Petro 424 Section I PI, IPR Curves Productivity Index - J - The ratio of the production rate of a well to its drawdown pressure. bpd/psi Drawdown Pressure -Δp- The pressure drop between the reservoir ( p s ) and the flowing bottom hole pressure (p wf ). wf s p p q J − = (1-1) note: q here is the gross liquid production rate in bpd. Ideally p wf for a q is measured using a bottom hole pressure gauge. A build up or drawdown test is used to calculate p s along with other parameters such as a skin factor (s). Equation 1-1 uses these values to get a productivity index for the well. Using the radial flow equation ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = w o wf r r kh q p p ln 007082 . μ β (1-2) which gives J r r kh p p q w o wf = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − ln 007082 . μ β (1-3) J or other parameters can be calculated depending on the known data.
Transcript
Petro 424
Section I PI, IPR Curves
Productivity Index - J - The ratio of the production rate of a well to its drawdown pressure. bpd/psi Drawdown Pressure -Δp- The pressure drop between the reservoir ( ps) and the flowing bottom hole pressure (pwf).
wfs ppqJ−
= (1-1)
note: q here is the gross liquid production rate in bpd. Ideally pwf for a q is measured using a bottom hole pressure gauge. A build up or drawdown test is used to calculate ps along with other parameters such as a skin factor (s). Equation 1-1 uses these values to get a productivity index for the well. Using the radial flow equation
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
w
owf r
rkh
qpp ln
007082.μβ (1-2)
which gives J
rr
khpp
q
wo
wf
=⎟⎠⎞⎜
⎝⎛
=− ln
007082.
μβ (1-3)
J or other parameters can be calculated depending on the known data.
Example 1-1 A Permo-Penn well is flowing 156 bopd from perforations at 10,454 to 10,468 ft. The bottom hole flowing pressure is recorded at 250 psi and the reservoir pressure is of the field has been recorded at 3410 psi. Find J. Using equation 1-1 J = 156 / 3410 - 250 J=.049 bpd/psi Example 1-2 A field is drilled up on rectangular 80 acre spacing. The reservoir pressure is 1000 psi, the permeability 50 md, the net sand thickness 20 feet, the oil viscosity 3 cp, and the oil formation factor 1.25. The wells are completed with 7 inch casing. Calculate the J for this well. The distance between wells on 80 acre spacing is 1864 ft, so r can be assumed to be 932 ft, ln(932/.292) = 8.06. Using equation 1-3 J = .007082 * 50 * 20 / 1.25 * 3 *8.06 J = .234 bpd/psi
Skin Effect s The pressure drop caused by the near wellbore skin effect is defined by the equation s
khqps πμ
2=Δ (1-4)
Add the pressure loss from skin to equation 1-2
⎟⎠⎞⎜
⎝⎛ ++= sr
rkh
qpp
w
owf ln
007082.μβ (1-5)
So equation 1-3 becomes J
srr
khpp
q
wo
wf
=⎟⎠⎞⎜
⎝⎛ +
=− ln
007082.
μβ (1-6)
Ideal PI – JI
skinwfsI ppp
qJ−−
= bpd/psi (1-7)
Actual PI – JA
wfsa pp
qJ−
= bpd/psi (1-8)
Specific PI - Js
)( wfss pph
qJ−
= bpd/psi/ft (1-9)
Effective wellbore radius ft (1-10) s
ww err −=/
Productivity Index in Horizontal Wells
Drainage pattern formed around a horizontal well
A schematic of horizontal well drainage area. Generally the length of a horizontal well is much greater than the reservoir thickness. In this case the flow in the well can be described by
( )⎥⎦⎤
⎢⎣
⎡Δ=
Lrphkq
eh
hh /4ln
1007078.βμ
(1-11)
This gives:
( )⎥⎦⎤
⎢⎣
⎡=
LrhkJ
eh
hh /4ln
1007078.βμ
(1-12)
In many cases there is a horizontal-to-vertical permeability anisotropy which effects the flow into the wellbore. For this case
( )( ) ⎥
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
+⎟⎠⎞
⎜⎝⎛+
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −+
Δ=
)1(ln
2/)2/(
ln
1007078.22
aniw
aniani
h
IrhI
LhI
LLaa
phkqβμ
(1-13)
where
v
hani k
kI = 5.5.4
2/25.5.
