Tarek Ahmed. Reservoir engineering handbook
Подождите немного. Документ загружается.
where e
w
= water influx rate, bbl/day
Q
o
= oil flow rate, STB/day
B
o
= oil formation volume factor, bbl/STB
Q
g
= free gas flow rate, scf/day
B
g
= gas formation volume factor, bbl/scf
Q
w
= water flow rate, STB/day
B
w
= water formation volume factor, bbl/STB
Equation 10-1 can be equivalently expressed in terms of cumulative
production by introducing the following derivative terms:
where W
e
= cumulative water influx, bbl
t = time, days
N
p
= cumulative oil production, STB
GOR = current gas-oil ratio, scf/STB
R
s
= current gas solubility, scf/STB
B
g
= gas formation volume factor, bbl/scf
W
p
= cumulative water production, STB
dN
p
/dt = daily oil flow rate Q
o
, STB/day
dW
p
/dt = daily water flow rate Q
w
, STB/day
dW
e
/dt = daily water influx rate e
w
, bbl/day
(GOR − R
s
)dN
p
/dt = daily free gas flow rate, scf/day
Example 10-1
Calculate the water influx rate e
w
in a reservoir whose pressure is sta-
bilized at 3000 psi.
Given: initial reservoir pressure = 3500 psi
dN
p
/dt = 32,000 STB/day
B
o
= 1.4 bbl/STB
GOR = 900 scf/STB
R
s
= 700 scf/STB
B
g
= 0.00082 bbl/scf
dW
p
/dt = 0
B
w
= 1.0 bbl/STB
e
dW
dt
B
dN
dt
(GOR – R )
dN
dt
B
dW
dt
B (10 - 2)
w
e
o
p
s
p
g
p
w
== + +
638 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 10 2001-10-25 14:22 Page 638
Solution
Applying Equation 10-1 or 10-2 gives:
e
w
= (1.4) (32,000) + (900 − 700) (32,000) (0.00082) + 0
= 50,048 bbl/day
Outer Boundary Conditions
The aquifer can be classified as infinite or finite (bounded). Geologi-
cally all formations are finite, but may act as infinite if the changes in the
pressure at the oil-water contact are not “felt” at the aquifer boundary.
Some aquifers outcrop and are infinite acting because of surface replen-
ishment. In general, the outer boundary governs the behavior of the
aquifer and, therefore:
a. Infinite system indicates that the effect of the pressure changes at the
oil/aquifer boundary can never be felt at the outer boundary. This
boundary is for all intents and purposes at a constant pressure equal to
initial reservoir pressure.
b. Finite system indicates that the aquifer outer limit is affected by the
influx into the oil zone and that the pressure at this outer limit changes
with time.
Flow Regimes
There are basically three flow regimes that influence the rate of water
influx into the reservoir. As previously described in Chapter 6, those flow
regimes are:
a. Steady-state
b. Semisteady (pseudosteady)-state
c. Unsteady-state
Flow Geometries
Reservoir-aquifer systems can be classified on the basis of flow geom-
etry as:
a. Edge-water drive
b. Bottom-water drive
c. Linear-water drive
Water Influx 639
Reservoir Eng Hndbk Ch 10 2001-10-25 14:22 Page 639
In edge-water drive, as shown in Figure 10-1, water moves into the
flanks of the reservoir as a result of hydrocarbon production and pressure
drop at the reservoir-aquifer boundary. The flow is essentially radial with
negligible flow in the vertical direction.
Bottom-water drive occurs in reservoirs with large areal extent and
gentle dip where the reservoir-water contact completely underlies the
reservoir. The flow is essentially radial and, in contrast to the edge-water
drive, the bottom-water drive has significant vertical flow.
In linear-water drive, the influx is from one flank of the reservoir. The
flow is strictly linear with a constant cross-sectional area.
RECOGNITION OF NATURAL WATER INFLUX
Normally very little information is obtained during the exploration-devel-
opment period of a reservoir concerning the presence or characteristics of
an aquifer that could provide a source of water influx during the depletion
period. Natural water drive may be assumed by analogy with nearby pro-
ducing reservoirs, but early reservoir performance trends can provide clues.
A comparatively low, and decreasing, rate of reservoir pressure decline with
increasing cumulative withdrawals is indicative of fluid influx.
640 Reservoir Engineering Handbook
Figure 10-1. Flow geometries.
Reservoir Eng Hndbk Ch 10 2001-10-25 14:22 Page 640
Successive calculations of barrels withdrawn per psi change in reser-
voir pressure can supplement performance graphs. If the reservoir limits
have not been delineated by developed dry holes, however, the influx
could be from an undeveloped area of the reservoir not accounted for in
averaging reservoir pressure. If the reservoir pressure is below the oil sat-
uration pressure, a low rate of increase in produced gas-oil ratio is also
indicative of fluid influx.
