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Determination of transient drainage across lease boundaries
PETROLEUM SCIENCE & ENGELSEVIER
Journal of Petroleum Science and Engineering 12 ( 1995) 269-276
Determination of transient drainage across lease boundaries Eric S. Carlson, Philip W. Johnson Department of Mineral Engineering, The University of Alabama, Tuscaloosa, AL 35487-0207, USA
Received 10 March 1994; accepted 23 August 1994
Abstract A technique which simplifies the determination of how much oil or gas flows across a linear segment during the transient flow period of the reservoir is presented. The technique is based on two definite integrals which result from application of the linesource solution of flow in an infinite-acting reservoir combined with the fundamental definition of flow across a plane. The theory, the outline of a methodology, and several examples are provided. The examples, one which determines the flow across a lease line for transient flow in an oil reservoir, one using the method for a bounded lease, and one using the method to assess the well spacing in a tight-gas reservoir, show that the technique can be applied to a variety of practical problems. Graphical solutions to the definite integrals are given to facilitate application of the method.
1. Introduction
I. I. Background and assumptions
The scenario is quite common. A wildcat well drilled adjacent to a lease boundary has just discovered a new reservoir. The well begins sustained production, and the adjacent lease holder asks the inevitable question, “How much of that production is coming from my property?” The question is easy enough to pose, but the answer has been far more complicated to determine. The theory for transient flow across a linear segment is summarized in this paper and a simple method for determining the instantaneous rate and the cumulative flow across the segment for any time during the infiniteacting period is presented. The concept of transient flow across a linear segment has many important applications, besides the wildcat problem described above. The methods presented here can be very useful for drainage evaluations in tight gas systems, in addition to waterfloods in saturated reservoirs. The examples shown illustrate the versatility of the method.
Hurst (1975, 1984) presented several studies regarding flow across lease boundaries in steady-state and pseudo steady-state systems. In one of his papers, he mentions the velocity in a transient flow system but does not develop the transient relations. Johnson ( 1988) presented a method for transient flow of radial systems, but the method only applied to circular boundaries. Details of the theory of transient flow across linear segments are presented in the Appendix. In this section, only the assumptions used for the theory and the relations required for its application are presented. The assumptions are the same as those used for infinite-acting flow in well test analysis. A homogeneous reservoir with uniform permeability, thickness and porosity is assumed. The reservoir is infinite in extent, and the well flows at a constant rate. If the reservoir is not infinite acting, the methods of Hurst should be used. Fig. 1 illustrates the basic system ideas and nomenclature. A well is arbitrarily located at the point (x,,
0920-4105/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SsDIO920-4105(94)00049-2
ES. Carlson, P. W. Johnson /Journal of Petroleum Science und Engineering 12 (1995) 269-276
270
for the cases where m and 1lm are not equal to 0. When y2=yl,m=0,and (x,,yo) =(x,,yI). Whenx,=x,, 11 m=Oand(x,,y,)=(x,,y,).
The distance, r,,,,, between the well and (x,,, y,,) is: Boundary Line rhase
=
J(Yw-Yo)2+(G+~“)2
(4)
The distance, r,, between (xi, y, ) and (x,, y,,) is: The Well at point (h, Yw)
B
x
TI =&y,
I
“r
r = r,,,
\
7\ 8 \\
Y t
l--L I
/ cos (e)
0, = arctan L i rha 3
\
I
X
If----rb,s, I
\\
a
_I
I
y,) of a Cartesian grid. A straight line segment away from the well has endpoints of (x,, yi) and (x2, y?). A perpendicular from the well intersects the segment at a point denoted (x0, y,). Given that the segment has a slope: Yz--Y1
(1)
x2-x1
can be given by:
xo~Yw-YI+~l+u~mh m+(llm)
(2)
x,-xl+my,+ m+(llm)
(llm)y,
t
‘lh -
0.02532 ft2 CP mD psi day
kt +pc,ri&
(7)
where k is the permeability, t is the time, 4 is the porosity, and c, is the total compressibility. It is shown in the appendix that the rate at any dimensionless time can be given by: qhH=qwqhr(bh8)
(8)
for flow across the segment between the angles 0 and 80r - 19.Figs. 2,3 and 4 show qhr ( fi)b, 0) as a function Of tot, for VariOUSScales Of fbh. To CValUate flOW aCrOSS the segment between 8, and e2, qb, the above equation is applied to give: qh = qh#2 + qh@l
(9)
if 13,and 0, have opposite signs (that is, the point (x0, y,) lies between (nz, y2) and (xi, yI> on the segment), and:
and y0 can be given by: yo=
(6)
The angle 0, is defined comparably for (x2, y2). The signs of the angles are arbitrary, but should be consistent. That is, if (..x~,y2) and (xi, yi) are on opposite sides of (x,, y,) , the angles should have opposite signs; if they are on the same side, they should have the same sign. Transient flow across the linear segment depends on the dimensionless time, f,),,, defined by:
Fig. 1. (A) The coordinate system and the flow vectors used in deriving the equations for transient flow across boundaries. (B) Schematic of the arrangement of the producing well and the boundary, including a definition of terms.
