Well Testing. 2 3 4 5 6 7 8 9 10 11 12.
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Well Testing
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Transcript of Well Testing. 2 3 4 5 6 7 8 9 10 11 12.
Well Testing
2
3
flow D-one (6)
small are and gradients pressure (5)
constant is (4)
)(isotropicdirection allin same theandconstant is (3)
pressure oft independen is (2)
ilitycompressibconstant and small has flowing liquid phase-single The (1)
assuch s,assumptionimportant severalon based derived is
Eq.(A.9), equation,y Diffusivit
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ydiffusivit hydraulic The10634.2
equationy diffusivit thecalled isequation This
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10634.2
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10634.2
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4
42
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42
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qB
ppkhpwhere
t
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rrwhere
t
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kt
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reservoir) (oil ---- Eq.(1.1) ofequation y diffusivit of form essDimensionl
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42
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22
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9
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11
assuch Eq.(A.5) From
const.)z(for fluid lecompressibFor
4
4
4
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MP
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have weEq.(A.5a), into Eq.(A.5b) ngSubstituti
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222
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ct
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11
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t
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r
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12
)5.()(
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state ofEquation From
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..
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11
assuch Eq.(A.5) From
gas ideal-nonFor
4
4
bAz
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RT
M
zRT
MP
zRTM
mmp
nzRTpV
dAtkr
pr
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constconstkFor
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13
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10634.2
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Eq.(A.5d) into Eq.(A.5b) ngSubstituti
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4
tp
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waterand gas oil, of flow ussimultaneoFor
4
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t
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phasesindividualtheofmobilitiestheofsumthe
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ilitycompressibsystemtotalthecwhere
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16
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kkk
Summation
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assuch Eq.(1.2a) From
17
1.3 Solution to Diffusivity Equation
)1.1(10634.2
1
ydiffusivit ofEquation
42
2
t
p
k
c
r
p
rr
p
• There are four solutions to Eq.(1.1) that are particularly useful in well testing:
(1) The solution for a bounded cylindrical reservoir
(2) The solution for an infinite reservoir with a well considered to
be a line source with zero wellbore radius,
(3) The pseudo steady-state solution
(4) The solution that includes wellbore storage for a well in an
infinite reservoir
18
• The assumptions that were necessary to develop Eq.(1.1)(1) Homogeneous and isotropic porous medium of uniform thickness,(2) Pressure-independent rock and fluid properties,(3) Small pressure gradient,(4) Radial flow(5) Applicability of Darcy’s law ( sometimes called laminar flow )(6) Negligible gravity force.
)1.1(10634.2
1
ydiffusivit ofEquation
42
2
t
p
k
c
r
p
rr
p
19
DDD
D
D
rD
DD
rr
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rr
i
wD
iD
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D
D
DD
D
tallforratp
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por
r
pr
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Bq
kh
Bq
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rallfortatp
formessDimensionl
tallforraspp
tforkh
Bq
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pr
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conditionsInitialandboundary
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ppkhp
t
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t
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r
p
rr
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D
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0)3(
11
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10634.2
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casereservoir Infinite rate,Constant
1
2
4
2
2
42
2
20
begins. production before , pressure, uniformat isreservoir the(4)
) as that (i.e. area infinitean drains well the(3)
radius, zero has well the(2)
, rateconstant aat produces wella (1)
thatAssume
WellSource-Linewith Reservoir lCylindrica Infinite
i
i
p
rpp
qB
21
x
nnu
i
wD
wD
iD
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DiDDD
tii
nn
xxxxxdu
u
exE
trp
r
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ktt
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ppkhpwhere
at
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at
rErtpor
kt
rcE
kh
qBpp
t
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r
p
rr
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!