2 ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛++=
LrLa eH for L/2 <0.9reH
You can find the equation for Jh. To include skin effect
( )( ) ⎥
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
++
⎟⎠⎞
⎜⎝⎛+
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −+
Δ=
'22
)1(ln
2/)2/(
ln
1007078.
eqaniw
aniani
h
sIr
hIL
hIL
Laa
phkqβμ
(1-14)
( ) ⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛++
+⎟⎟⎠
⎞⎜⎜⎝
⎛−= 1
34
11ln1 ,
2
2max,'
w
MAXH
w
H
aniseq r
ar
aIk
ks (1-15)
aH,max is the largest horizontal axis of the cone of damage.
Homework #1 Problem #1 For a well that has the following; spacing = 40 acres casing 5 1/2” ps = 2000 psi βo = 1.25 μ = 2.5cp k = 25 md h = 12 ft Find J and Js for this well. Calculate the pwf for a q of 100 bopd and 12 bwpd and also for 200 bopd and 25 bwpd. Problem #2 Find the J and kh for a well with the following; spacing = 40 acres casing 4 ½” μ = 2.8 cp β = 1.15 ps = 3400 psi pwf = 1250 psi qo = 250 bopd qw = 500 bwpd GOR = 1200 ft3/bbl
Equation 1-1 can be rewritten as q = J Δp (1-16) It can be seen that the relationship between q & Δp is a straight line that passes through the origin.
020406080
100120140160180
Drawdown psi figure 1 Equation 1-1 can also be rewritten as pwf = ps - q/J (1-17) With ps and J constant for any particular instant the plot of q vs BHFP will be a straight line as shown in figure 2. The point when BHFP is zero or at the greatest Δp is called the well’s potential. It is the maximum rate that the well could produce, it is noted that physically a BFP of 0 psi is basically not attainable.
0
500
1000
1500
2000
2500
3000
3500
q bpd figure 2 In the olden days this potential was to calculate the allowable for the well in many states.
Effects of Water Production Relative permeabilities can be used to calculate separately the flow of both oil and water. ko = kkro kw = kkrw (1-18) kro relative perm to oil krw relative perm to water Since the relative permeabilities are function of the oil and water saturations of the reservoir so are the flow rates of the oil and water.
( )
⎥⎦⎤
⎢⎣⎡ +⎟
⎠⎞⎜
⎝⎛
−=
srr
pphkq
woo
owfoo
ln
007082.
μβ
( )
⎥⎦⎤
⎢⎣⎡ +⎟
⎠⎞⎜
⎝⎛
−=
srr
pphkq
www
wwfww
ln
007082.
μβ (1-20)
Pseudo-Steady State Flow For wells that have a no-flow boundaries. These boundaries can be caused by the production of adjacent wells or a natural boundary, fault or pinchout . These are the equations that are used on older wells. The pressure at the boundary can be calculated by using
⎟⎟⎠
⎞⎜⎜⎝
⎛−+= 5.ln2.141
w
ewfe r
rkh
qpp βμ (1-21)
A much more helpful equation would be one that uses the average reservoir pressure which can be obtain in the field by means of a pressure test. This equation is
⎟⎟⎠
⎞⎜⎜⎝
⎛−=− 75.ln2.141_
w
ewf r
rkh
qpp βμ (1-22)
⎥⎦
⎤⎢⎣
⎡−
=75.ln
007082.
w
e
rr
khJβμ
(1-21)
With the addition of skin
⎥⎦
⎤⎢⎣
⎡−+
=75.ln
007082.
srr
khJ
w
eβμ (1-22)
In Flow Performance Relationship - IPR Curves
he Inflow Performance Relationship (IPR) for a well is the relationship .
Tbetween the flow rate of the well q and the flowing pressure of the well pwfIn single phase flow this is a straight line but when gas is moving in the reservoir, at a pressure below the bubble point, this in not a linear relationship.
