1
WATER INFLUX
Many reservoirs are bound on a portion or all of their peripheries
by water-bearing rocks called aquifers. The aquifer may be so
large compared to the reservoir size as to appear infinite, and it
may be so small as to be negligible in its effect on reservoir
performance. The aquifer may be entirely bound by
impermeable rocks so that the reservoir and aquifer together
form a closed or volumetric system. On the other hand, the
reservoir may outcrop at one or more places where it may be
replenished by surface waters. The aquifer may be horizontal
with the reservoir it adjoins or it may rise considerably above the
reservoir to provide some sort of artesian flow to the reservoir.
Hassan S. Naji,
Professor,
2
Aquifers retard reservoir pressure decline by providing a source of water influx by:
water expansion
expansion of other hydrocarbon accumulations in the aquifer rock
compressibility of the aquifer rock and
artesian flow which occurs when the aquifer rises to a level above the
reservoir.
To determine the effect of an aquifer on reservoir production, it is necessary to
calculate the amount of water influx, 𝑊𝑒. This calculation can be made using the
material balance equation when the initial hydrocarbon in place and the production
history are known. If correct values of 𝑊𝑒 are placed in the material balance equation
as a function of reservoir pressure, then the equation should plot as a straight line. To
obtain an estimate for both the initial hydrocarbon in place and water influx, then a
model for 𝑊𝑒 as a function of pressure is assumed. If a straight line is not obtained,
then a new model for 𝑊𝑒 is assumed and the procedure repeated. Models for
calculating 𝑊𝑒 are categorized on a time dependent basis to:
Steady-state models:
1. Pot aquifer model
2. Schilthuis model
Pseudosteady-state models:
3. Fetkovitch model
Unsteady-State Models:
4. Van Everdingen and Hurst model
5. Hurst simplified model
6. Carter-Tracy model
The basic concept for water influx calculation is:
𝑊𝑒(𝑡) = 𝑈 𝑆(𝑝, 𝑡)
Where:
𝑊𝑒(𝑡) is water influx,
𝑈 is the aquifer constant, and
𝑆(𝑝, 𝑡) is the aquifer function
3
1. The Pot Aquifer Model
The simplest model that can be used to estimate water influx into a gas or oil
reservoir is the pot aquifer model. The assumptions for the pot aquifer formulation
are:
1. finite closed aquifer,
2. large aquifer permeability such that aquifer expansion is complete within the
time step,
3. reservoir cannot be too large. Otherwise it is hard to satisfy assumption 2,
and
4. variable compressibility is permissible.
The aquifer pore volume compressibility sets the basis for the pot aquifer formulation.
Since aquifer compressibility is given by:
𝑐 = −1
𝑉𝑃𝑖
𝜕𝑉𝑃
𝜕𝑝≈ −
1
𝑉𝑃𝑖
∆𝑉𝑃
∆𝑝=
1
𝑉𝑃𝑖
𝑉𝑃𝑖 − 𝑉𝑃(𝑡)
𝑝𝑖 − 𝑝(𝑡)
𝑉𝑝𝑖 − 𝑉𝑃(𝑡) = 𝑐𝑉𝑝𝑖[𝑝𝑖 − 𝑝(𝑡)]
Thus the constant-compressibility pot aquifer model is written as:
𝑊𝑒 = 𝑐𝑡 𝑉𝑃𝑖 [𝑝𝑖 − 𝑝(𝑡)] = (𝑐𝑤 + 𝑐𝑓) 𝑉𝑃𝑖 [𝑝𝑖 − 𝑝(𝑡)] = 𝑈 𝑆(𝑝, 𝑡)
where:
𝑊𝑒(𝑡) is the cumulative water influx, bbl,
𝑐𝑡 is the aquifer total compressibility, psia-1
𝑐𝑤 is the aquifer water compressibility, psia-1
𝑐𝑓 is the aquifer rock compressibility, psia-1
𝑉𝑃𝑖 is the initial aquifer pore volume = [7758 𝐴ℎ∅] = [𝜋(𝑟𝑎𝑞
2 −𝑟𝑒2)ℎ∅
5.6146], bbl, and
𝑈 is the aquifer constant = (𝑐𝑤 + 𝑐𝑓) 𝑉𝑃𝑖, bbl/psi
𝑆(𝑝, 𝑡) is the aquifer function = 𝑝𝑖 − 𝑝(𝑡), psia
𝑝𝑖 is the initial aquifer (reservoir: pressure at the oil-water contact) pressure, psia
𝑝(𝑡) is the aquifer (reservoir: pressure at the oil-water contact) current pressure, psia
Aquifer pressures are approximated by reservoir pressures at the water-oil contact.