x,
(5)
\
I
m=-
(Xl +x”j2
The distance, r,, between (x2, y2) and (x,,, yo) can be obtained in a similar manner. The angle 0i is the angle between the line which goes through the well and (?I,,, y,) and the line that goes through the well and (x,, y,), as shown in Fig. 1. The magnitude of the angle can be given by:
“x
I
-Yo)*+
(3)
(10)
ES. Carlson, P. W. Johnson /Journal
of Petroleum Science and Engineering 12 (1995) 269-276
if 8, and 0, have the same sign. For cumulative flow:
Qw= QwQd tm ‘3
(11)
where Q,, is the cumulative flow across the segment between 0 and 0 or - 0, Q, is the cumulative production at the well, and Qbr (tDb, 0) is presented in Figs. $6 and 7. Just as in the rate figures, the only difference
271
between these cumulative production curves is the dimensionless time scale and the cumulative production across the entire segment can be given by:
QL,= Qm + Qbs,
(12)
0 25 e(dQS.)
1
e (deg.) ,65-90 65
0
1.0
0.5
0.0
1.5
2.0
10
Fig. 4. Dimensionless
20
production
30
t Db
40
rate at late dimensionless
50
times.
t,
Fig. 2. Dimensionless
production
rate at early dimensionless
e (deg.1
times.
-65-90
0.1.
.
.
i.
,.
.i
. . -ew0.l
0.12
0.10
2
d
0.05
j
0.08
0.04
0.02
0.0
0.5
1.0
1.5
2.0
tc+
Fig. 3. Dimensionless times.
production
rate at intermediate
dimensionless
Fig. 5. Dimensionless sionfess times.
cumulative
production
rate at early dimen-
E.S. Carlson, P. W. Johnson /Journal of Petroleum Science and Engineering I2 (1995) 269-276
272
Q (deg 1
60X
1
55
Examination of the late-time curves, Figs. 4 and 7, shows that the flow rates and cumulative flow rates across the linear segments approach the value of O/ (27~). This flow rate is the steady-state solution of the problem, which implies (correctly) that the transient behavior eventually approaches the steady behavior. These equations and assumptions are the only requirements for effective application of the method to be presented below.
2. Methodology
(2)
Fig. 6. Dimensionless cumulative production rate at intermediate dimensionless times. 0.20
(3) (4) (5) (6)
-
9 (deg.) 7
-% 0.15-
(7) (8)
a_
Look up qbr (tDb, 01) 9 Qbr( fDb, ‘% > 3 qbr ( fDb, 0,) 1 and (& (t,,, 6,) on the appropriate charts. Calculate the flow across the segment using Eqs. (9) 9 and 12 if 0, and 0, have opposite signs; otherwise, use Eqs. 10 and 13. ( 10) If more than 1 time is desired, go back to Step 7.
.-s 1 t .g
O.lO-
;ii z ; P ‘S s $
Determine the coordinates of the well and the endpoints of the segment (lease line, spacing boundary, gas cap, etc.) in Cartesian coordinates. Determine the point on the segment closest to the well (use Eqs. 2 and 3). Calculate r,,,, using Eq. 4. Calculate r, and r, using Eq. 5. Calculate the magnitude of Oi and 0, using Eq. 6. Determine the relative signs of 0, and 0,. If x, > X, > x, (that is, X, is on the interval between x1 and xz) and y2 >y” > y,, then OXand 0, have opposite signs. Otherwise, they have the same signs. Calculate toi, using Eq. 7.
0.05-
3. Sample applications
0
10
20
30 t Db
40
50
Fig. 7. Dimensionless cumulative production rate at late dimensionless times.
Three sample applications are presented to illustrate the procedure. The first example deals with the scenario posed in the introduction. The second shows how to use the method in bounded regions. The final example deals with drainage in a tight gas system. 3.1. Example I: Oil reservoir drainage across lease boundary
if 0, and 0, have opposite signs and:
QL,= Q, I Qdb,W
- Qdati>O,> 1
if 0, and 0, have the same sign.
(13)
In this case, the well is situated next to a lease boundary as shown in Fig. 8. The owner of the lease adjacent
E.S. Carlson, P. W. Johnson /Journal of Petroleum Science and Engineering 12 (199s) 269-276
213
3.2. Example 2: Bounded lease
I
II
Adjacent Lease
Producing Well
III
I
0
IV b
t
1320
Ii
6604
Fig. 8. How much oil flows across lease boundary 110 days of production (see Example l)?
