+11

2.0E+11

2.5E+11

0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09

tD

r iD2

α = 4.00 (ΔpD=1.095*10-1)

α = 10.39 (ΔpD=1.095*10-2)

α = 51.22 (ΔpD=10-7)

α = 59.84 (ΔpD =10-8)

α = 71.15 (ΔpD=10-9)

α = 26.06 (ΔpD=10-4)

α = 34.28 (ΔpD=10-5)

α = 42.69 (ΔpD=10-6)

α = 17.82 (ΔpD=1.095*10-3)

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0.0E+00

2.0E+10

4.0E+10

6.0E+10

8.0E+10

1.0E+11

0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09

tD

r iD2

Δ pD=0.1095

Δ pD=0.01095

Δ pD=0.001095

Radius coefficients from numerical solution with constant flow rate in an infinite reservoir

α = 3.986 (ΔpD=1.095*10-1)

α = 10.363 (ΔpD=1.095*10-2)

α = 17.799 (ΔpD=1.095*10-3)

28

Radius investigation equation from analytical and numerical solution -- constant flow rate case

Constant flow rate test

(q=100 stb/day)

riD criteria (I)

for ΔpD=0.1095

riD criteria (II)

for ΔpD=0.01095

riD criteria (III)

for ΔpD=0.001095

Analytical solution

riD2= 4.00tD

R2=1

riD2= 10.39tD

R2=1

riD2= 17.82tD

R2=1

Numerical solution

riD2= 3.986tD

R2=0.9999

riD2= 10.363tD

R2=0.9999

riD2= 17.799tD

R2=0.9999

Different criteria for pressure front will obtain different radius coefficient (α)

The smaller the ΔpD, the larger the radius coefficient (α), i.e., the faster the pressure front propagation.

29

Results and Discussions (2)

Effect of variable flow rates

30

Ei solution with superposition – variable flow rate

)}][4

()()4

({6.70

)(1

2

21

2

1

iD

Dn

iii

D

Dtotal ttt

rEiqq

t

rEiq

kh

Bp

)}][4

()()4

({2

1

1

2

2 1

12

1

iD

Dn

i

ii

D

DD ttt

rEi

q

qq

t

rEip

Bq

pkhp total

D 11 2.141

)(

21

1

)(000264.0][

w

iiD

rc

ttkttt

or

where

31

(a) Increasing flow rate (two-rates) test

32

Radius of investigation equations from analytical solution and numerical solution with increasing flow rate test

Increasing flow rate test

q1=100 stb/day

q2=150 stb/day

riD criteria (I)

for ΔpD=0.1095

riD criteria (II)

for ΔpD=0.01095

riD criteria (III)

for ΔpD=0.001095

Analytical solution

riD2= 4.101tD

R2=0.9991

riD2= 10.411tD

R2=0.9999

riD2= 17.819tD

R2=0.9999

Numerical solution

riD2= 4.082tD

R2=0.9991

riD2= 10.381tD

R2=0.9999

riD2= 17.804tD

R2=0.9999

Note: Radius coefficient(α) increase slightly for smaller ΔpD

33

(b) Decreasing flow rate (two-rates) test

34

Radius of investigation equations from analytical solution and numerical solution with decreasing flow rate test

Decreasing flow rate test

q1=100 stb/day

q2= 50 stb/day

riD criteria (I)

for ΔpD=0.1095

riD criteria (II)

for ΔpD=0.01095

riD criteria (III)

for ΔpD=0.001095

Analytical solution

riD2= 3.887tD

R2=0.9983

riD2= 10.374tD

R2=0.9999

riD2= 17.814tD

R2=0.9999

Numerical solution

riD2= 3.868tD

R2=0.9983

riD2= 10.344tD

R2=0.9999

riD2= 17.799tD

R2=0.9999

Note: Radius coefficient(α) decrease slightly for smaller ΔpD

35

(c) Middle flow rate increasing test

36

Radius of investigation equations from analytical solution and numerical solution with middle flow rate increasing test

