pressure transient analysis - houston university

8/7/2019 Pressure Transient Analysis - Houston University

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Petroleum Engineering

UniversityofHouston

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

Horners 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 ((P)) Equation Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Pseudopressure or Real Gas Potential ((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). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Conducting Well Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Wellbore Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Pressure Transient Analysis

Spring 2001


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Introduction Page 2


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 Houston

Dr. Christine Ehlig-Economidesemail: [email protected]

M.S.: Chemical Engineering, University of KansasPh.D.: Petroleum Engineering, Stanford University


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Description of a well test: Page 3


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 qs (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


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Types of tests Page 4


Types of tests

Drawdown test (DD)

difficult to maintain constant rate

this introduces scatter in mea-sured FBHP (flowing bottom holepressure)

Pressure buildup test (PBU)

advantage: rate is known, i.e. q=0

disadvantage: lost production

Injection test

advantage: injection rates areeasily 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


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Why we do transient testing Page 5


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 thesteady-state solutions are not valid this is known as transient flow. This is the period thatis 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

kh

------ hct

P 141.2qkh

--------------------------rerw-----ln S+

=


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Flow States Page 6


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

Pt------- 0=

Pt------- cons ttan=

Time

P LinearP

rw re

t3

t2

t1

Fig. 7. Pseudo steady-state flow regime


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Development of Flow Equations for Flow in Porous Media Page 7


Transient, , pres-

sure in reservoir/wellbore arechanging as a function of both

time and location.

Development of Flow

Equations for Flow in Porous MediaNote: there is a good writeup in Appendix A of Lee.

Whats needed to derive the diffusivity equation is:

A. Conservation of Mass (Continuity equation)

B. Darcys Law

C. Equation of State (EOS)

A. Continuity equation, cylindrical coordinates (r, z, )

P

rw re

t3

t2

t1

Fig. 8. Transient flow regime

P

t------- f x y z t , , ,( )=

vzr r dd

z v

z( ) zd r rd d+

v r zdd

vrr zdd r

rvr( ) d r zdd+

dr

dz

d

rd

vrr zdd

vzr r dd

z v

z( ) zd r rd d+

vzr r dd

Fig. 9. Cylindrical coordinate system


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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 , the last two terms are 0

.... therefore, for a fully perforated interval, the RADIAL DIFFUSIVITY EQUATIONis

B. Darcys Law

Isotropic: k=kr=k=kz

Assume single slightly compressible fluid - compressibility, c= constant

By integration:

v lbm

ft3

----------ft

s--- lbm

ft2

s--------------= =

t

r r zddd( ) r r zddd t

( )=

vrr zdd

r rv

r( ) d r zdd+ v

rr zdd[ ] ....r direction

v r zdd v( ) d r zdd+ v r zdd[ ] .... direction

vzr r dd

z v

z( ) zd r rd d+ v

zr r dd[ ] ....z direction

r r zt

( )dddr

rvr( ) r zddd v( ) r zddd z

vz( )r r zddd+ + + 0=

r r zddd

t ( ) 1

r---

r rvr( )

1

r---

v( ) z

vz( )+ + + 0=

t ( ) 1

r---

r rvr( )+ 0=

k

--- P=

rkr----

rd

dP=

k-----

ddP=

zkz-----

zd

dP=

t ( ) 1

r---

r r

kr----

rd

dP

+ 0=

or

1

r---

r r

kr----

rd

dP

t ( )=

c1

Vol---------

Pd

dVol 1

---

Pd

dVol

1

---=;


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Note:

1. Since doesnt change wrt time,

2. Also, since the pressure gradient is small,

Canceling s, and dividing through by

.... therefore, for a fully perforated interval, the RADIAL DIFFUSIVITY EQUATIONincluding Darcys law is

To solve this you need two boundary conditions and one initial condition. For a closedsystem:

Initial condition: P = Pi @ t=0

Boundary condition 1: No flow -

0ec P P0( )

c PdP0

P

------ 0 base ;0

= = =

r

c0ec P P0( )

rP

cr

P= =

r

c0ec P P0( )

rP

cr

P= =

1

r---

r rk

---

rP

t ( )=

1

r

---

r

Prk

---

r

P k

---

r

P rk

---

r2

2

P+ +

t

t

+=

k

---

1

r--- cr

rP

2

rP r

r

2

P

+ +

ct

P=

t

0

rP

2

1 crr

P

2

0;

r---

k

--- rP

r r

2

P

+

c t

P=

k

---

1

r---

rP r

r

2

P

+ c

k----------

tP

=

1

r---

r

rr

P 1

---

tP

where kc----------==

rP

re

0=


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Boundary condition 2:

For an infinite reservoir, BC1 becomes as .