I during the first
to the well wants to know how much oil has crossed the lease boundary after 110 days of production. The parameters used for the problem and the results are presented in the first column of Table 1. The perpendicular distance to the flow boundary is r,,,, = 660 ft, giving a dimensionless producing time of fDb= 14.2 1. The angle 0, = 45” and the angle 0, = - 45”, for a net total angle of 90”. From Fig. 4 we read that 23% of the flow is coming across the boundary after 110 days, and from Fig. 7 we read that 18.4% of the cumulative production has come across lease Boundary I. Table I Parameter magnitudes
This case uses parameters identical to those for Example 1 above. This time, however, all four of the rectangular bounds will be considered and the question is, “How much oil has been produced off the adjacent lease after 110 days?” In this case, a mass balance must first be calculated for the region. As shown in Fig. 2, the lease boundaries have been labelled as Segments I-IV. Flow across Segment I takes oil off the lease, while flow across segments II, III and IV bring oil into the lease. The mass balance shows that the total drainage off the lease is:
AQ,,=
Qw - ( Qm + Qm + Qm 1
and the total flow rate from the lease is: A qiease= qbl - ( qh*l + %1n+ qbI”l where the subscripts I, II, etc., refer to the property at the boundary. The parameters for this problem and the results are presented in columns 1-4, Table 1. The dimensionless
for example problems Example
Example 2 Side II
Example 2 Side IV
Example 2 Side III
Example 3 640 acres
Example 3 120 acres
10 110 2 1.5OE-05
10 110 2
IO 110 2
1.50E- 05
1.50E- OS
0.15
0.15
0.15
10 110 2 ISOE-05 0.15
0.01 10950 0.015 5.OOE-04 0.04
0.01 10950 0.015 5.OOE-04 0.04
(LYW) (ft) (x1. Y,) (ft) (G Yd (fi) ReSUltS
(0.0) (660,660) (660, - 660)
(0,O) ( 660,660) ( 1980,660)
(030) (660, - 660) (1980, -660)
(0.0) (1980,660) (1980, 1660)
(0.0) (2640.2640) (2640, - 2640)
(0.0) (1320, 1320) (1320, - 1320)
(x0. Y”) (ft) rb,, (ft) toll rl (ft) r2 (ft)
(660.0) 660 14.21 660 660 45.00 - 45.00 0.115 0.115 0.230 0.092
(0,660) 660 14.21 660 1980 45.00 71.57 0.115 0.168 0.053 0.127 0.092
(0, -660) 660 14.21 660 1980 45.00 71.57 0.115 0.168 0.053 0.127 0.092
( 1980.0)
(2640,O) 2640 1.33 2640 2640 45.00 -45.00 0.048 0.048 0.096 0.022
( 1320.0)
1980 1.58 660 660 18.43 - 18.43 0.026 0.026 0.052 0.013
0.184
0.035
0.035
0.025
0.044
0.124
Parameters k (mD) r (days) m (CPS) ct (psi-l) f
Ql Q2 %r1 %r? %r :::
Qbl
1
1320 5.30 1320 1320 45.00 -45.00 0.098 0.098 0.196 0.062
214
ES. Carlson, P. W. Johnson/Journal
of Petroleum Science and Engineering 12 (1995) 269-276
producing time for sides II and IV is exactly the same as for side I (Example 1) since r,,, has the same value, and angle 19,= 45”, as before; but now 8, = 71.57”, hence the net cumulative production through side II is QbI,=Qbr (14.21, 71.57) -Qt,,( 14.21, 45), or in this case, reading from Fig. 7, Qbr, = Q,,iv = 0.035 or 3.5% of the total production. There is very little flow across boundary III. Since the boundary is far from the well, r,,,, is large ( 1980 ft) and as a result the dimensionless time is small, f,),,= 1.58. Likewise, 13, and 0, are smaller ( + 18.43 and - l&43”, respectively) and the cumulative flow across this side is just 2.5% of the total production. Finally, the net cumulative production off the lease is: A Q,ease = 0.184 - (0.035 + 0.025 + 0.035) = 0.089 or 8.9% of the total production from the lease in question.
the wells drain only 120 acres, the net cumulative production across the boundaries will be about 49.6% after 30 years. Thus, for 640-acre spacing, 50.4% of the production comes from the center 120 acres and 45.2% comes from the outer 520 acres. The lease will not be drained in 30 years at 640 acre spacing, in fact the drawdown has barely reached this boundary. 3.4. Summary and conclusions The method presented is strictly limited to infiniteacting systems. The method is useful for wells with extended infinite-acting periods, which means either the compressibility is very high or the permeability is very low. For larger times, production across a linear segment approaches the steady-state solution.
from the well has come 4. Nomenclature
3.3. Example 3: Tight-gas reservoir It is proposed that 640-acre spacing be used to drain a tight-gas reservoir, with recovery over thirty years. Is 640-acre spacing adequate? Fig. 9 illustrates the problem layout. The parameters used for the problem and the results are presented in the fifth column of Table 1. Note that the combination of a high total compressibility, a large value for r,,,, and low permeability more than offset the effects of the low gas viscosity and the long time, giving a dimensionless producing time of tDb= 1.33. At this low dimensionless time the cumulative production across one of the four lease boundaries is 4.4% of the total production after 30 years, therefore about 17.6% of the production has moved across the boundaries. If the interwell distance is reduced by half SO that
k m 47 Cl h qbr ‘?h 4w
2
Qt./ rhase
rl 12
b
2640’4
bh
t II
Fig. 9. Will a tight gas well producing from the center of a 640 acre reservoir block drain the block in 30 years (see Example 2)?
