radial flow chap1v1
DESCRIPTION
reservoirTRANSCRIPT
1
DownholeMemory
SurfaceRecording
Casing
Tubing
Testing Valve(operated by
annulus pressure)
Packer(set by weight on
string)
PressureTransducer
Tailpipe
Figure 2.1.1
Drillstem
Testing
Assembly
Fig 1.1.1
Well Test Surface Hardware
Choke
Gas
Oil
Surface Choke providesRate Control
Orifice PlateFlow Measurement
qo
Q
Test Rate Limited bySeparator Capacity
Fig 2.1.1b
Test Separator
Fig 1.1.1b
2
InitialFlow
In i t i alS hu t in
Afterflow
FinalShutin
Final Flow
Time
Prod.Rate
Initial Res. Pressure
Drawdown BuildupBHP
Time Figure 2.1.2
Dual Flow - Dual Shutin Test
Fig 1.1.2
Fig 2.1.3
4 k hπ
q Bsµ
Transient Well Testing
Buildup Analysis - Horner (Theis) Plot
Semilog Analysisln
t + tp ∆∆t
BuildupAffected
byWellboreStorage Intercept
givesskin
factorS
Affectedby
Boundaries
ETR MTR LTR
slope = −
p*pws
Nomenclature due to
D. Pozzi (EPR)
Fig 1.1.30
3
alteredzone
pw
pw f
∆ps
UnalteredPermeability
k
rw rs re
wellboreradius
altered zoneradius
externalradius
∆p = Incremental Skin Pressure Drops
Fig2.1.4a
ks
Near Wellbore Altered Zone Fig 1.1.4a
Fig 2.1.4b
n = Number of perforations per foot
l = Length of penetration
= Phase angle
s
p
θ rw rs
ks
lp
Altered Zone
k
Combined SkinDamage + Perforation
Perforated Completion
S f n l k r ks p s s= , , , , ,θe j
Fig 1.1.4b
4
E T R M T R L T R
slope
p*
q µ4 khπ
m = −
l n t + tsi a ∆∆t
p
pws
∆pMBH
Fig
2.1.5Horner
Plot
0
Determination of Average Pressure
Fig 1.1.5
WELL PRESSURESTARTS TO BEAFFECTED BYBOUNDARIES
MTR LTR
TRANSIENTI.A. FLOW TRANSITION
LATETRANSIENT
SEMI-STEADY-STATEFLOW
pi
pwf
0 TIME
SCHEMATIC PLOT OF PRESSURE DECLINE AT
PRODUCING WELL
CONSTANT RATE WELL BOUNDED RESERVOIR
Figure 2.1.6
d pd t
q Bc r h
s
t e
= −φ π 2
Cartesian Plot Flow Regimes Fig 1.1.6
5
MTR LTR
lnt + tp ∆∆t
depletion
buildupdrawdown
HornerPlot
pws
p*p**
Closed "tank" ofpore volume, V
Figure 2.1.7
pi
0 Time, t q
Detection of Depletion
Fig 1.1.7
Skinre
k
Homogeneous Finite Reservoir
d d
Well Image
No FlowBoundary
k1
k2
r1
Composite InfiniteReservoir
d2
d1
Single Linear Fault Multiple Faults
µ2
µ1
θ
Some Well Test Models
Fig 1.1.8
6
T(t=0)
x
x
Heated Bar
Thermocouples in thermowells
Transient HeatConduction Equation
Inner B.C.
i.e.
. . . specified gradient(2 kind BC)nd
Linear Flow
t
Penetration Depth
Initial Temp.
T
FaceTemp.
