Transient Flow Equation and Solutios

AP Dr Muhannad Talib ShukerAP. Dr. Muhannad Talib ShukerGPE Department

Well Test Analysis, © UTP – MAY 2011

Transient Flow EquationIn the course of development of the transient flow pequation three independent equations will be used:

1 Continuity equation: material balance equation which 1. Continuity equation: material balance equation which states conservation of mass

E i f i D ’ i hi h d fi 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


pressure

Continuity Equation (1/6)

Well

Reservoir boundary

Formation Formation thickness

Schematic of reservoir


Continuity Equation (2/6)Continuity Equation (2/6)

Flow Element (control volume)

Under the steady‐state flow conditions, the same amount of fluid enters and leaves the flow

h

Mass inMass out

element. However they are not equal to each other during unsteady‐state (transient) flow conditions. Nevertheless, the

rwr

Δrh

Making a mass balance over the volume element during a time

mass must be conserved in both cases.

r+Δrvolume element during a time period of Δt

Mass entering Mass leaving Mass accumulated = (1)


volume element

during Δt

volume element

during Δt

in the volume

element during Δt

(1)

Continuity Equation (3/6)Continuity Equation (3/6)MASS IN MASS OUT MASS =

ACCUMULATED

( ) [ ] tAvMass rrin Δ= Δ+ρ (2)

where;

ν = velocity of flowing fluid

fl id d i

( ) [ ] rrin Δ+ρ

( ) ( ) ( ) thM ΔΔ2

ρ = fluid density at r+Δr

A = area at r+Δr

Δt = time interval

The area of the volume element at the entry:

A = 2π(r+Δr)h

( ) ( ) ( ) tvhrrMass rrin ΔΔ+= Δ+ρπ2 (3)

( ) ( )similarly;


( ) ( ) tvrhMass rout Δ= ρπ2 (4)


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

On the other hand;

(5)( ) ( ) tttt rrhMass Δ+Δ+ Δ= φρπ2

( ) ( )tt rrhMass φρπ Δ= 2 (6)

( ) ( ) ( )[ ]rrhMass φρφρπ Δ= 2

Substituting in above definition:

( )


( ) ( ) ( )[ ]tttAcc rrhMass φρφρπ −Δ= Δ+2. (7)

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

( )( )[ ] ( )[ ] ( ) ( )[ ] (8)( )( )[ ] ( )[ ] ( ) ( )[ ]tttrrr rhrtvhrtvrrh φρφρπρπρπ −Δ=Δ−ΔΔ+ Δ+Δ+ 222

Rearranging equation 8:

( )( ) ( )[ ] ( ) ( )[ ]tttrrr rhrvrvrrth φρφρπρρπ −Δ=−Δ+Δ Δ+Δ+ 22 (9)

Dividing the both sides of the equation 9 by 2πhΔr Δt :g q 9 y

(10)( )( ) ( )[ ] ( ) ( )[ ]

trhrhr

trhvrvrrth tttrrr

ΔΔ−Δ

=ΔΔ

−Δ+Δ Δ+Δ+

πφρφρπ

πρρπ

22

22

Hence finally:

( )( ) ( )[ ] ( ) ( )[ ]vrvrr tttrrr −−Δ+ Δ+Δ+ φρφρρρ1


(11)( )( ) ( )[ ] ( ) ( )[ ]

trrtttrrr

Δ=

ΔΔ+Δ+ φρφρρρ


Let’s take limits as both Δr and Δt approaches zero;

( )( )( ) ( )[ ] ( ) ( )[ ]vrvrr tttrrr −−Δ+ Δ+Δ+ φρφρρρ lim1lim (12)

or:

( )( ) ( )[ ] ( ) ( )[ ]trr

ttt

t

rrr

r Δ=

ΔΔ+

→Δ

Δ+

→Δ

φρφρρρ00

limlim

(13)( )[ ] [ ]φρρt

vrrr ∂

∂=

∂∂1

Continuity equation

where;

ν = velocity of flowing fluid




φ = porosity

Equation of MotionEquation of MotionDarcy’s law;Darcy s law;

(14)PkAq∂∂

=where;

k = permeability

fl id i itr∂μdefinition of velocity;

μ = fluid viscosity

(15)

Aqv =

Substituting in equation 14;g q 4

Pkv∂∂

= (16)


r∂μ

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

k ∂⎤⎡ ∂∂1(17)

E di th i ht h d id f ti

[ ]φρμ

ρtr

Pkrrr ∂

∂=⎥

⎦

⎤⎢⎣

⎡∂∂

∂∂1

Expanding the right hand side of equation 13:

(18)[ ]

ttt ∂∂

+∂∂

=∂∂ φρρφφρ

Porosity is related to the formation compressibility by:

c ∂=

φ1P

cr ∂=φ (19)

Applying the chain rule of differentiation to ∂φ/∂t:

P∂∂∂ φφ


(20)tP

Pt ∂∂

∂∂

=∂∂ φφ

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

P∂∂φ(21)

substituting this into equation 18 :

tPc

t r ∂∂

=∂∂ φφ

(22)

[ ]tPc

tt r ∂∂

+∂∂

=∂∂ ρφρφφρ

Finally substituting equation 22 into equation 17:

(23)PcPkr r ∂∂

+∂∂

=⎥⎦

⎤⎢⎣

⎡∂∂

∂∂ ρφρφρ1

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

ttrrr r ∂∂⎥⎦

⎢⎣ ∂∂ μ


the flow of any type of fluid in porous medium.