4 “1
Permeability (mD) Slope of the flow boundary Oil formation volume factor (reservoir bbls/ stock tank bbl) Total compressibility (psi ’ ) Reservoir thickness (ft) Dimensionless flow rate Flow rate across the boundary (bbls/day) Flow rate from the well (bbls/day) Dimensionless cumulative production Cumulative production across the boundary (bbls) Cumulative production from the well (bbls) Perpendicular distance from the well to the flow boundary (ft) Distance from (x,,, yo> to (xi, yI) (ft) Distance from (x,, y,) to (x2, y2) ( ft) Dimensionless time Time (days) Radial flow velocity Component of the flow velocity perpendicular to the boundary x-coordinate of the producing well (ft) x-coordinate of the perpendicular intersection of a line from the well to the flow boundary (ft). x-coordinate (ft)
of end one of the flow boundary
ES. Carlson. P. W. Johnson /Journal
x2
YW Y”
YI Y2
4
ii,
Q2
of Petroleum Science and Engineering 12 (1995) 26%276
x-coordinate of end two of the flow boundary (ft. y-coordinate of the producing well (ft) y-coordinate of the perpendicular intersection of a line from the well to the flow boundary (ft) y-coordinate of end one of the flow boundary (ft) y-coordinate of end two of the flow boundary (ft) Porosity Viscosity (cps) Angle between rb,,, and r, Angle between rb,,, and r2
215
The radial velocity, uI, can be written in terms of pressure gradient as: k 8~ p ar
k
ur=---=----
apau
p auar
(A3)
The parameter u (r,t) = ( C&U,?) / [ 4( 0.006329kt) 1, where t is the fixed time in days, and the other parameters have standard oil field units. With this definition for u:
at.4 2~ -_=-_=2 ar r
ucos (e) (A4) rbme
where the relation r = rbaC/cOs( 0) was substituted. The above reduces the flow integral to:
(A51 Appendix 1
When the reservoir is infinite acting, and the flow rate from the single well is constant, then:
1.1.1. Derivation ofjow across a bouna’ary
(A6)
Hurst (1975, , 1984) presented several studies of flow across lease boundaries in steady-state and pseudo steady-state systems. He alludes to the velocity in a transient flow system, and to the component of velocity normal to the lease line (Hurst, 1984). While the velocity component is critical to the problem, it must be integrated over the length of the linear segment to determine the overall flow behavior. Flow across the boundary, qb, at any fixed time can be given by the integral:
ap 1 e-U -=-mau 2 u
(A7)
where m = ( qwpB,) / (27&/z). The above leads to: 62
(‘48) When Eq. A8 is simplified, and it is noted that u = u,/ cos* ( 6) = u ( rb,,,, t) /cos2( 0) , it reduces to:
Y2 B&b = h uxdy I vi
and:
(Al) (A9)
where h is the formation height, and Fig. 1A shows the remaining notation. By noting that v, = - V~COS( 0) (when v, is positive, it acts in the negative x direction for the coordinate system shown), y = &&an ( 13))and that dy= (rb,,/cos28)d& the integral above can be converted to polar coordinates. This leads to:
(AZ)
Eq. A9 gives the flow rate across the boundary in terms of reservoir barrels. The flow rate in STB can be obtained by dividing this reservoir rate by B,. The cumulative flow across the boundary is just Eq. A9 integrated with respect to time:
(AlO)
ES. Carbon, P. W. Johnson /Journal of Petroleum Science and Engineering 12 (1995) 269-276
216
A more convenient form can be derived, however, by using the dimensionless parameter 1/ur, = tDband using the relations for changes of variables in integrals. If tDh= da, then dtldt,,, = a, and:
R)b(l)
Qdtd -=-4J
Hz
II
where actual flow across the segment can be obtained by multiplying the relative rate by the well rate. A relative cumulative flow may be defined as the cumulative flow across the linear segment divided by the cumulative flow from the well. Eq. All may be combined with definition of tn,, to give:
exp[ ( - l/(t,,
cos2@]dOdtD,
(All)
2%. t
mh(r)@
rntl(O)HI
exp[ - l/(t,,cos*8)]dfMt,, II 0 01
or:
Qb(bb>
1 a
=%a
or:
2rr
Qbr(tDb) m?(l)62
II
=y=-& w
exp[ ( - l/(tn,,)
(A15)
cos*@]dfkh,,,,
(A12)
il
0 01
Db
roh(J)bz
exp[ - ll(r,,
cos*8)]d<,,
(A16)
0 HI
where:
lw&i.se a = ‘4(0.006329k)
(Af3)
days. A relative oil flow rate across the linear segment, qbr, can be defined as the flow rate across the segment divided by the well rate. Using this definition and Eq. A9 gives:
=hT
exp[ - l/(t,cos*8)]dO I HI
(A14)
where the actual cumulative flow across the segment can be obtained by multiplying the relative cumulative production by the well cumulative production. Using the real-gas pseudo pressure, gas reservoirs can be shown to behave approximately the same.
References Hurst, W., 1975. How to figure oil drainage across a lease line. Oil Gas J., 73(3): 66-68. Hurst, W., 1984. How to calculate oil and gas drainage across lease lines. Oil Gas J., 82(22): 110-l 12. Johnson, P., 1988. The relationship between radius of drainage and cumulative production. SPE Form. Eval., March, 3 ( 1) : 267-270.
