JQCV123
2. Dictat (de 100 paraules aproximadament) ( 20 punts) Correcció
Some of the most important reservoir boundary models comprise multiple linear boundaries. These reservoir models include the infinite channel reservoir model, the intersecting sealing fault model, and the rectangular res-ervoir model. With the exception of the intersecting sealing fault model where the faults intersect at an arbitrary angle, solutions to the diffusivity equation for these models may be obtained using the method of images.
7.5.1 Infinite Channel Reservoir. Many reservoir bodies are long and narrow and may be modeled by a well between two parallel linear no-flow boundaries, often called an infinite channel reservoir model, as shown in Fig. 7.13 (Nutakki and Mattar 1982; Ehlig-Economides and Economides 1985).
Many different geologic processes can create narrow elongated reservoirs that may be modeled by the well in a channel model. Examples include stratigraphic traps, such as fluvial channel deposits, from which the model derives its name, point-bar deposits and offshore sand bars, and structural features such as closely spaced, parallel sealing faults.
Levorsen (1967) gives two especially interesting examples of geologic features that might give rise to linear flow. The South Ceres pool in Oklahoma produces from the Red Fork sand, which appears to be an offshore sand bar 10.5 miles long with an average thickness of 25 ft and an average width of 1,000 ft (pp. 300,301). The Scipio-Albion field in Michigan produces from a dolomitized zone in the Trenton limestone; dolomitization seems to be controlled by fracturing and minor faulting over a deeper fault (pp. 123,124). The trend averages 3,500 ft wide and is more than 25 miles long.
Fig. 7.14 shows the pressure response for a well centered in an infinite channel reservoir, while Fig. 7.15 shows the pressure response for wells at several different positions in the same reservoir.
L1
L2 W
Fig. 7.13—Well in an infinite-channel reservoir.
1E-01 1E+00 1E+01 1E+02 1E+03
IARF CLF
tD
1E+02 1E+03 1E+04 1E+05 1E+06 1E+07 1E+08
pD
Fig. 7.14—Pressure response for a well centered in an infinite-channel reservoir.
For a well centered in a channel, Fig. 7.14, IARF lasts until the sides of the channel are felt. CLF then begins following a very brief transition period. IARF ends at a time given by
t c w
k
c L
eIARF=256φµ t 2 =1 024, kφµ t 2, . . . (7.21) where w is the width of the channel and L is the distance from the well to either side of the channel. CLF begins at a time given by
t c w
k
c L
bCLF=350φµ t 2 =1 400, φµk t 2. . . . (7.22) As may be seen from Fig. 7.14, the transition from IARF to CLF is very short (slightly more than one-eighth log cycle) when the well is centered in the channel.
For a well off-center in a channel reservoir, Fig. 7.15, IARF lasts until the nearer side of the channel is seen.
If the well is much closer to one boundary than to the other, L w1 ≤ .0 1, HRF will then appear following a rela-tively long (1½ log cycles) transition. If 0 1. <L w1 <0 5. , HRF will not occur; instead, the derivative during the transition from IARF to CLF will deviate from that for IARF once the closer boundary begins to affect the pressure response. CLF will develop after the farther boundary begins to affect the pressure response. The time to end of IARF may be estimated by
t
where L1 is the distance to the closer boundary. The time to beginning of CLF may be estimated from
t
7.5.2 Intersecting No-Flow Boundaries. Two no-flow boundaries may intersect at an angle q, as in Fig. 7.16, creating a so-called “wedge” reservoir model (Van Poollen 1965; Prasad 1975). This model is especially useful in
0.5 L1/W = 0.05 0.1 0.25
tD
1E+02 1E+03 1E+04 1E+05 1E+06 1E+07 1E+08
1E-01
Fig. 7.15—Pressure response for a well at an arbitrary position in an infinite-channel reservoir.
areas with extensive faulting, when sealing faults provide the no-flow boundaries. Fig. 7.17 shows a portion of a structure map for a well in a reservoir bounded by intersecting sealing faults.
