Petroleum Engineering
University of H
ouston
© 2000-2001 M. Peter Ferrero, IX
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Description of a well test: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Types of tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Why we do transient testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Flow States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Development of Flow Equations for Flow in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Solutions to the Diffusivity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Skin Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Wellbore Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Wellbore Storage (WBS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Radius of Investigation (ROI). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Pseudo Steady-State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Shape Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Principle of Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Horner’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Buildup Test Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Derivative Analysis (Drawdown case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Ideal vs. Actual PBU/DD Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Flow Regimes & Model Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Gas Well Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Gas Tests - Pseudo (Ψ(P)) Equation Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Pseudopressure or Real Gas Potential (Ψ(P)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Determination of Skin and D for Gas Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Multiple Rate Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Odeh-Jones Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Flow Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Horizontal wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Pressure level in surrounding reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Drill Stem Tests (DST) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Conducting Well Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Wellbore Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Pressure Transient Analysis
Spring 2001
Pressure Transient Analysis
Introduction
Instructors:
Grading:
• 20% homework
• 40% midterm
• 40% final
Textbook:
• Well Testing by John Lee
Jeff Appemail: [email protected]
B.S.: Civil Engineering, Rice UniversityM.S.: Chemical Engineering, University
Currently completing Ph.D. in Chemical Engineering, University of Houston
of H ouston
Dr. Christine Ehlig-Economidesemail: [email protected]
M.S.: Chemical Engineering, University of KansasPh.D.: Petroleum Engineering, Stanford University
Introduction Page 2© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Description of a well test:
Schematic:
Process:
• flow well at single or multiple rates for time, tp.
• shut well in for pressure buildup (PBU), ∆∆∆∆t.
• measure P, T, and q’s (pressure, temperature, and flow rates, respectively).
Information gained:
• reservoir fluids [BHS (bottom hole sample), separator samples for PVT analysis]
• reservoir temperature and pressure (from gauge)
• permeability and skin (completion efficiency)
• reservoir description, qualitative (faults, changes in permeability, oil/water contact)
Fig. 1. Schematic of well test set-up
��������������������������������
Oil
Gas
Water
Choke
Pressure gauge
Packer
Perforations
Separator
Description of a well test: Page 3© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Types of tests
Drawdown test (DD)
• difficult to maintain constant rate
• this introduces scatter in mea-sured FBHP (flowing bottom hole pressure)
Pressure buildup test (PBU)
• advantage: rate is known, i.e. q=0
• disadvantage: lost production
Injection test
• advantage: injection rates are easily controlled
• disadvantage: analysis is compli-cated by multiphase effects and possible fracturing
T im e
q
P
Fig. 2. Drawdown test
T im e
q
P
Fig. 3. Pressure buildup test
T im e
q
P
Fig. 4. Injection test
Types of tests Page 4© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Falloff test
Interference/pulse test
• Tests connectivity of wells using a producers and observation wells
• Used to estimate transmissibility , and storativity
Drillstem test (DST)
• Way to go for exploration
• Utilize downhole shut-in which greatly reduces wellbore storage (WBS)
• Accurate production rate measurement
• on site production facilities
Why we do transient testingWhen we make a rate change, the system goes through a transition state during which the steady-state solutions are not valid – this is known as transient flow. This is the period that is the basis for well testing or pressure transient analysis.
• Steady-state equations do not yield “unique” values for k, h, & s:
• Log derived/core kh values are not always representative of system/reservoir kh.
• Well testing yields macroscopic , average system kh.
T im e
q
P
Fig. 5. Falloff test
khµ------ φhct
∆P 141.2qµβkh
--------------------------re
rw-----ln S+
=
Why we do transient testing Page 5© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Flow States
• Steady-state, , pressures
in reservoir/wellbore do not vary with time.
• Pseudo steady state,
, pressures in reservoir/wellbore are changing in a constant (linear) man-
ner
P
rw re
For all time
Fig. 6. Steady-state flow regime
∂P∂t------- 0=
∂P∂t------- cons ttan=
Time
P LinearP
rw re
t3
t2
t1
Fig. 7. Pseudo steady-state flow regime
Flow States Page 6© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
• Transient, , pres-
sure in reservoir/wellbore are changing as a function of both time and location.
Development of Flow Equations for Flow in Porous Media
Note: there is a good writeup in Appendix A of Lee.
What’s needed to derive the diffusivity equation is:
• A. Conservation of Mass (Continuity equation)
• B. Darcy’s Law
• C. Equation of State (EOS)
A. Continuity equation, cylindrical coordinates (r, z, θθθθ)
P
rw re
t3
t2
t1
Fig. 8. Transient flow regime
∂P∂t------- f x y z t, , ,( )=
ρvzr r θddz∂
∂ ρvz( ) zd r rd θd+
ρvθ r zdd
ρvrr θ zddr∂
∂ ρrvr( ) θd r zdd+
dr
dz
dθθθθ
rdθθθθ
ρvrr θ zdd
ρvzr r θddz∂
∂ ρvz( ) zd r rd θd+
ρvzr r θdd
Fig. 9. Cylindrical coordinate system
Development of Flow Equations for Flow in Porous Media Page 7© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
mass flux,
[Rate of mass accumulation] + [Rate of mass outflow] - [Rate of mass inflow] = 0
.... divide by
.... note that since there is no z or θ, the last two terms are 0
.... therefore, for a fully perforated interval, the RADIAL DIFFUSIVITY EQUATION is
B. Darcy’s Law
Isotropic: k=kr=kθ=kz
Assume single slightly compressible fluid - compressibility, c= constant
By integration:
ρv lbm
ft3---------- ft
s---× lbm
ft2 s⋅--------------= =
t∂∂ ρθr θ r zddd( ) r θ r zddd
t∂∂ ρθ( )=
ρvrr θ zddr∂
∂ ρrvr( ) θd r zdd+ ρvrr θ zdd[ ] ....r direction–
ρvθ r zddθ∂
∂ ρvθ( ) θd r zdd+ ρvθ r zdd[ ] ....θ direction–
ρvzr r θddz∂
∂ ρvz( ) zd r rd θd+ ρvzr r θdd[ ] ....z direction–
r θ r zt∂
∂ ρθ( )dddr∂
∂ ρrvr( ) θ r zdddθ∂
∂ ρvθ( ) θ r zdddz∂
∂ ρvz( )r θ r zddd+ + + 0=
r θ r zddd
t∂∂ ρθ( ) 1
r---
r∂∂ ρrvr( ) 1
r---
θ∂∂ ρvθ( )
z∂∂ ρvz( )+ + + 0=
t∂∂ ρθ( ) 1
r---
r∂∂ ρrvr( )+ 0=
ν kµ---– ∆P=
νrkr
µ----
rddP–=
νθkθµ-----
θddP–=
νzkz
µ-----
zddP–=
t∂∂ ρφ( )∴ 1
r---
r∂∂ ρr
kr
µ----
rddP–
+ 0=
or
1r---
r∂∂ ρr
kr
µ----
rddP
t∂∂ ρφ( )=
c 1Vol---------–
Pdd Vol≡ 1
ρ---
Pddρ Vol 1
ρ---=;→
Development of Flow Equations for Flow in Porous Media Page 8© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Note:
1. Since φ doesn’t change wrt time,
2. Also, since the pressure gradient is small,
Canceling ρ’s, and dividing through by
.... therefore, for a fully perforated interval, the RADIAL DIFFUSIVITY EQUATION including Darcy’s law is
To solve this you need two boundary conditions and one initial condition. For a closed system:
Initial condition: P = Pi @ t=0
Boundary condition 1: No flow -
ρ ρ0ec P P0–( )
c PdP0
P
∫ ρ∂ρ------ ρ0 base ρ≡;
ρ0
ρ
∫= = =
r∂∂ρ cρ0e
c P P0–( )
r∂∂P cρ
r∂∂P
= =
r∂∂ρ cρ0e
c P P0–( )
r∂∂P cρ
r∂∂P
= =
1r---
r∂∂ ρrk
µ---
r∂∂P
t∂∂ ρφ( )=
1r---
r∂∂Prk
µ---
r∂∂P ρk
µ---
r∂∂P ρrk
µ---
r2
2
∂∂ P+ +
φt∂
∂ρ ρt∂
∂θ +=
kµ---1
r--- crρ
r∂∂P
2
r∂∂P ρr
r
2
∂∂ P+ +
cφρt∂
∂P=
ρt∂
∂φ 0→
r∂∂P
21 crρ
r∂∂P
20→∴;«
ρr--- k
µ---
r∂∂P ρr
r
2
∂∂ P+
cφρt∂
∂P=
kµ---
1r---
r∂∂P ρr
r
2
∂∂ P+
cφµ
k----------
t∂∂P=
1r---
r∂∂ r
r∂∂P
1
η---
t∂∂P where η k
φµc----------==
r∂∂P
re
0=
Development of Flow Equations for Flow in Porous Media Page 9© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Boundary condition 2:
For an infinite reservoir, BC1 becomes as .
Darcy’s law came from Darcy’s investigation of the sewers in Paris. He conducted his experiments on flow through gravel.
Steady-state linear flow:
Darcy velocity in Cylindrical coordinates
Examples of tests:
• In transient flow, pressure will decrease wrt time at constant flow rate.
