PET 504Advanced Well Test Analysis

Lecture 5

Spring 2015, ITU

2Wellbore Storage

Gas

Valve

Oil

Occurs in both oil andgas wells.

Suppose Well is perforated Packered Shut-in at surface High pressure at well

head. Perforations are

plugged. If we open the well,

will it flow?

3Gas Well The tubing string acts like a very large tank of high-

pressure gas When the surface valve is opened, gas in the tubing

expands and escapes through the valve Production may occur for a very long time several

hours to several days. Gas is very compressible The tubing string is very long Large volume of gas stored.

4Pumping (oil) Well No packer Initial static fluid level at depth

DD. Perforations plugged

No flow from reservoir When pump is started we see

production at the surface Fluid is being produced from

the casing-tubing annulus

DD

Static Fluid Level

Sucker Rod

h

5Wellbore Storage Period The time period when surface production is primarily

due to fluids flowing out of the tubing or tubing-casing annulus is called the Wellbore StorageDominated Flow Period.

This period would exist even if the perforations wereopen to flow

During this period, the reservoir is not producingfluids, and pressure versus time data do not containreservoir information

6Surface Rate vs. Sandface Ratepr

essu

re,p w

f

Time, tqsc

qsf

drawdownbuildup

0

qsf sandface rateqsc surface rate

dttdp

BCqtq wfscsf

)(24)(

7Wellbore Storage Coefficient C is called wellbore storage coefficient and its unit

is bbl/psi. It gives the volume of wellbore fluid that will be

produced if the bottom hole flowing pressure is reduced1 psia.

ww VcC

Compressibilityof wellbore fluid, 1/psi

wellbore volume, bbl

Wellbore storage coefficient due tocompressibility

8Wellbore Storage Coefficient Wellbore storage coefficient due to changing liquid

level is given by

615.5144 cAC

Wellbore fluid density,lbm/ft3

Cross-sectional area where theliquid level changes, ft2.

9Wellbore Storage Coefficient, Example

Suppose we have 1000 ft deep well with 2 inch ODtubing in 7-5/8 inch ID casing. Without packer, theliquid will be pumped down the annular space. Thedensity of wellbore fluid and its compressibility are: 58lbm/ft3 and co = 1.5x10-5 psi-1. Compute C?

DDStatic Fluid Level

Sucker Rod

h

10

Example (Contd)

DDStatic Fluid Level

Sucker Rod

h

psibblAC c /131.058615.5295.0144

615.5144

2

22

295.0

1441

22

2625.7

ft

Ac

11

Example 2 Suppose now that we have a 1000 ft deep well with positive

pressure at well head. The fluid is stored in a 7-5/8 inch IDcasing. The density and compressibility of wellbore fluid are:58 lbm/ft3 and co = 1.5x10-5 psi-1 . Calculate C.

bbl

Vw

5.56615.51

1441000

2625.7 2

psibblVcC ww /105.85.56105.1 45 If we had gas instead of liquid, how would the value of Cchange?

12

Log-Log Diagnostic Plot for Storage

10-4 10-3 10-2 10-1 100 101 10210-1

100

101

102

Vertical well in circular/no-flow boundary

+1 slope line(wellbore storage)

pan

dp

' , ps

i

Time (h)

13

Pressure Behavior for Storage

At early times when we have storage dominatedflow:

Log-log plots of p and p' will be equal anddisplay straight lines with unit slope during thisperiod.

tCBqptpt

CBq

tpptp sciwfscwfi 24)(

24)()(

)(24

)( tptCBq

dtpd

ttp sc

14

Identification of Storage on Log-LogPlot

Why do we see unit slope line on a log-log plot?

tCBq

tp sc24

)(

tCBq

tp sc24

)(

CBq

ttp sc24

log)log(1)(log

CBq

ttp sc24

loglog*1)(log

15

Determination of C and pi A Cartesian plot of pwf vs t

0 t

CBq

mslope scw 24

pwf

ip

tCBqptp sciwf 24

)(

w

sc

m

BqC24

16

Note on Wellbore Storage In classical models, wellbore storage is treated as

constant. This is Ok if we have liquid system andpressure does not change much in the wellbore.

However, there are many cases where the wellborestorage coefficient varies significantly with pressuresuch as gas wells or wells with multi-phase flow inthe wellbore.

Also, there are tests that we often observe combinedeffects of both compressive and changing liquid typestorage phenomena.

17

Some Examplesdrawdown buildup

Buildup (phase segregation)To minimize such effects, we shouldPlace gauge near the perforations and usedownhole shut-in.

Oil Well Oil Well

18

Skin In practice, skin may be due to a variety of factors

Damage to formation due to invasion of mud filtrate andmud solids

Partial penetration Migration of fines Asphaltenes

Treatment of skin will depend on the specific cause.

