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Well test analysis
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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
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