• Separation of log-log and derivative plot indicates skin (larger separation=larger skin)
1. Derived diffusivity equation based on:
r∂∂P
rw
qµ2πhrw----------------=
P Pi→ r ∞→
velocity, u 0.001127–k
µβ-------
lddP⋅ ⋅=
q 0.001127–kAµβ-------
lddP⋅ ⋅= l
P 2
P 1
q
q
P e rm e a b i l i ty , kW a te r v isco si ty , µµµµ w
Fig. 10. Steady-state linear flow
velocity, u 0.001127–kµ---
rddP⋅ ⋅=
q 0.001127–2πrwk
µ----------------
rddP⋅ ⋅=
qdrr
-----rw
r2
0.00708–2πrwk
µ---------------- dP
Pw
P2⋅ ⋅=
q 0.00708khµβ-------
P2 Pw–( )r2
rw-----
ln
-------------------------⋅ ⋅–=
h
r w
A re a , A = 2 ππππ r w h (a re a o f c y lin d e r)
Fig. 11. Darcy velocity in cylindrical coordinates
Development of Flow Equations for Flow in Porous Media Page 10© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
• Continuity equation
• Darcy’s law
• EOS
2. Assumptions:
a. Radial flow over entire net thickness
b. Homogeneous and isotropic porous media (kr=kθ=kz)
c. Uniform net thickness
d. q and k are constant (independant of pressure)
e. Fluid is of small and constant compressibility
f. Constant µg. Small pressure gradients ( )
h. Negligible gravity forces
Solutions to the Diffusivity Equation3. Develop solutions to diffusivity equation.
• “Exact solution” - Van Everdingen & Hurst terminal rate solution (center, bounded, cir-cular system!). (We won’t use this!)
• Infinite reservoir, line source well
- constant rate, q- unbounded (infinite acting) reservoir
a. Initial condition: P=Pi at t=o for all radius, r
b. Boundary condition (BC) #1: for t>0...constant rate condition
c. BC #2: as for all t
Replace BC#1 to obtain “line source” approximation
for t>0
Line source solution:
r∂∂P
21«
Pwf Pi141.2qµβ
kh--------------------------
2tD
reD2
-------- reDln 34---–+⋅ 2
eα2tD–
J12 αηreD( )
αη2 J1
2
αηreD J12αη–( )
----------------------------------------------------
η 1=
∞
∑+–=
1r---
r∂∂ r
r∂∂P
1
η---
t∂∂P=
rr∂
∂P
rw
qµ2πkh--------------=
f
P Pi→ r ∞→
rr∂
∂P
rwr 0→lim
qµ2πkh--------------=
P r t( , ) Pi 70.6qµβkh
----------Eir2
–4ηt---------
where Ei x–( )– e µ–
µ-------- µd
x
∞
∫=
;+=
Solutions to the Diffusivity Equation Page 11© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
• DRAWDOWN ONLY
• Constant rate
• Unbounded reservoir
Limitations of the line source solution (E i)
a. Check to insure that Ei solution is valid
- for , the assumption of zero wellbore radius limits the accuracy of the
solution
- for , effects of boundaries are felt, Ei solution no longer
valid.
b. If Ei solution is valid, check applicability of ln approximation.
For wellbore, Pw (if Ei is valid, then it’s always valid at the wellbore)
ln approximation but for Ei
- If Ei function is valid at the wellbore, then ln approximation will always be valid at the wellbore!- Even if though the Ei function may be valid at radius, r (rw < r < re), the ln approxi-mation won’t always be valid.
Skin DevelopmentSkin, S, refers to a region near the wellbore of improved or reduced permeability compared to the bulk formation permeability.
Impairment (+S):
• Overbalanced drilling (filtrate loss)
• Perforating damage
• Unfiltered completion fluid
• Fines migration after long term production
• Non-darcy flow (predominantly gas well)
• Condensate banking- acts like turbulence
Stimulation (-S)
• Frac pack (0 to -0.5)
• Acidizing
100rw2
η--------------- t
re2
4η-------≤ ≤
t100rw
2
η---------------<
tre2
4η------->
P r t( , ) Pi 70.6qµβkh
----------Eir2
–4ηt---------
+=
Ei x( ) 1.781x( ) x 0.02≤,ln=
Eir2
–4ηt---------
0.445r2
ηt--------------------
r2
4ηt--------- 0.02≤,ln=
rw2
4ηt--------- 0.02≤
rw2
4ηt--------- 0.01
4-----------≤
Skin Development Page 12© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
• Hydraulic fracturing
Generally S>5 is considered bad; S= -3.5 to -4 is excellent.
Flow efficiency , FE, is the ratio of flow without skin to the flow with skin,
Combine with Darcy’s law:
Darcy w/o SDarcy /w S
-------------------------------- or FE8
S 8+-------------≈,
Radius
Pre
ssur
e
k of formationk including skin
∆∆∆∆Pk
∆∆∆∆Ps
rw rs
∆∆∆∆Pks
Fig. 12. Skin pressure drop
∆Ps ∆Pks ∆Pk–=
∆Ps 141.2qµβksh----------
rs
rw-----
ln 141.2qµβkh
----------rs
rw-----
ln–=
∆Ps 141.2qµβkh
---------- kks----- 1–
rs
rw-----
ln=
We define kks----- 1–
rs
rw-----
ln S=
∆Ps∴ 141.2qµβkh
----------S=
∆Ptotal ∆PS 0= ∆PS+=
∆Ptotal 141.2qµβkh
----------re
rw-----
ln 141.2qµβkh
----------S+ 141.2qµβkh
----------re
rw-----
ln S+= =
S 0> Damaged ks k<∴→
S 0< Stimulated ks k>∴→
S 0= Undamaged ks k=∴→
Skin Development Page 13© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
SEM examples of various clays which can cause formation damage
Fig. 13. Smectite (left) and kaolinite (right) coat grains and fill a pore. Note distinct differences in morphology of each clay ("honeycomb" smectite; vermicular booklets of kaolinite (x2000)(image courtesey of Westport Technology Center)
Fig. 14. Delicate wisps of "hairy" illite project into a pore. Note that the fibers not only form a highly rugose surface within the pore, but the fibers could break and migrate under extreme fluid pressures (x2500)(image courtesey of Westport Technology Center)
Fig. 15. Well-formed chlorite platelets form partial rosettes adjacent to, and coating quartz overgrowths (x2500)(image courtesey of Westport Technology Center)
Fig. 16. Well-formed, but rather randomly oriented kaolinite booklets post-date quartz overgrowths (x700)(image courtesey of Westport Technology Center)
Skin Development Page 14© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
SEM examples of formation damage and stimulation
Fig. 17. SEM image of perforation damage with percussion fines(x305)
Fig. 18. SEM image of completion damage with polymer filament (x105)
Fig. 19. SEM image of pre-acid treatment (x3100) Fig. 20. SEM image of post-acid treatment (x3100)
Skin Development Page 15© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Wellbore Solutions1. Ideal reservoir (no skin)
2. Solution at sandface (including skin)
Wellbore Storage (WBS)• Unit slope on log-log plot of ∆P vs. time
• Straight line on cartesian,
Storage between the sandface and shut-in valve allow the formation to continue to flow when we affect a shut-in. This is due to fluid compressibility.
We will consider two cases:
1. A well with a gas-liquid interface
2. A liquid filled well
Pw r t( , ) Pi 70.6qµβkh
----------0.445rw
2
ηt--------------------
where η 2.6374–×10 k
φµct---------------------------------=;ln+=
Pw t( ) Pi 70.6qµβkh
----------1688φµctrw
2
ktp-------------------------------
from Lee;ln+=
∆Pwf Pi Pwf– ∆Pk ∆Pskin+ 70.6qµβkh
----------0.445rw
2
ηt--------------------
ln– 141.2qµβkh
----------S+= = =
∆Pwf 70.6qµβkh
----------0.445rw
2
ηt-------------------- 2S–
ln–=
Pwf Pi 70.6qµβkh
----------0.445rw
2
ηt--------------------
2S–ln η 2.6374–×10 k
φµct---------------------------------=;+=
b 0≠
Wellbore Solutions Page 16© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
General definitions
Vwb = volume of liquid in well (ft3)
Awb = cross-sectional area of well (ft2)
ρl = density of wellbore fluid (lbm/ft3)
h = height of liquid column in wellbore (ft)
Gas-liquid interface
• pumping wells, gas lift wells
• injection wells (on vacuum)
• an approximation for most natu-rally flowing oil wells (except highly undersaturated oils, P >Pb)
Wellbore mass balance
[Mass inflow] - [Mass outflow] = Accumulation of Mass
Assume constant density, ρl
Note:
Vwb
Awbh liquid
dh
Pt
qβ β β β ((((RB/D)
qSFβ β β β ((((RB/D) Pw + Pt + ρρρρlgh144
Fig. 21. Wellbore storage definitions
qSFβ qβ–( )ρ 245.615---------------
tdd ρvWB( )=
bblD
-------- lbm
ft3----------•
ft3
bbl-------- 24
5.615--------------- lbm
ft3---------- ft3•
=
qSF q–( )β 245.615---------------
td
dvWB where vWB AWBhtd
dvWB AWB tddh=;=;=∴
qSF q–( )β 245.615---------------AWB td
dh=
h144 Pw Pt–( )
ρg----------------------------------=
tddh 144
ρg----------
td
dPw assumetd
dPt =→;=
qSF q–( )∴ )β 245.615---------------
144AWB
ρg----------------------
td
dPw=
Wellbore Storage (WBS) Page 17© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Definition: Wellbore storage coefficient for a gas-liquid interface
Example:
3.5” tubing, AWB = 0.041 ft2
ρo = 50 lbm/ft3
vwb = 100 bbldepth = 17,000 ft
Solution
(note that for a gas-liquid interface the cs is independent of well depth!)