19

Note on Skin

rw rs

Undamaged case

Simulatedpermeability

ps0pwf

pwf

s > 0 s < 0

Bqpkh

ssc

s

2.141

ks < k ks > k

20

Effective Wellbore Radius Concept

w

s

s r

r

kk

s ln1 ks, permeability of region with rs

srr ww exp Effective wellbore radius0 sifkks 0 sifkks

ww rrifs 0 ww rrifs 0

21

Pseudo-Skin or Geometrical Skin, sp

It is skin effect due to well geometryand completion.

For limited-entry wells, sp is positive (sp >0).

For hydraluic fractured wells, horizontaland slantented wells, sp is negative (sp

22

Wellbore Storage Type CurvesIn 1983, Bourdet et al. Developed type curves for an fullypenetrating active well producing in an infine acting reservoir:

23

Wellbore Storage Type Curves

Dimensionless variables

Ctkh

Ct

D

D

000295.0

Bq

pkhpsc

D 2.141 Bq

pkhpsc

D 2.141

22615.5

wtD hrc

CC S

DeC2

4

22.64 10

Dt w

ktt

c r

24

pan

dp'

,psia

25

Manual Type-Curve Matching The use of Bourdet et al. type curves:

Step 1: Determine kh (md-ft) from pressure match points:

Step 2: Determine wellbore storage coefficient C (bbl/psi)from time-match points:,

M

MDsc p

pBqkh 2.141

MDD

M

CttkhC/

000295.0

26

Manual Type-Curve Matching

Step 3: Compute dimensionless wellbore storage coefficientfrom

Step 4: Finally compute skin from

22615.5

wtD hrc

CC

D

MS

D

CeC

s2

ln21

Integrating WS and Skin in pressureequation

Van Everdingen and Hursts 1949 paper was one of the first applications ofLaplace transforms in petroleum reservoir engineering. In their paper theyshowed that:

0

0

2

0 0 0

1

5.6152

solution including WS and Skin

solution without WS and Skin

DD

D D

Dt w

D D D

D D D

s p Sps sC p S

CCc hr

S skin

p L p p

in p L p p

28

Other Models

29

Limited-Entry Vertical Wells Three different flow regimes

(a) Early radial (b1) Hemi-spherical flow

(c) Late- (or pseudo-) radial flow (b2) spherical flow

30

Log-Log Diagnostic PlotLimited-entry Vertical Well

1E-4 1E-3 1E-2 1E-1 1E+0 1E+1 1E+21

10

100

1000

pan

dp

'

t, hour

-1/2 slope line

Hemi-spherical

Spherical

Pressure-Derivative

Late-radial

31

Spherical Flow Regime

If spherical flow is observed, then

3/2

245370.6 1 2 1sc tsch s w s

q B cq B spk r h k t

kb km

tkcBq

ps

tsc 15.12262/3

(-1/2 slope line on log-log plot)

(Cartesian plot)

32

Spherical Permeability (Contd)

tkcBq

hs

rkBqp

s

tsc

wsh

sc 12453)21(6.70 2/3

vhsvhs kkkkkk 2/33 2 Spherical permeability

1

2

2

/215.05.0

/215.05.0ln

h

vww

h

vww

ws

kkhr

kkhr

hrEffective sphericalradius

33

Effective Spherical Radius

1E-3 1E-2 1E-1 1E+0 1E+1 1E+2 1E+3Anisotropy ratio, kh/kv

1

10

100

1000hw = 3.2 ft

34

Spherical Flow Analysis

0t

10

kb

k

tscvhs

m

cBqkkk 24532/3tmbp kk1

slope= mk 22/3

h

sv k

kk

ssc

khw

rBqbkh

s1

6.702For hemi-spherical flow,divide mk by 2

35

Limited-Entry Late-Time Radialp w

f, ps

i

t, hour

p1hr slope, mr

1E-5 1E-4 1E-3 1E-2 1E-1 1E+0 1E+1 1E+23000

3500

4000

4500

5000

5500

36

Limited-Entry Late-Time Radial During late-time radial flow (if observed)

t

wt

h

h

scwfi s

rc

kt

hkBqppp 87.023.3loglog6.162 2

rg

sch

h

scr

m

qhkhk

Bqslopem

6.1626.162

23.3log151.1 21

wt

h

r

hrt

rc

km

ps

hrihr ppp 11

37

Pseud Skin, sp We can compute damage skin s if the computed value of

total skin st after computing pseudo-skin due to limited-entry geometry:

ptwpw

t sshh

ssshh

s

/ 4 / 41 ln ln

2 / 4 / 42

w

w w w whp

ww w v w w w w w

hz h h z hh h k h hs hh r k h z h h z h

h

Papatzacos formula

38

Vertical Well in Channel

Channel

L1

L2

Channel

Radial Flow

Linear flow

No flow boundary

No flow boundary

39

Vertical Well in a Channel

Linear flow regime

40

Linear Flow

Pressure during linear flow is described by

1 28.133 sc

l l lt

q Bp m t b mk ch L L

chscl sskhBqb 2.141

21

121 sinln2

lnLL

Lr

LLs

w

sc

41

Identification of Linear Flow Regime

tsc

llckLLh

Bqmtm

tdpdp

21

133.8ln

(1/2 slope line on log-log plot)

42

Pressure Behavior of a Well in aChannel

Kanal iersinde bir kuyu

Presure-derivative

43

Linear Flow Analysis

tl

sc

ckmhBqLL

133.821

sBq

bkhs

sc

lch 2.141

0 t0

lb

slope = ml

chw

sr

LLLL

Lexp

2arcsin1 21

21

1

Cartesian plot

44

Well near a Sealing Fault

L

45

Fault Problem

It is solved using the method of superposition in space.