Figure 3
Factors influencing the shape of the IPR are the pressure drop and relative k
across the reservoir.
table 1
It can be seen that the majority of the pressure drop caused by production is
the
near the wellbore. This is confirmed by the radial flow equation. In this situation even if the average reservoir pressure is above the bubble point, area around the wellbore is not, which causes the gas to come out of solution
in this area causing the relative permeability (which is based on fluid saturation) of the liquids to change. As the pwf is lower for a greater flrate the greater this effect has on the well which causes the IPR Curve to bend down.
ow
2 Stratified Formation or Zones
figure 4
When zones of varying kh are opened in a well, the one with the highest kh
rt
well contribute more to the production of the well, then the lower kh zones will contribute, thus the average reservoir pressure of the high kh zones drops faster than the other zones in the well. This causes the zones to staflowing at different flowing bottom hole pressures. At the lower rates or higher flowing pressures it is the zone with the lowest kh that have the highest average pressure so that it produces first and then as the flowingpressure drops below the average pressure of the other zones that start to contribute to the flow. The PI of the well improves as more of the zones contribute, so the PI improves with the lowering of the flowing pressure.
figure- 5
Vogel’s Method Vogel developed an empirical equation for the shape of the IPR curve.
2
__ 8.2.1' ⎟⎟
⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−=
p
p
p
pqq wfwf (1-23)
where q’ is the potential of the well or flow at 0 pwf. Using the productivity index J we get
( )_1
(' p
pJp
ppJqq wf
s
wfs −=−
= (1-24)
assuming ps and average reservoir pressure approximately the same. Hence the difference between the value of q derived from the Vogel equation and the straight line method is
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=− __ 1'8.
p
p
p
pqqq wfwf
slvm (1-25)
The value is always positive, and at the end points, pwf = p and pwf = 0 it is 0. Standing rewrote the equation
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
pp
pp
qq wfwf 8.11'
(1-26)
this gives
)8.1('p
ppqJ wf+= (1-27)
as pwf goes to p
pqJ '8.1* = (1-28)
combining
⎟⎠⎞
⎜⎝⎛ +
=
ppJJwf8.1
8.1* (1-29)
using the pseudo-steady state radial flow equation
⎥⎦⎤
⎢⎣⎡ −
=75.ln
007082.*
w
eoo
o
rr
hkJμβ
(1-30)
by canceling out constant terms
poo
o
foo
o
p
f
k
k
JJ
⎟⎠⎞⎜
⎝⎛
⎟⎠⎞⎜
⎝⎛
=
μβ
μβ*
*
(1-31)
by using q’= J*p/1.8 in Vogel’s equation we get
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
2*
8.2.18.1 p
pp
ppJq wfwf (1-32)
Homework #2 1)Take the data from problem 1, homework on and calculate the J of a horizontal well with a 1000’ horizontal section and a formation thickness of 25’ and effective radius of 450’ from the well bore. Change the L to 1500’ and find J. Calculate J if the well has a vertical k of 15 md and 30 md.
2) Well #2A is flowing at 1120 bopd through 2 7/8” tubing. There is zero water cut, and the GLR is 820 scf/bbl. A pressure survey on the well shows that the flowing pressure at 6470’ is 675 psig, while the pressure build up shows a static pressure of 2080 psig at a datum level of 6500’. Using Vogel’s method, draw the IPR curve, and estimate the well’s potential. Reservoir analysis indicates that the ratio of the value of kro/βoμo @ 2080 psi to its value at the static pressure of 1500 psig is 1.57. Estimate what the well’s potential rate will be when the static pressure dropped to 1500 psig.
Fetkovich’s Approximation Since Vogel’s method is not always in accordance with field data, Fetkovich suggested ( )n
wfo ppCq 22 −= (1-33)
if i
w
eoo
ro
prr
khkJ
2)ln(
007082.'
