Assuming no water or gas injection, the linear form of the MBE as expressed by Equation 11-25 can be written as:
F =N [Eo+m Eg+Ef,w] +We (11-32)
Several terms in the above relationship may disappear when imposing the conditions associated with the assumed reservoir-driving mechanism.
For a volumetric and undersaturated reservoir, the conditions associated with a driving mechanism are:
• We=0, since the reservoir is volumetric
• m =0, since the reservoir is undersaturated
• Rs=Rsi=Rp, since all produced gas is dissolved in the oil Applying the above conditions on Equation 11-32 gives:
F =N (Eo+Ef,w) (11-33)
or
where N =initial oil-in-place, STB
F =NpBo+WpBw (11-35)
Eo =Bo−Boi (11-36)
Δp=pi−–p
r
pi=initial reservoir pressure –p
r=volumetric average reservoir pressure
When a new field is discovered, one of the first tasks of the reservoir engineer is to determine if the reservoir can be classified as a volumetric reservoir, i.e., We=0. The classical approach of addressing this problem is to assemble all the necessary data (i.e., production, pressure, and PVT) that are required to evaluate the right-hand side of Equation 11-36. The term F/(Eo+Ef,w) for each pressure and time observation is plotted versus cumulative production Npor time, as shown in Figure 11-16. Dake (1994) suggests that such a plot can assume two various shapes, which are:
• All the calculated points of F/(Eo+Ef,w) lie on a horizontal straight line (see Line A in Figure 11-16). Line A in the plot implies that the reser- voir can be classified as a volumetric reservoir. This defines a purely depletion-drive reservoir whose energy derives solely from the expan- sion of the rock, connate-water, and the oil. Furthermore, the ordinate value of the plateau determines the initial oil-in-place N.
• Alternately, the calculated values of the term F/(Eo+Ef,w) rise, as illus- trated by the curves B and C, indicating that the reservoir has been
E B c S c
S p
f w oi
w w f
wi
, = +
−
⎡
⎣⎢
⎤
⎦⎥
1 Δ (11-37)
N F
Eo Ef w
= + ,
(11-34)
energized by water influx, abnormal pore compaction, or a combination of these two. Curve C in Figure 11-16 might be for a strong water-drive field in which the aquifer is displacing an infinite acting behavior, whereas B represents an aquifer whose outer boundary has been felt and the aquifer is depleting in unison with the reservoir itself. The downward trend in points on curve B as time progresses denotes the diminishing degree of energizing by the aquifer. Dake (1994) points out that in water-drive reservoirs, the shape of the curve, i.e., F/(Eo+Ef,w) vs. time, is highly rate dependent. For instance, if the reservoir is pro- ducing at a higher rate than the water-influx rate, the calculated values of F/(Eo+Ef,w) will dip downward revealing a lack of energizing by the aquifer, whereas, if the rate is decreased, the reverse happens and the points are elevated.
Similarly, Equation 11-33 could be used to verify the characteristic of the reservoir-driving mechanism and to determine the initial oil-in- place. A plot of the underground withdrawal F versus the expansion term (Eo+Ef,w) should result in a straight line going through the origin with N being the slope. It should be noted that the origin is a “must”
point; thus, one has a fixed point to guide the straight-line plot (as shown in Figure 11-17).
Figure 11-16.Classification of the reservoir.
This interpretation technique is useful in that, if the linear relationship is expected for the reservoir and yet the actual plot turns out to be non-linear, then this deviation can itself be diagnostic in determining the actual drive mechanisms in the reservoir.
A linear plot of the underground withdrawal F versus (Eo+Ef,w) indi- cates that the field is producing under volumetric performance, i.e., no water influx, and strictly by pressure depletion and fluid expansion. On the other hand, a nonlinear plot indicates that the reservoir should be characterized as a water-drive reservoir.
Example 11-3
The Virginia Hills Beaverhill Lake field is a volumetric undersaturated reservoir. Volumetric calculations indicate the reservoir contains 270.6 MMSTB of oil initially in place. The initial reservoir pressure is 3685 psi. The following additional data are available:
Swi =24% cw =3.62 ×10−6psi−1 cf =4.95 ×10−6psi−1 Bw =1.0 bbl/STB pb =1500 psi
Figure 11-17.Underground withdrawal vs. Eo+ Efw.
The field production and PVT data are summarized below:
Volumetric No. of Bo Np Wp
Average Pressure Producing Wells bbl/STB MSTB MSTB
3685 1 1.3102 0 0
3680 2 1.3104 20.481 0
3676 2 1.3104 34.750 0
3667 3 1.3105 78.557 0
3664 4 1.3105 101.846 0
3640 19 1.3109 215.681 0
3605 25 1.3116 364.613 0
3567 36 1.3122 542.985 0.159
3515 48 1.3128 841.591 0.805
3448 59 1.3130 1273.530 2.579
3360 59 1.3150 1691.887 5.008
3275 61 1.3160 2127.077 6.500
3188 61 1.3170 2575.330 8.000
Calculate the initial oil-in-place by using the MBE and compare with the volumetric estimate of N.
Solution
Step 1.Calculate the initial water and rock expansion term Ef,w from Equation 11-37:
Ef,w=10.0 ×10−6(3685 −–p
r) Step 2.Construct the following table:
F, Mbbl Eo, bbl/STB p–
r, psi Equation 10-35 Equation 10-36 Δp Ef, w Eo+ Ef, w
3685 — — 0 0 —
3680 26.84 0.0002 5 50 ×10−6 0.00025
3676 45.54 0.0002 9 90 ×10−6 0.00029
3667 102.95 0.0003 18 180 ×10−6 0.00048
3664 133.47 0.0003 21 210 ×10−6 0.00051
3640 282.74 0.0007 45 450 ×10−6 0.00115
Ef w, . . ( . ) . p
= × . + ×
−
⎡
⎣⎢ ⎤
⎦⎥
− −
1 3102 3 62 10 0 24 4 95 10 1 0 24
6 6
Δ
(table continued on next page)
F, Mbbl Eo, bbl/STB p–
r, psi Equation 10-35 Equation 10-36 Δp Ef, w Eo+ Ef, w
3605 478.23 0.0014 80 800 ×10−6 0.00220
3567 712.66 0.0020 118 1180 ×10−6 0.00318
3515 1105.65 0.0026 170 1700 ×10−6 0.00430 3448 1674.72 0.0028 237 2370 ×10−6 0.00517 3360 2229.84 0.0048 325 3250 ×10−6 0.00805 3275 2805.73 0.0058 410 4100 ×10−6 0.00990 3188 3399.71 0.0068 497 4970 ×10−6 0.01170
Step 3.Plot the underground withdrawal term F against the expansion term (Eo+Ef,w) on a Cartesian scale, as shown in Figure 11-18.
Step 4.Draw the best straight line through the points and determine the slope of the line and the volume of the active initial oil-in-place as:
N =257 MMSTB
It should be noted that the value of the initial oil-in-place as deter- mined from the MBE is referred to as the effective or active initial oil-in- place. This value is usually smaller than that of the volumetric estimate due to oil being trapped in undrained fault compartments or low-perme- ability regions of the reservoir.
Figure 11-18.F vs. (Eo+ Efw) for Example 11-3.