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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
Well Test Analysis, © UTP – MAY 2011
pressure
Continuity Equation (1/6)
Well
Reservoir boundary
Formation Formation thickness
Schematic of reservoir
Well Test Analysis, © UTP – MAY 2011
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)
Well Test Analysis, © UTP – MAY 2011
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;
Well Test Analysis, © UTP – MAY 2011
( ) ( ) tvrhMass rout Δ= ρπ2 (4)
Continuity Equation (4/6)Continuity Equation (4/6)
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:
( )
Well Test Analysis, © UTP – MAY 2011
( ) ( ) ( )[ ]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
Well Test Analysis, © UTP – MAY 2011
(11)( )( ) ( )[ ] ( ) ( )[ ]
trrtttrrr
Δ=
ΔΔ+Δ+ φρφρρρ
Continuity Equation (6/6)Continuity Equation (6/6)
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
ρ = fluid density at r+Δr
Well Test Analysis, © UTP – MAY 2011
ρ = fluid density at r+Δr
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)
Well Test Analysis, © UTP – MAY 2011
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
Well Test Analysis, © UTP – MAY 2011
(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
Well Test Analysis, © UTP – MAY 2011
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
Well Test Analysis, © UTP – MAY 2011
PttPrrrr r ∂∂∂⎥⎥⎦⎢
⎢⎣ ∂
⎟⎠
⎜⎝ ∂∂∂⎟⎟
⎠⎜⎜⎝
φρφρμ 2
Transient Flow Equation for Slightly Compressible Fluids (2/6)
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
Well Test Analysis, © UTP – MAY 2011
ttrrrr frf ∂∂⎥⎥⎦⎢
⎢⎣
⎟⎠
⎜⎝ ∂∂∂⎟⎟
⎠⎜⎜⎝
φφμ 2
Transient Flow Equation for Slightly Compressible Fluids (3/6)
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
Well Test Analysis, © UTP – MAY 2011
Transient Flow Equation for Slightly Compressible Fluids (4/6)
(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.
Well Test Analysis, © UTP – MAY 2011
Transient Flow Equation for Slightly Compressible Fluids (5/6)
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
Well Test Analysis, © UTP – MAY 2011
Transient Flow Equation for Slightly Compressible Fluids (6/6)
Diffusivity equation is generally is shown as:
( )PcP ∂⎟⎞
⎜⎛ ∂∂ φμ1 (34)
tP
kc
rPr
rrt
∂∂
=⎟⎠⎞
⎜⎝⎛
∂∂
∂∂ φμ1
⎠⎝
Well Test Analysis, © UTP – MAY 2011
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
Well Test Analysis, © UTP – MAY 2011
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)
Well Test Analysis, © UTP – MAY 2011
( )
Initial and Boundary ConditionsInitial and Boundary Conditions for Constant Production Rate, No‐Flow 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)
d 0⎟⎞
⎜⎛ ∂P
Well Test Analysis, © UTP – MAY 2011
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:
Initial Condition: ( ) iPrP =0, (35)( ) i,
Boundary Conditions:
I B dPrkhq ⎟⎞
⎜⎛ ∂
=π2
Inner Boundarywr
rrq ⎟
⎠⎜⎝ ∂
=μ
(36)
d ( ) PtrrP
Well Test Analysis, © UTP – MAY 2011
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 ∂⎠⎝ ∂∂
Well Test Analysis, © UTP – MAY 2011
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)
Well Test Analysis, © UTP – MAY 2011
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.
Well Test Analysis, © UTP – MAY 2011
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)
Well Test Analysis, © UTP – MAY 2011
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
Well Test Analysis, © UTP – MAY 2011
( )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
Well Test Analysis, © UTP – MAY 2011
( )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
Well Test Analysis, © UTP – MAY 2011
( )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)
Well Test Analysis, © UTP – MAY 2011
⎞⎛⎟⎟⎞
⎜⎜⎛ −
−= 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.
Well Test Analysis, © UTP – MAY 2011
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)
Well Test Analysis, © UTP – MAY 2011
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.
Well Test Analysis, © UTP – MAY 2011
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
Well Test Analysis, © UTP – MAY 2011