μβ=
(1-34) the equation becomes ( )n
wfo ppJq 22' −= (1-35) ( )n
o pJq 2' '= (1-36) Fetkovich assumed that the log log plot of qo vs Δp2 is a straight line with a unity slope, n=1. Using Fekovich
1) plot the q vs Δp2 2) Find the slope of line 3) Calculate J’ using one of the flow rates 4) Using J’ calculate the well’s potential and pwf for any other rate. 5) If only rate and pressure is known assume a slope of 1
Potential of Gas Wells Calculated Open Absolute Flow Multi-point Test Back Pressure Test Four Point Test By definition COAF is the flow when pws is equal to 0 psi. Starting with the radial flow for gas
qkh p p
Tz rr
scs ws
w
=−. (
ln( )
703 2 2
μ
)
)
(1-37)
taking μ and z as constants at the values of the average reservoir pressure and rewriting equation 1-10 in the general form (1-38) (q C p psc s wf= −2 2
where
Ckh
T z rrw
= − − −
.
ln( )
703
μ (1-39)
taking the log of both sides ( )log log logq C p psc s wf= + −2 2 (1-40) it can be seen that this is a straight line on a log-log plot with unity slope. Rawlins and Schellhardt modified the equation 1-11 to (1-41) (q C p psc s wf
n= −2 2 )
to fit observed field data in 1936.
Procedures for Evaluating Multi-point Tests
1. Plot q vs (ps
2 - pwf2) on log-log paper
2. Draw a straight line through the points
3. find the angle between the line and the (ps2 - pwf
2) axis a) the tangent of this angle is n b) n can also be found by taking the slope of the line.
4. Taking a flow point and solving for the potential using
( )q qp
p ps
s wf
n
' =−
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
2
2 2
5. The potential can also be found graphically by extrapolating the line to the point at which pwf equals 0; the q at that point is the potential of the well The potential can also be calculated by find C and n by simultaneously solving the equation ( )n
wfs ppCq 22 −= using two of the flow points. Also by getting n from the plot and solving for C at a point on the line and then making pwf equal to 0 in the equation.
Homework #3 Given: PI = 1410 psia # qo pwf bpd psia 0 0 1410 1 72 1170 2 118 1050 3 155 888 4 208 632 Find the potential of the well. Write the equation for the flow rate determining the value for J’ and n. Find the potential using Vogel Method for tests 1 and 4. Find the potential using linear PI for tests 1 and 4. Find the potential using pseudo steady state PI for tests 1 and 4.
Skin Factor To include skin equation 1-37 can be written as
)75.ln(
)(703. 22
−+−
=sr
rTzppkhq
w
wsssc μ
(1-42)
so C will be
)75.ln(
703.
−+= −−−
srrzT
khC
wμ
(1-43)
Non-Darcy Flow Aronofsky and Jenkins used the Forchheeimer flow equation for a more exact solution
( )( )[ ]DqsrrzT
ppkhq
wd
wf
++−
=/ln1424
22
μ (1-44)
where p, μ, z are at average reservoir pressure and D is the Nondarcy coefficient. rd is the Aronofsky and Jenkins effective drainage radius, it is time dependent until rd = .472re. Otherwise D
w
d trr 5.1=
where 2
000264.
wtD rc
kttφμ
=
Equation can be rearranged as 222 142475.ln1424 q
khzTDqsr
rkh
zTppw
ewf
μμ+⎟
⎠⎞⎜
⎝⎛ ++=− (1-45)
This can be written as (1-46) 222 bqaqpp wf +=−
a and b can be found by plotting Δp2/q vs q on Cartesian coordinates, for 4 flow rates. The intercept of the line is a and the slope is b. D can then be calculated from b. D is usually a very small number. An empirical relationship for D
2
1.05106
perfw
s
hrhkD
μγ −−×
=
γ is gas gravity, ks is the perm around the wellbore. It can be seen that the Δp2 vs q on log log paper will curve up for Non-Darcy flow.