4
Example #1
Water Influx Calculations for a Pot Aquifer Model
Semester: Homework :
Name: SS:
A wedge-shaped reservoir is suspected of having a fairly strong natural water drive.
The geometry of the reservoir-aquifer system is shown by the following figure.
The following aquifer data are given:
Thickness 100 ft
Permeability 200 md
Porosity 0.25
Compressibility 4.0 x 10-6
Aquifer/reservoir radius ratio re/rw 5.0
Water viscosity 0.55 cp
Water compressibility 3.0 x 10-6
Water formation volume factor 1.0 RB/STB
5
The following reservoir data are given:
Time
years
P @ OWC, psia
Np
MM STB
Rp
SCF/STB
Bo
RB/STB
Rs
SCF/STB
Bg
RB/SCF
0
1
2
3
4
5
6
7
8
9
10
2740
2500
2290
2109
1949
1818
1702
1608
1535
1480
1440
0
7.88
18.42
29.15
40.69
50.14
58.42
65.39
70.74
74.54
77.43
650 (Rsi)
760
845
920
975
1025
1065
1095
1120
1145
1160
1.404
1.374
1.349
1.329
1.316
1.303
1.294
1.287
1.280
1.276
1.273
650 (Rsi)
592
545
507
471
442
418
398
383
371
364
0.00093
0.00098
0.00107
0.00117
0.00128
0.00139
0.00150
0.00160
0.00170
0.00176
0.00182
Calculate the amount of water influx if a pot aquifer model is applicable.
Solution:
re 5.0 x (9200.0) 46000 ft
VPi π(46,0002-9,2002)(140/360)(100)(0.25)/5.6146 11.05054 MMM bbl
𝑈 = (𝑐𝑤 + 𝑐𝑓) 𝑉𝑃𝑖 7.0 x 10-6 x 11.05054 x 109 77353.77829 bbl/psia
Time
years
P @ OWC, psia
(pi - p)
psia
We U x (pi - p)
MM bbl
0
1
2
3
4
5
6
7
8
9
10
2740
2500
2290
2109
1949
1818
1702
1608
1535
1480
1440
0
240
450
631
791
922
1038
1132
1205
1260
1300
0.000
18.565
34.809
48.810
61.187
71.320
80.293
