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Contents Chapter 3 Water Influx Calculation (pp. 149-185) ......................................................................... 2

3.1 Classification of aquifers ...................................................................................................... 2 3.2 Water influx models ......................................................................................................... 6

3.2.1 The pot aquifer model ............................................................................................... 6 3.2.2 The Schilthuis steady-state model ............................................................................ 8 3.2.3 The Hurst modified steady-state model .................................................................. 11 3.2.4 The van Everdingen and Hurst unsteady-state model ............................................. 12 3.2.5 The Fetkovich method ................................................................................................. 43

Chapter 3 Water Influx Calculation (pp. 149-185)

Water-bearing rocks called aquifers surround nearly all hydrocarbon reservoirs. These aquifers

maybe substantially larger than the oil or gas reservoirs they adjoin as to appear infinite in size,

and they may be so small in size as to be negligible in their effect on reservoir performance.

The objective of this chapter, however, concern those reservoir–aquifer systems in which the size

of the aquifer is large enough and the permeability of the rock is high enough that water influx

occurs as the reservoir is depleted. This chapter is designed to provide the various water influx

calculation models and a detailed description of the computational steps involved in applying

these models.

3.1 Classification of aquifers

Hydrocarbon production from the reservoir and the subsequent pressure drop prompt a response

from the aquifer to offset the pressure decline. This response comes in the form of a water influx,

commonly called water encroachment, which is attributed to:

• Expansion of the water in the aquifer;

• Compressibility of the aquifer rock;

• Artesian flow where the water-bearing formation outcrop is located structurally higher

than the pay zone.

Aquifers can be classified based on different criteria

• Degree of pressure maintenance

• Outer boundary conditions

• Flow regimes

• Flow geometries

Degree of pressure maintenance

�water influx rate � = � oil flow

rate � + � free gas flow rate

� + � water

production rate

�

ew = QoBo + QgBg + QwBw (3.1)

Where:

ew = water influx rate, bbl/day

Qo = oil flow rate,STB/day

Bo = oil formation volume factor, bbl/STB

Qg = free gas flow rate, scf/day

Bg = gas formation volume factor, bbl/scf

Qw = water flow rate, STB/day

Bw = water formation volume factor, bbl/STB

Express in terms of cumulative production:

ew = dWe dt

= Bo dNp dt

+ (GOR − Rs) dNp dt

Bg + dWp dt

Bw (3.2)

Where:

We = cumulatvie water influx, bbl

t = time, days

Np = cumulative oil production, STB

GOR = current gas − oil ratio, scf/STB

Rs = current gas solubility, scf/STB

Bg = gas formation volume factor, bbl/scf

Wp = cumulative water production, STB dNp dt

= daily oil flow rate Qo, STB/day

dWp dt

= daily water flow rate Qw, STB/day

dWe dt

= daily water influx rate ew, bbl/day

(GOR − Rs) dNp dt

= daily free gas rate, scf/day

Example 3.1

Calculate the water influx rate ew in a reservoir whose pressure is stabilized at 3000 psi.

Given: initial reservoir pressure = 3500 psi, dNp dt

= 32 000 STB/day , Bo = 1.4 bbl STB

, GOR =

900 scf STB

, Rs = 700 scf STB

, Bg = 0.00082 bbl scf

, dWp dt

= 0, Bw = 1.0 bbl/STB

Solution

ew = dWe

dt = Bo

dNp dt

+ (GOR − Rs) dNp dt

Bg + dWp

dt Bw

= (1.4)(32 000) + (900 − 700)(32 000)(0.00082) + 0

= 50 048 bbl/day

Outer boundary conditions

The outer boundary governs the behavior of the aquifer and can be classified as follows:

• 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.

• 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

• Steady state flow

�∂p ∂t � i

= 0 (3.3)

• Pseudo steady state flow

�∂p ∂t � i

= constant (3.4)

• Unsteady state flow

�∂p ∂t � = f(i, t) (3.5)

Figure 3.1. Flow regimes.

Flow geometries

• Edge-water Drive

• Bottom-water Drive

• Linear-water Drive

Recognition of natural water influx

• A comparatively low, and decreasing, rate of reservoir pressure decline with increasing

cumulative withdrawals is indicative of fluid influx.

• If the reservoir pressure is below the oil saturation pressure, a low rate of increase in

produced GOR is also indicative of fluid influx.

3.2 Water influx models

The mathematical water influx models that are commonly used in the petroleum industry

include:

• pot aquifer;

• Schilthuis steady state;

• Hurst modified steady state;

• van Everdingen and Hurst unsteady state:

 edge-water drive;

 bottom-water drive;

• Fetkovich method:

 radial aquifer;

 linear aquifer

3.2.1 The pot aquifer model

c = 1 V ∂V ∂p

= 1 V ∆V ∆p

(3.6) or:

∆V = cV∆p (3.7) Applying the above basic compressibility definition to the aquifer gives:

Water influx = (aquifer compressivility) × (initial volume of water)(pressure drop) Or:

We = ctWi(pi − p) (3.8)

ct = cw + cf

Where: We = cumulative water influx, bbl ct = aquifer total compressibility, psi−1 cw = aquifer water compressibility, psi−1 cf = aquifer rock compressibility, psi−1 Wi = initial volume of water in the aquifer, bbl pi = initial reservoir pressure, psi p = current reservoir pressure (pressure at oil − water contact), psi

