Dake L.P. Fundamentals of reservoir engineering

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NATURAL WATER INFLUX 311

140°

’

Fig. 9.10 Aquifer-reservoir geometry, exercise 9.2

Properties common to the reservoir

and aquifer

Reservoir properties

h = 100 ft

k = 200 mD

= 0.55 cp

=0.25

=3.0×10

-6

/psi

=4.0×10

-6

/ psi

=1.0

N = 312×10

stb

=0.05

(PVT data are plotted in figs. 9.12

and 9.13 and listed in table 9.3)

TIME (years)

8910

1000

800

600

2800

2400

2000

1600

1200

800

400

Psia

PRESSUR

MMstb

scf/stb

Pressure

100

Fig. 9.11 Reservoir production and pressure history; exercise 9.2

NATURAL WATER INFLUX 312

PRESSURE (psia)

0 1000 2000 3000

500

400

300

200

100

scf/stb

rb/scf

0.005

0.004

0.003

0.002

0.001

0.006

600

Fig. 9.12 R

and B

as functions of pressure; exercise 9.2

PRESSURE (psia)

0 1000 2000 300

rb/stb

1.4

1.3

1.2

1.1

Fig. 9.13 B

as a function of pressure; exercise 9.2

Initially the reservoir was at bubble point pressure but apparently there was no initial

gas cap (m=0). The graphs of cumulative oil production (N

), cumulative GOR (R

) and

the average reservoir pressure (p), as functions of time, are shown in fig. 9.11, for the

first 10 years of production. It may be assumed that the pressure decline at the oil

water contact is the same as the average pressure decline in the reservoir.

It was initially suspected, based on seismic and geological evidence that the value of

= r

was about 10. Do you consider this to be the correct value of r

based on

10 years of production history, and if not, what is your estimate of the correct value of

this ratio?

NATURAL WATER INFLUX 313

PRESSURE

(psia)

2700

2500

2300

2100

1900

1700

1500

1300

8910

TIME

ears

)

2620

p =2500

2395

p =2290

2199

p =2109

2029

p =1949

1883

p =1818

1760

p =1702

1655

p =1608

1571

p =1535

1507

p =1480

1460

p =1440

p =2740

Fig. 9.14 Reservoir pressure decline approximated by a series of discrete pressure

steps; exercise 9.2

Interpreted Data

Fig. 9.14 shows a more detailed graph of the continuous pressure decline which has

been approximated by 10 annual pressure steps, as described in the text. The

magnitude of the ∆p's can be calculated, using equs. (9.15) and (9.16), as shown in

table 9.2.

NATURAL WATER INFLUX 314

Time

(years)

Pressure at the

O.W.C.

(psia)

Plateau Pressure

levels

(psia)

∆p

(psi)

0 2740 (p

) 120

1 2500 2620 225

2 2290 2395 196

3 2109 2199 170

4 1949 2029 146

5 1818 1883 123

6 1702 1760 105

7 1608 1655 84

8 1635 1571 64

9 1480 1507 47

10 1440 1460

TABLE 9.2

The "end of the year values" of N

, the cumulative oil production, R

, the cumulative

GOR and PVT properties, corresponding to the pressure points given in column 2 of

table 9.2,are listed in table 9.3.

Time

(years)

(MM/stb)

(scf/stb)

(rb/stb)

(scf/stb)

(rb/scf)

0 650 (R

) 1.404(B

) 650 (R

) .00093 (B

)

1 7.88 760 1.374 592 .00098

2 18.42 845 1.349 545 .00107

3 29.15 920 1.329 507 .00117

4 40.69 975 1.316 471 .00128

5 50.14 1025 1.303 442 .00139

6 58.42 1065 1.294 418 .00150

7 65.39 1095 1.287 398 .00160

8 70.74 1120 1.280 383 .00170

9 74.54 1145 1.276 381 .00176

10 77.43 1160 1.273 364 .00182

TABLE 9.3

NATURAL WATER INFLUX 315

EXERCISE 9.2 SOLUTION

The procedure in tackling this type of problem is as follows.

1) Calculate the water influx using the unsteady state influx theory of Hurst and van

Everdingen for r

= 10.