87.564
93.211
97.466
100.560
6
For variable-compressibility pot aquifer, the above equation is written as:
𝑊𝑒(𝑡) = ∑ ∆𝑡𝑊𝑒
𝑗+12
𝑛
𝑗=0
= ∑ 𝑉𝑃
𝑗+12(𝑐𝑓 + 𝑐𝑤)
𝑗+12∆𝑡𝑝𝑗+1
2
𝑛
𝑗=0
= 𝐴ℎ ∑ ∅𝑗+12(𝑐𝑓 + 𝑐𝑤)
𝑗+12∆𝑡𝑝𝑗+1
2
𝑛
𝑗=0
Where:
∅𝑗+12 =
∅𝑗 + ∅𝑗+1
2
(𝑐𝑓 + 𝑐𝑤)𝑗+1
2 =(𝑐𝑓 + 𝑐𝑤)
𝑗+ (𝑐𝑓 + 𝑐𝑤)
𝑗+1
2
∆𝑡𝑝𝑗+12 = 𝑝𝑗 − 𝑝𝑗+1
y = -0.8487x2 + 18.398x + 0.785
0
20
40
60
80
100
120
0 2 4 6 8 10 12
We
, MM
bb
l
Time, years
We = U x (pi - p)
We = U x (pi - p) Poly. (We = U x (pi - p))
7
2. Schilthuis Steady-State Aquifer Model
Schilthuis used Darcy's Law to start deriving his model as follows:
𝑞 = 0.00708𝑘ℎ
𝜇𝐵
(𝑝𝑖 − 𝑝)
ln𝑟𝑒
𝑟𝑤
Including the skin effect, the above equation is written as follows:
𝑞 = 0.00708𝑘ℎ
𝜇𝐵
(𝑝𝑖 − 𝑝)
[ln𝑟𝑒
𝑟𝑤− 0.75 + 𝑠]
𝑞 =𝑑𝑉
𝑑𝑡=
0.00708𝑘ℎ
𝜇𝐵 [ln𝑟𝑒
𝑟𝑤− 0.75 + 𝑠]
𝑑𝑝
𝑑𝑡
𝑞 = ∫𝑑𝑉
𝑑𝑡𝑑𝑡
𝑡
0
= 𝑊𝑒 = ∫0.00708𝑘ℎ
𝜇𝐵 [ln𝑟𝑒
𝑟𝑤− 0.75 + 𝑠]
𝑑𝑝
𝑝
0
= 𝐾𝑠 ∫ 𝑑𝑝
𝑝
0
𝑑𝑊𝑒
𝑑𝑡= 𝑞 = 0.00708
𝑘ℎ
𝜇𝑤𝐵𝑤
(𝑝𝑖 − 𝑝)
[ln𝑟𝑒
𝑟𝑤− 0.75 + 𝑠]
= 𝐾𝑠(𝑝𝑖 − 𝑝)
Thus we write:
𝑊𝑒(𝑡) = 𝐾𝑠 ∫𝑑𝑝
𝑑𝑡
𝑡
0
𝑑𝑡 = 𝐾𝑠 ∫ ∆𝑝𝑡
𝑡
0
𝑑𝑡 = 𝐾𝑠 ∑ ∆𝑝𝑡
𝑡
0
∆𝑡
8
3. Fetkovitch Pseudosteady-State Aquifer Model
Fetkovitch (1973) started derivation of his aquifer model with Darcy’s equation.
The assumptions for Fetkovitch aquifer formulation are:
1. finite closed aquifer,
2. large aquifer permeability such that aquifer expansion is complete within the
time step,
3. Water influx rate 𝑑𝑊𝑒
𝑑𝑡 is controlled by aquifer permeability via aquifer J.
4. reservoir can be too large depending on the magnitude of aquifer permeability,
and
5. constant aquifer compressibility.
