Pseudo steady state

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By:

1. Yoanita Warapsari - 22212011

2. Singgih Suganda - 22212015

3. Subihi Eka Prasetya - 22212049

4. Ghadafi Maksum - 22212079

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INTRODUCTION:

When well is producing, there is a pressure drop along the wellbore, and then propagate through the reservoir radius with function of time

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Occured when Pressure propagation hasn’t reached the reservoir boundary.P = Pi at t = 0, for all rP = Pi at r = infinite, for all t(solution will be explained in ch. 7) [Ei solution]

Stabilized Flow ConditionStabilized Flow Condition

Flow condition based on the Pressure Propagation

Stabilized flow

Unstabilized flowUnstabilized flow

Unstabilized Flow ConditionUnstabilized Flow Condition

Will be describe in the next slide

What is Stabilized flow Condition?

Recall:When well is producing, there is a pressure drop along the wellbore, and then propagate through the reservoir radius with function of time

When the ‘pressure propagation’ has reached the reservoir boundary, and continues to flow, it called “Stabilized flow Conditions”.

1 PI equation

Knowing the Skin effect (wellbore damage)

Knowing the steam soaking effect

Effect of Stimulation to PI ratio

Knowing the Volumetric Res. (ch. 7)

Knowing the reservoir variabel (ch. 7)

23456

7 Knowing the Reservoir shape (ch. 7)

Type of Flow:

-Semi-steady state condition

- Steady-state condition

Steady-stateSemi-steady state

Stabilized flow

Applications

Steady stateSemi-steady state

Stabilized flow

Applications

Semi-steady state conditions

Figure of Pressure distribution and geometry appropriate for the solution of the radial diffusivity

equation under semi-state conditions

Semi steady state :

constant ,

Chain Rule :

q

t

pt

Vq tcons

tan

t

p

p

V

t

V

t

pVC

t

Vt

Vt

p

p

V

V

1

Vc

q

t

p

t

hrc

q

t

p

e2

Mathematic Equation

Radial Flow

Integrating this equation

At the outer no-flow boundary

integrating

t

p

k

c

r

pr

rr

.1

hrk

q

e2

12

2

2C

khr

rq

r

pr

e

0r

p

2

1

1

2

,2

er

r

rkh

q

r

p

sokh

qC

r

rwe

pp r

rr

kh

qp r

wf

2

2

2ln

2

Mathematic Equation cont’d

negligible

In the case when r = re

PI relationship :

And then determination

or then

2

2

2

2

22ln

2 eewfr r

rw

r

r

rw

r

kh

qpp

Srw

r

kh

qpp ewfe 2

1ln

2

S

rwrekh

pp

qPI

wfe

21

ln

2

re

rw

re

rw

dV

pdVp

hrr

drrhpp

we

re

rw22

2

re

rwwe

prdrrr

p22

2

2er

re

rwe

prdrr

p2

2

Mathematic Equation cont’d

Combine with previous equation, then

Inflow equation with average pressure :

re

rwee

wf drr

r

rw

rr

kh

q

rpp

2

2

2 2ln

2

2

4ln

2ln

22ee

re

rw

r

rw

rerdr

rw

rr 82

2

2

3e

re

rw e

rdr

r

r

Srw

r

kh

qpp ewf 4

3ln

2

Mathematic Equation cont’d

Steady-state condition

Steady state solution well flow for production when P/ t =0

Steady-state condition Darcy’s Law for radial flow of single phase oil

STEADY STATE SOLUTION

dr

dpkAq

From Darcy’s Law for radial flow of single phase oil :

rhA 2

dr

dprkhq

2

drrkh

qdp

r

r

P

P wwf

1

2

wwf r

rIn

kh

qPP

2

Where

STEADY STATE SOLUTION

Equation for steady state :

wwf r

rIn

kh

qPP

2

Since the outer boundary pressure cannot be measured directly, therefore need P average within drainage volume.