• If the aquifer shape is radius:

𝑊𝑊𝑖𝑖 = � 𝜋𝜋(𝑟𝑟𝑎𝑎2 − 𝑟𝑟𝑒𝑒2)ℎ∅

5.615 �

(3.9) Where: ra = radius of the aquifer, ft r𝑒𝑒 = radius of the reservoir, ft h = thickness of the aquifer, ft ∅ = porosity of the aquifer

One of the simplest modifications is to include the fractional encroachment angle f in the equation, as illustrated in Figure 3.2.1, to give:

We = (𝑐𝑐𝑤𝑤 + ct)Wi𝑓𝑓(pi − p) (3.10) Where the fractional encroachment angle f is defined by:

𝑓𝑓 = (𝑒𝑒ncroachment angle)°

360° =

𝜃𝜃 360°

(3.11) This 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. Example 3.2 Calculate the cumulative water influx that result 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:

Solution: Step 1. Calculate the initial volume of water in the aquifer from Equation 3.9:

𝑊𝑊𝑖𝑖 = � 𝜋𝜋(𝑟𝑟𝑎𝑎2 − 𝑟𝑟𝑒𝑒2)ℎ∅

5.615 �

Figure 3.2. Radial aquifer geometries.

= � 𝜋𝜋(10 0002 − 26002)(25)(0.12)

5.615 � = 156.5 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀

Step 2. Determine the cumulative water influx by applying Equation 3.10: We = (𝑐𝑐𝑤𝑤 + ct)Wi𝑓𝑓(pi − p)

= (4.0 + 3.0)10−6(156.5 × 106) � 80

360 � (200) = 48689 bbl

3.2.2 The 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 ew can then be determined by applying Darcy’s equation:

𝑑𝑑𝑊𝑊𝑒𝑒 𝑑𝑑𝑑𝑑

= 𝑒𝑒𝑤𝑤 = � 0.00708𝑘𝑘ℎ

𝜇𝜇𝑤𝑤 ln � 𝑟𝑟𝑎𝑎 𝑟𝑟𝑒𝑒 � � (𝑝𝑝𝑖𝑖 − 𝑝𝑝)

(3.12) This relationship can be more conveniently expressed as:

𝑑𝑑𝑊𝑊𝑒𝑒 𝑑𝑑𝑑𝑑

= 𝑒𝑒𝑤𝑤 = 𝐶𝐶(𝑝𝑝𝑖𝑖 − 𝑝𝑝) (3.13) Where: ew = rate of water influx, bbl/day k = permeability of the aquifer, md h = thickness of the aquifer, ft ra = radius of the aquifer, ft rw = radius of the reservoir, ft t = time, days C = water influx constant, bbl/day/psi Example 3.3 The data given in Example 3.1 is used in this example: 𝑝𝑝𝑖𝑖 = 3500 𝑝𝑝𝑝𝑝𝑝𝑝,𝑝𝑝 = 3000 𝑝𝑝𝑝𝑝𝑝𝑝,𝑄𝑄𝑜𝑜 = 32 000 STB/day,𝐵𝐵𝑜𝑜 = 1.4 bbl/STB,𝐺𝐺𝐺𝐺𝐺𝐺 = 900 scf/STB, 𝐺𝐺𝑠𝑠 = 700 scf/STB,𝐵𝐵𝑔𝑔 = 0.00082 bbl/scf,𝑄𝑄𝑤𝑤 = 0,𝐵𝐵𝑤𝑤 = 1.0 bbl/STB. Calculate the Schilthuis water influx constant. Solution Step 1. Solve for the rate of water influx ew by using Equation 3.1: ew = QoBo + QgBg + QwBw = (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 3.13:

𝑑𝑑𝑊𝑊𝑒𝑒 𝑑𝑑𝑑𝑑

= 𝑒𝑒𝑤𝑤 = 𝐶𝐶(𝑝𝑝𝑖𝑖 − 𝑝𝑝) or:

𝐶𝐶 = 𝑒𝑒𝑤𝑤 (𝑝𝑝𝑖𝑖−𝑝𝑝)

= 50 048 3500−3000

= 100 bbl/day/psi In terms of the cumulative water influx We, Equation 3.13 is integrated to give the common Schilthuis expression for water influx as:

� 𝑑𝑑𝑊𝑊𝑒𝑒 = � 𝐶𝐶(𝑝𝑝𝑖𝑖 − 𝑝𝑝)𝑑𝑑𝑑𝑑 𝑡𝑡

𝑊𝑊𝑒𝑒

(3.14) or:

𝑊𝑊𝑒𝑒 = 𝐶𝐶 � (𝑝𝑝𝑖𝑖 − 𝑝𝑝)𝑑𝑑𝑑𝑑 𝑡𝑡

(3.15) Where: We = cumulative water influx, bbl C = water influx constant,bbl/day/psi t = time, days pi = initial reservoir pressure, psi p = pressure at the oil − water contact at time t, psi The area under the curve in Figure 3.3 is the integral ∫ (𝑝𝑝𝑖𝑖 − 𝑝𝑝)𝑑𝑑𝑑𝑑