Fig. 7.18 shows the pressure response for a well centered between two no-flow boundaries intersecting at an angle of 60°. This models shows IARF until the boundaries are encountered, a very long transition period (more than 2 log cycles), followed by fractional radial flow (FRF). The derivative during FRF will be larger than the derivative during IARF, by a factor 360/q.
The smaller the angle between the boundaries, the longer the transition region, as shown in Fig. 7.19 showing the pressure response for a boundaries intersecting at 10°. For an angle this small, the transition from IARF to FRF is very long, exhibiting a period during which the derivative has a slope almost exactly ½, characteristic of linear
L1
L2
θ
L
Fig. 7.16—Well between intersecting no-flow boundaries.
Fig. 7.17—Well between intersecting sealing faults.
1E-01 1E+00 1E+01 1E+02 1E+03
360 xθ
IARF FRF
tD PD, tD*PD'
1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 1E+09
θ= 60º
Fig. 7.18—Pressure response for a well centered between intersecting no-flow boundaries.
flow. This flow period might be called pseudolinear flow (PLF); during PLF, the pressure transient is affected by both boundaries of the wedge, but the transient has not reached the apex of the wedge. Thus, the pressure response during PLF is the same as that for a well in a channel with sides that are slightly out of parallel. This implies that even non-ideal channels may exhibit a pressure response that can be analyzed using techniques developed for parallel no-flow boundaries.
In Fig. 7.20, the distance L1 from the well to the closer boundary, is one tenth the distance L from the well to the apex of the wedge. Not surprisingly, the pressure response shows IARF, followed by a transition into HRF, followed by a longer transition into FRF.
7.5.3 Rectangular Reservoir Model. One of the most frequently used (and one of the most frequently abused) bounded reservoir models is that of a well at an arbitrary location in a rectangular reservoir. Because of its flexibility in matching many different pressure responses, this model is often used even when there is no external information to justify its use.
Geologically, the rectangular reservoir model is most directly applicable to wells in fault blocks bounded by sealing faults. The rectangular reservoir model also finds application in fluvial channel, point bar, and offshore bar deposits where the ends of the reservoir affect the test response. Finally, the rectangular reservoir model is often
2
1
PLF FRF
IARF 1E-01
1E+00 1E+01 1E+02 1E+03
tD PD, tD*PD'
1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 1E+09
θ= 10º
Fig. 7.19—Pressure response for a well between no-flow boundaries intersecting at a small angle.
L1/L= 0.1
HRF FRF
IARF 1E-01
1E+00 1E+01 1E+02 1E+03
tD PD, tD*PD'
1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 1E+09
360 xθ
θ= 60º
Fig. 7.20—Pressure response for a well between intersecting no-flow boundaries, well much closer to one boundary than to the other.
used for analyzing tests on development wells drilled in a regular pattern. In the latter application, the model will be rigorously justified only if all offset wells are produced in synchronization with the test well.
Figs. 7.22 through 7.26 show just a few of the wide range of pressure responses possible with the rectangular reservoir model with no-flow boundaries.
Ly1
Ly2
W Lx2 Lx1
L
Fig. 7.21—Well in a closed rectangular reservoir.
IARF PSSF
1E+03
1E+02
1E+01
1E+00
1E-01
tD
1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 1E+09
pD
Fig. 7.22—Drawdown pressure response for a well centered in a square reservoir.
IARF CLF PSSF
1E+03
1E+02
1E+01
1E+00
1E-01
tD
1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 1E+09
pD
Fig. 7.23—Drawdown pressure response for a well centered in a 10 × 1 closed rectangular reservoir.
IARF QRF PSSF 1E+03
1E+02
1E+01
1E+00
1E-01
tD
1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 1E+09
pD
Fig. 7.24—Drawdown pressure response for a well in one corner of a square reservoir.