)1(
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and (hours), at time well thefrom (feet) distanceat (psi) pressure is Where
000264.0
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)7.1(42
1,
)7.1(948
6.70
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1
is Eq.(1.1) osolution t the,conditions eUnder thos
32
2
2
2
2
42
2
22
80907.0ln2
1
5772.04
ln2
1
42
1
0025.05772.0)ln()(
solution Theis or the
solution source-line or the
solution integral-lexponentia thecalled is Eq.(1.7)][or Eq.(1.7a)
2
22
D
DD
D
D
D
DiD
i
r
tp
t
r
t
rEp
xforxxE
)7.1(100
0025.0
2c
r
t
xfor
D
D
23
24
25
)1(1079.3
)7.1(100
)7.1.(
)7.1(80907.0ln2
1
)7.1(42
1
assuch Eq.(1.7a) From
25
2
2
2
Dwt
D
D
D
D
D
DiD
rk
rctor
cr
t
whenusebemaybEq
br
t
at
rEp
26
)7.1(1688
ln6.70
109289.5ln
6.70
1064.22458.2ln
2
1
)2458.2ln(1064.2
ln2
1
80907.01064.2
ln2
1
2.141
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80907.0ln2
1)(
1i.e., , eAt wellbor
80907.0ln2
1
assuch Eq.(1.7b) From
ry.satisfacto is integral lexponentia ery when thsatisfacto is integral
lexponentia theion toapproximat log thepurpose, practicalfor Thus,
5 when 2%about only is Eq.(1.7b) and Eq.(1.7a)between difference But the
2
2
4
2
4
2
4
2
4
2
2
dkt
rc
kh
qB
rc
kt
kh
qBpp
rc
kt
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kt
rc
kt
qB
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r
r
tp
r
t
wt
wtwi
wt
wt
wt
wi
DDD
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27
)7.1(1688
ln6.70
109289.5ln
6.70
assuch Eq.(1.7b) From
2
2
4
dkt
rc
kh
qB
rc
kt
kh
qBpp
rr
t
ti
28
tkh
qB
k
rc
kh
qBpp
tk
rc
kh
qBpp
tk
rc
kh
qB
dkt
rc
kh
qB
rc
kt
kh
qBpp
wtiw
wtiw
wt
wt
wtwi
ln6.701688
ln6.70
ln1688
ln6.70
ln1688
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)7.1(1688
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109289.5ln
6.70
2
2
2
2
2
4
Question:Why does pw > pi for certain t ?
29
equation. flow radial state-steady by the modeled becan
) ( zone thisacross drop pressure additional the), ( radiusouter and
) (ty permeabili uniform of zone alteredan toequivalent considered
is zone stimulatedor damaged theifout that pointed Hawkins
.fracturing hydraulicor n acidizatioby stimulated are sother wellMany
.operations completionor drilling from resulting wellborethe
near (damage)ty permeabili reduced have most wells practice,In
ss
s
pr
k
30
)1(80907.01064.2
ln1
ln6.70
80907.011064.2
ln2
1
2.141
)(
At (1)
zone) (damaged For
80907.0ln2
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orenear wellb zone damaged-nonFor
4
2
2
22
4
2
t
s
ssrssi
w
swt
srssis
w
sDs
s
D
DD
c
tk
rhk
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rrrc
tk
qB
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r
rrorrr
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r
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ln1
ln6.70
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4
2
2
4
t
s
wswsi
wt
swsis
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wDw
c
tk
rhk
qBpp
rc
tk
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r
rrwellborerr
31
)4(ln2.141
ln6.70
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qB
r
r
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zonedamagenokkforSimilarity
r
r
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qB
r
r
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qBpp
c
tk
rhk
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c
tk
rhk
qBpp
32
0For
. n torather tha toalproportioninversely
is zone altered in the drop pressure that thestatesequation This
)9.1(ln12.141
,
Eq.(4)]-[Eq.(3) zone damage across drop pressure Additional
ss
s
w
s
s
wrswsrsss
wswssrss
pkk
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r
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kh
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33
87lim0,,
lim0,,
)10.1(ln1
)11.1(21688
ln6.70
ln121688
ln6.70
ln12.1411688
ln6.70
1688ln
6.70
Eq.(1.9)] and Eq.(1.7d) [from is wellboreat the drop pressure The
).(stimulatety permeabili enhancedor (damage)ty permeabili reduced aWith
2
2
2
2
oritlowerskkstimulatediswellif
sforituppernoskkdamagediswellIf
r
r
k
kswhere
skt
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kh
qBppor
r
r
k
k
kt
rc
kh
qB
r
r
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k
kh
qB
kt
rc
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pkt
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s
s
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s
s
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s
s
wt
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s
wtwfi
swt
wfi
34
80907.0ln2
1
1064.2
42
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1688ln6.70
1688ln6.70
ln1688
ln6.70
21688
ln6.70
Eq.(1.11) From
2
2
42
2
2
22
2
D
DD
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DD
DiD
sws
st
swt
swt
wtwfi
r
tpor
erc
kttwhere
t
rEp
radiuswellboreeffectiveerrwhere
kt
rc
kh
qB
kt
erc
kh
qB
ekt
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skt
rc
kh
qBpp
35
hrstandftratp
hrstandftratp
hrstandftratpCalculate
sftr
psip
fthpsic
STBRBBmdk
ftrcp
DSTBq
oilonlypoductingiswellTheGiven
solutionfunctionEithe
guwellborethebeyondpressuresofnCalculatioExample
e
i
t
o
w
3100?)3(
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(2
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31.010637.210637.2
2
2
25
