2-Solution of the Diffusivity Equation

Solution of the

Diffusivity Equation

Dr. Mohammed Abdalla Ayoub

U n i v e r s i t i Te k n o l o g i P E T R O N A S , B a n d a r S e r i I s k a n d a r,

3 1 7 5 0 Tr o n o h , P e r a k , M a l a y s i a | Te l : + 6 0 5 3 6 8 7 0 8 6 |

F a x : + 6 0 5 3 6 5 5 6 7 0

E - ma i l :

a b d a l l a . a y o u b @ u t p . e d u . my

Reservoir Engineering II, PDB(3023)

Learning objectives

Recap on the Diffusivity Equation

Contents

Pressure distribution as a function of time

Transient (unsteady-state) flow

Derivation of the Diffusivity Equation

Radial Flow of Slightly Compressible Fluids

Solutions to the Diffusivity Equation

Solutions to the Diffusivity Equation

The diffusivity equation:

Where,

Diffusivity constant

Solutions to the Diffusivity Equation

The three most common flow conditions are:

◦ Steady state

◦ P=const. at all r and t. ∂P/ ∂t=0

◦ This is the case where natural water influx or injection of some fluid

◦ Semi-steady state

◦ The effect of the outer boundary has been felt ◦ ∂P/ ∂r=0 at r=re

◦ ∂P/ ∂t ≈ const. for all r and t

◦ Transient – early time

◦ No boundary effect (infinite acting reservoir) ◦ P = f(r, t) and ∂P/ ∂r=q(r, t)

Radial Flow of Slightly Compressible Fluids cont’d

The diffusivity equation as represented by Equation 18 is essentially

designed to determine the pressure as a function of time t and position r.

◦

P=f(r, t)

Assumptions and limitations used in developing Equation 18:

◦

Homogeneous and isotropic porous medium

◦

Uniform thickness

◦

Single phase flow

◦

Laminar flow

Radial Flow of Slightly Compressible Fluids cont’d

For steady state flow:

◦ ∂P/∂t=0

Hence, eq. (18) reduces to:

Equation 19 is

Laplace’s equation

for steady-state flow.

Solutions to the Diffusivity Equation

To obtain a solution to the diffusivity equation (Equation 18), it

is necessary to specify:

◦

An initial condition

◦ Uniform pressure P_i when production begins.

◦

Impose two boundary conditions. (outer and inner boundary)

◦ The well is producing at a constant production rate.

Solutions to the Diffusivity Equation cont’d

Based on the boundary conditions imposed on,

there are two generalized solutions to the

diffusivity equation:

◦

Constant-terminal-pressure solution

◦

Constant-terminal-rate solution

Constant-terminal-pressure solution

The pressure is known to be constant at some particular radius and the

solution is designed to provide the cumulative fluid movement across

the specified radius (boundary).

It is widely used in water influx calculations.

A detailed description of the solution and its practical reservoir

engineering applications will be discussed in the

water influx chapter

.

Solution to Diffusivity Equation

Constant-terminal-pressure solution

is designed to provide the cumulative

flow at any particular time for a reservoir in which the pressure at one

boundary of the reservoir is held constant. This technique is frequently used

in water influx calculations in gas and oil reservoirs.

Constant-terminal-rate solution

of the radial diffusivity equation solves for

the pressure change throughout the radial system providing that the flow

rate is held constant at one terminal end of the radial system, i.e., at the

producing well.

These are two commonly used forms of the constant-terminal-rate solution:

The EI-function solution



Infinite acting reservoir



well is producing at a constant flow rate.



uniform reservoir pressure, P

_i

, when production begins.



wellbore radius of r

_w

, is centered in a cylindrical reservoir of radius r

_e

.



No flow across the outer boundary.

