3- transient flow equation and solutions

Well Test Analysis, © UTP – MAY 2011

AP. Dr. Muhannad Talib Shuker GPE Department


Transient Flow Equation In the course of development of the transient flow equation three independent equations will be used:

1. Continuity equation: material balance equation which

states conservation of mass

2. Equation of motion : Darcy’s equation which defines fluid flow through porous media

3. Equation of state : Compressibility equation which describes changes in the fluid volume as a function of pressure

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Continuity Equation (1/6)

Schematic of reservoir

Well

Formation thickness

Reservoir boundary



rw

r

r+r

r

h

Flow Element (control volume)

Mass in Mass out

Making a mass balance over the volume element during a time period of t

Mass entering

volume element

during t

Mass leaving

volume element

during t

Mass accumulated

in the volume

element during t

=

Under the steady-state flow conditions, the same amount of fluid enters and leaves the flow element. However they are not equal to each other during unsteady-state (transient) flow conditions. Nevertheless, the mass must be conserved in both cases.

(1)

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tvhrrMass rrin 2

MASS IN MASS OUT MASS

ACCUMULATED

=

where;

= velocity of flowing fluid

= fluid density at r+r

A = area at r+r

t = time interval

The area of the volume element at the entry:

A = 2(r+r)h

tAvMass rrin

(3)

(2)

tvrhMass rout 2

similarly;

(4)


tttAcc rrhMass 2.

Mass accumulated = mass at time t – mass at time t

(5)

tt rrhMass 2 (6)


On the other hand;

Substituting in above definition:

tttt rrhMass 2

(7)


Continuity Equation (5/6) Substituting Equations 3, 4 and 7 in equation 1:

(8) tttrrr rhrtvhrtvrrh 222

tttrrr rhrvrvrrth 22

Rearranging equation 8:

(9)

Dividing the both sides of the equation 9 by 2hr t :

(10)

trh

rhr

trh

vrvrrth tttrrr

2

2

2

2

Hence finally:

(11)

tr

vrvrr

r

tttrrr

1



Let’s take limits as both r and t approaches zero;

(12)

or:

(13)

tr

vrvrr

r

ttt

t

rrr

r

00lim

1lim

t

vrrr

1

where;

= velocity of flowing fluid

= fluid density at r+r

= porosity

Continuity equation


Equation of Motion

Darcy’s law;

(14)

(15)

r

PkAq

definition of velocity;

A

qv

Substituting in equation 14;

r

Pkv

(16)

where;

k = permeability

= fluid viscosity


Transient Flow Equation (1/2) Substituting equation 16 in equation 13;

(17)

Expanding the right hand side of equation 13:

(18)

tr

Pkr

rr

1

ttt

Porosity is related to the formation compressibility by:

Pcr

1

(19)

(20)

Applying the chain rule of differentiation to /t:

t

P

Pt


Transient Flow Equation (2/2) Substituting equation 19 in equation 20;

(21)

substituting this into equation 18 :

(22)

t

Pc

ttr

Finally substituting equation 22 into equation 17:

(23)

Equation 23 is the general partial differential equation that describes the flow of any type of fluid in porous medium.

t

Pc

tr

t

Pc

tr

Pkr

rrr

1

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Transient Flow Equation for Slightly Compressible Fluids (1/6)

Let us simplify equation 23 by assuming permeability and viscosity are constants with respect to pressure, time and distance;

(24)

Expanding above equation gives :

(25)

Applying the chain rule in the the above equation:

(26)

t

Pc

tr

Pr

rr

kr

tt

Pc

rr

P

r

P

r

P

r

kr

2

2

Pt

P

t

Pc

Pr

P

r

P

r

P

r

kr

2

2

2



Dividing the both sides of the above equation by ;

(27)

Remembering fluid compressibility is related to its density by:

(28)

Combining equations 27 and 28:

(29)

Pt

P

t

Pc

Pr

P

r

P

r

P

r

kr

1112

2

2

Pc f

1

t

Pc

t

Pc

r

Pc

r

P

r

P

r

kfrf

2

2

21



The square of pressure gradient over distance can be considered very small and negligible which yields;

(30)

Defining the total compressibility ct:

(31)

Substituting equations 31 in 30 and rearranging:

(32)

t

Pcc

r

P

r

P

r

kfr

2

21

frt ccc

t

P

k

c

r

P

rr

P t

12

2



Equation 32 is called as DIFFUSIVITY EQUATION and is considered one of the most important and widely used mathematical expression in Petroleum Engineering. The diffusivity equation can be rearranged with the inclusion of field units and is used in the analysis of well testing data where time is commonly in hours.

(32)

t

P

k

c

r

P

rr

P t

12

2

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Where; k = permeability, md r = radial position, ft P = pressure, psia ct = total compressibility, psi-1

t = time, hours = porosity, fraction = viscosity, cp

(33)

t

P

k

c

r

P

rr

P t

0002637.0

12

2

Assumptions inherent in equation 33 (2,3,4): 1. Radial flow into well opened entire thickness of

formation 2. Laminar flow (Darcy) 3. Homogeneous and isotropic porous medium 4. Porous medium has constant permeability and

compressibility 5. Gravity effects are negligible 6. Isothermal conditions 7. Fluid has small and constant compressibility 8. Fluid viscosity is constant

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Diffusivity equation is generally is shown as:

(34)

t

P

k

c

r

Pr

rr

t

1


Solutions to Diffusivity Equation

There are three basic cases of interest towards the solution of Diffusivity Equation:

1. Constant production rate, Infinite Reservoir

2. Constant production rate, no-flow at the outer boundary

3. Constant production, constant pressure at the outer boundary

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Initial and Boundary Conditions for Constant Production Rate, Infinite Boundary

(34)

t

P

k

c

r

Pr

rr

t

1Equation:

Initial Condition: iPrP 0, (35)

Boundary Conditions:

Inner Boundary

wrr

Pr

khq

2(36)

Outer Boundary iPtrP , (37)

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Pressure at any location = Pi (Initial Pressure) More to do with time rather than radius.