Journal of Petroleum Science and Engineering 12 ( 1995) 269-276
Determination of transient drainage across lease boundaries Eric S. Carlson, Philip W. Johnson Department of Mineral Engineering, The University of Alabama, Tuscaloosa, AL 35487-0207, USA
Received 10 March 1994; accepted 23 August 1994
Abstract A technique which simplifies the determination of how much oil or gas flows across a linear segment during the transient flow period of the reservoir is presented. The technique is based on two definite integrals which result from application of the linesource solution of flow in an infinite-acting reservoir combined with the fundamental definition of flow across a plane. The theory, the outline of a methodology, and several examples are provided. The examples, one which determines the flow across a lease line for transient flow in an oil reservoir, one using the method for a bounded lease, and one using the method to assess the well spacing in a tight-gas reservoir, show that the technique can be applied to a variety of practical problems. Graphical solutions to the definite integrals are given to facilitate application of the method.
1. Introduction
I. I. Background and assumptions
The scenario is quite common. A wildcat well drilled adjacent to a lease boundary has just discovered a new reservoir. The well begins sustained production, and the adjacent lease holder asks the inevitable question, “How much of that production is coming from my property?” The question is easy enough to pose, but the answer has been far more complicated to determine. The theory for transient flow across a linear segment is summarized in this paper and a simple method for determining the instantaneous rate and the cumulative flow across the segment for any time during the infiniteacting period is presented. The concept of transient flow across a linear segment has many important applications, besides the wildcat problem described above. The methods presented here can be very useful for drainage evaluations in tight gas systems, in addition to waterfloods in saturated reservoirs. The examples shown illustrate the versatility of the method.
Hurst (1975, 1984) presented several studies regarding flow across lease boundaries in steady-state and pseudo steady-state systems. In one of his papers, he mentions the velocity in a transient flow system but does not develop the transient relations. Johnson ( 1988) presented a method for transient flow of radial systems, but the method only applied to circular boundaries. Details of the theory of transient flow across linear segments are presented in the Appendix. In this section, only the assumptions used for the theory and the relations required for its application are presented. The assumptions are the same as those used for infinite-acting flow in well test analysis. A homogeneous reservoir with uniform permeability, thickness and porosity is assumed. The reservoir is infinite in extent, and the well flows at a constant rate. If the reservoir is not infinite acting, the methods of Hurst should be used. Fig. 1 illustrates the basic system ideas and nomenclature. A well is arbitrarily located at the point (x,,
0920-4105/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SsDIO920-4105(94)00049-2
ES. Carlson, P. W. Johnson /Journal of Petroleum Science und Engineering 12 (1995) 269-276
270
for the cases where m and 1lm are not equal to 0. When y2=yl,m=0,and (x,,yo) =(x,,yI). Whenx,=x,, 11 m=Oand(x,,y,)=(x,,y,).
The distance, r,,,,, between the well and (x,,, y,,) is: Boundary Line rhase
=
J(Yw-Yo)2+(G+~“)2
(4)
The distance, r,, between (xi, y, ) and (x,, y,,) is: The Well at point (h, Yw)
B
x
TI =&y,
I
“r
r = r,,,
\
7\ 8 \\
Y t
l--L I
/ cos (e)
0, = arctan L i rha 3
\
I
X
If----rb,s, I
\\
a
_I
I
y,) of a Cartesian grid. A straight line segment away from the well has endpoints of (x,, yi) and (x2, y?). A perpendicular from the well intersects the segment at a point denoted (x0, y,). Given that the segment has a slope: Yz--Y1
(1)
x2-x1
can be given by:
xo~Yw-YI+~l+u~mh m+(llm)
(2)
x,-xl+my,+ m+(llm)
(llm)y,
t
‘lh -
0.02532 ft2 CP mD psi day
kt +pc,ri&
(7)
where k is the permeability, t is the time, 4 is the porosity, and c, is the total compressibility. It is shown in the appendix that the rate at any dimensionless time can be given by: qhH=qwqhr(bh8)
(8)
for flow across the segment between the angles 0 and 80r - 19.Figs. 2,3 and 4 show qhr ( fi)b, 0) as a function Of tot, for VariOUSScales Of fbh. To CValUate flOW aCrOSS the segment between 8, and e2, qb, the above equation is applied to give: qh = qh#2 + qh@l
(9)
if 13,and 0, have opposite signs (that is, the point (x0, y,) lies between (nz, y2) and (xi, yI> on the segment), and:
and y0 can be given by: yo=
(6)
The angle 0, is defined comparably for (x2, y2). The signs of the angles are arbitrary, but should be consistent. That is, if (..x~,y2) and (xi, yi) are on opposite sides of (x,, y,) , the angles should have opposite signs; if they are on the same side, they should have the same sign. Transient flow across the linear segment depends on the dimensionless time, f,),,, defined by:
Fig. 1. (A) The coordinate system and the flow vectors used in deriving the equations for transient flow across boundaries. (B) Schematic of the arrangement of the producing well and the boundary, including a definition of terms.