Dynamic Temperature Distribution
Fig 2.1.9
∂∂ ρ
∂∂
Tt
kC
Txp
=2
2
qA
k Tx
x
= −=
∂∂
0
∂∂Tx
qkA
x=
= −0
Constant heat flux
Fig 1.1.9
h
re
rw
qWell in theCentre of a
CircularReservoir
RadialFlow
Fig 2.2.1
Model Reservoir
Fig 1.2.1
7
ln r
t
p(r ,t)
pi
Well-Bore
Trans ien t Deve lopment o fthe Format ion Pressure
Dis t r ibu t i on
Fig 2.2.2Fig 1.2.2q
104 105 106 107 108 109
10
1
10-1
10-2
10-1 1 10 102 103 104
pD
t /rD D2
Figure 2.2.3Single Well in an Infinite Reservoir (No Skin)
Exponential Integral Solution
Fig 1.2.3
8
∆ps
pw
pwf
"SKIN"
tPRESSURE PROFILEIN THE FORMATION
rw
NEGATIVE SKINFACTOR
i.e. STIMULATION
−∆p s
pwf
pw
k > ka
PRESSURE PROFILEIN FORMATION
RIGOROUS SKINCONCEPT PROFILE
t
STIMULATED ZONE
Figure 2.2.4
DimensionlessSkin
S pq B
kh
s
s
=∆
µπ2
Positive Skin Factor
i.e. Damage
Fig 1.2.4
C R Dpi
0
qpw f
TIME, t0
CARTESIAN PLOT
INTERCEPT
SLOPE, m
ln t0
pw f
SEMILOG PLOT
Figure 2.2.5
= −qµ
4 khπ
p (t=1)wf
Rate Schedule
Ideal (CSFR) Drawdown
Fig 1.2.5
9
104
105 5x105
106
2 x10 6
t = 3x10D6
0
1
2
3
4
5
6
7
8
91 200 400 600 800 1000
rD
pD
r =10De3
Figure2.2.6
Dimensionless Pressure Distributions in Radial Flow
SSS
I A
Well in aClosed
Reservoir
Fig 1.2.6
HENCE E (x) IS DENOTED -Ei(-x)1
1
2
3
0.4 1.0 1.6
E (x)1
-Ei(-x)
0
x Fig 2.2.7
Ei x eu
u
x
e j = −−
−
∞z
Exponential Integral Function
Fig 1.2.7
10
103 5x103 t =10D4
1 100 200rD
PRESSUREDISTURBANCE
FRONT Fig 2.2.8
Radius of Influence
p = 0.1D
Fig 1.2.8
102
q
ACTIVEWELL
OBSERVATIONWELL
MINIMUM OBSERVABLE pDEPENDS ON GAUGE RESOLUTION
∆
rD
"ARBITRARY"CRITERION
Ei SOLUTION
pi
pwo
OBSWELL
PRESSURE
0 t Fig 2.2.9
p Ei rtDD
D
=FHGIKJ
12 4
2
p p khqD = =
∆ 2 0 1πµ
.
Depth of
Investigation
Fig 1.2.9
11
1
2
3
4
5
6
pD
0 1 2 3 4 5 6 7ln rD
Radiusof
Drainage
rD
Fig 2.2.10
I.-A. TransientPressure Profile
at t = 10D
5
Steady-StatePressure Profile
for Same p (1,t )D D
rDd rD i
r = classical depth of investigation
D i
Fig 1.2.10
t = 0.3De
r = 1000De
4
2
0
t = 10D t = 25D
pwD
ln tD
1 10 102 103 104 105 106 0 2 x 103 4 x 103 6 x 103 8 x 103 10 x 103
0
1
2
3
4
5
pwD
PRODUCTION
SHUT-IN0
q
0 tD
0 2 4 6 8 10 12 14
6
8
10
Cartesian Graph
tD tD
Semilog Plot
Pressure Drawdown at the Wellbore Fig 1.2.11
12
rD = 1.0
2.0
1.2
20
EXPONENTIAL INTEGRALSOLUTION
10-2 10-1 1 10 102 10310-2
10-1
1
10
pD
t /rD2D
Finite Wellbore Radius Solution
Finite Wellbore Radius (FWR) Solution
Fig 1.2.12
pp
e
pw f
StabilisedPressure
Distribution
SSS Depletion Wel l in Cent re o f a
Closed Circu lar Reservoi r
rerw
Fig 2.2.13Por e Vo lume = hAφ
qs
Fig 1.2.13
13
TIME, t
Pressure Drawdown Testing
RATEq
0
0
SHUT-IN
PRODUCING
TIME, t
p = pws i
0 Fig2.3.1
Bottom Hole
Pressurepwf Fig
1.3.1
0
pt=1
ln tNOTE : ln t = 0 corresponds to t = 1
Fig 2.3.2
Deviation from straight linecaused by damage andwellbore storage effects
slope, m = − 4 khπq Bs µ
Drawdown Semilog Plot
Bottom Hole
Pressurepwf
Fig 1.3.2
14
•o
Well 1q
1
Well 2q
2
r1
r2
Well 3
Observation Well
Active Well
Three Well System Figure 2.4.1
Principle of Superposition