2) Apply the technique of Havlena and Odeh in interpreting the material balance as the

equation of a straight line (refer Chapter 3, sec. 7). For a reservoir with no initial gas

cap the material balance is

()

ooi sisg

wwc f

po p sg oi e

oi wc

(B B ) (R R )B

(c S c )

NB R RB NB p W

B1S

−+−

æö

éù

+− = + ∆+

ç÷

ëû

−

èø

where W

is the cumulative water influx (rb). If the entire pressure drop over the

10 year period is considered (∆p = 1300 psi) the magnitude of the compressibility term

wwc f

(c S c )

(3 .05 4)

p 10 1300 0.006

1S .95

−

×+

∆= × × ∼

−

This is only 2% of the maximum value of the term accounting for the expansion of the

oil plus its originally dissolved gas (refer table 9.6; (E

)

max

= .3895/1.404 = .277) and

therefore, the pore and water compressibility effects will be neglected. The material

balance is then reduced to

+ (R

- R

] = N [(B

- B

) + (R

- R

] + W

or F = NE

and a plot of F/E

versus W

should be linear with intercept F/E

= N when W/E

= 0

and with unit slope.

3) If such a plot is non-linear for r

= 10 , vary the value of this parameter until a straight

line is obtained.

1) Calculation of W

for r

= 10

Since annual time steps have been selected, the dimensionless time coefficient can

most conveniently be expressed, with t in years and all other parameters in field units,

2.309kt

2.309 200 t

5.67 t

.25.55710 (9200)

φµ

−

××

××× ×

(9.7)

Similarly, the value of the aquifer constant, in field units, is

U1.119fhcr

= (9.8)

where f =

140°/360° = 0.3889

NATURAL WATER INFLUX 316

i.e. U = 1.119×.3889×.25×100×7×10

-6

×(9200)

U = 6446 bbls/psi

Time

(years)

dimensionless

time

∆p

(psi)

= 10)

= 5)

0 0 120

1 5.67 225 4.95 4.88

2 11.34 196 8.12 7.46

3 17.01 170 10.90 9.10

4 22.68 146 13.50 10.09

5 28.35 123 15.90 10.83

6 34.02 105 18.10 11.27

7 36.69 84 20.20 11.52

8 45.36 64 22.20 11.69

9 51.03 47 24.00 11.81

10 56.70 25.70 11.89

TABLE 9.4

The water influx can now be calculated, using values of ∆p

and W

listed in table 9.4,

as shown in table 9.5.

(years)

= U

−

∆

- t

)

(MMrb)

1 6446 (120×4.95) 3.829

2 6446 (120×8.12 + 225×4.95) 13.460

3 6446 (120×10.90 + 225×8.12 + 196×4.95) 26.462

4 6446 (120×13.50 + 225×10.90 + 196×8.12 + 170×4.95) 41.935

5 6446 (120×15.90 + 225×13.50 + 196×10.90 + 170×8.12

+ 146×4.95)

59.207

6 6446 (120×18.10 + 225×15.90 + 196×13.50 + 170×10.90

+ 146×8.12 + 123×4.95)

77.628

7 6446 (120×20.20 + 225×18.10 + 196×15.90 + 170×13.50

+ 146×10.90 + 123×8.12 + 105×4.95)

96.805

8 6446 (120×22.20 + 225×20.20 + 196×18.10 + 170×15.90

+ 146×13.50 + 123×10.90 + 105×8.12 + 84×4.95)

116.284

9 6446 (120×24.00 + 225×22.20 + 196×20.20 + 170×18.10

+ 146×15.90 + 123×13.50 + 105×10.90 + 84×8.12

+ 64×4.95)

135.601

10 6446 (120×25.70 + 225×24.00 + 196×22.20 + 170×20.20

+ 146×18.10 + 123×15.90 + 105×13.50 + 84×10.90

+ 64×8.12 + 47×4.95)

154.401

TABLE 9.5

NATURAL WATER INFLUX 317

2) Material Balance Calculation

F = N

+ (R

- R

)

= ((B

o -

) + (R

si -

)

Time

(years)

(MMrb)

(rb/stb)

F/E

(MMstb)

= 10)

= 5)

1 12.124 .0268 452.4 142.9 140.9

2 30.761 .0574 535.9 234.5 223.8

3 52.826 .0.923 5.72.3 286.7 260.3

4 79.798 .1411 565.5 297.2 253.5

5 105.964 .1881 563.3 314.8 251.3

6 132.292 .2380 555.8 326.2 243.8

7 157.080 .2862 548.8 338.2 236.8

8 179.177 .3299 543.1 352.5 231.2

9 196.654 .3630 541.7 373.6 229.7

10 210.743 .3895 541.1 396.4 229.1

TABLE 9.6

The plot of F/E

versus W

, fig. 9.15, shows that the trend deviates below the

theoretical 45° line after the first year. This indicates that the correct value of r

should

be somewhat smaller. The calculation has therefore been repeated with r

=5.

3) Calculation of W

for r

= 5.

Values of We for r

=5 are listed in table 9.7. These have been calculated using the W

values listed in column 5 of table 9.4.