x,
(5)
\
I
m=-
(Xl +x”j2
The distance, r,, between (x2, y2) and (x,,, yo) can be obtained in a similar manner. The angle 0i is the angle between the line which goes through the well and (?I,,, y,) and the line that goes through the well and (x,, y,), as shown in Fig. 1. The magnitude of the angle can be given by:
“x
I
-Yo)*+
(3)
(10)
ES. Carlson, P. W. Johnson /Journal
of Petroleum Science and Engineering 12 (1995) 269-276
if 8, and 0, have the same sign. For cumulative flow:
Qw= QwQd tm ‘3
(11)
where Q,, is the cumulative flow across the segment between 0 and 0 or - 0, Q, is the cumulative production at the well, and Qbr (tDb, 0) is presented in Figs. $6 and 7. Just as in the rate figures, the only difference
271
between these cumulative production curves is the dimensionless time scale and the cumulative production across the entire segment can be given by:
QL,= Qm + Qbs,
(12)
0 25 e(dQS.)
1
e (deg.) ,65-90 65
0
1.0
0.5
0.0
1.5
2.0
10
Fig. 4. Dimensionless
20
production
30
t Db
40
rate at late dimensionless
50
times.
t,
Fig. 2. Dimensionless
production
rate at early dimensionless
e (deg.1
times.
-65-90
0.1.
.
.
i.
,.
.i
. . -ew0.l
0.12
0.10
2
d
0.05
j
0.08
0.04
0.02
0.0
0.5
1.0
1.5
2.0
tc+
Fig. 3. Dimensionless times.
production
rate at intermediate
dimensionless
Fig. 5. Dimensionless sionfess times.
cumulative
production
rate at early dimen-
E.S. Carlson, P. W. Johnson /Journal of Petroleum Science and Engineering I2 (1995) 269-276
272
Q (deg 1
60X
1
55
Examination of the late-time curves, Figs. 4 and 7, shows that the flow rates and cumulative flow rates across the linear segments approach the value of O/ (27~). This flow rate is the steady-state solution of the problem, which implies (correctly) that the transient behavior eventually approaches the steady behavior. These equations and assumptions are the only requirements for effective application of the method to be presented below.
2. Methodology
(2)
Fig. 6. Dimensionless cumulative production rate at intermediate dimensionless times. 0.20
(3) (4) (5) (6)
-
9 (deg.) 7
-% 0.15-
(7) (8)
a_
Look up qbr (tDb, 01) 9 Qbr( fDb, ‘% > 3 qbr ( fDb, 0,) 1 and (& (t,,, 6,) on the appropriate charts. Calculate the flow across the segment using Eqs. (9) 9 and 12 if 0, and 0, have opposite signs; otherwise, use Eqs. 10 and 13. ( 10) If more than 1 time is desired, go back to Step 7.
.-s 1 t .g
O.lO-
;ii z ; P ‘S s $
Determine the coordinates of the well and the endpoints of the segment (lease line, spacing boundary, gas cap, etc.) in Cartesian coordinates. Determine the point on the segment closest to the well (use Eqs. 2 and 3). Calculate r,,,, using Eq. 4. Calculate r, and r, using Eq. 5. Calculate the magnitude of Oi and 0, using Eq. 6. Determine the relative signs of 0, and 0,. If x, > X, > x, (that is, X, is on the interval between x1 and xz) and y2 >y” > y,, then OXand 0, have opposite signs. Otherwise, they have the same signs. Calculate toi, using Eq. 7.
0.05-
3. Sample applications
0
10
20
30 t Db
40
50
Fig. 7. Dimensionless cumulative production rate at late dimensionless times.
Three sample applications are presented to illustrate the procedure. The first example deals with the scenario posed in the introduction. The second shows how to use the method in bounded regions. The final example deals with drainage in a tight gas system. 3.1. Example I: Oil reservoir drainage across lease boundary
if 0, and 0, have opposite signs and:
QL,= Q, I Qdb,W
- Qdati>O,> 1
if 0, and 0, have the same sign.
(13)
In this case, the well is situated next to a lease boundary as shown in Fig. 8. The owner of the lease adjacent
E.S. Carlson, P. W. Johnson /Journal of Petroleum Science and Engineering 12 (199s) 269-276
213
3.2. Example 2: Bounded lease
I
II
Adjacent Lease
Producing Well
III
I
0
IV b
t
1320
Ii
6604
Fig. 8. How much oil flows across lease boundary 110 days of production (see Example l)?