𝑡𝑡 0 . This area at time t can be

determined numerically by using the trapezoidal rule (or any other numerical integration method) as:

� (𝑝𝑝𝑖𝑖 − 𝑝𝑝)𝑑𝑑𝑑𝑑 𝑡𝑡

0 = 𝑎𝑎𝑟𝑟𝑒𝑒𝑎𝑎𝐼𝐼 + 𝑎𝑎𝑟𝑟𝑒𝑒𝑎𝑎𝐼𝐼𝐼𝐼 + 𝑎𝑎𝑟𝑟𝑒𝑒𝑎𝑎𝐼𝐼𝐼𝐼𝐼𝐼 + ⋯

= � 𝑝𝑝𝑖𝑖 − 𝑝𝑝1

2 � (𝑑𝑑1 − 0) +

(𝑝𝑝𝑖𝑖 − 𝑝𝑝1) + (𝑝𝑝𝑖𝑖 − 𝑝𝑝2) 2

(𝑑𝑑2 − 𝑑𝑑1) + (𝑝𝑝𝑖𝑖 − 𝑝𝑝2) + (𝑝𝑝𝑖𝑖 − 𝑝𝑝3)

2 (𝑑𝑑3 − 𝑑𝑑2) + ⋯

Equation 3.15 can then be written as:

𝑊𝑊𝑒𝑒 = 𝐶𝐶�(∆𝑝𝑝)∆𝑑𝑑 𝑡𝑡

(3.16)

Figure 3.2. Radial aquifer geometries.

Example 3.4 The pressure history of a water drive oil reservoir is given below: The aquifer is under a steady-state flowing condition with an estimated water influx constant of 130 bbl/day/psi. Given the initial reservoir pressure is 3500 psi, calculate the cumulative water influx after 100, 200, 300, and 400 days using the steady-state model. Solution Step 1. Calculate the total pressure drop at each time t: Step 2. Calculate the cumulative water influx after 100 days:

𝑊𝑊𝑒𝑒 = 𝐶𝐶 �� 𝑝𝑝𝑖𝑖 − 𝑝𝑝1

2 � (𝑑𝑑1 − 0)� = 130 �

50 2 � (100 − 0)

= 325 000 𝑀𝑀𝑀𝑀𝑀𝑀 Step 3. Determine We after 200 days:

𝑊𝑊𝑒𝑒 = 𝐶𝐶 �� 𝑝𝑝𝑖𝑖 − 𝑝𝑝1

2 � (𝑑𝑑1 − 0) + �

(𝑝𝑝𝑖𝑖 − 𝑝𝑝1) + (𝑝𝑝𝑖𝑖 − 𝑝𝑝2) 2

� (𝑑𝑑2 − 𝑑𝑑1)�

= 130 �� 50 2 � (100 − 0) + �

50 + 90 2

� (200 − 100)� = 1 235 000 𝑀𝑀𝑀𝑀𝑀𝑀 Step 4. We after 300 days:

𝑊𝑊𝑒𝑒 = 𝐶𝐶 �� 𝑝𝑝𝑖𝑖 − 𝑝𝑝1

2 � (𝑑𝑑1 − 0) + �

(𝑝𝑝𝑖𝑖 − 𝑝𝑝1) + (𝑝𝑝𝑖𝑖 − 𝑝𝑝2) 2

� (𝑑𝑑2 − 𝑑𝑑1)

+ (𝑝𝑝𝑖𝑖 − 𝑝𝑝2) + (𝑝𝑝𝑖𝑖 − 𝑝𝑝3)

2 (𝑑𝑑3 − 𝑑𝑑2)�

= 130 �� 50 2 � (100 − 0) + �

50 + 90 2

� (200 − 100) + (120 + 90)

2 (300 − 200)�

= 2 600 000 𝑀𝑀𝑀𝑀𝑀𝑀 Step 5. Similarly, calculate We after 400 days:

𝑊𝑊𝑒𝑒 = 130 �2500 + 7000 + 10 500 + � 160 + 120

2 � (400 − 300)�

= 4 420 000 𝑀𝑀𝑀𝑀𝑀𝑀

3.2.3 The Hurst modified steady-state model One of the problems associated with the Schilthuis steadystate model is that as the water is drained from the aquifer, the aquifer drainage radius ra will increase as the time increases. Hurst (1943) proposed that the “apparent” aquifer radius ra would increase with time and, therefore, the dimensionless radius ra/re may be replaced with a time-dependent function as given below:

𝑟𝑟𝑎𝑎 𝑟𝑟𝑒𝑒

= 𝑎𝑎𝑑𝑑

(3.17) Substituting Equation 3.17 into Equation 3.12 gives:

𝑒𝑒𝑤𝑤 = 𝑑𝑑𝑊𝑊𝑒𝑒 𝑑𝑑𝑑𝑑

= � 0.00708𝑘𝑘ℎ 𝜇𝜇𝑤𝑤 ln(𝑎𝑎𝑑𝑑)

� (𝑝𝑝𝑖𝑖 − 𝑝𝑝)

(3.18) Simplified form:

𝑒𝑒𝑤𝑤 = 𝑑𝑑𝑊𝑊𝑒𝑒 𝑑𝑑𝑑𝑑

= 𝐶𝐶(𝑝𝑝𝑖𝑖 − 𝑝𝑝) 𝜇𝜇𝑤𝑤 ln(𝑎𝑎𝑑𝑑)

(3.19)

𝑊𝑊𝑒𝑒 = 𝐶𝐶 � (𝑝𝑝𝑖𝑖 − 𝑝𝑝) 𝜇𝜇𝑤𝑤 ln(𝑎𝑎𝑑𝑑)

𝑡𝑡

0 𝑑𝑑𝑑𝑑

(3.20)

𝑊𝑊𝑒𝑒 = 𝐶𝐶�� ∆𝑝𝑝

ln(𝑎𝑎𝑑𝑑) � ∆𝑑𝑑

𝑡𝑡

(3.21) There are two unknown constant in Hurst modified steady-state equation, a and C. The procedure for determining the constants a and C is based on expressing Equation 3.19 as a linear relationships:

� 𝑝𝑝𝑖𝑖 − 𝑝𝑝 𝑒𝑒𝑤𝑤

� = 1 𝐶𝐶

ln(𝑎𝑎𝑑𝑑)

or: 𝑝𝑝𝑖𝑖 − 𝑝𝑝 𝑒𝑒𝑤𝑤

= � 1 𝐶𝐶 � ln(𝑎𝑎) + �

1 𝐶𝐶 � ln(𝑑𝑑)

(3.22)

3.2.4 The van Everdingen and Hurst unsteady-state model The dimensionless form of diffusivity equation for the transient flow behavior in reservoirs or aquifers is:

𝜕𝜕2𝑃𝑃𝐷𝐷 𝜕𝜕𝑟𝑟𝐷𝐷

2 + 1 𝑟𝑟𝐷𝐷

𝜕𝜕𝑃𝑃𝐷𝐷 𝜕𝜕𝑟𝑟𝐷𝐷

= 𝜕𝜕𝑃𝑃𝐷𝐷 𝜕𝜕𝑡𝑡𝐷𝐷

(3.23)

Van Everdingen and Hurst (1949) proposed solutions to the dimensionless diffusivity equation for the following two reservoir– aquifer boundary conditions:

• constant terminal rate; • constant terminal pressure

They applying the Laplace transformation to the equation to determine the water influx in the following systems:

• edge-water drive system (radial system); • bottom-water drive system; • linear-water drive system.

Figure 3.3. Graphical determination of C and a.

Figure 3.4. Water influx into a cylindrical reservoir.

Edge-water drive Van Everdingen and Hurst assumed that the aquifer is characterized by:

• uniform thickness; • constant permeability; • uniform porosity; • constant rock compressibility; • constant water compressibility.

Van Everdingen and Hurst proposed a solution to the dimensionless diffusivity equation that utilizes the constant-terminal-pressure condition in addition to the following initial and outer boundary conditions: Initial conditions:

𝑝𝑝 = 𝑝𝑝𝑖𝑖 for all values of radius r Outer boundary conditions:

• For an infinite aquifer: 𝑝𝑝 = 𝑝𝑝𝑖𝑖 at r = ∞

• For a bounded aquifer 𝜕𝜕𝑝𝑝 𝜕𝜕𝑟𝑟

= 0 at r = ra

The authors expressed their mathematical relationship for calculating the water influx in the form of a dimensionless parameter called dimensionless water influx WeD. They also expressed the dimensionless water influx as a function of the dimensionless time tD and dimensionless radius rD; thus they made the solution to the diffusivity equation generalized and it can be applied to any aquifer where the flow of water into the reservoir is essentially radial. The solutions were derived for the cases of bounded aquifers and aquifers of infinite extent. The authors presented their solution in tabulated and graphical forms as reproduced here in Figures 3.6 through 3.8 and Tables 3.1 and 3.2. The two dimensionless parameters tD and rD are given by:

Figure 3.5. Idealized radial flow system (edge-water drive reservoir).

𝑑𝑑𝐷𝐷 = 6.328 × 10−3 𝑘𝑘𝑑𝑑

∅𝜇𝜇𝑤𝑤𝑐𝑐𝑡𝑡𝑟𝑟𝑒𝑒2

(3.24) 𝑟𝑟𝐷𝐷 =

𝑟𝑟𝑎𝑎 𝑟𝑟𝑒𝑒

(3.25) ct = cw + cf

(3.26)

Where: t = time, days k = permeability of the aquifer, md ∅ = porosity of the aquifer µw = viscosity of wain the aquifer, cp ra = radius of the aquifer, ft re = radius of the reservoir, ft ct = total compressibility coefficient, psi−1 cw = compressibility of the water, psi−1 cf = compressibility of the aquifer, psi−1 The water influx is then given by:

𝑊𝑊𝑒𝑒 = 𝐵𝐵∆𝑝𝑝𝑊𝑊𝑒𝑒𝐷𝐷 (3.27) with:

B = 1.119∅𝑐𝑐𝑡𝑡𝑟𝑟𝑒𝑒2ℎ (3.28) Where: We = cumulative water influx, bbl B = water influx constant, bbl/psi ∆p = pressure drop at the boundary, psi WeD = dimensionless water influx One of the simplest modifications is to introduce the encroachment angle, as a dimensionally parameter f, to the water influx constant B, as follows:

𝑓𝑓 = 𝜃𝜃

360

(3.29) B = 1.119∅𝑐𝑐𝑡𝑡𝑟𝑟𝑒𝑒2ℎ𝑓𝑓

(3.30)

Figure 3.6. Dimensionless water influx WeD for several values of re/rR.