Fig 1.4.1
q2
q
t T1
q - q2 1
Two-Rate Flow Schedule
Fig 2.4.2
Inject ion wellrate q q
2 1−
Production wellrate q
1
Superposition of RatesFig
1.4.2
q1
0
15
Total Response
ExtrapolatedPressure
Principle of Superposition
Injection Wellat Rate q - q2 1
Well atRate q1
0 T1 t
0
p pi w−
∆pDD
∆pDD
Fig 1.4.3
RATE
t p ∆
∆
t
t
FLOWING
SHUT-IN
t p
BHP
pws
p ( t=0)wf∆
Figure2.5.1
Schematic Flow-Rate and Pressure Behaviour for an Ideal Buildup
q
Fig 1.5.1a
16
pws
ln t + tp ∆∆ t
p*
Deviation from StraightLine caused by
Afterflow and Skin
0
slope, m = −4 π k hq Bs
µ
Semilog (Horner) Plot for a Buildup
Fig 2.5.1bFig 1.5.1b
rD0
5
1 200
q
0tpD
∆ t D
t = 10p D
4
Pressure Build-Up in a Reservoir
Figure 2.5.2
t = 10D4
∆
10
50
200
2 10 3
psD
Fig 1.5.2
17
0
40 7
ln ∆∆
tD
tDt +pD0
5
ps D
∆ t D
10 103
××10 3 5 10 3
t = 10p D4
Pressure Build-Up
at Wellbore
Semilog
Cartesian
DimensionlessResponse
(Horner)
Fig 1.5.3
psD
t + tp ∆∆t
135
2
6
∆tD
t = 10pD
4
∆t < 10D
Ei Function notRepresented by
Log Approximation
Fig 2.5.4
pD
Dimensionless Build-up Semilog (Horner) Plot
Fig 1.5.4
18
t + tp ∆∆t
Effect of Afterflow on a Horner Plot
Data Affectedby Wellbore
Storage
Correct SemilogStraight Line
pws
0
p*
slope m
Fig 2.5.5
ETR MTR
Fig 1.5.5
1
1000
0.001 100t
∆p(psi)
UnitSlope
Da ta o f Co rr ec tS emi log S lo pe
(hr)Fig 2.5.6
MTR
Log - Log Diagnostic Plot for Afterflow
Fig 1.5.6
19
Logt + tp ∆∆t
xx
xx
x x x x x x x
p*p1 hr
slope m
0
Determination of p on the Horner Plot1hr
Fig 2.5.7
MTRStraight
Line
∆t = 1 hr
S p pm
tt
kc r
wf hr p
p t w
=−
++
− +LNMM
OQPP11513
13 22751
2. .log log
φµ
pws
Fig 1.5.7
Determine and
very accurately
p ( t=0 t( t=0)wf ∆ ∆)
+
t( t=0)∆p ( t=0)
wf∆
End ofDrawdown
Buildup
∆t
Stabilise flow-rate before shutin
q
Q = cumulative volume
Flow-Rate
∆t
Shutin
Afterflow∆ − ∆p = p p ( t=0)BU ws wf
∆ − ∆t = t t( t=0) t =pQq
Test Precautions
Fig 1.5.8
20
Time
Pressure Flow period
ReservoirDisturbance
Shut-in periodReservoirRecovery
Horner Plot
Largem
Smallm
Samepr
pws
Log t + tp ∆∆t
Small ReservoirDisturbance
Low flow-rateSmall viscosityHigh permeability
Large ReservoirDisturbance
High flow-rateViscous fluidLow permeability
Permeability of Reservoir Rock from a DST
After Matthews and RussellLarge mSmall m
Scribed Tin Chart from Amerada Gauge
Fig 1.5.9
3424
3423
3422
pws
8 7 6 5 4
Horner Plot
Early Piper Well(HP Gauge) slope
m = 0.7465 psi−
kh = 1.067*10 md.ftS = 3.08
6
q = 11750 bbl/d
B = 1.28 = 0.75 cpr = 0.362 ft = 0.237
c = 1.234*10 psi
s
w
t
µ
φ-5 -1
(psia)
Fig 2.5.10 lnt t
tp + ∆
∆
3425
Fig 1.5.10
21
0tp
∆t
Pressure Build-Up in a Reservoir
∆ t 1
∆ t 2
∆ t 3
∆ t 4
∆ t 5
p (t ,r )r p p1
p (t ,r )r p p2
p (t ,r )r p p3
p (t ,r )r p p4
p (t ,r )r p p5
pr
rrw
Reservoir pressure distributionat moment of shut-in, p (t )r p
Peaceman ProbeRadius Concept
Fig 2.5.11
p pw r r rw
==
Fig 1.5.11
pwf
pi
Time, t
tp2 tp1tp1 tp3
tp3
tp4
Transient Productivity Index, Jt
p (t )wf p3
pws
or
J is strongly time dependentt
pwf
J qp p tt
s
i wf p
=− d i
Fig 1.5.12