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(34)

t

P

k

c

r

Pr

rr

t

1Equation:



Inner Boundary

wrr

Pr

khq

2(36)

Outer Boundary 0

err

P(38)

Initial and Boundary Conditions for Constant Production Rate, No-Flow Boundary

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Pressure at any radius = Pi Has to do more with time, rather than radius.

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Pressure difference across the boundary is equal to zero. Therefore, no flow across boundary.


(34)

t

P

k

c

r

Pr

rr

t

1Equation:



Inner Boundary

wrr

Pr

khq

2(36)

Outer Boundary (39) ie PtrrP ,

Initial and Boundary Conditions for Constant Production Rate, Constant Pressure Boundary

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Pressure at any location is equal to initial pressure. Has to do more with time rather than radius.

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Pressure reaching the boundary radius (re, location) at any time will be equal to Pi.


Dimensionless Form of Diffusivity Equation

(34)

t

P

k

c

r

Pr

rr

t

1

Most of the time dimensionless groups are used to express Diffusivity equation more simply. Many well test analysis techniques use dimensionless variables to depict general trends rather than working with specific parameters (like k, t, rw, re and h).

One must define dimensionless groups to be able to convert the diffusivity equation below to its dimensionless form.


Dimensionless Groups for Diffusivity Equation

(40)

Dimensionless Pressure:

PPqB

khP iD

Dimensionless Radius:

w

Dr

rr (41)

Dimensionless time:

2

wt

Drc

ktt

(42)


Dimensionless form of Diffusivity Equation

The diffusivity equation then can be expressed in dimensionless form by utilizing the dimensionless groups as:

(43)

D

D

D

DD

DD t

P

r

Pr

rr

1

Now it is needed to express the boundary and initial conditions in dimensionless forms.

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Dimensionless Boundary and Initial Conditions for the Diffusivity Equation for Constant Rate, Infinite Reservoir

(44)

Initial Condition:

00, DDD trP

Outer Boundary:

(45)

Inner Boundary:

1

1

DrD

D

r

P(46)

0, DDD trP

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Dimensionless Boundary and Initial Conditions for the Diffusivity Equation for Constant Rate, No-Flow Boundary

(47)

Initial Condition:

00, DDD trP

Outer Boundary:

(48)

Inner Boundary:

(49)

0

eDrD

D

r

P

1

1

DrD

D

r

P

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Dimensionless Boundary and Initial Conditions for the Diffusivity Equation for Constant Rate, Constant Pressure Boundary

(50)

Initial Condition:

00, DDD trP

Outer Boundary:

(51)

Inner Boundary:

(52)

1

1

DrD

D

r

P

0, DeDDD trrP

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ASSIGNMENT

Prove that the below partial differential equation is the dimensionless form of Diffusivity Equation.

Prove also that the below initial and boundary conditions are the dimensionless forms of Constant Rate Infinite Boundary case.

Initial Condition: 00, DDD trP

Outer Boundary:

Inner Boundary:

1

1

DrD

D

r

P

0, DDD trP

D

D

D

DD

DD t

P

r

Pr

rr

1


Solution to Diffusivity Equation for Constant Line Source

Production Rate Infinite Boundary Case Diffusivity Equation:

Initial and Boundary Conditions

Initial Condition: 00, DDD trP

Outer Boundary:

Inner Boundary: 1lim0

D

DrD

DD

r r

Pr

0, DDD trP

D

D

D

DD

DD t

P

r

Pr

rr

1(43)

(44)

(45)

(46)


D

DiD

t

rEP

42

1 2

This is the line source solution of the Diffusivity Equation for constant production rate and infinite reservoir case.


Solution to Diffusivity Equation for Constant Line Source Production Rate Infinite Boundary Case

For

(83)

Exponential integral can be approximated as

80907.0ln

2

12

D

DD

r

tP (84)

01.04

2

D

D

t

r


Solution to Diffusivity Equation for Constant Line Source Production Rate Infinite Boundary Case

And the dimensionless pressure at the wellbore

(85)

Exponential integral can be approximated as

80907.0ln2

1 DwellboreD tP (86)

1Dr

This is the solution for dimensionless bottom hole well pressure for constant production rate infinite reservoir case.


References 1. Dominique Bourdet, “Well Test Analysis: The Use of Advanced Interpretation

Models”, Handbook of Petroleum Exploration and Production, 3. Elsevier, 2002 (Chapter 1)

2. Tarek Ahmed, and Paul D. McKinney, “Advanced Reservoir Engineering”, Elsevier, 2005 (Chapter 1)

3. John Lee, John B. Rollins, and John P. Spivey, “Pressure Transient Testing”, SPE Textbook series Vol. 9.

4. C. S. Matthews, and D. G. Russell, “Pressure Buildup and Flow Tests in Wells”, SPE Monograph Vol. 1

3- transient flow equation and solutions

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