NATURAL WATER INFLUX 318

(years)

= U

−

∆

- t

)

(MMrb)

1 6446 (120×4.88) 3.775

2 6446 (120×7.46 + 225×4.88) 12.848

3 6446 (120×9.10 + 225×7.46 + 196×4.88) 24.024

4 6446 (120×10.09 + 225×9.10 + 196×7.46 + 170×4.88) 35.775

5 6446 (120×10.83 + 225×10.09 + 196×9.10 + 170×7.46

+ 146×4.88)

47.276

6 6446 (120×11.27 + 225×10.83 + 196×10.09 + 170×9.10

+ 146×7.46 + 123×4.88)

58.035

7 6446 (120×11.52 + 225×11.27 + 196×10.83 + 170×10.09

+ 146×9.10 + 123×7.46 + 105×4.88)

67.778

8 6446 (120×11.69 + 225×11.52 + 196×11.27 + 170×10.83

+ 146×10.09 + 123×9.10 + 105×7.46 + 84×4.88)

76.259

9 6446 (120×11.81 + 225×11.69 + 196×11.52 + 170×11.27

+ 146×10.83 + 123×10.09 + 105×9.10 + 84×7.46

+ 64×4.88)

83.398

10 6446 (120×11.89 + 225×11.81 + 196×11.69 + 170×11.52

+ 146×11.27 + 123×10.83 + 105×10.09 + 84×9.10

+ 64×7.46 + 47×4.88)

89.225

TABLE 9.7

The corresponding values of W

, required for the material balance calculations, are

listed in table 9.6 and plotted in fig. 9.15. As can be seen, in this case, all the points lie

on the straight line of unit slope which intercepts the ordinate at N

= 312 MMstb, thus

confirming that r

= 5 is the correct value of the ratio of the aquifer/reservoir radius.

Also apparent from this plot is the fact that after the third year there occurs a reversal in

the trend of plotted points with both W

/ E

and F/E

decreasing with time. Havlena and

Odeh have pointed out

4,5

that this reversal is to be expected for a finite acting aquifer

but nevertheless all points, for the correct value of r

, must lie on the linear trend. For

the incorrect aquifer model (r

= 10), however, the calculated values of W

continually increase as F/E

decrease, as illustrated in fig. 9.15.

NATURAL WATER INFLUX 319

9,10

600

550

500

450

400

350

N =312

300

F/E (MMstb)

100 200 300 40

W/E (MMstb)

1,2,3...etc. Time (years)

r=5

r=10

Fig. 9.15 Aquifer fitting using the interpretation technique of Havlena and Odeh

9.4 THE APPROXIMATE WATER INFLUX THEORY OF FETKOVITCH FOR FINITE

AQUIFERS

The unsteady state influx theory of Hurst and van Everdingen provides the correct

method for calculating the cumulative water influx, under practically all circumstances,

for radial and linear aquifers. Unfortunately, it has the disadvantage that calculations

performed using the method are rather tedious, due to the complexity of superposing

solutions for each time step. This drawback is exaggerated by the fact that influx

calculations, when history matching, usually require a trial and error approach.

Because of this, many attempts have been made in the past to find a more direct

computational method of performing water influx calculations which would duplicate

results obtained with the Hurst and van Everdingen method and remove the necessity

of superposition.

The most successful of the methods is that proposed by Fetkovitch in 1971

. In this

approach the flow of aquifer water into a hydrocarbon reservoir is modelled in precisely

NATURAL WATER INFLUX 320

the same way as the flow of oil from a reservoir into a well. An inflow equation of the

form

qJ(pp)

==−

(9.18)

is used where,

= water influx rate

J = aquifer productivity index

p = reservoir pressure, i.e. pressure at the oil or gas water contact.

p = average pressure in the aquifer.

The latter is evaluated using the simple aquifer material balance

= c W

- p

) (9.19)

in which p

is the initial pressure in the aquifer and reservoir. This balance can be

alternatively expressed as

pp1 p1

cWp

æö

=− =−

ç÷

èø

(9.20)

where W

= c W

is defined as the initial amount of encroachable water and

represents the maximum possible expansion of the aquifer. Differentiating equ. (9.20)

with respect to time gives

eeia

dW W

dt p dt

=−

(9.21)

and substituting equ. (9.21) into equ. (9.18) and separating the variables gives

=−

−

This equation will now be integrated for the initial condition that at t=0 (W

= 0,

p = p

)

a pressure drop ∆p = p

- p is imposed at the reservoir boundary. Furthermore, the

boundary pressure p remains constant during the period of interest so that

()

Jp t

ln p p C

−=− +

where C is an arbitrary constant of integration which can be evaluated from the initial

conditions as C = In(p

- p), and therefore

()

iei

Jp t / W

pp ppe

−

−= −

(9.22)