Horizontal Gas Well For steady state:
( )( )
( ) ⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
++
⎟⎠⎞
⎜⎝⎛+
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −+
−×=
−
DqIr
hIL
hIL
LaazTpphk
q
aniw
aniani
wfh
)1(ln
2/)2/(
ln
110703.22
223
μ (1-47)
For pseudo-steady state;
( )( )
( ) ⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
+−+
⎟⎠⎞
⎜⎝⎛+
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −+
−×=
−
DqIr
hIL
hIL
LaazTpphk
q
aniw
aniani
wfh
75.)1(
ln2/
)2/(ln
110703.22
223
μ (1-48)
q is mscfpd Gas Productivity Index 22
wfgas pp
qJ−
= mscf/psi2 (1-49)
Effects of Drawdown on WOR When more than one productive stringers are open in a well, these can be in one zone; the stringers may water out at differing rates. By using IPR curves it is possible to determine if the pressure of the water source is higher than that of the oil zones and if there will be interflow in the well if shut in. By interflow we mean water flowing into the oil zones from the water zones. This can be done by plotting the gross, oil and water IPR curves from production test data. From the plot determine the static pressure for the oil and water zones as well as the well’s average static pressure. The point on the water IPR curve that corresponds to the well’s average static pressure is the water inflow into the oil zone when the well is shut in. This can also be found by multiplying the difference in the water and the well’s average static pressures by the water PI. Example
Gross rate Water Cut Water rate Oil rate pwf bpd % bpd bpd psi
47 85 40 7 1300 90 60 54 36 920
125 48 60 65 630 162 45 73 89 310
figure 6 The shape of the water cut curve is typical of a high-pressure water.
figure 7 The shape of a water cut curve that is typical of a low-pressure water
Homework #4
1) A well completed with perforations from 9897 – 9932 feet from the surface in 5 ½ casing, has an initial reservoir pressure of 3572 psig and a fluid gradient of .35 psi/ft. The well was tested using at 130 bopd and no water with a bottom hole flowing pressure of 500 psig, using 2” tubing.
Is it possible for this well to produce without a pump? Possible rate? Keeping a fluid level 200’ above the pump for efficient pump loading what is the maxium rate that can be produced? Use both straight line and Vogel. The fluid properties, 1.15 stb/rb, viscosity of 2.6cp. Reservoir properties, perm of 5.7 md, re of 1980 ft. Would further treatment help this well? 2) The following data was collected on the J. J. Fed #1
TD 4501’ Perfs 4448-4456’ in 4 ½” casing 2 2/8” tubing set at 4400’ with no packer tubing casing BHP Rate psig psig psia mscd 884 847 897.2 0 860 827 871.6 299 810 780 825.8 649 750 727 770.2 825 700 674 718.0 1026 What is the slope of the line, what is n? Calculate the CAOF of this well. ` What would the well produce if the line pressure was 200 psi?
PI and IPR Summary
The ability of a well to produce fluids. The uses of the Productivity Index and IPR Curves Find the well’s potential, q’, the maximum production rate.
Predict production rates for planning production schedules and sizing production equipment Reference point for the comparison of wells in a field. Find Flow Efficiency of the well to plan or verify completion techniques. During production monitoring to help diagnose production problems if any. Equipment or reservoir problem Selecting testing procedures to identify production problems. Comparing the PI of the field test to a calculated PI to verify reservoir properties or an indication of skin in the well.
Productivity Index J
Simplest of the methods, one production and pressure point and a straight line. But the least accurate for calculating the well’s potential, greater error as pwf is lower. Very good for calculating flow efficiencies, FE.
IPR Curves
A very high accuracy if obtained by using a multipoint production test data. For one point tests Vogel Method
Reasonable accuracy for the plot and the potential. Problem is that good mobility data is needed for a calculated ideal curve and for future average reservoir plots. Fetkovich Method Not as accurate as Vogel. But because of the straight forward method easier for a quick calculation.
Gas Wells
Multi-point tests are the norm for gas wells. One-point test assumes a straight line at 45 degrees or an assumed value for the field. Pressure depletion gas wells have a constant productivity index, calculated using the square of the pressures.
Flow Efficiency
The Flow Efficiency of the well which is the ratio of the actual PI to the ideal PI is used to check if the well is a candidate for a work over to remove damage. Also it can be sued to verify a stimulation job. If FE < 1 possible damage, FE > 1 a stimulated zone.
Nomenclature
h thickness ft J Productivity Index bpd/psi k Permeability md L Length of Horizontal section ft p Pressure psi pwf Bottom hole pressure flowing psi ps Bottom hole pressure shut in psi Δpskin Delta pressure skin psi q Flow rate bpd qsc Flow rate gas mscfpd q’ Well potential bopd, mscfpd re radius effective ft reh radius horizontal effective ft rw radius wellbore ft s skin s’eq skin effective horizontal T Reservoir Temp Ro β formation volume factor rb/stb
