pressure transient analysis

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


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 © 2000-2001 M. Peter Ferrero, IX


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: © 2000-2001 M. Peter Ferrero, IX


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 © 2000-2001 M. Peter Ferrero, IX


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 © 2000-2001 M. Peter Ferrero, IX


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 © 2000-2001 M. Peter Ferrero, IX


• 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 © 2000-2001 M. Peter Ferrero, IX


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---=;→



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=



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



• 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 © 2000-2001 M. Peter Ferrero, IX


• 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 © 2000-2001 M. Peter Ferrero, IX


• 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=∴→



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)



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)



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 © 2000-2001 M. Peter Ferrero, IX


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) © 2000-2001 M. Peter Ferrero, IX


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=



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==



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≤



= 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) © 2000-2001 M. Peter Ferrero, IX


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 © 2000-2001 M. Peter Ferrero, IX


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 © 2000-2001 M. Peter Ferrero, IX


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≈



Intersecting faults (90 degree)

Need three image wells


(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



Intersecting faults (45 degree)

Need seven image wells


(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



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)



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 © 2000-2001 M. Peter Ferrero, IX


• 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 © 2000-2001 M. Peter Ferrero, IX


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) © 2000-2001 M. Peter Ferrero, IX


Skin

a. DD


----------------------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

-----------------------------------------–=



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



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 © 2000-2001 M. Peter Ferrero, IX


Flow Regimes & Model Recognition

Radial flow

homogeneous, infinite acting system

single fault

ETR

MTR

∆P

∆P’

∆t

WBS dominates


---------- 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 © 2000-2001 M. Peter Ferrero, IX


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( )′=



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



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!



- 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 © 2000-2001 M. Peter Ferrero, IX


- 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 © 2000-2001 M. Peter Ferrero, IX


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 © 2000-2001 M. Peter Ferrero, IX


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)) © 2000-2001 M. Peter Ferrero, IX


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+( )++=



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+( )++=



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------



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 © 2000-2001 M. Peter Ferrero, IX


• 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→


----------ηgtP

0.445rw2

--------------------

log 0.87 Sm D q+( )+=


----------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 © 2000-2001 M. Peter Ferrero, IX


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------->


----------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=



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=



• 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



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=



• 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)



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=



• 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+=



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≈



• 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...



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 © 2000-2001 M. Peter Ferrero, IX


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 © 2000-2001 M. Peter Ferrero, IX


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 © 2000-2001 M. Peter Ferrero, IX


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 © 2000-2001 M. Peter Ferrero, IX


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 © 2000-2001 M. Peter Ferrero, IX


• 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 © 2000-2001 M. Peter Ferrero, IX


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

--------



, 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= =



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



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) © 2000-2001 M. Peter Ferrero, IX


• 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 © 2000-2001 M. Peter Ferrero, IX


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 © 2000-2001 M. Peter Ferrero, IX


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 © 2000-2001 M. Peter Ferrero, IX

pressure transient analysis

Documents

partial penetration

van everdingen

fully perforated

westport technology

t1 t2 t3

pressure transient

flow regime

real gas potential