𝑞𝑜 = (𝑘ℎ
141.22 [ln (𝑟𝑒𝑟𝑤
) − 0.75 + 𝑠]) ∫
𝑘𝑟𝑜
𝜇𝑜𝐵𝑜𝑑𝑝
𝑝𝑡
𝑝
Since the productivity index of a well, denoted by J, is a measure of the ability of the well to
produce. It is given by:
𝐽 =𝑞𝑜
𝑃𝑖 − 𝑃𝑡
Where:
J = Wellbore productivity index, STB/day/psig
rP = Average (static) reservoir pressure, psig
oQ = Wellbore stabilized oil flow rate, STB/day
wfP = Wellbore stabilized bottom-hole flowing pressure, psig
𝐽 =0.00708 𝑘ℎ
𝜇𝑤𝐵𝑤 [ln (𝑟𝑒𝑟𝑤
) − 0.75],
𝑅𝐵
𝑑𝑎𝑦/𝑝𝑠𝑖
𝑊𝑒 = ∫𝑑𝑊𝑒
𝑑𝑡𝑑𝑡
𝑡
0
𝐽 =0.00708 𝑘ℎ
𝜇𝑤𝐵𝑤 {𝑟𝑒
2
𝑟𝑒2 − 𝑟𝑤
2 [ln (𝑟𝑒𝑟𝑤
) − 0.75 +𝑟𝑤
2
𝑟𝑒2 (1 −
𝑟𝑤2
4𝑟𝑒2)]}
𝑊𝑒(𝑡) = 𝑐𝑡 𝑉𝑃𝑎𝑞 (𝑝𝑖 − �̅�𝑎𝑞)
9
Derivation of Fetkovitch aquifer model starts with:
𝑞𝑤𝐵𝑤 =𝑑𝑊𝑒
𝑑𝑡= 𝐽[�̅�𝑎𝑞 − 𝑝(𝑡)]
(1)
Where:
𝐽 =0.00708 𝑘ℎ
𝜇𝑤𝐵𝑤 {𝑟𝑒
2
𝑟𝑒2 − 𝑟𝑤
2 [ln (𝑟𝑒𝑟𝑤
) − 0.75 +𝑟𝑤
2
𝑟𝑒2 (1 −
𝑟𝑤2
4𝑟𝑒2)]}
,𝑅𝐵
𝑑𝑎𝑦/𝑝𝑠𝑖
𝑊𝑒(𝑡) = 𝑐𝑡 𝑉𝑃𝑎𝑞 (𝑝𝑖 − �̅�𝑎𝑞)
�̅�𝑎𝑞 = 𝑝𝑖 −𝑊𝑒(𝑡)
𝑐𝑡 𝑉𝑃𝑎𝑞
(2)
𝑑𝑊𝑒
𝑑𝑡= −𝑐𝑡 𝑉𝑃𝑎𝑞
𝑑�̅�𝑎𝑞
𝑑𝑡
(3)
Plugging (3) into (1) yields:
𝐽[�̅�𝑎𝑞 − 𝑝(𝑡)] = −𝑐𝑡 𝑉𝑃𝑎𝑞 𝑑�̅�𝑎𝑞
𝑑𝑡
(4)
𝑑�̅�𝑎𝑞
[�̅�𝑎𝑞
− 𝑝(𝑡)]= −
𝐽𝑑𝑡
𝑐𝑡 𝑉𝑃𝑎𝑞
(5) Integrating both sides yields:
∫𝑑�̅�𝑎𝑞
[�̅�𝑎𝑞
− 𝑝(𝑡)]
𝑝
𝑝𝑖
= − ∫𝐽𝑑𝑡
𝑐𝑡 𝑉𝑃𝑎𝑞
𝑡
0
(6) Assuming that 𝑝(𝑡) is constant yields:
10
ln[�̅�𝑎𝑞 − 𝑝] = −𝐽𝑡
𝑐𝑡 𝑉𝑃𝑎𝑞+ ln[𝑝𝑖 − 𝑝]
(7) Rearranging yields:
�̅�𝑎𝑞 − 𝑝 = (𝑝𝑖 − 𝑝)𝑒−𝐽𝑡
𝑐𝑡 𝑉𝑃𝑎𝑞
(8)
Substituting (8) into (1) yields:
𝑊𝑒 = (𝑝𝑖 − 𝑝)𝑐𝑡 𝑉𝑃𝑎𝑞 (1 − 𝑒−𝐽𝑡
𝑐𝑡 𝑉𝑃𝑎𝑞)
(9)
4. van Everdingen and Hurst Unsteady-State Aquifer Model
van Everdingen and Hurst (19XX) started derivation of their aquifer model with
Darcy’s equation. The assumptions for van Everdingen and Hurst aquifer formulation
are:
1. Finite closed aquifer for linear case or infinite for radial case,
2. The system is compressible since we have 𝐶𝑡 in the denominator of 𝑡𝐷 with
constant compressibility 𝐶𝑡 = 𝐶𝑤 + 𝐶𝑓,
3. Both aquifer size and reservoir size via 𝑟𝑒
𝑟𝑤 as well as aquifer diffusivity via
0.006328 𝑘
∅ 𝜇 𝐶𝑡,
𝑓𝑡2
𝑑𝑎𝑦 and the outer boundary condition affect water influx rate.