Governing Equation (WBS)
qsf = sandface flowrate, STB/D
q = surface flowrate, STB/D
cs = WBS coefficient, bbl/psi
β = formation volume factor, RB/STB
= change in BHP wrt time
BIG NOTE: Using downhole shut-in eliminates most WBS
Pure Wellbore Storage
B - Unit slope on log-log plotA - straight line on cartesian plot
Why?A - 100% WBS, q=0 (PBU)
•
• Therefore, cs can be calcu-lated from the slope of a straight line (intercept must be zero!)
B - Log-log plot, 100% WBS, q=0 @surface (PBU)
cs144ABW
5.615ρl----------------------
25.65ABW
ρl---------------------------= = bbl
psi--------
cs 25.65AWB
ρl------------ 25.65 0.041
50---------------
0.02 bblpsi--------= = =
qSF q–( ) 24cs
β-----
td
dPw=
td
dPw
∆∆∆∆ t
∆∆∆∆ P βqSF
24cs------------- m=
Fig. 22. c s from cartesian plot
qSF 24cs
β-----
td
dPw=
Wellbore Storage (WBS) Page 18© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Estimate cs from any ( )
pair on unit slope line
Completely liquid filled wellboreWellbore mass balance
[Mass inflow] - [Mass outflow] = Accumulation of Mass
log ∆∆∆∆t
log ∆∆∆∆Pwm = 1
βqSF
24cs-------------
Fig. 23. c s from log-log plot. Estimate c s from any ( ∆∆∆∆Pw, ∆∆∆∆t) pair on unit slop eline
qSF 24cs
β-----∆P
∆t--------=
∆PwβqSF
24cs-------------∆t=
∆Pw( )logβqSF
24cs-------------∆tlog=
∆Pw( )log m ∆t( )logβqSF
24cs-------------
log+=
∆Pw ∆t,
tlndd x( )
tdd x( )
tdd t( )ln
t( )lndd x( )⋅ t
t( )lndd x( )= =
tdd x( ) t
t( )lndd x( )=
∆PW∴td
d ∆PW( ) tβqSF
24cS-------------⋅= =
Take log of both sides[ ]
tdd ∆PW( )log t( )log
βqSF
24cS-------------
log+=
m∴ 1 intercept βqSF
24cS------------- for ∆P==
Wellbore Storage (WBS) Page 19© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Example:
vWB = 100 bblc = 1X10-5 psi-1
Solution
Note: for cs < 0.003 there is basically no WBS
Determining the end of WBS
Drawdown case (100% WBS)
Buildup case (100% WBS)
q = rate prior to a PBU
qSFβ qβ–( )ρ 245.615---------------
tdd ρvWB( )=
Note vWB AWBh=→[ ]
qSF q–( )βρ 245.615---------------vWB td
dρ =
by chain rule c 1ρ---
P∂∂ρ≡
tddP
Pwddρ
td
dPw cρ
td
dPw= =→
qSF q–( )βρ 245.615---------------vWBcρ
td
dPw=
qSF q–( )β 245.615---------------vWBc
td
dPw=
cs
vWBc
5.615---------------≡ bbl
psi-------- where c = average fluid compressibility
csvWBc
5.615--------------- 100 1 5–×10( )
5.615--------------------------------- 0.0002= = = bbl
psi--------
qSF q– 24cs
β-----
td
dPw=
qSF 0 initially as open to rate q=
q 24cs
β-----
td
dPw=
q 0 initially as the well is shut in=
qSF fixed=
qSF 24cs
β-----
td
dPw=
WBS is over when 24cs
β-----
td
dPw 0.01q≤
Wellbore Storage (WBS) Page 20© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
= production rate for a drawdown test
Radius of Investigation (ROI)This is one of the basic concepts to well test analysis.
From the error function:
R feett hou
k mD
f frac
m cpc psi -
q1
q2
q3
t
PWF
t
P
rw re
t3
t2
t1
Pi
r1 r3r2
Fig. 24. Illustration of ROI
Ri 4ηt= η 2.6374–×10 k
φµct---------------------------------=;
Radius of investigation is INDEPENDENT of q
Radius of Investigation (ROI) Page 21© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Pseudo Steady-StateDepletion of a closed system
Pseudo steady-state occurs when the pressure transient has reached all boundaries in a closed system.
The solution, based on the Van Everdingen & Hurst terminal exact solution of a bounded, cylindrical reservoir is
This is very difficult to apply!
Shape Factorsp. 9-10 of Lee text
Principle of SuperpositionThe diffusivity equation is a linear homogeneous equation (with homogeneous BC’s).
Therefore, linear combinations of solutions are also solutions. The combined linear solution eliminates the following restrictions:
• Single well
• Reservoir boundaries
• Constant rate
PWF Pi141.2qµβ
kh-------------------------- 2ηt
re2
---------re
rw-----
ln 0.75–+ for –= tre2
4η-------≥
t∂∂PWF 141.2qµβ
kh--------------------------
2ηre2
-------–=∴ η 2.6374–×10 k
φµct---------------------------------=;
t∂∂PWF 141.2qµβ
kh--------------------------
2 2.637 4–×10( )k
re2φµct
-----------------------------------------–0.0744qβ–
φcthre2
---------------------------- Note:Vp πre2φh reservoir volume== =
t∂∂PWF 0.234qβ
ctVp----------------------– ∆P
∆t--------= =
1r---
r∂∂ r
r∂∂P
1
η---
t∂∂P=
Pseudo Steady-State Page 22© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Multi-well solution
Determine
Check for if
B
A
C
rAB
rAC
q
t
qA
q B
qC
∆PA
∆PTOTAL A∆PA ∆PB ∆PC+ +=
∆P r t( , ) Pi 70.6qµβkh
---------- 0.445r2
ηt--------------------ln 2S–+=
∆P Pi P r t( , )– 70.6– qµβkh
---------- 0.445r2
ηt--------------------
ln 2S–= =
∆PTOTAL A∴ 70.6–
qAµβkh
-------------- 0.445r2
ηt--------------------
ln 2SA– 70.6qBµβ
kh-------------- Ei
rAB2
–
4ηt------------
70.6qCµβ
kh-------------- Ei
rAC2
–
4ηt------------
––=
1.781x( )lnr2
4ηt--------- 0.02<
Principle of Superposition Page 23© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Boundaries
Single fault
For long time,
For not totally sealing faults use FOG FACTORS (for q of image well):
• 1 = sealing
• 0 = no fault
• -1 = water drive (constant P)
Geologic model Mathematical Model
use image well)
q
L
Fig. 25. Single fault geologic model
qactual
qimage
L L
no flow boundary
Fig. 26. Single fault geologic model
∆Ptotal ∆Pactual ∆P eimag+=
Pi PWF– 70.6– qµβkh
----------0.445rw
2
ηt--------------------
ln 2S– 70.6qµβkh
---------- Ei2L( )2
–4ηt
----------------- –==
Ei4L2
4ηt---------
0.445 2L( )2
ηt------------------------------
ln≈
Principle of Superposition Page 24© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Intersecting faults (90 degree)
Need three image wells
Geologic model Mathematical Model
(use e well)
q
L
L
Fig. 27. 90 degree intersecting fault geologic model
L 2
L 2
qactual
qimage
qimage
qimage
L
L
L
L
no flow boundary
Fig. 28. 90 degree intersecting fault mathematical model
Principle of Superposition Page 25© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Intersecting faults (45 degree)
Need seven image wells
Geologic model Mathematical Model
(use image well)
q
Fig. 29. 45 degree intersecting fault geologic model
qactual
qimage
qimage
qimage
qimage
qimage
qimage
qimage
Fig. 30. 45 degree intersecting fault mathematical model
Principle of Superposition Page 26© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Variable rateSingle well producing at variable rates (ideal, infinite reservoir)
t1 t2
q1
q2
q3
t0
q1
-q1
+
+
q2
+
-q2
+
q3
q1
+
q2-q1
+
q3-q2
=
OR
∆P = f(q,t)
Principle of Superposition Page 27© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
General Solution
Horner’s Approximation• Avoids the use of superposition to model variable rates
• Can replace the need for multiple function evaluation each representing a rate
change, with a single function ( ) that contain a single rate and producing time.