I during the first
to the well wants to know how much oil has crossed the lease boundary after 110 days of production. The parameters used for the problem and the results are presented in the first column of Table 1. The perpendicular distance to the flow boundary is r,,,, = 660 ft, giving a dimensionless producing time of fDb= 14.2 1. The angle 0, = 45” and the angle 0, = - 45”, for a net total angle of 90”. From Fig. 4 we read that 23% of the flow is coming across the boundary after 110 days, and from Fig. 7 we read that 18.4% of the cumulative production has come across lease Boundary I. Table I Parameter magnitudes
This case uses parameters identical to those for Example 1 above. This time, however, all four of the rectangular bounds will be considered and the question is, “How much oil has been produced off the adjacent lease after 110 days?” In this case, a mass balance must first be calculated for the region. As shown in Fig. 2, the lease boundaries have been labelled as Segments I-IV. Flow across Segment I takes oil off the lease, while flow across segments II, III and IV bring oil into the lease. The mass balance shows that the total drainage off the lease is:
AQ,,=
Qw - ( Qm + Qm + Qm 1
and the total flow rate from the lease is: A qiease= qbl - ( qh*l + %1n+ qbI”l where the subscripts I, II, etc., refer to the property at the boundary. The parameters for this problem and the results are presented in columns 1-4, Table 1. The dimensionless
for example problems Example
Example 2 Side II
Example 2 Side IV
Example 2 Side III
Example 3 640 acres
Example 3 120 acres
10 110 2 1.5OE-05
10 110 2
IO 110 2
1.50E- 05
1.50E- OS
0.15
0.15
0.15
10 110 2 ISOE-05 0.15
0.01 10950 0.015 5.OOE-04 0.04
0.01 10950 0.015 5.OOE-04 0.04
(LYW) (ft) (x1. Y,) (ft) (G Yd (fi) ReSUltS
(0.0) (660,660) (660, - 660)
(0,O) ( 660,660) ( 1980,660)
(030) (660, - 660) (1980, -660)
(0.0) (1980,660) (1980, 1660)
(0.0) (2640.2640) (2640, - 2640)
(0.0) (1320, 1320) (1320, - 1320)
(x0. Y”) (ft) rb,, (ft) toll rl (ft) r2 (ft)
(660.0) 660 14.21 660 660 45.00 - 45.00 0.115 0.115 0.230 0.092
(0,660) 660 14.21 660 1980 45.00 71.57 0.115 0.168 0.053 0.127 0.092
(0, -660) 660 14.21 660 1980 45.00 71.57 0.115 0.168 0.053 0.127 0.092
( 1980.0)
(2640,O) 2640 1.33 2640 2640 45.00 -45.00 0.048 0.048 0.096 0.022
( 1320.0)
1980 1.58 660 660 18.43 - 18.43 0.026 0.026 0.052 0.013
0.184
0.035
0.035
0.025
0.044
0.124
Parameters k (mD) r (days) m (CPS) ct (psi-l) f
Ql Q2 %r1 %r? %r :::
Qbl
1
1320 5.30 1320 1320 45.00 -45.00 0.098 0.098 0.196 0.062
214
ES. Carlson, P. W. Johnson/Journal
of Petroleum Science and Engineering 12 (1995) 269-276
producing time for sides II and IV is exactly the same as for side I (Example 1) since r,,, has the same value, and angle 19,= 45”, as before; but now 8, = 71.57”, hence the net cumulative production through side II is QbI,=Qbr (14.21, 71.57) -Qt,,( 14.21, 45), or in this case, reading from Fig. 7, Qbr, = Q,,iv = 0.035 or 3.5% of the total production. There is very little flow across boundary III. Since the boundary is far from the well, r,,,, is large ( 1980 ft) and as a result the dimensionless time is small, f,),,= 1.58. Likewise, 13, and 0, are smaller ( + 18.43 and - l&43”, respectively) and the cumulative flow across this side is just 2.5% of the total production. Finally, the net cumulative production off the lease is: A Q,ease = 0.184 - (0.035 + 0.025 + 0.035) = 0.089 or 8.9% of the total production from the lease in question.
the wells drain only 120 acres, the net cumulative production across the boundaries will be about 49.6% after 30 years. Thus, for 640-acre spacing, 50.4% of the production comes from the center 120 acres and 45.2% comes from the outer 520 acres. The lease will not be drained in 30 years at 640 acre spacing, in fact the drawdown has barely reached this boundary. 3.4. Summary and conclusions The method presented is strictly limited to infiniteacting systems. The method is useful for wells with extended infinite-acting periods, which means either the compressibility is very high or the permeability is very low. For larger times, production across a linear segment approaches the steady-state solution.
from the well has come 4. Nomenclature
3.3. Example 3: Tight-gas reservoir It is proposed that 640-acre spacing be used to drain a tight-gas reservoir, with recovery over thirty years. Is 640-acre spacing adequate? Fig. 9 illustrates the problem layout. The parameters used for the problem and the results are presented in the fifth column of Table 1. Note that the combination of a high total compressibility, a large value for r,,,, and low permeability more than offset the effects of the low gas viscosity and the long time, giving a dimensionless producing time of tDb= 1.33. At this low dimensionless time the cumulative production across one of the four lease boundaries is 4.4% of the total production after 30 years, therefore about 17.6% of the production has moved across the boundaries. If the interwell distance is reduced by half SO that
k m 47 Cl h qbr ‘?h 4w
2
Qt./ rhase
rl 12
b
2640’4
bh
t II
Fig. 9. Will a tight gas well producing from the center of a 640 acre reservoir block drain the block in 30 years (see Example 2)?