The drawdown pressure response for a well in the center of a square, Fig. 7.22, is virtually identical to that for a well in a closed circle, Fig. 7.9, having the same drainage area, exhibiting first IARF, followed by pseudosteady-state flow after a brief transition period.
Fig. 7.23 shows the drawdown pressure response for a well centered in a 10 × 1 rectangle. The pressure response shows IARF until the pressure transient reaches the nearer pair of sides, then there is a short transition into CLF, which lasts until the pressure transient reaches the farther pair of sides and there is another short transition into pseudosteady-state flow.
Fig. 7.24 shows the drawdown pressure response for a well in a corner of a square reservoir. The IARF regime lasts until the pressure transient reaches the nearer pair of intersecting boundaries. There is a long transition into fractional or quarter radial flow within the 90° angle formed by the adjacent sides. After the pressure transient reaches the opposite sides of the reservoir, there is a brief transition period, followed by pseudosteady-state flow.
Fig. 7.25 shows the pressure response for a well in a long, narrow reservoir closed at one end, sometimes called a semi-infinite channel or a U-shaped reservoir. This reservoir model exhibits two linear flow regimes, the first linear flow in both directions along the channel (LF1) before the pressure response reaches the end of the channel.
LF1 LF2
IARF 1E+03
1E+02
1E+01
1E+00
1E-01
tD
1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 1E+09
pD
Fig. 7.25—Pressure response for a well in a semi-infinite channel with a no-flow boundary at the end of the channel.
The first linear flow period is followed by a long transition into a second linear flow period, LF2, in one direction toward the open end of the channel. The derivative follows a half-slope line for both linear flow periods that for the second linear flow period is shifted upward from that for the first flow period by a factor of two.
Fig. 7.26 shows the drawdown pressure response for a well off-center in a long, narrow reservoir. This example is constructed specifically to show the flexibility of the rectangular reservoir model and is not intended to represent a pressure response expected to be observed in practice. The reservoir geometry and the well placement are chosen so that each boundary is 10× further from the well than the previous boundary; thus, the effects of each boundary on the pressure response can be identified separately. This model shows IARF, followed by HRF after the pressure transient reaches the closest boundary. After the transient reaches the second boundary, quarter radial flow appears. When the transient reaches the third boundary, the flow pattern becomes linear as the transient moves toward the remaining boundary. Finally, pseudosteady-state flow begins once the transient reaches the farthest boundary.
HRF PSSF
IARF QRF CLF
1E+05
1E+04
1E+03
1E+02
1E+01
1E+00
1E-01
tD
1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 1E+09
pD
Fig. 7.26—Drawdown pressure response for a well located so each boundary is 10 times further away than the next closer boundary.
Linear stabilization Slope = -½ CLF
IARF 1E+03
1E+02
1E+01
1E+00
1E-01
tD
1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 1E+09
pD
Fig. 7.27—Drawdown pressure response for a well in a semi-infinite channel with a single constant-pressure boundary at the end of the channel.
Fig. 7.27 shows the pressure response for a well in a semi-infinite channel reservoir with a constant-pressure boundary at one end. As with the pressure response for the semi-infinite channel in Fig. 7.25, this model shows IARF, followed by CLF. However, after the pressure transient reaches the constant-pressure boundary, the pressure asymptotically approaches a constant value while the derivative decreases with a negative half-slope.
This behavior has the same slope as spherical flow but is caused by pressure stabilization in an infinite linear system, or simply linear stabilization.
Fig. 7.28 shows the pressure response for a well centered in a 10 × 1 rectangular reservoir with a constant-pressure boundary at one end. In contrast to the semi-infinite channel case in Fig. 7.27, after the constant-pressure transient reaches the constant-pressure boundary, the pressure stabilizes very quickly while the derivative decreases almost exponentially. Interestingly, the derivative behaves very similar to that for the well in a reservoir with a circular constant-pressure boundary (Fig. 7.12), even though the constant-pressure boundary comprises less than 5% of the total length of the perimeter of the rectangle.