e

w

e

w

r

r

r

r

dV

pdV

P drrhdV 2

e

w

e

w

r

r

r

r

drrh

drrhp

P

2

2

hrr

drrhp

Pwe

r

r

e

w

22

2

e

w

r

rwe

prdrrr

P22

2

STEADY STATE SOLUTION

wwf r

rIn

kh

qPP

2

e

w

r

re

prdrr

P2

2

222222 )/1( eewewe rrrrrr

e

w

r

rwe

prdrrr

P22

2

Substituting

e

w

r

r wewf rdr

r

rIn

kh

q

rPP

2

22

e

w

r

r we

drr

rInr

kh

q

r

2

22

drr

rw

kh

q

r

e

w

e

w

r

r

r

r

e rr

Inr

2

12

2

2 2

2

2

2

1

2 w

ewf r

rIn

kh

qPP

422

2 22

2e

w

ee

e

r

r

rIn

r

kh

q

r

General Solutions for Stabilized flow Condition

wwf r

rIn

kh

qPP

2

w

ewfe r

rIn

kh

qPP

2

2

1

2 w

ewf r

rIn

kh

qPP

4

3

2 w

ewf r

rIn

kh

qPP

2

2

22 ewwf r

r

r

rIn

kh

qPP

2

1

2 w

ewfe r

rIn

kh

qPP

Steady State Semi Steady State

General relationship between P and r

Inflow equation,P = Pe at r = re

Inflow equation, average pressure

Steady State Solutions

kh

q

2 kh

Bq o2.141

4

3

2 w

ewf r

rIn

kh

qPP

s

ww err

Applications of Stabilized flow solution

1 Stimulating Well by steam soaking

Flow simulating in wellbore damage condition

Skin prediction

PI ratio

2

3

4

Stimulated well dy steam soaking

Viscosity Difference

Temperature Difference

Steam Soaking Stimulated Well

A well is stimulated by steam soaking.For rw < r < rh Ts is uniform = Steam temperatureFor r > rh Apply Tr as reservoir temperatureWhere Viscosity of the oil at Ts

and Viscosity of the oil at Tr

oh

oc

Stabilized Flow Equation for Steam Soaking

w

ohwfr r

rIn

kh

qPP

2

h

ochr r

rIn

kh

qPP

2

Inflow equation under steady state flow condition :

hw rrr eh rrr

w

hohwfh r

rIn

kh

qPP

2

h

eoche r

rIn

kh

qPP

2

h

eoc

w

hohwfe r

rIn

r

rIn

kh

qPP

2

h

e

w

h

oc

ohocwfe r

rIn

r

rIn

kh

qPP

2 For stimulated well

hh

w

h

e

w

h

oc

ohocwfe r

rwIn

r

rIn

r

rIn

r

rIn

kh

qPP

2

w

oc

r

reInS

kh

q

2

ww

h

w

h

oc

ohoc

r

reIn

r

rIn

r

rIn

kh

q

2

w

h

w

h

oc

oh

r

rIn

r

rInS

w

eocwfe r

rIn

kh

qPP

2

Inflow equation for Unstimulated well :

welledunstimulat PI

wellstimulated PI increase ratio PI

w

eoc

h

e

w

h

oc

ohoc

rr

Inkh

qq

rr

Inrr

Inkh

q

q

2

2

Then the effect on productivity index due to steam soaking :

h

e

w

h

oc

oh

w

e

r

rIn

r

rIn

r

rIn

Stabilized Flow Equation for Steam Soaking

Wellbore Damage

Wellbore Damage

Skin FactorSkin Factor

There will be a skin factor due to wellbore damage

There will be a skin factor due to wellbore damage

PI stimulatedPI stimulated

In order to increase permebility value besause of skin factor, stimulating well applied. And this method will increase the PI value

In order to increase permebility value besause of skin factor, stimulating well applied. And this method will increase the PI value

APPLICATION OF THE STABILIZED INFLOW EQUATIONS

IN WELLBORE DAMAGE

The inflow equations appropriate for the pressure distribution shown

In particular,

a. Skin Factor

i.e.

Skin Factor

b. PI ratio

Case: During drilling, a well is damaged out to a radius of 4 ft from the wellbore so that the permeability within the damaged zone is reduced to 1/100 th of the undamaged effective permeability. After completion the well is stimulated so that the permeability out to a distance of 10 ft from the wellbore is increased to ten times the undamaged permeability. What will be the PI ratio increase if the wellbore radius is 0.333 ft and the drainage radius 660 ft?

since,

2 General solution for Steady state flow:

Conclusions

1 General solution for Semi-steady state flow:

4

3

2 w

ewf r

rIn

kh

qPP

2

1

2 w

ewf r

rIn

kh

qPP

3 Solution form stabilized flow eq, is used to

-Determining the skin effect-PI equation and its ratio-Well test applications (ch. 7)

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Questions?

THANK YOU…