Procedure
• Single rate used is most recent non-zero rate, qlast
• Producing time is cumulative production (Np) divided by qlast
Buildup Test Solutions(Chapter 2 - Lee)
Ideal pressure buildup test
• Infinite acting reservoir (no boundaries have been felt by transient)
• Formation and fluid properties are uniform (Ei and ln function apply)
∆P Pi PWF– 70.6µβkh------- qi qi 1––( )
i 1=
m
∑0.445rw
2
η t ti 1––( )--------------------------
ln 2S––= =
Can incorporate dozens of rates
Ei xln( )
Ei
tP 24Production from well∑Most recent rate
-------------------------------------------------------------NP
qlast----------= =
∆P Pi PWF– 70.6qlastµβ
kh------------------
0.445rw2
ηtP--------------------
ln 2S––= =
Note: tlast 2 tnext to last⋅>
qlast
qnext PBU
q=0
Horner’s Approximation Page 28© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
• Use superposition to model variable rates
q is the rate prior to PBU. Use Horner’s approximation with multiple rates
P* is always taken as the extrapolation from the MTR irregardless of whether boundaries or late time effects are seen. If late time effects are observed, P* may not correspond to Pi
or
tP
q
∆t
-q
∆P ∆Pqtp ∆t+∆Pq∆ t
– DD PBU– Pi PWF–( ) PWS PWF–( )– Pi PWS–= = = =
∆P Pi PWS– 70.6– qµβkh
----------0.445rw
2
η tp ∆t+( )-------------------------
ln 2S– 70.6 q–( )µβkh
------------------0.445rw
2
η ∆t( )--------------------
ln 2S––= =
Pi PWS– 70.6– qµβkh
----------0.445rw
2
η tp ∆t+( )-------------------------
ln 2S0.445rw
2
η ∆t( )--------------------
ln 2S+––=
PWS Pi 70.6qµβkh
----------0.445rw
2
η tp ∆t+( )-------------------------
ln0.445rw
2
η ∆t( )--------------------
ln–+=
Pi 70.6qµβkh
----------∆t
tp ∆t+( )---------------------
ln+=
Note: xln 2.302 xlog=
P∴ WS Pi 162.6qµβkh
----------tp ∆t+( )
∆t---------------------
log–=
m ∆y∆x-------
P2 P1–
tP ∆t2+
∆t2--------------------
logtP ∆t1+
∆t1--------------------
log–
--------------------------------------------------------------------------= =
PWS
1000 100 10 1
tP ∆t+
∆t------------------
Pi = P* (infinite shut-in)m 162.6qµβ
kh--------------------------=
P2 P1–
10( )log 100( )log–-------------------------------------------------
P2 P1–
1 2–------------------- P1 P2–= ==
tP ∆t+
∆t-----------------
∆t ∞→lim 1=Note:
P
Buildup Test Solutions Page 29© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Derivative Analysis (Drawdown case)Bourdet derivative
Drawdown solution
tlndd 1
2.302---------------
tlogdd⋅=
By chain rule
d ( )dt
----------td
d tlnd ( )d tln----------⋅ 1
t--- d ( )
d tln----------⋅= =
d ( )d tln---------- t d ( )
dt----------⋅=
PWF Pi70.6qµβ
kh----------------------
0.445rw2
ηt--------------------
ln 2S–+=
Pi PWF– 70.6qµβkh
----------------------0.445rw
2
ηt--------------------
ln 2S––70.6qµβ
kh----------------------– tln–
0.445rw2
η--------------------
ln 2S–+= =
Take Bourdet Derivative
tlndd Pi PWF–( ) t 70.6qµβ
kh----------------------–
1t---–
70.6qµβkh
----------------------= =td
d tln 1t---–=;
tlndd Pi PWF–( )
tlndd ∆P( ) m 70.6qµβ
kh----------------------= = =
PWF
10001
m 162.6qµβ
kh--------------------------=
tPlog
10001
m 70.6qµβkh
------------------------=
kh∴ 70.6qµβd ∆P( )d tln
----------------@ stabilization
-----------------------------------------------------------=
tlog
d ∆P( )d tln
----------------logMTR
Derivative Analysis (Drawdown case) Page 30© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Skin
a. DD
Pi PWF– 70.6qµβkh
----------------------0.445rw
2
ηt--------------------
ln 2S––162.6qµβ
kh--------------------------–
0.445rw2
ηt--------------------
log 0.87S–= =
Pi PWF– m ηt
0.445rw2
--------------------
log 0.87S+ m tlogη
0.445rw2
--------------------
log 0.87S+ += =
Pi PWF–
m----------------------- tlog
η
0.445rw2
--------------------
log 0.87S+ +=
S 1.151Pi PWF–
m----------------------- 2.25η
rw2
---------------log– tlog–=∴
Take t = 1 hour
SDD∴ 1.151Pi PWF1hr
–
m---------------------------- 2.25η
rw2
---------------log–=
m 162.6qµβkh
--------------------------=
tPlogtp=1
PWF1hr
Semi-log MTR!
tP
tps
∆P Pi PWF–=
∆P′ kh 70.6qµβd ∆P( )d tln
-----------------------------------------–=
Derivative Analysis (Drawdown case) Page 31© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
b. PBUThe instant a well is shut-in, PWF :
PWF Pi162.6qµβ
kh--------------------------
0.445rw2
ηtP--------------------
log 0.87S–+=
PWF Pi mηtP
0.445rw2
--------------------
log– 0.87S–+=
PWF Pi m tPlog 2.25ηrw2
---------------log 0.87S+ + ………………1, from Drawdown–=
Shut-in pressure (during PBU),
PWS Pi mtP ∆t+
∆t-----------------………………………………… 2log–=
Subtract 1 from 2
PWS PWF– mtP ∆t+
∆t-----------------
log– m tPlog m 2.25ηrw2
---------------
log 0.87S+ + +=
PWS PWF–
m-----------------------------
tP ∆t+
tP∆t-----------------
log–k
φµctrw2
-----------------
log 3.23– 0.87S+ +=
SPBU∴ 1.151PWS PWF∆ t 0=
–
mHorner semi-log MTR-------------------------------------------------
k
φµctrw2
-----------------
log– 3.23tP ∆t+
tP∆t-----------------
log ∆tlog–+ +=
∆P PWS PWF∆ t 0=–=
∆tlog∆∆∆∆t s
m′ 70.6qµβkh
------------------------=
∆P
PWSskin
d ∆P( )
dtP ∆t+
∆t-----------------
ln
--------------------------------
m 162.6qµβkh
--------------------------=
tP ∆t+
∆t-----------------log
PWS
Derivative Analysis (Drawdown case) Page 32© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Ideal vs. Actual PBU/DD Testsa. Drawdown case
b. Drawdown: log-log plot
m 162.6qµβkh
--------------------------=
tPlog
PWF
tPlog
ETRWBS
MTRkh, SInfinite actingRadial flow
LTRTransient reaches boundariesReservoir heterogeneity
PWF
Ideal (no WBS or LTR)
Actual
tPlog
∆P
∆P’
∆P
∆P’
tPlog
ETR MTR LTR
∆P
∆P’
∆P
∆P’
Ideal
Actual
Ideal vs. Actual PBU/DD Tests Page 33© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Flow Regimes & Model Recognition
Radial flow
homogeneous, infinite acting system
single fault
ETR
MTR
∆P
∆P’
∆t
WBS dominates
Pi PWF– 162.6qµβkh
---------- tlog constant+=
d ∆P( )d tln
---------------- 70.6qµβkh
----------=
kh 70.6qµβ∆P′stabilized-----------------------------=
Using superposition and image wells
∆Ptotal ∆Pwell ∆P eimag+=
Pi PWF– 70.6– qµβkh
----------0.445rw
2
ηt--------------------
ln 2S– 70.6qµβkh
----------0.445 2L( )2
ηt------------------------------ln
–==
70.6– qµβkh
---------- 20.445
ηt---------------
ln rw2
ln 2L( )2ln 2S–+ +=
PWF Pi 162.6qµβkh
---------- 20.445
ηt---------------
log rw2
log 2L( )2log 2S–+ ++=
Note: td
d tln( ) 1t---–=
d ∆P( )d tln
---------------- td ( )dt
----------=d ∆P( )d tln
---------------- t 70.6qµβkh
---------- 21t---–=→
slope doubles∴2 faults, slope x4
3 faults, slope x8, etc.
tPlog
ETR MTR LTR∆P
∆P’ ∆P
∆P’m2m
PWS
10001
m
2m
ETRMTRLTR
tP ∆t+
∆t-----------------log
Flow Regimes & Model Recognition Page 34© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
increase/decrease in kh or decrease in kh
contacts
constant pressure boundary
aquifer (strong)
gas cap (high compressibility)
water/gas support (pressure support)
tPlog
ETR MTR LTR∆P
∆P’
∆P’
∆P
(kh)inner
(kh)outer
Concentric model:
inner
outer
Radius for k inner :
t is where slope becomes negative
[For ROI’s in outer zone, use k of outer zone! No matter if the k is higher or lower]
increase
decrease
ROI 4ηt ; η2.637 4–×10 ki
φµct----------------------------------==
tPlog
ETR MTR LTR∆P
∆P’
∆Po’
∆P
µµµµw < µµµµo
µµµµw > µµµµo
Same kh!
khµ------
o
khµ
------
w
----------------
70.6qµβ∆Po( )′
----------------------
70.6qµβ∆Pw( )′
---------------------------------------------= same kh!→
∆Pw( )′µw
µo------- ∆Po( )′=∆Pw’
variable kh!