Middle flow rate increasing test:

q1=100 stb/day

q2=150 stb/day

q3=100 stb/day

riD criteria (I)

for ΔpD=0.1095

riD criteria (II)

for ΔpD=0.01095

riD criteria (III)

for ΔpD=0.001095

Analytical solutionriD

2= 4.246tD

R2=0.9981

riD2= 10.494tD

R2=0.9999

riD2= 17.849tD

R2=0.9999

Numerical solutionriD

2= 4.227tD

R2=0.9981

riD2= 10.464tD

R2=0.9999

riD2= 17.833tD

R2=0.9999

Note: Radius coefficient(α) increase for smaller ΔpD

37

(d) Middle flow rate decreasing test

38

Radius of investigation equations from analytical solution and numerical solution with middle flow rate decreasing test

Middle flow rate decreasing test:

q1=100 stb/day

q2= 50 stb/day

q3=100 stb/day

riD criteria (I)

for ΔpD=0.1095

riD criteria (II)

for ΔpD=0.01095

riD criteria (III)

for ΔpD=0.001095

Analytical solutionriD

2= 3.698tD

R2=0.9956

riD2= 10.280tD

R2=0.9998

riD2= 17.782tD

R2=0.9999

Numerical solutionriD

2= 3.679tD

R2=0.9956

riD2= 10.249tD

R2=0.9998

riD2= 17.767tD

R2=0.9999

Note: Radius coefficient(α) decrease for smaller ΔpD

39

The results of the dimensionless radius of investigation at the criterion ΔpD= 0.1095

0.0E+00

2.0E+09

4.0E+09

6.0E+09

8.0E+09

1.0E+10

1.2E+10

1.4E+10

0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09

tD

r iD2

q1=100 q2=150

q1=100 q2=50

q1=100 q2=150 q3=100

q1=100 q2=50 q3=100

q=100

Note: Radius coefficient(α) is affected by rate changes for larger ΔpD

40

The results of the dimensionless radius of investigation at the criterion ΔpD= 0.01095

0.0E+00

5.0E+09

1.0E+10

1.5E+10

2.0E+10

2.5E+10

3.0E+10

3.5E+10

0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09

tD

r iD2

q1=100 q2=150

q1=100 q2=50

q1=100 q2=150 q3=100

q1=100 q2=50 q3=100

q=100

Note: Radius coefficient(α) is slightly affected by rate changes for small ΔpD

41

The results of the dimensionless radius of investigation at the criterion ΔpD= 0.001095

0.0E+00

1.0E+10

2.0E+10

3.0E+10

4.0E+10

5.0E+10

6.0E+10

0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09

tD

r iD2

q1=100 q2=150

q1=100 q2=50

q1=100 q2=150 q3=100

q1=100 q2=50 q3=100

q=100

Note: Radius coefficient(α) is very slightly affected by rate changes for smaller ΔpD

42

Results and Discussions (3)

Effect of skin factor

43

The effect of skin factor to the radius coefficients in simulation studies (constant flow

rate test)

0.0E+00

1.0E+10

2.0E+10

3.0E+10

4.0E+10

5.0E+10

6.0E+10

0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09 3.5E+09 4.0E+09 4.5E+09 5.0E+09

tD

riD2

α = 17.799 (s=0, 2, 5, 8, 10 for ΔpD=1.095*10-3)

α = 10.363 (s=0, 2, 5, 8, 10 for ΔpD=1.095*10-2)

α = 3.986 (s=0, 2, 5, 8, 10 for ΔpD=1.095*10-1)

The radius coefficient (α) is independent of skin factor

44

Results and Discussions (4)

Effect of wellbore storage volume

45

The effect of wellbore storage volume (constant flow rate test)