13

The Exponential integral (Ei) can be used to estimate the pressure change throughout the radial system of the reservoir. However, the following assumptions must be considered:

--- (20)

where p (r,t) = pressure at radius r from the well after t hours

t = time, hrs

k = permeability, md. Q_o= flow rate, STB/day

























kt

r

c

E

kh

B

Q

p

t

r

p

_i o o o _i o t 2

948

6

.

70

)

,

(





This equation can be used to calculate pressures at

any distance r, from the wellbore when the well is

opened for production of:

The reservoir is assumed to be infinite-acting if:

15 --- (20)

(x)

























kt

r

c

E

kh

B

Q

p

t

r

p

_i o o o _i o t 2

948

6

.

70

)

,

(





When X is less than 0.02, the exponential integral E_i can be calculated using the following Equation:

• when x > 10.0, the E_i (−x) can be considered zero when x is in the range of 0.01 < x < 3.0

Example

17 17

A single well is producing oil at a constant rate of 5500 STB/D. Calculate the pressure at a location 500 feet away from the well after 5 days, and 10 days of production. The drainage boundary of the well is located 1000 feet away. Other rock and fluid properties of the reservoir are as follows:

Step 1. use the following equation to check that the well has been on production for sufficient length of time so that the accuracy is not affected.

The length of time the well is on production must be greater than the time required.

Solution

Next, it is important to check that the reservoir is infinite-acting after 10 days of production. This time of production must be less than the time calculated from the following equation:

The reservoir will cease to be infinite-acting after about 19.3 days of production. Hence, after 10 days of production, the reservoir is still infinite-acting.

19 Step 2: Calculate reservoir pressure after 5 days, and 10 days of

production at 500 feet from well. Substituting Eq. (20) with the data provided gives:

























kt

r

c

E

kh

B

Q

p

t

r

p

o t i o o o i 2

948

*

6

.

70

)

,

(































t

E

t

r

p

_i

*

24

*

7

)

500

(

*

10

*

4

*

5

.

4

*

19

.

0

*

948

*

345

*

7

5

.

4

*

1

.

1

*

5500

*

6

.

70

19250

)

,

(

2 6













t

E

t

r

p

(

,

)

19250

795

.

89

*

_i

4

.

8246

At 5 days of production, the Ei (-x) is:

(

0

.

96

)

5

8246

.

4

8246

.

4

_

_











 











i i i

E

E

t

E

Because x is higher than 0.1, so the value of Ei(-x) can be obtained from the table which is equal to 0.24.

24

.

0

*

89

.

795

19250

)

,

(

r

t





p

psia

t

r

p

(

,

)



19

,

059

)

48

.

0

(

10

8246

.

4

8246

.

4

_

_











 











i i i

E

E

t

E

Because x is higher than 0.1, so the value of Ei(-x) can be obtained from the table which is equal to 0.585.

585

.

0

*

89

.

795

19250

)

,

(

r

t





p

psia

t

r

p

(

,

)



18

,

784

An oil well is producing at a constant flow rate of 300 STB/day under unsteady-state flow conditions. The reservoir has the following rock and fluid properties:

1. Calculate pressure at radii of 0.25, 5, 10, 50, 100, 500, 1000, 1500, 2000, and 2500 feet, for 1 hour.

Plot the results as:

A. Pressure versus logarithm of radius B. Pressure versus radius

21

























kt

r

c

E

kh

B

Q

p

t

r

p

_i o o o _i o t 2

948

6

.

70

)

,

(





Example

Solution

• Step 1. From Equation (20) :

• Step 2. Perform the required calculations after one hour in the following tabulated form:

0.01 < x < 3.0, use Ei chart

x >10.9,

• Step 3. Show results of the calculation graphically as illustrated in Figures below:

Pressure profiles as a function of time

Pressure profiles as a function of time on a semi-log scale

23

• When the parameter x in the E_i-function is less than 0.01, Eq. (21) in the form of the following equation can also be used to calculate the P_wf:

--- (21)

• For most of the transient flow calculations, engineers are primarily concerned with the behavior of the bottom-hole flowing pressure at the wellbore, i.e., r = r_w

25

--- (22)

where k = permeability, md t = time, hr

c_t = total compressibility, psi−1

• Equations (21) and (22) cannot be used until the flow time t exceeds the limit imposed by: --- (23) where t = time, hr k = permeability, md































162

.