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br
t
at
rErtp
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Solution
D
D
D
DiDDD
wt
D
36
psip
pp
r
tpb
psippkh
qBpp
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PorTablefrom
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rrftrAt
i
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38
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322
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r
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pp
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r
t
r
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40
wD
wD
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ktt
Bq
ppkhpwhere
t
p
r
p
rr
p
t
p
k
c
r
p
rr
p
2
2
2
42
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yDiffusivit ofEquation
41
0 21
21
20101
2
2
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)()(
)()()()()1(4
solution wellboreFinite (b)
80907.0)ln(2
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4(
2
1
solution source Line (a)
reservoir Infinite (1)
Solutions
2
uYuJu
duurJuYurYuJep
r
t
t
rEp
DDtu
D
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42
0 2
1
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2
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1,
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uYuJu
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rwellboreAt
uYuJu
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solutionwellboreFinite
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VEH
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43
44
w
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e
rr
rr
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r
rrat
r
p
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r
pr
rallfortatp
formessDimensionl
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tforkh
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r
pr
rallfortatpp
conditionsboundaryandInitial
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r
p
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p
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:
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10634.2
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casereservoir Circular Bounded rate,Constant
1
2
2
42
2
45
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64
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infinite reservoir
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65
Boundary effect time estimated from radius of investigation equation
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6.5
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7.5
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( II )
( III )
closed circular reservoir with reD = 3000 case
66
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73
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103
InterferenceTest• Consider three wells, well
A, B, and C that start to produce at the same time from infinite reservoir (Fig. 1.8). Application of the principle of superposition shows that
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• In Eq.(1.49), there is a skin factor for well A, but does not include skin factors for wells B and C. Because most wells have a nonzero skin factor and because we are modeling pressure inside the zone of altered permeability near well A, we must include its skin factor.
• However, the pressure of nonzero skin factors for wells B and C affects pressure only inside their zones of altered permeability and has no influence on pressure at Well A if Well A is not within the altered zone of either Well B or Well C.
106
Bounded reservoir• Consider the well (in fig. 1.9)
a distance, L, from a single no-flow boundary. Mathematically, this problem is identical to the problem of a two-well system; actual well and image well.
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• Extensions of the imaging technique also can be used, for example, to model
(1) pressure distribution for a well between two boundaries intersecting at 90°; (2) the pressure behavior of a well between two parallel boundaries; and (3) pressure behavior for wells in various locations completely surrounded by no-flow boundaries in rectangular-shape reservoirs.
• [ Matthews, C. S., Brons, F., and Hazebroek, P.: “A method for determination of average pressure in a bounded reservoir,” Trans, AIME (1954) 201, 182-191
108
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• Proceeding in a similar way, we can model an actual well with dozens of rate changes in its history
• we also can model the rate history for a well with a continuously changing rate (with a sequence of constant-rate periods at the average rate during the period).
110
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1.6 Horner’s Approximation• In 1951, Horner reported an approximation that can be used in
many cases to avoid the use of superposition in modeling the production history of a variable-rate well.
• With this approximation, we can replace the sequence of Ei functions, reflecting rate changes, with a single Ei function that contains a single producing time and a single producing rate.
• The single rate is the most recent nonzero rate at which the well was produced; we call this rate qlast for now.