∆Pw( )′
khµ------
o
khµ------
w
---------------- ∆Po( )′=
Flow Regimes & Model Recognition Page 35© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Spherical (Partial Penetration Completions)
∆P 70.6– µβkh------- q
0.445rw2
ηt--------------------
ln q 0.445 2L( )2
ηt------------------------------
ln–==
70.6– µβkh------- q 0.445
ηt---------------
ln q 0.445ηt
--------------- ln– q
rw2
2L( )2--------------ln+=
∆P 70.6– qµβkh
----------rw
2L-------
2
ln=
PWS
10001
m=0
tP ∆t+
∆t-----------------log ∆t
rea lity
goes to zero (in theory)
∆t
m=0.5
early radial: khp, mechanical skin(usually masked by WBS)
late radial: khT, Sglobal=Smech+Spartial penetration
Sglobal can be very large (maybe 400-500)
spherical - t-0.5
transition region between early radial and late radial
- can estimate kv/kh ratio
early radial late radial
hThp
Flow Regimes & Model Recognition Page 36© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Linear flow (Infinite conductivity fractures)
• linear flow region (0.5 slope) represents stimulated well
• fracture conductivity > 10,000 mD-ft
• time transition between linear and radial flow corresponds to the frac. length (half length
• kh and skin are calculated from the radial flow region (need kh to estimate frac length). Therefore, to estimate the frac. length, for a large frac. into a “low” permeability zone, you may need a pre-frac. test.
Bi-linear flow (finite conductivity fractures)
• fracture conductivity < 10,000 mD•ft
• pressure drop in fracture is not negligible
• almost never happens
tPlog
Linear
∆P
m=0.5
flowRadialflow
tPlog
Linear
∆P
m=0.5
flowRadialflow
m=0.25
Bi-linearflow
The bi-linear flow is very fast, need a very longfracture to distinguish!
Flow Regimes & Model Recognition Page 37© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
- if you can see the bi-linear region, it can be used to estimate frac-conductivity (if the matrix permeability is known)
- linear region is used to estimate frac. half length
- radial flow region is used to estimate kh, S
Gas Well TestingSame analysis procedure as for oil well testing with the following exceptions:
• Gas properties (transport), µ, z, cg vary as a function of pressure. Gas is considered a highly compressible fluid whereas oil is considered a slightly compressible fluid.
• Non-darcy flow, or turbulence, can exist in gas wells which shows up as a skin due to extra pressure drop. Therefore, differentiation between true mechanical skin and skin due to non-darcy flow is important
- non-darcy flow signifies that Darcy’s law does not properly predict the ∆P due to flow of gas in porous media
- in porous media, non-darcy flow develops when Re > 50 ( )
- low µ and high velocities (close to the wellbore) are the contributing factors to non-darcy flow
Gas tests - Diffusivity Equation Development
a. EOS for gas:
For gases: µ and z may vary considerably as a function of pressure. Therefore, to account for this, the pseudo-pressure function was developed.
Gas Tests - Pseudo ( Ψ(P)) Equation Developmenta. Continuity equation
b. Darcy’s law
c. EOS
Reρνd
µ----------=
MWRT-----------
Pz----
= P ρz RMW-----------T=→
ψ P( ) 2 Pµz------ Pd
PB
P
∫=
x∂∂ ρux( )
y∂∂ ρuy( )
z∂∂ ρuz( )+ +
t∂∂ ρφ( )–=
u kµ--- P∇= ux
kx
µ-----
x∂∂P=; uy
ky
µ-----
y∂∂P=; uz
kz
µ-----
z∂∂P=;
Gas Well Testing Page 38© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
- oil and water: slightly compressible fluids
- For gases
(b) + (c) into (a)
Differentiating Ψ(P) wrt x, y, z, and t
Input Darcy’s law into Continuity equation:
Input EOS:
assume isotropic conditions
ρ ρoec ρ ρo–( )
=
ρ MWRT-----------
Pz----
=
isotropic kx ky kz= =→
cx∂
∂P
2
y∂∂P
2
z∂∂P
2+ +
x2
2
∂
∂ P
y2
2
∂
∂ P
z2
2
∂
∂ P+ + +φµct
2.637 4–×10 k---------------------------------
t∂∂P=
ψ P( ) 2 Pµz------ Pd
PB
P
∫=
x∂∂ψ 2P
µz-------
x∂∂P=
x∂∂P µz
2P-------
x∂∂ψ=;
y∂∂ψ 2P
µz-------
y∂∂P=
y∂∂P µz
2P-------
y∂∂ψ=;
z∂∂ψ 2P
µz-------
z∂∂P=
z∂∂P µz
2P-------
z∂∂ψ=;
t∂∂ψ 2P
µz-------
t∂∂P=
t∂∂P µz
2P-------
t∂∂ψ=;
x∂∂ ρ
kx
µ-----
x∂∂P
y∂∂ ρ
ky
µ-----
y∂∂P
z∂∂ ρ
kz
µ-----
z∂∂P
+ +
t∂∂ φρ( )=
ρ MWRT-----------
Pz----
=
MWRT-----------
x∂∂ kx
µ-----P
z----
x∂∂P
MW
RT-----------
y∂∂ ky
µ-----P
z----
y∂∂P
MW
RT-----------
z∂∂ kz
µ-----P
z----
z∂∂P
+ + MW
RT-----------φ
t∂∂ P
z----
=
k kx ky kz= = =
Gas Tests - Pseudo ( ΨΨΨΨ(P)) Equation Development Page 39© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
t∂∂ P
z----
P∂∂ P
z----
t∂
∂PÞ= ρ MWRT-----------
Pz----
=
cg1ρ---
P∂∂ρ 1
ρ---MW
RT-----------
P∂∂ P
z----
RTMW-----------
zP----
MWRT-----------
P∂∂ P
z----
= = =
cgzP----
P∂∂ P
z----
=P∂∂ P
z----
⇒ Pz----
cg=
t∂∂ P
z----
Pz----
cg t∂∂P= cg ct for gas reservoir≈
Substituting x∂
∂Ψy∂
∂Ψz∂
∂Ψt∂
∂ Pz----
;;;
x∂∂ P
µz------
µz2P-------
x∂∂Ψ
y∂∂ P
µz------
µz2P-------
y∂∂Ψ
z∂∂ P
µz------
µz2P-------
z∂∂Ψ
+ +
φcgPz----
k-------------------
t∂∂P=
12---
x2
2
d
d Ψy2
2
d
d Ψz2
2
d
d Ψ+ +φcg
Pz----
k------------------- µz
2P-------
t∂∂Ψ=
x2
2
d
d Ψy2
2
d
d Ψz2
2
d
d Ψ+ + 1ηg------
t∂∂Ψ=
In radial coordinates:
1r---
r∂∂ r
r∂∂Ψ
1
ηg------
t∂∂Ψ= where ηg
2.637 4–×10 kφµgcg
---------------------------------=
Gas Tests - Pseudo ( ΨΨΨΨ(P)) Equation Development Page 40© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Pseudopressure or Real Gas Potential ( ΨΨΨΨ(P))
a. Ψ(P) (good for all pressures)
Transient development
Drawdown equation
where Psc = atmospheric pressure (usually 14.7 psia)
Tsc = 520 R
T = R, reservoir temperature
S = mechanical skin
D = turbulence factor (non-Darcy flow)
OR,
Buildup equation
Linea
r
Constant
µµµµz
P
P2 PΨ(P)
30002000
Gas
µµµµz
P
Liquid
ρ ρoec P Po–( )
=
Liquid: slightly compressible system
0 P 2000≤ ≤
2000 P 3000≤ ≤
3000 P<
Approximation to ΨΨΨΨ(P)
P2
Ψ(P)
P
is good for all pressuresNote:
Ψ Pwf( ) Ψ Pi( ) 50300PscqgT
Tsckh------------------- 1.151
1688φµctrw2
ktp-------------------------------
log S– D qg++=
°
°
Ψ Pwf( ) Ψ Pi( )1637qgT
kh-----------------------
0.445rw2
ηtp--------------------
log η S D qg+( )–+= where η 2.6374–×10 k
φµgct---------------------------------=
Note: no µg βg because Ψ P( ), 2 Pµz------ Pd∫=
Pseudopressure or Real Gas Potential ( ΨΨΨΨ(P)) Page 41© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Pseudo-steady state equation (PSS): when transient reaches all boundaries of reservoir - must be a closed system.
b. P2- valid for low pressures (P<2000psi) where µz is constant. Gas properties µ, z, Bg, etc. are evaluated at static pressure or initial pressure
q
time
tp ∆∆∆∆t
Ψ Pws( ) Ψ Pi( )1637qgT
kh-----------------------
tp ∆t+
∆t----------------
log+=
Ψ Pwf( ) Ψ Pi( ) 50300PscqgT
Tsckh-------------------
re
rw-----
ln 0.75– S D qg+( )++=
OR
Ψ Pwf( ) Ψ Pi( ) 1422qgT
kh----------
re
rw-----
ln 0.75– S D qg+( )++=
Pseudopressure or Real Gas Potential ( ΨΨΨΨ(P)) Page 42© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
c. P- valid for high pressures (P>3000psi) where uz/P is constant. Gas properties evalu-ated a initial/static pressure. Can use P for tests where Pi and lowest Pware greater than 3000 psi.