4. True unsteady state process.
van Everdingen and Hurst started their derivation as follows:
𝑊𝑒(𝑡𝑛+1) = 𝐵 ∑ ∆𝑝𝑗𝑊𝑒𝐷(𝑡𝐷𝑛+1 − 𝑡𝐷
𝑗)
𝑛
𝑗=0
Where:
𝐵 =2𝜋
5.6146∅ 𝐶𝑡 ℎ 𝑟𝑤
2𝜃
360
11
𝐵 =0.006328 𝑘𝑡
∅ 𝜇 𝐶𝑡𝑟𝑤2
∆𝑝0 =(𝑝0 − 𝑝1)
2
∆𝑝1 =(𝑝0 − 𝑝2)
2
∆𝑝2 =(𝑝1 − 𝑝3)
2
∆𝑝3 =(𝑝2 − 𝑝4)
2
∆𝑝𝑖 =(𝑝𝑖−1 − 𝑝𝑖+1)
2, 𝑓𝑜𝑟 𝑖 ≥ 1
12
Summary of Aquifer Models
Model Formula Pot Aquifer 𝑊𝑒
𝑛 = (𝑐𝑤 + 𝑐𝑓) 𝑉𝑃𝑎𝑞0 (𝑝𝑖 − 𝑝𝑛)
Schilthuis 𝑊𝑒(𝑡) = 𝐾𝑠 ∑(∆𝑃𝑡)∆𝑡
𝑛
𝑗=0
Schilthuis – Model
∆𝑡𝑊𝑒𝑛 = (�̅�𝑎𝑞
𝑛−1 − �̅�𝑎𝑞𝑛 )
𝑊𝑒𝑖
𝑝𝑖[1 − 𝑒
(−𝐽𝑝𝑖∆𝑡𝑛
𝑊𝑒𝑖)]
�̅�𝑎𝑞𝑛−1 = 𝑝𝑖 (1 −
∑ ∆𝑡𝑊𝑒𝑗𝑛−1
𝑗=0
𝑊𝑒𝑖)
𝑊𝑒𝑖 = 𝑐𝑡𝑉𝑃𝑎𝑞𝑝𝑖
�̅�𝑛 =𝑝𝑛+1 + 𝑝𝑛
2
∆𝑡𝑛 = 𝑡𝑛 − 𝑡𝑛−1
Van Everdingin & Hurst
𝑊𝑒(𝑡𝑛+1) = 𝐵 ∑(∆𝑝𝑗) 𝑊𝑒𝐷 (𝑡𝐷𝑛+1 − 𝑡𝐷
𝑗)
𝑛
𝑗=0
𝐵 =2𝜋
5.6146∅𝑐𝑡ℎ𝑟𝑤
2𝜃
360
13
Hurst-Modified
𝑊𝑒(𝑡) =𝑘ℎ
70.6𝜇 ∑(∆𝑝𝑡)𝑗
𝑡𝑗+1 − 𝑡𝑗
𝑎 + ln 𝑡𝑗+1
𝑛
𝑗=0
(∆𝑝𝑡)𝑗 = 𝑝𝑖 −𝑝𝑗 + 𝑝𝑗+1
2
𝑎 = ln (𝑘
70.6∅𝜇𝑐𝑡𝑟𝑤2
)
Carter-Tracy
∆𝑡𝑊𝑒𝑛+1 = (
𝐵∆𝑝𝑡 − 𝑊𝑒𝑛𝑝𝐷
′ (𝑡𝐷𝑛+1)
𝑝𝐷(𝑡𝐷𝑛+1) − 𝑡𝐷
𝑛𝑝𝐷′ (𝑡𝐷
𝑛+1)) (𝑡𝐷
𝑛+1 − 𝑡𝐷𝑛)
𝑝𝐷 =2
√𝜋√𝑡𝐷