• This single producing time is found by dividing cumulative production from the well by the most recent rate; we call this producing time tp, or pseudoproducing time
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(1) The basis for the approximation is not rigorous, but intuitive, and is founded on two criteria:
(a) Use the most recent rate, such a rate, maintained for any significant period
(b) Choose an effective production time such that the product of the rate and the production time results in the correct cumulative production. In this way, material balance will be maintained accurately.
115
• (2) If the most recent rate is maintained sufficiently long for the radius of investigation achieved at this rate to reach the drainage radius of the tested well, then Horner’s approximation is always sufficiently accurate.
• We find that, for a new well that undergoes a series of rather rapid rate changes, it is usually sufficient to establish the last constant rate for at least twice as long as the previous rate.
• When there is any doubt about whether these guidelines are satisfied, the safe approach is to use superposition to model the production history of the well.
116
Example 1.6 – Application of Horner’s Approximation
• Given: the Production history was as follows:
simulated? be well
for thishistory production theshould how not, If
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? t(1) :Find p
117
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119
Reference Books• (A) Lee, J.W., Well Testing, Society of petroleum Engineers of AIME, Dallas, Texas,,
1982.
• (B) Earlougher, R.C., Jr., Advances in Well Test Analysis, Society of Petroleum Engineers, Richardson, Texas, 1977, Monograph Series, Vol. 5.
• (1) Carlson, M.R., Practical Reservoir Simulation: Using, Assessing, and Developing Results, PennWell Publishing Co., Houston,TX, 2003.
(2) FANCHI, J.R., Principles of Applied Reservoir Simulation, Second Edition, PennWell Publishing Co., Houston,TX, 2001.
(3) Ertekin, T., Basic Applied Reservoir Simulation, PennWell Publishing Co., Houston,TX, 2003.
(4) Koederitz, L.F., Lecture Notes on Applied Reservoir Simulation, World Scientific Publishing
• Company, MD, 2005
120
Introduction
• This course intended to explain how to use well pressures and flow rates to evaluate the formation surrounding a tested well , by analytical and numerical methods.
• Basis to this discussion is an understanding of
(1) the theory of fluid flow in porous media, and
(2) pressure-volume-temperature (PVT) relations for fluid systems of practical interest.
121
Introduction (cont.)• One major purpose of well testing is to determine the ability of
a formation to produce fluids.
• Further, it is important to determine the underlying reason for a well’s productivity.
• A properly designed, executed, and analyzed well test usually can provide information about
FORMATION PERMEABILITY, extent of WELLBORE DAMAGE (or STIMULATION),
RESERVOIR PRESSURE, and (perhaps) RESERVOIR BOUNDARIES and HETEROGENEITIES.
122
Introduction (cont.)
• The basic test method is to create a pressure drawdown in the wellbore, this causes formation fluids to enter the wellbore.
• If we measure the flow rate and the pressure in the wellbore during production or the pressure during a shut-in period following production, we usually will have sufficient information to characterize the tested well.
123
Introduction (cont.)
• This course discusses(1) basic equations that describe the unsteady-state flow of fluids in
porous media,(2) pressure buildup tests,(3) pressure drawdown tests,(4) other flow tests,(5) type-curve analysis,(6) gas well tests,(7) interference and pulse tests, and(8) drillstem and wireline formation tests
• Basic equations and examples use engineering units (field units)
124
Chapter 1 Fluid Flow in Porous Media
125
1.1 Introduction
(a) Discussion of the differential equations that are used most often to model unsteady-state flow.
(b) Discussion of some of the most useful solutions to these equations, with emphases on the exponential-integral solution describing radial, unsteady-state flow.
(c) Discussion of the radius-of-investigation concept
(d) Discussion of the principle of superposition Superposition, illustrated in multiwell infinite reservoirs, is used
to simulate simple reservoir boundaries and to simulate variable rate production histories.
(e) Discussion of “pseudo production time”.
126
1.2 The ideal reservoir model
• Assumptions used (1) Slightly compressible liquid (small and constant compressibility) (2) Radial flow (3) Isothermal flow (4) Single phase flow
• Physical laws used (1) Continuity equations (mass balances) (2) Flow laws (Darcy’s law)
127
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128
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