Ψ P( ) 2 Pµz------ Pd
PB
P
∫=
- µz is a constant
Ψ P( ) 2µz------ P Pd
PB
P
∫ P2
µz------= =
- Drawdown transient equation: replaceΨ P( ) with P2
µz------
Pwf2
µz------------
Pi2
µz--------
1637qgT
kh-----------------------
0.445rw2
ηtp--------------------
log 0.87 S D qg+( )–+=
for P 02000 µz is constant→
Pwf2 Pi
2 1637qgµzT
kh-------------------------------
0.445rw2
ηtp--------------------
log 0.87 S D qg+( )–+=
- Buildup transient equation:
Pws2 Pwi
2 1637qgµzT
kh-------------------------------
tp ∆t+
∆t----------------
log+=
PSS equation:
Pwf2 Pi
21422
qgµzT
kh-----------------
re
rw-----
ln 0.75– S D qg+( )++=
Pseudopressure or Real Gas Potential ( ΨΨΨΨ(P)) Page 43© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Summary
1. Buildup and drawdown analysis are conducted on gas wells in the same manner as for oil wells.
2. Choose Ψ(P), P2, or P depending upon the pressure range during test period
• Ψ(P) - valid for all pressures ranges. Gas properties for diffusivity, , are evaluated at the static or initial pressure.
• P2 - valid for low pressures (below 2000 psi) where µz is constant. Gas properties µ, z, βg, etc. are evaluated at static or initial pressure.
• P - valid for high pressures (above 3000 psi) where is constant. Gas properties are
Assume Pµz------ constant
Pi
µizi--------- at initial reservoir pressure→= =
Ψ P( ) 2 Pµz------ Pd
PB
P
∫ 2Pi
µizi--------- Pd
PB
P
∫2 Pi( )P
µizi------------------= = =
Drawdown transient equation:
2 Pi( )µizi
-------------Pwf
µz---------
2 Pi( )µizi
-------------Pi
µz------
1637qgT
kh-----------------------
0.445rw2
ηgtp--------------------
log 0.87 S D qg+( )–+=
Pwf Pi1637qgµiziT
kh2Pi---------------------------------
0.445rw2
ηgtp--------------------
log 0.87 S D qg+( )–+=
Consider real gas law:
PVzT--------
sc
PVzT--------
res
=
βgi
Vr
Vsc---------
Psc
Tsc---------
ziTr
Pi----------
1000ScfMcf----------
5.615Scfbbl---------
------------------------------14.7
520-----------
ziTr
Pi---------- 5.035
ziTr
Pi---------- in
RBMcf----------
= = = =
Pwf Pi
162.6qgµβgi
kh--------------------------------
0.445rw2
ηgtp--------------------
log 0.87 S D qg+( )–+= where µ is at end of drawdown
Buildup equation in terms of P:
Pws Pi
162.6qgµβgi
kh--------------------------------
tp ∆t+
∆t----------------
log+=
SG 1.151 ∆Pm--------
k
φµctrw2
-----------------
log 3.23+–=
PSS equation:
Pwf Pi 141.2qgµiβgi
kh------------------
re
rw-----
ln 0.75– S D qg+( )++=
η
Pµz------
Pseudopressure or Real Gas Potential ( ΨΨΨΨ(P)) Page 44© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
evaluated at static or initial pressure. Can use P for tests where Pi and lowest Pwf are greater than 3000 psi.
3. For critical systems or systems where large variation in gas properties occur across the range of test pressures, use Ψ(P).
Determination of Skin and D for Gas WellsGlobal skin, Sg, calculated from gas well tests:
where Sm is mechanical skin and D is a turbulence factor.
Well deliverability or potential is not linear with P, but is dependent upon rate if . For , Sg increases as a function of rate.
PSS Equation:
Example of the effect of turbulence
Multiple Rate Testing• Method for discriminating between Sm and non-Darcy skin.
Sm=5D=1x10-5(MCF/D)q=40,000MCF/D
Sg= Sm + D|q| = 5 + (40000)(1x10 -5) = 5.4
Sm=5D=1x10-4(MCF/D)q=40,000MCF/D
Sg= Sm + D|q| = 5 + (40000)(1x10 -4) = 9
Sg Sm D q+=
D 0≠D 0≠
P
q
D 0≠
D 0=
Pwf Pi141.2qµβ
kh--------------------------
re
rw-----
ln 0.75 S D q+[ ]+––=
Determination of Skin and D for Gas Wells Page 45© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
• kh and Sg are evaluated in standard fashion through PBU’s.
a. Theoretical method
b. Empirical method
Multi-rate test types:
• Flow after flow tests - usually with increasing flow rate
• Isochronal
• Modified isochronal - most popular
• Multi-flows followed by one PBU
a. Theoretical method
The flow equation can be written in the form (Deliverability equations):
Consider P>3000 psi,
1. Transient flow equation (DD)
Multi-rate test (say 4 points) - flow times must be equal
ψ Pi( ) ψ Pwf( )– aq bq2+=
Pi2 Pwf
2– aq bq2+=
Pi Pwf– aq bq2+=
ψ P( ) P→
Pi PWF– 162.6qµβkh
----------ηgtP
0.445rw2
--------------------
log 0.87 Sm D q+( )+=
Pi PWF– 162.6qµβkh
----------ηgtP
0.445rw2
--------------------
log 0.87Sm+ 141.2µβkh-------Dq2
+=
Pi PWF–
q----------------------- 162.6µβ
kh-------
ηgtP
0.445rw2
--------------------
log 0.87Sm+ 141.2µβkh-------Dq+=
Pi PWF–
q----------------------- a t( ) bq+=
Multiple Rate Testing Page 46© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
l
From intercept, mechanical skin, Sm:
From slope, turbulence coefficient, D:
2. Pseudo-steady state flow attained ( for well centered in circular drainage area)
q
Pi Pwf–
q--------------------
b (turbulence)
a(t)
a t( ) 162.6µβkh-------
ηgtP
0.445rw2
--------------------
log 0.87Sm+=
Sm
a t( ) 162.6µβkh-------
ηgtP
0.445rw2
--------------------
log–
141.2µβkh-------
--------------------------------------------------------------------------------------=
Sma t( )kh
141.2µβ---------------------- 1.151
ηgtP
0.445rw2
--------------------
log–=
b 141.2µβkh-------D=
D bkh141.2µβ----------------------=
MSCFD
------------------ 1–
tPre2
4η------->
Pi PWF– 141.2qµβkh
----------re
rw-----
ln 0.75– Sm D q+( )+=
Pi PWF–
q----------------------- 141.2µβ
kh-------
re
rw-----
ln 0.75– Sm+ 141.2µβkh-------Dq+=
a∴ 141.2µβkh-------
re
rw-----
ln 0.75– Sm+=
b 141.2µβkh-------D=
Multiple Rate Testing Page 47© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
from intercept, a, calculate Sm
from slope b, calculate turbulence coefficient D
This yields the stabilized flow equation: . Use this to estimate flow rates
as a function of . Therefore, given “a” and “b”, you can estimate a drawdown for a specified rate, or a rate for a specified drawdown.
NOTE: This development is possible only if PSS is reached during all rates in the multi-rate test.
Same methodology is used for P2 and Ψ(P) analysis:
P2:
• Transient flow
q
Pi Pwf–
q--------------------
b
a
Pi PWF– aq bq2+=
∆P
Pi2 PWF
2– 1637µzT
kh------------------------q
ηtP
0.445rw2
--------------------
log 0.87Sm+ 1422µzTkh
------------------------Dq2+=
Pi2 PWF
2–
q------------------------ a t( ) bq+=
a t( ) 1637µzTkh
------------------------qηtP
0.445rw2
--------------------
log 0.87Sm+=
b 1422µzTkh
------------------------D=
Multiple Rate Testing Page 48© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
• PSS (all rates need to reach PSS)
Deliverability equations:
Now, say we want a deliverability equation of the form , but cannot flow
each rate to PSS. Alternative - flow 3 rates at transient conditions and final rate to PSS.
q
Pi2 Pwf
2–
q---------------------
b
a(t)
Sm 1.151a t( )kh
1637µzT------------------------
ηtP
0.445rw2
--------------------
log–=
Dbkh
1422µzT------------------------=
(Flow times must be equal)
Pi2 PWF
2– 1422µzT
kh------------------------q
re
rw-----
ln 0.75– Sm+ 1422µzTkh
------------------------Dq2+=
Pi2 PWF
2–
q------------------------ a t( ) bq+=
a t( ) 1422µzTkh
------------------------qre
rw-----
ln 0.75– Sm+=
b 1422µzTkh
------------------------D=
Pi PWF– aq bq2+=
q
Pi Pwf–
q--------------------
b
ab
a(t)
PSS
Transient
Multiple Rate Testing Page 49© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Note that the slope, , is the same irregardless of whether flow is transient or
pseudo steady state. However, the intercept, “a”, is different as shown on preceding graph. The intercept from the stabilized or PSS flow is required for the deliverability
equation (“a” in this equation IS NOT a function of time).
c. Empirical method
• AOF - absolute open (hole) flow - ( )
• based on historical observation that a log-log plot of vs. q is approximately a
straight line.