Figure 3.7. Dimensionless water influx WeD for several values of re/rR.

Figure 3.7. Dimensionless water influx WeD for infinite aquifer.

Figure 3.8. Dimensionless water influx WeD for infinite aquifer.

Table 3.1 Dimensionless water influx WeD for infinite aquifer.

Table 3.2 Dimensionless water influx WeD for several values of re/rR.

Example 3.5 Calculate the water influx at the end of 1, 2, and 5 years into a circular reservoir with an aquifer of infinite extent, i.e.,reD = ∞. The initial and current reservoir pressures are 2500 and 2490 psi, respectively. The reservoir aquifer system has the following properties. Solution Step 1. Calculate the aquifer total compressibility coefficient ct from Equation 3.26:

ct = cw + cf = 0.7(10−6) + 0.3(10−6) = 1 × 10−6 𝑝𝑝𝑝𝑝𝑝𝑝−1

Step 2. Determine the water influx constant from Equation 3.30: B = 1.119∅𝑐𝑐𝑡𝑡𝑟𝑟𝑒𝑒2ℎ𝑓𝑓

= 1.119(0.2)(1 × 10−6)(2000)2(22.7) � 360 360

�

= 20.4 Step 3. Calculate the corresponding dimensionless time after 1, 2, and 5 years: 𝑑𝑑𝐷𝐷 = 6.328 × 10−3

𝑘𝑘𝑡𝑡 ∅𝜇𝜇𝑤𝑤𝑐𝑐𝑡𝑡𝑟𝑟𝑒𝑒2

= 6.328 × 10−3 100𝑑𝑑

(0.8)(0.2)(1 × 10−6)(2000)2

= 0.9888𝑑𝑑 Thus in tabular form: Step 4. Using Table 3.1, determine the dimensionless water influx WeD: Step 5. Calculate the cumulative water influx by applying Equation 3.27:

𝑊𝑊𝑒𝑒 = 𝐵𝐵∆𝑝𝑝𝑊𝑊𝑒𝑒𝐷𝐷 In calculating the cumulative water influx into a reservoir at successive intervals, it is necessary to calculate the total water influx from the beginning. This is required because of the different times during which the various pressure drops have been effective. The van Everdingen and Hurst computational procedure for determining the water influx as a function of time and pressure is summarized by the following steps and described conceptually in Figure 3.9:

Step 1. Assume that the boundary pressure has declined from its initial value of pi to p1 after t1

days. To determine the cumulative water influx in response to this first pressure drop ∆𝑝𝑝1 = 𝑝𝑝𝑖𝑖 − 𝑝𝑝1 can be simply calculated from Equation 3.27, or

𝑊𝑊𝑒𝑒 = 𝐵𝐵∆𝑝𝑝1(𝑊𝑊𝑒𝑒𝐷𝐷)𝑡𝑡1 where We is the cumulative water influx due to the first pressure drop ∆𝑝𝑝1 . The dimensionless water influx (𝑊𝑊𝑒𝑒𝐷𝐷)𝑡𝑡1 is evaluated by calculating the dimensionless time at 𝑑𝑑1 R days. This simple calculation step is shown by A in Figure 3.9.

Step 2. Let the boundary pressure decline again to 𝑝𝑝2 after 𝑑𝑑2 days with a pressure drop of

∆𝑝𝑝2 = 𝑝𝑝1 − 𝑝𝑝2. The total cumulative water influx after 𝑑𝑑2 days will result from the first pressure drop ∆𝑝𝑝1 and the second pressure drop ∆𝑝𝑝2, or:

We = water influx due to ∆p1 + water influx due to ∆p2 We = (We)∆p1 + (We)∆p2

Where: (We)∆p1 = B∆p1(WeD)t1

(We)∆p2 = B∆p2(WeD)t2− t1 The above relationships indicate that the effect of the first pressure drop ∆𝑝𝑝1 will continue for the entire time 𝑑𝑑2, while the effect of the second pressure drop will continue only for 𝑑𝑑2 − 𝑑𝑑1days as shown by B in Figure 3.9.

Figure 3.9. Illustration of the superposition concept.