0.0E+00

1.0E+10

2.0E+10

3.0E+10

4.0E+10

5.0E+10

6.0E+10

0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09 3.5E+09 4.0E+09 4.5E+09 5.0E+09

tD

riD2

α = 17.799 (CD=102, 103, 104, 105

for ΔpD=1.095*10-3)

α = 10.363 (CD=102, 103, 104, 105

for ΔpD=1.095*10-2)

α = 3.986 (CD=102, 103, 104, 105

for ΔpD=1.095*10-1)

Note: Radius coefficient(α) is independent of wellbore storage volume in late time

46

The effect of wellbore storage volume (constant flow rate test)

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E+06

1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08 1.0E+09 1.0E+10

tD

riD

cD=103

cD=0

cD=105

cD=104cD=102

ΔpD=0.1095

Note: Radius coefficient(α) is affected by wellbore storage volume in early time

47

Which criteria for defining pressure front is suitable in conjunction with pressure behavior affected by bounded reservoir?

48

Pressure response for a bounded reservoir

PDwf

Log (tD)

Infinite reservoir pressure response

Bounded reservoir pressure response

Dimensionless boundaryaffecting time, tD

*

Deviated point

re

49

Boundary affecting time equation

• From radius of investigation equation, such as

– When pressure front reaches boundary then back to the wellbore, i.e., pressure front propagates two-times of external boundary radius ( riD= 2reD ), is applied

*2 **4 DeD tr re(in terms of wellbore radius, rw)

riD2 = α tD

2* )/4( eDD rtor

50

(a) bounded circular reservoir with reD=3000

No-flow boundary

re = 1050 ft rw = 0.35 ft

300035.0

1050

ft

ft

r

rr

w

eeD

51

Boundary affecting time estimated from radius of investigation equation for the bounded circular reservoir with reD = 3000

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0

10.5

11.0

1.0E+05 1.0E+06 1.0E+07 1.0E+08

tD

p D

infinite reservoir

reD=3000 (re=1050 ft)

( I )

( II )

( III )

The visually deviated point from type curve analysis

2* 4eDD rt

52

(b) Bounded circular reservoir with reD=1000

No-flow boundary

re = 350 ft rw = 0.35 ft

100035.0

350

ft

ft

r

rr

w

eeD

53

4

6

8

10

12

14

1.0E+04 1.0E+05 1.0E+06 1.0E+07

tD

p D

infinite reservoir

reD=1000 (re=350 ft)

Boundary affecting time estimated from radius of investigation equation for the bounded circular reservoir with reD = 1000

( I )

( II )( III )

The visually deviated point from type curve analysis

2* 4eDD rt

54

Discussions of radius of investigation equation

• The study of radius of investigation in an infinite reservoir– Using different criterion ΔpD defined at the pressure front, we obtained

different radius coefficients (α) which vary from 4 to 71.15 for the pressure front varied from 0.1095 to 10-9, respectively.

• The study of boundary effect time in a bounded reservoir– The results of boundary effect time from the radius of investigation

equation with radius coefficient (α) of 17.82 (i.e., riD2=17.82tD for

ΔpD= 0.001095) are consistent with those from the deviated point of pressure type curves of the infinite and bounded reservoirs.

• The radius of investigation equation should be riD2=17.82tD,

where the radius coefficient (α) is 17.82.

55

Conclusions

• The relationship between the square of the dimensionless radius of investigation and the dimensionless time is linear (riD

2= αtD) for a constant flow rate but not necessarily linear for variable flow rates

• The radius coefficient (α) in constant flow rate cases varies from 4 to 71.15 as the defined dimensionless pressure at the pressure front of the radius of investigation is varies from 0.1095 to 10-9 , respectively

56

Conclusions (Cont.)

• The radius of investigation equation is independent of skin factor. The wellbore storage effect affects the propagation of the radius of investigation only at an early time and depending on the size of wellbore storage volume.

• The radius coefficient (α) is 17.82 and should be used in the equation of radius of investigation.

57

Thank you for your attention

58