4 “1
Permeability (mD) Slope of the flow boundary Oil formation volume factor (reservoir bbls/ stock tank bbl) Total compressibility (psi ’ ) Reservoir thickness (ft) Dimensionless flow rate Flow rate across the boundary (bbls/day) Flow rate from the well (bbls/day) Dimensionless cumulative production Cumulative production across the boundary (bbls) Cumulative production from the well (bbls) Perpendicular distance from the well to the flow boundary (ft) Distance from (x,,, yo> to (xi, yI) (ft) Distance from (x,, y,) to (x2, y2) ( ft) Dimensionless time Time (days) Radial flow velocity Component of the flow velocity perpendicular to the boundary x-coordinate of the producing well (ft) x-coordinate of the perpendicular intersection of a line from the well to the flow boundary (ft). x-coordinate (ft)
of end one of the flow boundary
ES. Carlson. P. W. Johnson /Journal
x2
YW Y”
YI Y2
4
ii,
Q2
of Petroleum Science and Engineering 12 (1995) 26%276
x-coordinate of end two of the flow boundary (ft. y-coordinate of the producing well (ft) y-coordinate of the perpendicular intersection of a line from the well to the flow boundary (ft) y-coordinate of end one of the flow boundary (ft) y-coordinate of end two of the flow boundary (ft) Porosity Viscosity (cps) Angle between rb,,, and r, Angle between rb,,, and r2
215
The radial velocity, uI, can be written in terms of pressure gradient as: k 8~ p ar
k
ur=---=----
apau
p auar
(A3)
The parameter u (r,t) = ( C&U,?) / [ 4( 0.006329kt) 1, where t is the fixed time in days, and the other parameters have standard oil field units. With this definition for u:
at.4 2~ -_=-_=2 ar r
ucos (e) (A4) rbme
where the relation r = rbaC/cOs( 0) was substituted. The above reduces the flow integral to:
(A51 Appendix 1
When the reservoir is infinite acting, and the flow rate from the single well is constant, then:
1.1.1. Derivation ofjow across a bouna’ary
(A6)
Hurst (1975, , 1984) presented several studies of flow across lease boundaries in steady-state and pseudo steady-state systems. He alludes to the velocity in a transient flow system, and to the component of velocity normal to the lease line (Hurst, 1984). While the velocity component is critical to the problem, it must be integrated over the length of the linear segment to determine the overall flow behavior. Flow across the boundary, qb, at any fixed time can be given by the integral:
ap 1 e-U -=-mau 2 u
(A7)
where m = ( qwpB,) / (27&/z). The above leads to: 62
(‘48) When Eq. A8 is simplified, and it is noted that u = u,/ cos* ( 6) = u ( rb,,,, t) /cos2( 0) , it reduces to:
Y2 B&b = h uxdy I vi
and:
(Al) (A9)
where h is the formation height, and Fig. 1A shows the remaining notation. By noting that v, = - V~COS( 0) (when v, is positive, it acts in the negative x direction for the coordinate system shown), y = &&an ( 13))and that dy= (rb,,/cos28)d& the integral above can be converted to polar coordinates. This leads to:
(AZ)
Eq. A9 gives the flow rate across the boundary in terms of reservoir barrels. The flow rate in STB can be obtained by dividing this reservoir rate by B,. The cumulative flow across the boundary is just Eq. A9 integrated with respect to time:
(AlO)
ES. Carbon, P. W. Johnson /Journal of Petroleum Science and Engineering 12 (1995) 269-276
216
A more convenient form can be derived, however, by using the dimensionless parameter 1/ur, = tDband using the relations for changes of variables in integrals. If tDh= da, then dtldt,,, = a, and:
R)b(l)
Qdtd -=-4J
Hz
II
where actual flow across the segment can be obtained by multiplying the relative rate by the well rate. A relative cumulative flow may be defined as the cumulative flow across the linear segment divided by the cumulative flow from the well. Eq. All may be combined with definition of tn,, to give:
exp[ ( - l/(t,,
cos2@]dOdtD,
(All)
2%. t
mh(r)@
rntl(O)HI
exp[ - l/(t,,cos*8)]dfMt,, II 0 01
or:
Qb(bb>
1 a
=%a
or:
2rr
Qbr(tDb) m?(l)62
II
=y=-& w
exp[ ( - l/(tn,,)
(A15)
cos*@]dfkh,,,,
(A12)
il
0 01
Db
roh(J)bz
exp[ - ll(r,,
cos*8)]d<,,
(A16)
0 HI
where:
lw&i.se a = ‘4(0.006329k)
(Af3)
days. A relative oil flow rate across the linear segment, qbr, can be defined as the flow rate across the segment divided by the well rate. Using this definition and Eq. A9 gives:
=hT
exp[ - l/(t,cos*8)]dO I HI
(A14)
where the actual cumulative flow across the segment can be obtained by multiplying the relative cumulative production by the well cumulative production. Using the real-gas pseudo pressure, gas reservoirs can be shown to behave approximately the same.
References Hurst, W., 1975. How to figure oil drainage across a lease line. Oil Gas J., 73(3): 66-68. Hurst, W., 1984. How to calculate oil and gas drainage across lease lines. Oil Gas J., 82(22): 110-l 12. Johnson, P., 1988. The relationship between radius of drainage and cumulative production. SPE Form. Eval., March, 3 ( 1) : 267-270.