Empirical equation:
Once slope is determined, , estimate c from measured data: . Then the
deliverability equation becomes:
• Flow after flow
• Isochronal
• Modified isochronal
b µβDkh
-----------≈
Pi PWF– aq bq2+=
PWS 14.7psia≈
Pi2 PWF
2–
q c Pi2 PWF
2–( )
n=
Pi2 PWF
2–( )
n qc---=
Pi2 PWF
2–( ) q
1n--- 1
c---
1n---
=
Pi2 PWF
2–( )log 1
n--- qlog 1
n--- 1
c---log+= where 1
n--- 1
c---log is constant
log (q)
Pi2 Pwf
2–( )log
slope = 1/n
AOF
Pi2
14.7( )2–( ) n = 1: Darcy flow
n = 0.5: non-Darcy flow
Therefore,
slope = 1: Darcy flow
slope = 2: non-Darcy flow
1n--- c q
Pi2 PWF
2–( )
n--------------------------------=
q c Pi2 PWF
2–( )
n=
Multiple Rate Testing Page 50© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
• Multi-flows followed by one PBU
a. Flow after flow (discussed)
Theoretical (equal flow times)
Lee’s book refers to stabilization or PSS for each rate, i.e. each rate must reach PSS. Generally this is never feasible and not necessary. Usually never possible to have even one rate reach stabilization.
b. Isochronal testing
• Applicable for any permeability - required for lower permeabilities
• Well is produced at four rates of equal time length
• Well is shut-in for PBU between each flow period until pressure builds back up to initial or static pressure before proceeding to next rate
• Flow time of last rate may be extended until stabilization (PSS). This is done only if fea-sible (need high permeability, small reservoir)
• Isochronal tests performed on wells where time to reach PSS too long
• Data recorded in isochronal tests is transient (except for last rate possibly)
• kh is estimated from PBU’s
Pi Pwf– a t( )q bq2+= - all rates in transient flow
Pi Pwf– a t( )q bq2+= - stabilized deliverability equation (1 rate in PSS)
Multiple Rate Testing Page 51© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Flow equation (transient period)
STANDARD: all 4 rates in transient flow
RARE: 3 rates transient flow, last rate in PSS
Comments:
• Estimation of D is independent of flow regime (transient/PSS)
• Calculation of intercept, “a”, is dependent upon flow regime which will impact deliver-ability equation.
- If final rate reaches stabilization, deliverability equation will be more accurate
- If all rates are in transient regime, extrapolated rates based on deliverability equation will be high (optimistic)
c. Modified isochronal
• Applicable to any permeability system
q1
q2
q3
t
q
q4
t
P
Pi PWF–
q----------------------- 162.6
µβg
kh---------
ηtP
0.445rw2
--------------------
log 0.87Sm+ 141.2µβg
kh---------D q+=
a t( ) 162.6µβg
kh---------
ηtP
0.445rw2
--------------------
log 0.87Sm+=
b 141.2µβg
kh---------D=
Multiple Rate Testing Page 52© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
• Reduced time required to conduct
• Well produced at 4 rates, PBU following each rate. Flowing periods/PBU’s all same time duration.
• Last pressure in each PBU is Pi for analysis of following flowing period (derivative, Odeh-Jones)
• As with isochronal testing, last rate can be extended to stabilization (if practical) to pro-vide more accurate deliverability equation.
• Same analysis procedure as for isochronal testing
• kh is estimated from PBU’s
Analysis procedure:
1. Analyze each PBU for
• kh, S
• kh should be roughly the same from each PBU. If not, most likely error is in rate mea-surement
2. Estimate Sm and D
• Plot Sg vs. q ( )
- if Sg is constant then there is no turbulence
- if Sg is linear with q then there no turbulence
q1
q2
q3
t
q
q4
t
P
Pi1 Pi2 Pi3 Pi4
Sg Sm Dq+=
Multiple Rate Testing Page 53© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
3. Develop deliverability equation:
• transient
- kh is calculated from PBU’s
- Sm and D are calculated intercept and slope, respectively, from a plot of Sg vs. q
- µ and Bg are from static (phase behavior - PVT) data
- t is from test data
• PSS
- can be developed if accurate estimates for kh, Sm and D are made from multi-rate/PBU testing.
- need estimate of reservoir size, re. However, this is normally not very sensitive to the
answer ( )
d. Multi-flows followed by one PBU
q
D
Sm
SG SG = Sm + Dq
Pi PWF– aq bq2+=
Pi PWF– 162.6µβkh-------
ηtP
0.445rw2
--------------------
log 0.87Sm+ 141.2µβg
kh---------Dq+=
Pi PWF– 141.2qµβg
kh-------------
re
rw-----
ln 0.75– Sm+ 141.2µβkh-------Dq2
+=
re
rw-----
ln 7.5≈
Multiple Rate Testing Page 54© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
• Measure BHP vs. time
• Analyze PBU based on multiple rates - superposition
- Guess values for kh, Sg, Pi, Sm, and D until match all flowing pressures.
- Perform non-linear regression on flowing data to estimate Sm and D.
- Use Odeh-Jones analysis to estimate turbulence (pertains only to flowing pressures)
The advantage of flow after flow followed by a PBU is that it saves time. It does not require multiple PBU’s. The disadvantage is that if a reliable kh value cannot be estimated from the final PBU, then the entire analysis can be in error.
t
P
q
D
Sm
SG SG = Sm + Dq
NOTE: SG = Sm + Dq IS NOT VALID
FOR FLOW AFTER FLOW! S G = Sm + Dq only works
for flow-PBU-flow-PBU...
Multiple Rate Testing Page 55© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Odeh-Jones AnalysisSkin analysis (Sm) for gas wells based on flowing pressures. Extension of theoretical development presented earlier.
Multi-rate Drawdown Test Analysis
Assume 2 rate test (both rates are non-zero) and apply superposition.
let
divide through by
where is the superposition time function, STF
Plot
PWF Pi 162.6µβkh-------
0.445rw2
ηt--------------------
log 0.87SG++=
t
q1
q2
t0 t1
Pi PWF–( ) ∆P=
162.6– µβkh------- q1
0.445rw2
ηt--------------------
0.87Sq1– q1
0.445rw2
η t t1–( )--------------------
log– 0.87Sq1 q2
0.445rw2
η t t1–( )--------------------
log 0.87Sq2–+ +log=
m′ 162.6µβkh-------=
Pi PWF–( )∴
m′– q1– t q1 t t1–( ) q2 t t1–( )log– q1
0.445rw2
ηt--------------------
log q1
0.445rw2
ηt--------------------
log–+log+log m′q2
0.445rw2
ηt--------------------
log 0.87S––=
q2
Pi PWF–( )q2
--------------------------- m′q2------ q1 tlog q2 q1–( ) t t1–( )log+[ ] m′ η
0.445rw2
--------------------
log 0.87S++=
q1 tlog q2 q1–( ) t t1–( )log+
Pi PWF–( )q2
--------------------------- vs. STFq2
------------
Odeh-Jones Analysis Page 56© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Now, if non-Darcy flow effects are present, skin increases with increasing rate. Therefore, intercept values, , increases as skin increases.
STF/q2
Pi Pwf–
q2--------------------
slope = m' = ;
intercept = b'
162.6µβkh------- kh 162.6µβ
m'-------=
b′ m′ η
0.445rw2
--------------------
log 0.87S+=
SG 1.151 b′m′------
η
0.445rw2
--------------------
log–=
b′
STF/q2
Pi Pwf–
q2-------------------- b2'
b1'
q2
q1
S2 1.151b2′m′--------
η
0.445rw2
--------------------
log–=
S1 1.151b1′m′--------
η
0.445rw2
--------------------
log–=
S2 > S1
Odeh-Jones Analysis Page 57© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Based on relation SG = Sm + Dq (if flow tests are performed followed by PBU’s)
Flow Regimes1. Radial flow - increase in separation of ∆P and ∆P' indicates increasing skin
2. Spherical flow (partial penetration completions)Flow regime sequence:
- early radial (khp, Sm) - hp is the thickness of the perforated zone
- spherical (kv/kh)
- late radial (kht, SG, Pi) - ht is the total zone thickness
3. Linear flow (hydraulically fractured wells)- Infinite conductivity (no ∆P in the fracture)
q
D
Sm
SG
∆P
∆P′cskh, S, Pi
hp hT
m=0.5
early radial: khp, Sm
(usually masked by WBS)
kv/kh khT, Sg, Pi
Flow Regimes Page 58© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Flow regime sequence:
- linear (fracture half-length)
- late radial (kh, S)
4. Bi-linear flow- Finite conductivity fracture (∆P in fracture accounted for)Flow regime sequence:
- bilinear - flow through fractures (usually masked- rarely seen )
- linear - flow from matrix to fractures
- late radial - radial flow in matrix (basically pure radial)
∆P
∆P'
m = 0.5 (linear region)- characteristic of stimulated wells
kh, S
log
∆ ∆∆∆ P, ∆ ∆∆∆
P'
∆∆∆∆t
Same rate q
fractured
unfractured
t
Pi
m = 0.25 (fracture conductivity, RARELY seen)
∆P
∆P '
m = 0.5 (linear, fracture length)
late radial: kh, S
log
∆ ∆∆∆ P, ∆ ∆∆∆
P'
∆∆∆∆t
Flow Regimes Page 59© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Horizontal wells
Flow regime sequence:
• Early radial - , perforation skin (Sm, if have khh)
kv and kh play role in early radial response. Estimate and, if khh is known, then you
can estimate the perforation skin, Sm.
• Transition region - estimate L (drainhole length) from beginning of transition. You need khh to estimate L.
h
L
kv, kh
transition
late radial
early radial
khhLL kvkh
Lh
L
h
L
h
Early radial:
Transition:
Late radial:
Lkv
kh-----
Lkv
kh-----
Horizontal wells Page 60© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
• Late radial - khh, SG
Need late radial to estimate the well’s productivity index (PI), Sm and drainhole length.