6

log

₂

3

.

23

w t o o o o i f w

r

c

kt

kh

B

Q

p

p





Solution, cont…

An oil well is producing at a constant flow rate of 300 STB/day under unsteady-state flow conditions. Estimate the bottom-hole flowing pressure after 10 hours of production.The reservoir has the following rock and fluid properties:

27 Equation 22 can be used to calculate pwf only if the time exceeds the time limit

Since the specified time of 10 hr is greater than 0.000267 hrs, the pwf can then be estimated by Equation 22.

The Dimensionless Pressure Drop (p

_D

) Solution

The solution describes the pressure drop as a function of time and radius for fixed values of external radius r_e and well bore radius r_w, rock and fluid properties. It is expressed in term of dimensionless variables as:

P

_D

= f(t

_D

, r

_D

, r

_eD

)

t

D = dimensionless time

r

_D = dimensionless radius

r

_eD = dimensionless external radius

This solution to the diffusivity equation at constant terminal rate under the stated initial and boundary conditions was reported by van Everdingen and Hurst. The solution is presented in terms of dimensionless variables based on Darcy’s equation in a radial form.

Rearrange Dimensionless external radius r_eD Dimensionless pressure P_D 29

The P_D is presented in terms of dimensionless variables based on Darcy’s equation for a radial flow system.

In Dimensionless format the previous equation can be written as:

--- (25)

Based on total drainage area, t_DA

In transient flow analysis, the dimensionless pressure

P

_D

= f(t

_D

, r

_D

, r

_eD

)

If the reservoir is fixed in size. i.e. r_eD is a particular value, then the dimensionless pressure drop P_D at the wellbore is only a function of dimensionless time, P_D = f(t_D)

The dimensionless pressure variable P_D is defined as: --- (24)

















kh

B

Q

p

p

P

o o o wf e D

00708

.

0



The dimensionless time variable

31

--- (25a)

where A = total drainage area = π r_e2

r_e = drainage radius, ft r_w = wellbore radius, ft

r_eD = dimensionless external radius

r_eD = dimensionless external radius

The Dimensionless Pressure Drop (p

_D

) Solution

        kh B Q p p P o o o wf e D 00708 . 0 

The dimensions units can be obtained by substituting the appropriate dimensions such as M for mass, L for length and T for the time for each of the quantities in the original Darcy low:















L L LT M L L T L L T M P_D * * 00708 . 0 / / / / 2 3 3 3 2  

After cancelation, the P_D is dimensionless

L

T

M

L

T

ML

area

force

p





_{2 2}



₂



T

L

Q

3



2

L

A



LT

M





2

L

K



L

h



3 3

L

L

B

_o



Dimensionless Diffusivity Equation

33 2

000264

.

0

w t D

r

c

kt

t











2

 

2 2

/

/

*

*

000264

.

0

L

M

LT

LT

M

T

L

t

_D





After cancelation, the t_D is dimensionless

--- (26)

Introduce p_D, t_D, and r_D into the diffusivity equation:

Assumptions:

 Perfectly radial reservoir system

 The producing well is in the center and producing at a constant production rate of Q  Uniform pressure pi throughout the reservoir before production

 No flow across the external radius r_e

Van Everdingen and Hurst (1949) proposed an analytical solution to

the above equation For an Infinite-acting reservoir as:

35 For an infinite-acting reservoir, i.e., re_D = ∞,

the dimensionless pressure drop function p_D is strictly a function of the dimensionless time t_D, or:

Chatas and Lee tabulated the p_D values for the infinite-acting reservoir as shown here.