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)

tPc

trPr

rrk

r ∂∂

+∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛ ρφρφρμ

Expanding above equation gives :

ttrrr ∂∂⎠⎝ ∂∂⎟⎠⎜⎝ μ

⎤⎡⎞⎛ 2

(25)

ttPc

rrP

rP

rP

rk

r ∂∂

+∂∂

=⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

+∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛ ρφρφρρρμ 2

2

Applying the chain rule in the the above equation:

(26)PPcPPPk

r∂∂

+∂

=⎥⎤

⎢⎡ ∂

⎟⎠⎞

⎜⎝⎛ ∂+

∂+

∂⎟⎟⎞

⎜⎜⎛ ρφρφρρρ 2

2

2


PttPrrrr r ∂∂∂⎥⎥⎦⎢

⎢⎣ ∂

⎟⎠

⎜⎝ ∂∂∂⎟⎟

⎠⎜⎜⎝

φρφρμ 2


Dividing the both sides of the above equation by ρ;

(27)⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+∂∂

=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎠⎞

⎜⎝⎛∂∂

+∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛Pt

PtPc

PrP

rP

rP

rk

rρ

ρφφρ

ρμ111 2

2

2

Remembering fluid compressibility is related to its density by:

⎠⎝⎥⎦⎢⎣ ⎠⎝⎠⎝⎠⎝ ρρμ

∂ρ1(28)

Combining equations 27 and 28:

Pc f ∂

∂=

ρρ1

g q

(29)PcPcPcPPk

frf∂

+∂

=⎥⎤

⎢⎡

⎟⎠⎞

⎜⎝⎛ ∂+

∂+

∂⎟⎟⎞

⎜⎜⎛ φφ

2

2

21


ttrrrr frf ∂∂⎥⎥⎦⎢

⎢⎣

⎟⎠

⎜⎝ ∂∂∂⎟⎟

⎠⎜⎜⎝

φφμ 2


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

(30)( )tPcc

rP

rP

rk

fr ∂∂

+=⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛φ

μ 2

21

Defining the total compressibility ct:

⎦⎣⎠⎝ μ

(31)

Substituting equations 31 in 30 and rearranging:

frt ccc +=

(32)

tP

kc

rP

rrP t

∂∂

=∂∂

+∂∂ φμ1

2

2



(32)PcPP t ∂=

∂+

∂ φμ12

2

tkrrr ∂∂∂ 2

Equation 32 is called as DIFFUSIVITY EQUATION and isconsidered one of the most important and widely usedmathematical expression in Petroleum Engineering.p g gThe diffusivity equation can be rearranged with the inclusion offield units and is used in the analysis of well testing data wheretime is commonly in hours.



2(33)

tP

kc

rP

rrP t

∂∂

=∂∂

+∂∂

0002637.01

2

2 φμ

Where;k = permeability, md

di l i i f

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

formationr = radial position, ftP = pressure, psiact = total compressibility, psi‐1t = time, hoursφ i f i

o at o2. Laminar flow (Darcy)3. Homogeneous and isotropic porous medium4. Porous medium has constant permeability and

compressibilityG it ff t li iblφ = porosity, fraction

μ = viscosity, cp5. Gravity effects are negligible6. Isothermal conditions7. Fluid has small and constant compressibility8. Fluid viscosity is constant



Diffusivity equation is generally is shown as:

( )PcP ∂⎟⎞

⎜⎛ ∂∂ φμ1 (34)

tP

kc

rPr

rrt

∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

∂∂ φμ1

⎠⎝


Solutions to Diffusivity EquationTh th b i f i t t t dThere are three basic cases of interest towardsthe solution of Diffusivity Equation:

1. Constant production rate, InfiniteReservoir

2. Constant production rate, no‐flow at theouter boundary

3. Constant production, constant pressure at3. Constant production, constant pressure atthe outer boundary


Initial and Boundary Conditions yfor Constant Production Rate, Infinite Boundary

(34)PcPr t ∂=⎟

⎞⎜⎛ ∂∂ φμ1

Equation: (34)

tkrr

rr ∂=⎟

⎠⎜⎝ ∂∂

Equation:

Initial Condition: ( ) iPrP =0, (35)( ) i,

Boundary Conditions:

I B dPrkhq ⎟⎞

⎜⎛ ∂

=π2

Inner Boundarywr

rrq ⎟

⎠⎜⎝ ∂

=μ

(36)

Outer Boundary ( ) iPtrP =∞→ , (37)


( )

Initial and Boundary ConditionsInitial and Boundary Conditions for Constant Production Rate, No‐Flow Boundary

(34)PcPr t ∂=⎟

⎞⎜⎛ ∂∂ φμ1

Equation: (34)

tkrr

rr ∂=⎟

⎠⎜⎝ ∂∂

Equation:



I B dPrkhq ⎟⎞

⎜⎛ ∂

=π2

Inner Boundarywr

rrq ⎟

⎠⎜⎝ ∂

=μ

(36)

d 0⎟⎞

⎜⎛ ∂P


Outer Boundary 0=⎟⎠

⎜⎝ ∂ er

r(38)

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

(34)PcPr t ∂=⎟

⎞⎜⎛ ∂∂ φμ1

Equation: (34)

tkrr

rr ∂=⎟

⎠⎜⎝ ∂∂

Equation:



I B dPrkhq ⎟⎞

⎜⎛ ∂

=π2

Inner Boundarywr

rrq ⎟

⎠⎜⎝ ∂

=μ

(36)

d ( ) PtrrP


Outer Boundary (39)( ) ie PtrrP == ,

Dimensionless Form ofDimensionless Form of Diffusivity EquationMost of the time dimensionless groups are used to express Diffusivity 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.

(34)

tP

kc

rPr

rrt

∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

∂∂ φμ1

tkrrr ∂⎠⎝ ∂∂


Dimensionless Groups forDimensionless Groups for Diffusivity Equation

Dimensionless Pressure:

(40)

Dimensionless Pressure:

( )PPqBkhP iD −=μ

Dimensionless Radius:

rr = ( )

wD r

r = (41)

Dimensionless time:

2wt

D rcktt

φμ= (42)


Dimensionless form ofDimensionless form of Diffusivity Equation

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

(43)

D

D

D

DD

DD tP

rPr

rr ∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂1

DDDD trrr ∂⎠⎝ ∂∂

d d h b d d l dNow it is needed to express the boundary and initial conditions in dimensionless forms.


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

Initial Condition:

(44)

Initial Condition:

( ) 00, ==DDD trP

Outer Boundary:

( )( ) 0∞→ trP (45)

Inner Boundary:

( ) 0, =∞→ DDD trP

( )1

1

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=DrD

D

rP

(46)


Dimensionless Boundary and Initial Conditions for the yDiffusivity Equation for Constant Rate, No‐Flow Boundary

Initial Condition:

(47)

Initial Condition:

( ) 00, ==DDD trP

Outer Boundary:

( 8)0=⎟⎟⎞

⎜⎜⎛ ∂ DP

(48)

Inner Boundary:

0=⎟⎟⎠

⎜⎜⎝ ∂ eDrDr

(49)

( )1

1

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=DrD

D

rP


( )D

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

Initial Condition:

(50)

Initial Condition:

( ) 00, ==DDD trP

Outer Boundary:

( )( ) 0== trrP (51)

Inner Boundary:

( ) 0, == DeDDD trrP

(52)

( )1

1

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=DrD

D

rP


( )D

ASSIGNMENT # 1Due Date: Monday 28‐2011Prove that the below partial differential equation is the dimensionless form of Diffusivity Equation.

D

D

D

DD

DD tP

rPr

rr ∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂1

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: ( ) 0=∞→ trPOuter Boundary:

Inner Boundary: 1−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂ DP

( ) 0, =∞→ DDD trP


( )1⎟⎠

⎜⎝ ∂ =DrDr

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

PP ∂⎟⎞

⎜⎛ ∂∂1

I i i l d B d C di i

D

D

D

DD

DD tP

rPr

rr ∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂1

(43)

Initial and Boundary Conditions

Initial Condition: ( ) 00, ==DDD trP (44)

Outer Boundary:

⎞⎛ ∂P

( ) 0, =∞→ DDD trP (45)

Inner Boundary: 1lim0

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

→D

DrD

DDr r

Pr (46)


⎞⎛⎟⎟⎞

⎜⎜⎛ −

−= DiD

rEP 1 2

⎟⎠

⎜⎝ D

iD t42Thi i h li l i f h Diff i i E i f d i d This is the line source solution of the Diffusivity Equation for constant production rate and infinite reservoir case.


S l ti t Diff i it E ti f C t t Li SSolution to Diffusivity Equation for Constant Line Source Production Rate Infinite Boundary Case

For

(83)01.04

2

<D

D

tr

Exponential integral can be approximated as

⎥⎤

⎢⎡

⎟⎞

⎜⎛

809070l1 DtP ⎥⎦

⎢⎣

+⎟⎟⎠

⎜⎜⎝

= 80907.0ln2 2

D

DD r

P (84)


S l ti t Diff i it E ti f C t t Li SSolution to Diffusivity Equation for Constant Line Source Production Rate Infinite Boundary Case

And the dimensionless pressure at the wellbore

(85)1=Dr

Exponential integral can be approximated as

) [ ]809070l1P ) [ ]80907.0ln2

+= DwellboreD tP (86)

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


References1. Dominique Bourdet, “Well Test Analysis: The Use of Advanced Interpretation

Models”, Handbook of Petroleum Exploration and Production, 3. Elsevier, 2002 (Chapter 1)T k Ah d d P l D M Ki “Ad d R i E i i ” 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 9Textbook series Vol. 9.

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


Transient Flow Equation and Solutios

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