Early water production from edge wells is indicative of water encroach-
ment. Such observations must be tempered by the possibility that the early
water production is due to formation fractures; thin, high permeability
streaks; or to coning in connection with a limited aquifer. The water pro-
duction may be due to casing leaks.
Calculation of increasing original oil-in-place from successive reser-
voir pressure surveys by using the material balance assuming no water
influx is also indicative of fluid influx.
WATER INFLUX MODELS
It should be appreciated that in reservoir engineering there are more
uncertainties attached to this subject than to any other. This is simply
because one seldom drills wells into an aquifer to gain the necessary
information about the porosity, permeability, thickness and fluid proper-
ties. Instead, these properties frequently have to be inferred from what
has been observed in the reservoir. Even more uncertain, however, is the
geometry and areal continuity of the aquifer itself.
Several models have been developed for estimating water influx that
are based on assumptions that describe the characteristics of the aquifer.
Due to the inherent uncertainties in the aquifer characteristics, all of the
proposed models require historical reservoir performance data to evalu-
ate constants representing aquifer property parameters since these are
rarely known from exploration-development drilling with sufficient accu-
racy for direct application. The material balance equation can be used to
determine historical water influx provided original oil-in-place is known
from pore volume estimates. This permits evaluation of the constants in
the influx equations so that future water influx rate can be forecasted.
The mathematical water influx models that are commonly used in the
petroleum industry include:
• Pot aquifer
• Schilthuis’ steady-state
Water Influx 641
Reservoir Eng Hndbk Ch 10 2001-10-25 14:22 Page 641
• Hurst’s modified steady-state
• The Van Everdingen-Hurst unsteady-state
- Edge-water drive
- Bottom-water drive
• The Carter-Tracy unsteady-state
• Fetkovich’s method
- Radial aquifer
- Linear aquifer
The following sections describe these models and their practical appli-
cations in water influx calculations.
The Pot Aquifer Model
The simplest model that can be used to estimate the water influx into a
gas or oil reservoir is based on the basic definition of compressibility. A
drop in the reservoir pressure, due to the production of fluids, causes the
aquifer water to expand and flow into the reservoir. The compressibility
is defined mathematically as:
or
∆V = c V ∆ p
Applying the above basic compressibility definition to the aquifer gives:
Water influx = (aquifer compressibility) (initial volume of water)
(pressure drop)
or
W
e
= (c
w
+ c
f
) W
i
(p
i
− p) (10 - 3)
where W
e
= cumulative water influx, bbl
c
w
= aquifer water compressibility, psi
−1
c
f
= aquifer rock compressibility, psi
−1
W
i
= initial volume of water in the aquifer, bbl
e
dW
dt
B
dN
dt
(GOR – R )
dN
dt
B
dW
dt
B (10 - 2)
w
e
o
p
s
p
g
p
w
== + +
642 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 10 2001-10-25 14:22 Page 642
p
i
= initial reservoir pressure, psi
p = current reservoir pressure (pressure at oil-water contact), psi
Calculating the initial volume of water in the aquifer requires the
knowledge of aquifer dimension and properties. These, however, are sel-
dom measured since wells are not deliberately drilled into the aquifer to
obtain such information. For instance, if the aquifer shape is radial, then:
where r
a
= radius of the aquifer, ft
r
e
= radius of the reservoir, ft
h = thickness of the aquifer, ft
φ=porosity of the aquifer
Equation 10-3 suggests that water is encroaching in a radial form from
all directions. Quite often, water does not encroach on all sides of the
reservoir, or the reservoir is not circular in nature.
To account for these cases, a modification to Equation 10-2 must be
made in order to properly describe the flow mechanism. One of the sim-
plest modifications is to include the fractional encroachment angle f in
the equation, as illustrated in Figure 10-2, to give:
W
e
= (c
w
+ c
f
) W
i
f (p
i
− p) (10-5)
where the fractional encroachment angle f is defined by:
The above model is only applicable to a small aquifer, i.e., pot aquifer,
whose dimensions are of the same order of magnitude as the reservoir
itself. Dake (1978) points out that because the aquifer is considered rela-
tively small, a pressure drop in the reservoir is instantaneously transmit-
ted throughout the entire reservoir-aquifer system. Dake suggests that for
large aquifers, a mathematical model is required which includes time
dependence to account for the fact that it takes a finite time for the
aquifer to respond to a pressure change in the reservoir.
f
encoachment angle
o
oo
=
(
)
=
360 360
θ
(10 - 6)
W
rrh
i
ae
=
−
(
)
πφ
22
5 615.
(10 - 4)
Water Influx 643
Reservoir Eng Hndbk Ch 10 2001-10-25 14:22 Page 643
Example 10-2
Calculate the cumulative water influx that results from a pressure drop
of 200 psi at the oil-water contact with an encroachment angle of 80°.
