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,

hnaji@kau.edu.sa

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