For long drainholes with low , it can take long times to reach late radial.
When do horizontal wells outperform vertical wells:
Physically this means thin reservoir sections with long drainholes with decent (0.05-0.1)
Note: If the deviation <65 degrees, then treat as a vertical well.
Pressure level in surrounding reservoir1. Infinite reservoir
• extrapolation of MTR for Pi
• semi-infinite LTR for Pi (faults, kh changes, etc.)
Horizontal well outperforms vertical well when:
Vertical well outperforms horizontal well when:
(Seen a number of times in Prudhoe Bay)
kv
kh-----
Lkv
kh----- khh» or
kv
kh-----
hL---»
kv
kh-----
Lkv
kh----- khh»
khh, S g
L kvkh , Sm
Lkv
kh----- khh«
khh, S g
L kvkh , Sm
PWS
10001 tP ∆t+
∆t-----------------log
MTR
ETR
P*=Pi
PWS
10001 tP ∆t+
∆t-----------------log
MTR
ETR
P*=Pi
LTR
Pressure level in surrounding reservoir Page 61© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
2. Depleted reservoir - average static drainage area pressure ( )
= stabilized pressure if well was SI in depleted field.
P* method: Mathews-Brohs-Hazeroch (MBH)(pp. 35-38 of Lee)
1. From Horner plot, extrapolate the MTR to P*,
2. Estimate drainage area shape and size (A) in ft2
3. Calculate - use same tp used to construct Horner plot
4. Choose appropriate curve from figure 2.17 A-G (Lee)
5. Enter plot on abscissa at , go up to appropriate curve, read value of
P P∗≠
P
P
L or distance
A B C
PP
∆t ∞tp ∆t+
∆t---------------- 1=,→
PWS
10001 tP ∆t+
∆t-----------------log
MTR
ETR
P*=Pi
LTR
P
ηtpA
--------
ηtpA
--------
Pressure level in surrounding reservoir Page 62© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
, calculate .
Note: where m = horner MTR slope ( ). Also, =
the derivative m.
Advantages
• does not require data beyond MTR. However, MTR MUST be present
• applicable to wide variety of drainage shapes (well need not be centered)
Disadvantages
• requires knowledge of drainage area size and shape
• not good for layered reservoirs
• requires knowledge of fluid properties and porosity and ct
Example 2.6 P* method
Use data from examples 2.2-2.4
Well centered in square drainage area
tp = 13630 hours
P* = 4577 psia
m = 70
k = 7.65 mD
A = (2640)2 = 6.97x106 ft2 (160 acres)
From figure 2.17A, p. 36
Modified Muskat Method(pp. 40-41, Lee)
2.302 P∗ P–( )m
------------------------------------- PDMBH= P
P∗ P–
70.6qµβkh
--------------------------------- 2.302 P∗ P–( )
m-------------------------------------= 162.6qµβ
kh---------- 70.6qµβ
kh----------
η 2.6374–×10( ) 7.65( )
0.039( ) 0.8( ) 1.75–×10( )
------------------------------------------------------------- 3800ft2
hr------= =
ηtPA
-------- 3800( ) 13630( )
6.976×10
---------------------------------------- 7.45= =
PDMBH5.45 2.302 P∗ P–( )
m-------------------------------------= =
P∴ 4577 5.45( ) 70( )2.302
----------------------------– 4411psia= =
Pressure level in surrounding reservoir Page 63© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Using superposition and PSS solution, the late time PBU can be approximated by:
Therefore, plot vs.∆t. If correct, will plot as a straight line. Data must be in
following time range:
Modified Muskat Method Procedure
1. Assume a value of
2. Plot vs.∆t
3. If line is straight - correct
If line curves upward - too large
If line curves downward - too small
4. Try another using above guidelines until line is straight
Restrictions
• method fails if well not centered in drainage area
• requires long shut-in (needs to reach PSS)
• difficult to pick correct straight line
P PWS– 118.6qµβkh
----------0.00388k∆t–
φµctre2
----------------------------------
exp=
P PWS–( )log 118.6qµβkh
---------- log 0.00168
k∆t
φµctre2
-----------------
–=
P PWS–( )log A B∆t+=
P PWS–( )log P
250φµctre2
k--------------------------- ∆t
750φµctre2
k---------------------------≤ ≤
or
0.51re( )2
4η------------------------ ∆t
0.89re( )2
4η------------------------≤ ≤
P
P PWS–( )log
P
P
P
∆∆∆∆t
P PWS–( )logToo large
Too small
P
Pressure level in surrounding reservoir Page 64© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Advantages
• requires no knowledge of reservoir properties (A, φ, ct, etc.)
• works for hydraulically fractured wells (assuming radial flow is established)
Drill Stem Tests (DST)• performed through dedicated test string (3.5-4.5)
• valves are annulus or tubing operated (perform poorly in mud due to solids)
• downhole shut-in a plus to minimize wellbore storage
• must kill well to recover string and gauges
• normally only done on exploration or appraisal wells
• requires rig to trip test string
• can perforate tubing conveyed perforators or wire line
• need cushion to bring well on (seawater, diesel, nitrogen)
Flow/PBU periods
• oil: 24 hour stable flow after cleanup (defined as basic sediment and water < 5%) 36 hour PBU
• gas: 3-4 rate test after cleanup, 8 hours per test (single rate, same as oil test, if D not required) 36 hour PBU
Pressure measurement
• memory gauges
• surface readout
Conducting Well TestsCompleted Wells (development scenario)
• wells completed with final tubing string
1. Run memory gauges (2) on SL or use surface readout gauges (PLT) on EL
• quartz gauges with lithium battery (temperatures to 350F)
• Run gauges with well flowing or prior to opening up well. Must record flowing pressures prior to PBU for skin calculation
• Obtain accurate rate measurement (history)
• Do not move gauge or wireline, or perform any well operation during PBU!
• Place gauge as close to perforations as possible to minimize phase segregation effects
• Make static gradient mm while pulling out of hole at conclusion of PBU to verify wellbore fluid composition
• Memory gauges must be programmed on surface prior to running in hole on SL
Drill Stem Tests (DST) Page 65© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
• Rate of pressure measurement can be modified with surface readout gauges (EL)
2. Softset gauges - not good with sand production!
• Run tandem gauges on SL and set on bottom
• Retrieve days, weeks, months later and download/analyze data
• Need accurate rate measurement to take full advantage of pressure measurement
• Many short PBU’s will occur over weeks, months
• Use gap capacitance/quartz gauges
3. Permanent gauge installation
• Install 2 gauges (quartz) in mandrels
• Need electrical cable run to surface (similar to ESP), as well as, data transmission cable (pressure, time temperature)
• Excellent for remote locations where wire line intervention is difficult. Negates the need to run wire line gauges
• Payout over life of well (cost ≅ U$150,000)
4. Gauge/flowmeter installation
• Exal/Expro permanent gauge/flowmeter
• Quartz gauge
• Flowmeter, venturi effect, estimate flow rate based on ∆P (Bernoulli’s principle)
• Remote locations, subsea applications where a dedicated flowline per well is not feasi-ble
• Only good for single phase flow
• An example of such an installation is the BP-Amoco/Shell/Marathon Troika project
Wellbore EffectsPhase segregation
• Need two or more phases
• If gradient changes between gauge and perforations during PBU due to phase segrega-tion (water falling/oil rising, water falling/gas rising), the pressure data will be corrupted until phase segregation is complete
• If gauge is above the interval and phase segregation occurs during PBU, the pressure is greater than pure reservoir response
• Gas humping: Water falling back/imbibing into formation during PBU. Can be especially severe in low permeability gas reservoirs. Remedy: Place gauges as close as possible to top of perforations or within perforated interval or below interval (within 50 feet should be OK)
Wellbore Effects Page 66© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Indexù
Buildup Test Solutions 28
÷Continuity equation (cylindrical coordinates) 7
õDarcy’s Law 8Derivative Analysis (Drawdown case) 30Drawdown test 4Drill Stem Tests 65Drillstem test (DST) 5
ñFalloff test 5Flow efficiency 13Flow Regimes & Model Recognition 34
íHorizontal wells 60Horner’s Approximation 28
ëInjection test 4Interference/pulse test 5Isotropic 8
ãMathews-Brohs-Hazeroch 62Modified Muskat Method 63Multiple Rate Testing 45Multi-rate Drawdown Test Analysis 56
!Odeh-Jones Analysis 56
#Pressure buildup test 4Pseudo (Y(P)) Equation Development 38
'Radius of Investigation 21
)Skin (drawdown) 31
Index Page 67© 2000-2001 M. Peter Ferrero, IX
Pressure Transient Analysis
Skin and D for Gas Wells 45Skin Development 12Solutions to the Diffusivity Equation 11
Exact solution 11Infinite reservoir, line source 11Line source solution 11Van Everdingen & Hurst terminal rate solution 11
Superposition 22
1Wellbore Effects 66Wellbore Solutions 16
Ideal reservoir (no skin) 16Solution at sandface (including skin) 16
Wellbore Storage 16Competely liquid filled wellbore 19Determining the end of WBS 20Gas-liquid interface 17
Index Page 68© 2000-2001 M. Peter Ferrero, IX