46

Pressure Behavior of Well near a Fault

radialHemi-radial

47

Semil-log Analysis

radial

Hemi-radial

t* (Intersection time)

48

Early-Radial Semi-log Analysis

Early Radial flow

s

rc

kt

khBqpp

wt

sciwf 87.023.3loglog

6.1622

r

scscr

m

qkhkh

Bqslopem

6.1626.162

23.3log151.1 21

wtr

hr

rc

km

ps

49

Hemi-radial Semi-log Analysis

sLc

krc

kt

khBqpp

twt

sciwf

435.062.14

log21log

21log

6.1622

22

r

scscr

m

qkhkh

Bqslopem

6.1626.162

tc

tkL*

01217.0 (distance to the fault)

50

Well near a Constant Pressure Boundary

Gas-Cap

Impermeable layer

Oil-zoneaquifer zone

High conductivity fault

L

100kLwk

F ffcD

Oil zone

51

Flow Regimes

Pressure-derivative

52

Distance to Constant Pressure Bdry

0t

10

tsc cBqhmkL

100194.0

tmp 11

slope = m-1

Cartesian plot

53

Vertically Fractured Wells

One option for increasing the productivity of a wellwith significant skin damage is to verticallyfracture the reservoir by pumping fracturing fluidsalong with proppants into the well at high pressure.

Well

54

Productivity Increase A vertical fracture increases a well's

productivity in two ways: It allows the reservoir fluids to bypass a near-

wellbore damaged zone and enter the wellbore viathe fracture system

it increases the wellbore area open to flow,which in turn reduces the pressure drawdown onthe reservoir for a specified production rate.

55

Transient Flow Regimes Fracturing a well changes the flow regimes visible in

pressure transient data. Before a well is fractured, flow in the reservoir is

essentially radial towards the wellbore for all times Reservoir flow during wellbore storage dominated flow and

the following transition period is radial, but at variablesandface rate.

56

Transient Flow Regimes After fracturing, (assuming that the fracture conductivity is

very high when compared with the reservoir conductivity),early time flow in the reservoir is essentially perpendicular tothe fracture - this is referred to as linear flow.

Eventually, flow in the reservoir at points far away from thefracture begins to affect the wellbore pressure response; thepressure derivative curve once again exhibits the signature ofradial flow - i.e., the derivative is flat: pseudoradial flow

57

Fracture Geometry

hfx

L

58

Infinite Conductivity Fracture Linear flow

Vertical fractureLxf

Well

ll btmp t

mp l2

59

Pressure Behavior

If there is damage on thefracture surface

Infinite-Conductivity Fractured Well

0.001 0.01 0.10 1.00 10.00 100.00 1000.000.01

0.10

1.00

10.00

pan

dp

'

time

1/2 slope line

Pseudo-radial flow

pp'=constant

60

Linear Flow Analysis

0 t0

lb

kcmhBqL

tl

scxf

06.4

ll btmp

0,lb if well or fracture is damaged

slope = ml

Cartesian plot

61

Finite-Conductivity Fractured Wells

Finite Conductivity

fracture

formation

xf

fffD p

pkL

wkC

300fDC nfinite-conductivity

62

Flow Regimes for FCF Wells

(a) Linear flow in fracture (b) bi-linear flow

(c) Linear flow perpendicularto fracture surface

(d) Pseudo-radial flow

63

Pressure BehaviorFinite Conductivity Fractured Well

Pressure derivative

64

Effect of ConductivityFinite conductivity Fractured Well

1

10

100

slope 1/4 (bi-linear)

Slope 1/2 (linear)

Slope zero (Radial)

p Dan

d

p D'

tD10-6 10-4 10-2 10010-3

10-1

101

xf

fffD kL

wkC

65

Bi-Linear Flow Regime

4

4/1 11.44kcwkh

Bqmbtmp

tff

scililil

4/1

4t

m

dtpd

tp il (1/4 slope line on log-log plot)

66

Bi-Linear Flow Analysis

0 4 t0

ilb

217.1945

ilsc

tff

mhBq

kcwk

ilil btmp 4

fractureindamagebil ,0

slope = mil6.1fDC

6.1fDC

Cartesian plot

67

Review-Model Identification Well with skin and storage

68

Model Identification Vertically Fractured Well

69

Model Identification Dual Porosity Reservoir

70

Model Identification Composite Reservoir

1

1

1

1 outer

inner

kk

mr

//

71

Model Identification Sealing Fault

72

Model Identification Constant Pressure Boundary