For t_D < 0.01 For 100 < t_D < 0.25 r2 eD For 0.02 < t_D < 1000 --- (28) --- (29)

Infinite-Acting Reservoir

The computational procedure of using the p_D-function in determining the bottom-hole flowing pressure:

Calculate the dimensionless time t_D

1

Calculate the dimensionless radius r_eD

2

Using the calculated values of t_D and r_eD, determine the corresponding pressure function p_D from the appropriate table or equation

3

Solve for the pressure at the desired radius, i.e., r_w

4

It should be pointed out that, for an infinite acting reservoir with t_D > 100, the p_D-function is related to

the Ei-function by the following relation:

_{--- (27)}

--- (27a)

A well is producing at a constant flow rate of 300 STB/day under unsteady-state flow condition. Assuming an infinite acting reservoir, calculate the bottom-hole flowing pressure after one hour of production by using the dimensionless pressure approach. Assume the reservoir boundary is 1000 ft and the reservoir has the following rock and fluid properties

Calculate the dimensionless time t_D

37

Solution, cont…

Since 100 < tD < 0.25 r2

eD, use Equation 29 to calculate the dimensionless pressure

drop function

Example 5

39 39

An oil well located at X field is supposed to produce at a constant production rate from a drainage boundary of 304.8 meters away from the wellbore. After 10 days of production, the bottom hole flowing pressure was found to be 8134 psia at a wellbore radius of 0.5 ft. Estimate the well production rate in bbl/day. other rock and fluid properties of the reservoir are as follows:

Solution

Step 1: Calculate the dimensionless time t_D using Eq. 25.

Step 2: Calculate the dimensionless radius re_D .

147

,

518

)

5

.

0

(

*

10

*

4

*

5

.

4

*

19

.

0

10

*

24

*

7

*

000263

.

0

000263

.

0

2 6 2







_ w t D

r

c

kt

t



2000

5

.

0

2808

.

3

*

8

.

304

_





w e eD

r

r

r

Step 3: Calculate the dimensionless pressure P_D using Eq. 29.



ln

0

.

80907



6

.

9835

5

.

0







_D D

t

P

Solution, cont…

41

Step 4: Calculate the oil production rate using Eq. 27.

day

STB

Q

P

kh

B

Q

P

P

_{wf i} o o o _D

5500

/

00708

.

0





















Step 5: convert the oil production rate from STB/day to bbl/day

day

bbl

STB

bbl

day

STB

Q



5500

/

*

1

.

1

/



6050

/

The arrival of the pressure disturbance at the well drainage boundary marks the end of the transient flow period and the beginning of the semi (pseudo)-steady state.

During this flow state, the reservoir boundaries and the shape of the drainage area influence the wellbore pressure response as well as the behavior of the pressure distribution throughout the reservoir.

There is a short period of time that separates the transient state from the semi-steady state that is called late-transient state. Due to its complexity and short duration, the late transient flow is not used in practical well test analysis.

43

Finite-Radial Reservoir

where

--- (30)

For 25 < t_D and 0.25 r2 eD < tD

--- (31)

when r2 eD >> 1

--- (32)

Example 6

A well is producing at a constant flow rate of 300 STB/day under unsteady-state

flow condition. The reservoir has the following rock and fluid properties

Assuming an infinite acting reservoir, i.e., r_eD = ∞, calculate the bottom-hole flowing pressure after one hour of production by using the dimensionless pressure approach.

Solution

Solution, cont…

45

Step 2. Since t_D > 100, use Equation:

 The main difference between the two formulations is that

the p

_D

-function can be only used to calculate the pressure at radius r

when the flow rate Q is constant and known. In that case, the

p

_D

-function application is essentially

restricted to the wellbore

radius because the rate is usually known.

 On the other hand, the Ei-function approach can be used to

calculate the pressure at any radius in the reservoir by using the

well flow rate Q.

47