Step 3. A third pressure drop of ∆𝑝𝑝3 = 𝑝𝑝2 − 𝑝𝑝3 would cause an additional water influx as illustrated by C in Figure 3.9. The total cumulative water influx can then be calculated from:

We = (We)∆p1 + (We)∆p2 + (We)∆p3 Where:

(We)∆p1 = B∆p1(WeD)t3 (We)∆p2 = B∆p2(WeD)t3− t1 (We)∆p3 = B∆p3(WeD)t3− t1

The van Everdingen and Hurst water influx relationship can then be expressed in a more generalized form as:

𝑊𝑊𝑒𝑒 = 𝐵𝐵�∆𝑝𝑝𝑊𝑊𝑒𝑒𝐷𝐷 (3.31) The authors also suggested that instead of using the entire pressure drop for the first period, a better approximation is to consider that one-half of the pressure drop, 1

2 (𝑝𝑝𝑖𝑖 − 𝑝𝑝1), is effective

during the entire first period. For the second period the effective pressure drop then is one-half of the pressure drop during the first period, 1

2 (𝑝𝑝𝑖𝑖 − 𝑝𝑝2), which simplifies to:

1 2

(𝑝𝑝𝑖𝑖 − 𝑝𝑝1) + 1 2

(𝑝𝑝1 − 𝑝𝑝2) = 1 2

(𝑝𝑝𝑖𝑖 − 𝑝𝑝2) Similarly, the effective pressure drop for use in the calculations for the third period would be one-half of the pressure drop during the second period, 1

2 (𝑝𝑝1 − 𝑝𝑝2), plus one-half of the pressure

drop during the third period, 1 2

(𝑝𝑝2 − 𝑝𝑝3) which simplifies to 1 2

(𝑝𝑝1 − 𝑝𝑝3)The time intervals must all be equal in order to preserve the accuracy of these modifications. Edwardson et al. (1962) developed three sets of simple polynomial expressions for calculating the dimensionless water influx WeD for infinite-acting aquifers.

• For 𝑑𝑑𝐷𝐷 < 0.01

𝑊𝑊𝑒𝑒𝐷𝐷 = � 𝑑𝑑𝐷𝐷 𝜋𝜋

(3.32) • For 0.01 < 𝑑𝑑𝐷𝐷 < 200

𝑊𝑊𝑒𝑒𝐷𝐷 = (1.2838�𝑑𝑑𝐷𝐷 + 1.19328𝑑𝑑𝐷𝐷 + 0.269872(𝑑𝑑𝐷𝐷) 3 2 + 0.00855294(𝑑𝑑𝐷𝐷)2)/(1

+ 0.616599�𝑑𝑑𝐷𝐷 + 0.0413008𝑑𝑑𝐷𝐷

(3.33)

• For 𝑑𝑑𝐷𝐷 > 200

𝑊𝑊𝑒𝑒𝐷𝐷 = −4.29881 + 2.02566𝑑𝑑𝐷𝐷

ln (𝑑𝑑𝐷𝐷)

(3.34)

Bottom- water drive

Coats (1962) presented a mathematical model that takes into account the vertical flow effects

from bottom-water aquifers. He correctly noted that in many cases reservoirs are situated on top

of an aquifer with a continuous horizontal interface between the reservoir fluid and the aquifer

water and with a significant aquifer thickness. He modified the diffusivity equation to account for

the vertical flow by including an additional term in the equation, to give:

𝜕𝜕2𝑝𝑝 𝜕𝜕𝑟𝑟2

+ 1 𝑟𝑟 𝜕𝜕𝑝𝑝 𝜕𝜕𝑟𝑟

+ 𝐹𝐹𝑘𝑘 𝜕𝜕2𝑝𝑝 𝜕𝜕𝑧𝑧2

= 𝜇𝜇∅𝑐𝑐 𝑘𝑘

𝜕𝜕𝑝𝑝 𝜕𝜕𝑑𝑑

(3.35)

where 𝐹𝐹𝑘𝑘is the ratio of vertical to horizontal permeability,

or:

𝐹𝐹𝑘𝑘 = 𝑘𝑘𝑣𝑣 − 𝑘𝑘ℎ

(3.36)

where:

kv = vertical permeability3.35

kh = horizontal permeability

Allardand Chen (1988): General solution for Equation 3.35:

𝑧𝑧𝐷𝐷 = ℎ

𝑟𝑟𝑒𝑒�𝐹𝐹𝑘𝑘

(3.37)

where:

zD = dimensionless vertical distance

h = aquifer thickness, ft

𝑊𝑊𝑒𝑒 = 𝐵𝐵�∆𝑝𝑝𝑊𝑊𝑒𝑒𝐷𝐷 (3.38)

B = 1.119∅𝑐𝑐𝑡𝑡𝑟𝑟𝑒𝑒2ℎ

(3.39)

Notes: the water influx constant B in bottom-water drive reservoir does not include the

encroachment angle 𝜽𝜽.

Allard and Chen tabulated the values of 𝑊𝑊𝑒𝑒𝐷𝐷 as a function of 𝑟𝑟𝐷𝐷, 𝑑𝑑𝐷𝐷, and 𝑧𝑧𝐷𝐷 . These values are

presented in Table 3.3 through 3.7.

Table 3.3 Dimensionless water influx WeD for infinite aquifer.

Table 3.4 Dimensionless water influx WeD for r𝐷𝐷 \ = 4.

Table 3.5 Dimensionless water influx WeD for r𝐷𝐷 \ = 6.

Table 3.6 Dimensionless water influx WeD for r𝐷𝐷 \ = 8.

Table 3.7 Dimensionless water influx WeD for r𝐷𝐷 \ = 10.