The reservoir-aquifer system is characterized by the following properties:
Reservoir Aquifer
radius, ft 2600 10,000
porosity 0.18 0.12
c
f
, psi
−1
4 × 10
−6
3 × 10
−6
c
w
, psi
−1
5 × 10
−6
4 × 10
−6
h, ft 20 25
Solution
Step 1. Calculate the initial volume of water in the aquifer from Equa-
tion 10-4.
W MMbbl
i
=
−
(
)
(
)
=
π(, ) .
.
.
10 000 2600 25 0 12
5 615
156 5
22
644 Reservoir Engineering Handbook
Figure 10-2. Radial aquifer geometries.
Reservoir Eng Hndbk Ch 10 2001-10-25 14:22 Page 644
Step 2. Determine the cumulative water influx by applying Equation 10-5.
Schilthuis’ Steady-State Model
Schilthuis (1936) proposed that for an aquifer that is flowing under the
steady-state flow regime, the flow behavior could be described by
Darcy’s equation. The rate of water influx e
w
can then be determined by
applying Darcy’s equation:
The above relationship can be more conveniently expressed as:
where e
w
= rate of water influx, bbl/day
k = permeability of the aquifer, md
h = thickness of the aquifer, ft
r
a
= radius of the aquifer, ft
r
e
= radius of the reservoir
t = time, days
The parameter C is called the water influx constant and is expressed in
bbl/day/psi. This water influx constant C may be calculated from the
reservoir historical production data over a number of selected time inter-
vals, provided that the rate of water influx e
w
has been determined inde-
pendently from a different expression. For instance, the parameter C may
be estimated by combining Equation 10-1 with 10-8. Although the influx
constant can only be obtained in this manner when the reservoir pressure
dW
dt
eCpp
e
wi
== −( ) (10 - 8)
dW
dt
e
kh
r
r
pp
e
w
w
a
e
i
==
−
0 00708.
ln
()
µ
(10 - 7)
W = (4 + 3) 10 (156.5 10 ) (200) = 48,689 bbl
e
66−
×
80
360
Water Influx 645
Reservoir Eng Hndbk Ch 10 2001-10-25 14:22 Page 645
stabilizes, once it has been found, it may be applied to both stabilized
and changing reservoir pressures.
Example 10-3
The data given in Example 10-1 is used in this example:
p
i
= 3500 psi p = 3000 psi Q
o
= 32,000 STB/day
B
o
= 1.4 bbl/STB GOR = 900 scf/STB R
s
= 700 scf/STB
B
g
= 0.00082 bbl/scf Q
w
= 0B
w
= 1.0 bbl/STB
Calculate Schilthuis’ water influx constant.
Solution
Step 1. Solve for the rate of water influx e
w
by using Equation 10-1.
e
w
= (1.4) (32,000) + (900 − 700) (32,000) (0.00082)+ 0
= 50,048 bbl/day
Step 2. Solve for the water influx constant from Equation 10-8.
If the steady-state approximation adequately describes the aquifer flow
regime, the calculated water influx constant C values will be constant
over the historical period.
Note that the pressure drops contributing to influx are the cumulative
pressure drops from the initial pressure.
In terms of the cumulative water influx W
e
, Equation 10-8 is integrat-
ed to give the common Schilthuis expression for water influx as:
or
WCppdt
ei
o
t
=−
∫
( ) (10 - 9)
dW C p p dt
e
o
We
i
o
t
∫∫
=−
(
)
C bbl day psi=
−
=
50 048
3500 3000
100
,
()
//
646 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 10 2001-10-25 14:22 Page 646
where W
e
= cumulative water influx, bbl
C = water influx constant, bbl/day/psi
t = time, days
p
i
= initial reservoir pressure, psi
p = pressure at the oil-water contact at time t, psi
When the pressure drop (p
i
− p) is plotted versus the time t, as shown in
Figure 10-3, the area under the curve represents the integral .
This area at time t can be determined numerically by using the trape-
zoidal rule (or any other numerical integration method), as:
Equation 10-9 can then be written as:
Example 10-4
The pressure history of a water-drive oil reservoir is given below:
t, days p, psi
0 3500 (p
i
)
100 3450
200 3410
300 3380
400 3340
The aquifer is under a steady-state flowing condition with an estimated
water influx constant of 130 bbl/day/psi. Calculate the cumulative water
influx after 100, 200, 300, and 400 days using the steady-state model.
WC pt
e
o
t
=
∑
()∆∆ (10 -10)
() . ()
()( )
()
()()
()
.
p
i
p
o
t
dt area
I
area
II
area
III
etc
p
i
p
t
p
i
pp
i
p
tt
p
i
pp
i
p
tt
etc
−=
∫
=+ + +=
−
−
+
−+−
−+
−+−
−
+
1
2
1
0
12
2
21
23
2
32
()ppdt
i
o
t
−=
∫
Water Influx 647
Reservoir Eng Hndbk Ch 10 2001-10-25 14:22 Page 647