Example 3.6

An infinite-acting bottom-water aquifer is characterized by the following properties:

The boundary pressure history is given below:

Calculate the cumulative water influx as a function of time by using the bottom-water drive solution and compare with the edge-water drive approach. Solution

Step 1. Calculate the dimensionless radius for an infinite-acting aquifer:

𝑟𝑟𝐷𝐷 =∝

Step 2. Calculate 𝑧𝑧𝐷𝐷 from Equation 3.37:

𝑧𝑧𝐷𝐷 = ℎ

𝑟𝑟𝑒𝑒�𝐹𝐹𝑘𝑘

= 200

2000√0.04 = 0.5

Step 3. Calculate the water influx constant B:

B = 1.119∅𝑐𝑐𝑡𝑡𝑟𝑟𝑒𝑒2ℎ

= 1.119(0.1)(8 × 10−6)(2000)2(200) = 716 𝑀𝑀𝑀𝑀𝑀𝑀/𝑝𝑝𝑝𝑝𝑝𝑝

Step 4. Calculate the dimensionless time 𝑑𝑑𝐷𝐷

𝑑𝑑𝐷𝐷 = 6.328 × 10−3 𝑘𝑘𝑑𝑑

∅𝜇𝜇𝑤𝑤𝑐𝑐𝑡𝑡𝑟𝑟𝑒𝑒2

= 6.328 × 10−3 50 𝑑𝑑

(0.1)(0.395)(8 × 10−6)(2000)2 = 0.2503 𝑑𝑑

Step 5. Calculate the water influx by using the bottomwater model and edge-water model.

𝑊𝑊𝑒𝑒 = 𝐵𝐵�∆𝑝𝑝𝑊𝑊𝑒𝑒𝐷𝐷

Linear-water drive

𝑊𝑊𝑒𝑒 = 𝐵𝐵𝐿𝐿��∆𝑝𝑝𝑛𝑛�𝑑𝑑 − 𝑑𝑑𝑛𝑛� (3.40) Where:

BL = linear − aquifer water influx constant,bbl/psi/√time

t = time (any convenient time units, e. g. , months, years)

∆p = pressure drop as defined previously for the radial edge − water drive

3.2.5 The Fetkovich method To simplify water influx calculations further, Fetkovich proposed a model that uses a

pseudosteady-state aquifer PI and an aquifer material balance to represent the system

compressibility.

PI equation for the aquifer:

𝑒𝑒𝑤𝑤 = 𝑑𝑑𝑊𝑊𝑒𝑒 𝑑𝑑𝑑𝑑

= 𝐽𝐽(𝑝𝑝𝑎𝑎�� − 𝑝𝑝𝑟𝑟)

(3.41)

Where:

ew = water influx rate from aquifer,bbl/day

J = productivity index for the aquifer,bbl/day/psi

pa�� = average aquifer pressure, psi

pr = inner aquifer boundary pressure, psi

Aquifer material balance equation for a constant compressibility:

𝑊𝑊𝑒𝑒 = 𝑐𝑐𝑡𝑡𝑊𝑊𝑖𝑖(𝑝𝑝𝑟𝑟 − 𝑝𝑝𝑎𝑎��)𝑓𝑓

(3.42)

Where:

Wi = initial volume of water in the aquifer, bbl

ct = total aquifer compressibility, cw + cf, psi−1

pi = initial pressure of the aquifer, psi

f = θ

360

Equation 3.42 suggests that the maximum possible water influx occurs if 𝑝𝑝𝑎𝑎�� = 0, 𝑜𝑜𝑟𝑟:

𝑊𝑊𝑒𝑒𝑖𝑖 = 𝑐𝑐𝑡𝑡𝑊𝑊𝑖𝑖𝑝𝑝𝑖𝑖𝑓𝑓

(3.43)

Where:

Wei = maximum water influx, bbl

Combining Equation 3.43 with 3.42 gives:

𝑝𝑝𝑎𝑎�� = 𝑝𝑝𝑖𝑖 �1 − 𝑊𝑊𝑒𝑒

𝑐𝑐𝑡𝑡𝑊𝑊𝑖𝑖𝑝𝑝𝑖𝑖 � = 𝑝𝑝𝑖𝑖 �1 −

𝑊𝑊𝑒𝑒 𝑊𝑊𝑒𝑒𝑖𝑖

�

(3.44)

Differentiating Equation 3.44 with respect to time gives: 𝑑𝑑𝑊𝑊𝑒𝑒 𝑑𝑑𝑑𝑑

= − 𝑊𝑊𝑒𝑒𝑖𝑖 𝑝𝑝𝑖𝑖

𝑑𝑑𝑝𝑝𝑎𝑎�� 𝑑𝑑𝑑𝑑

(3.45)

Fetkovich combined Equation 3.45 with 3.41and integrated to give the following form:

𝑊𝑊𝑒𝑒 = 𝑊𝑊𝑒𝑒𝑖𝑖 𝑝𝑝𝑖𝑖

(𝑝𝑝𝑖𝑖 − 𝑝𝑝𝑟𝑟)𝑒𝑒𝑒𝑒𝑝𝑝 � −𝐽𝐽𝑝𝑝𝑖𝑖𝑑𝑑 𝑊𝑊𝑒𝑒𝑖𝑖

�

(3.46)

Where:

We = cumulative water influx, bbl

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

t = time, days

Fetkovich suggested that, if the reservoir–aquifer boundary pressure history is divided into a

finite number of time intervals, the incremental water influx during the nth interval is:

(∆𝑊𝑊𝑒𝑒)𝑛𝑛 = 𝑊𝑊𝑒𝑒𝑖𝑖 𝑝𝑝𝑖𝑖

[(𝑝𝑝𝑎𝑎��)𝑛𝑛−1 − (𝑝𝑝𝑟𝑟��)𝑛𝑛−1] �1 − 𝑒𝑒𝑒𝑒𝑝𝑝 �− 𝐽𝐽𝑝𝑝𝑖𝑖∆𝑑𝑑𝑛𝑛 𝑊𝑊𝑒𝑒𝑖𝑖

��

(3.47)

where (𝑝𝑝𝑎𝑎��)𝑛𝑛−1 is the average aquifer pressure at the end of the previous time step. This average

pressure is calculated from Equation 3.44 as:

(𝑝𝑝𝑎𝑎��)𝑛𝑛−1 = 𝑝𝑝𝑖𝑖 �1 − (𝑊𝑊𝑒𝑒)𝑛𝑛−1 𝑊𝑊𝑒𝑒𝑖𝑖

�

(3.48)

The average reservoir boundary pressure (𝑝𝑝𝑟𝑟��)𝑛𝑛 is estimated from:

(𝑝𝑝𝑟𝑟��)𝑛𝑛 = (𝑝𝑝𝑟𝑟)𝑛𝑛 + (𝑝𝑝𝑟𝑟)𝑛𝑛−1

(3.49)

The productivity index J of the aquifer is given by the following expressions:

Where:

w = width of the linear aquifer, ft

L = length of the linear aquifer, ft

rD = dimensionless radius, ra/re

k = permeability of the aquifer, md

t = time, days

θ = encroachment angle

h = thicknes of the aquifer

f = θ

360

The following steps describe the methodology of using Fetkovich model in predicting the

cumulative water influx:

Step 1. Calculate the initial volume of water in the aquifer from:

𝑊𝑊𝑖𝑖 = 𝜋𝜋

5.615 (𝑟𝑟𝑎𝑎2 − 𝑟𝑟𝑒𝑒2)ℎ∅

Step 2. Calculate the maximum possible water influx Wei by applying Equation 3.43, or:

𝑊𝑊𝑒𝑒𝑖𝑖 = 𝑐𝑐𝑡𝑡𝑊𝑊𝑖𝑖𝑝𝑝𝑖𝑖𝑓𝑓

Step 3. Calculate the productivity index J based on the boundary conditions and aquifer

geometry

Step 4. Calculate the incremental water influx (∆𝑊𝑊𝑒𝑒)𝑛𝑛 from the aquifer during the nth time

interval by using Equation 3.47. For example, during the first time step ∆𝑑𝑑1:

(∆𝑊𝑊𝑒𝑒)1 = 𝑊𝑊𝑒𝑒𝑖𝑖 𝑝𝑝𝑖𝑖

[𝑝𝑝𝑖𝑖 − (𝑝𝑝𝑟𝑟��)1] �1 − 𝑒𝑒𝑒𝑒𝑝𝑝 �− 𝐽𝐽𝑝𝑝𝑖𝑖∆𝑑𝑑1 𝑊𝑊𝑒𝑒𝑖𝑖

��

With:

(𝑝𝑝𝑟𝑟��)1 = 𝑝𝑝𝑖𝑖 + (𝑝𝑝𝑟𝑟)1

For the second time interval ∆𝑑𝑑2:

(∆𝑊𝑊𝑒𝑒)2 = 𝑊𝑊𝑒𝑒𝑖𝑖 𝑝𝑝𝑖𝑖

[(𝑝𝑝𝑎𝑎��)1 − (𝑝𝑝𝑟𝑟��)2] �1 − 𝑒𝑒𝑒𝑒𝑝𝑝 �− 𝐽𝐽𝑝𝑝𝑖𝑖∆𝑑𝑑2 𝑊𝑊𝑒𝑒𝑖𝑖

��

where (𝑝𝑝𝑎𝑎��)1is the average aquifer pressure at the end of the first period and removing (∆𝑊𝑊𝑒𝑒)11

barrels of water from the aquifer to the reservoir. From Equation 3.48:

(𝑝𝑝𝑎𝑎��)1 = 𝑝𝑝𝑖𝑖 �1 − (∆𝑊𝑊𝑒𝑒)1 𝑊𝑊𝑒𝑒𝑖𝑖

�

Step 5. Calculate the cumulative (total) water influx at the end of any time period from:

𝑊𝑊𝑒𝑒 = �(∆𝑊𝑊𝑒𝑒)𝑖𝑖

𝑛𝑛

𝑖𝑖=1

Chapter 3 Water Influx Calculation (pp. 149-185)

3.1 Classification of aquifers
3.2 Water influx models

3.2.1 The pot aquifer model
3.2.2 The Schilthuis steady-state model
3.2.3 The Hurst modified steady-state model
3.2.4 The van Everdingen and Hurst unsteady-state model
3.2.5 The Fetkovich method