Darcys law came from Darcys investigation of the sewers in Paris. He conducted hisexperiments 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:

rP

rw

q2hrw----------------=

P Pi r

velocity, u 0.001127k

-------

ld

dP =

q 0.001127kA

-------

ld

dP = l

P 2

P 1q

q

Pe rm ea bi l i ty , k

W a ter viscosity, w

Fig. 10. Steady-state linear flow

velocity, u 0.001127k

---

rd

dP =

q 0.0011272rwk

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

rd

dP =

qdr

r-----

rw

r2

0.007082rwk

---------------- dP

Pw

P2 =

q 0.00708kh

-------

P2 Pw( )r2rw-----

ln

------------------------- =

h

rw

Area, A = 2 rw h (area o f cy l ind er )

Fig. 11. Darcy velocity in cylindrical coordinates


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Solutions to the Diffusivity Equation Page 11


Continuity equation

Darcys law

EOS

2. Assumptions:

a. Radial flow over entire net thickness

b. Homogeneous and isotropic porous media (kr=k=kz)

c. Uniform net thickness

d. q and k are constant (independant of pressure)

e. Fluid is of small and constant compressibility

f. Constant g. 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 wont 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:

rP

2

1

Pwf Pi141.2q

kh--------------------------

2tD

reD2

-------- reDln3

4---+ 2

e2tD

J12 reD( )

2

J12

reD J12( )

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

1=

+=

1

r---

r

rr

P 1

---

tP

=

rr

P

rw

q2kh--------------=

f

P Pi r

rr

P

rwr 0lim

q2kh--------------=

P r t( , ) Pi 70.6qkh

----------Eir2

4t---------

where Ei x( )e

-------- d

x

=

;+=


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Skin Development Page 12


DRAWDOWN ONLY

Constant rate

Unbounded reservoir

Limitations of the line source solution (Ei)

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 its 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 E i function may be valid at radius, r (rw < r < re), the ln approxi-

mation wont always be valid.

Skin DevelopmentSkin, S, refers to a region near the wellbore of improved or reduced permeabilitycompared 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

--------------- t

re2

4-------

t100rw

2

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

P r t( , ) Pi 70.6qkh

----------Eir2

4t---------

+=

Ei x( ) 1.781x( ) x 0.02,ln=

Eir2

4t---------

0.445r2

t--------------------

r2

4t--------- 0.02,ln=

rw2

4t--------- 0.02

rw2

4t---------

0.01

4-----------


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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 Darcys law:

Darcy w/o S

Darcy /w S-------------------------------- or FE

8

S 8+-------------,

Radius

Pressure

k of formationk including skin

Pk

Ps

rw rs

Pks

Fig. 12. Skin pressure drop

Ps Pks Pk=

Ps 141.2qksh----------

rsrw-----

ln 141.2qkh

----------rsrw-----

ln=

Ps 141.2qkh

----------k

ks----- 1

rsrw-----

ln=

We definek

ks----- 1

rsrw-----

ln S=

Ps 141.2qkh

----------S=

Ptotal PS 0= PS+=

Ptotal 141.2qkh

----------rerw-----

ln 141.2qkh

----------S+ 141.2qkh

----------rerw-----

ln S+= =

S 0> Damaged ks k

S 0= Undamaged ks k=


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SEM examples of various clays which can cause formation damage

Fig. 13. Smectite (left) and kaolinite (right) coat grains and fill apore. Note distinct differences in morphology of eachclay ("honeycomb" smectite; vermicular booklets ofkaolinite (x2000)(image courtesey of Westport Technology Center)

Fig. 14. Delicate wisps of "hairy" illite project into a pore. Notethat the fibers not only form a highly rugose surfacewithin the pore, but the fibers could break and migrateunder extreme fluid pressures (x2500)(image courtesey of Westport Technology Center)

Fig. 15. Well-formed chlorite platelets form partial rosettesadjacent to, and coating quartz overgrowths (x2500)(image courtesey of Westport Technology Center)

Fig. 16. Well-formed, but rather randomly oriented kaolinitebooklets post-date quartz overgrowths (x700)(image courtesey of Westport Technology Center)


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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 polymerfilament (x105)

Fig. 19. SEM image of pre-acid treatment (x3100) Fig. 20. SEM image of post-acid treatment (x3100)


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Wellbore Solutions Page 16


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 flowwhen 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 70.6qkh

----------0.445rw

2

t--------------------

where 2.637410 k

ct---------------------------------=;ln+=

Pw t( ) Pi 70.6qkh

----------1688ctrw

2

ktp-------------------------------

from Lee;ln+=

Pwf Pi Pwf Pk Pskin+ 70.6qkh

----------0.445rw

2

t--------------------

ln 141.2qkh

----------S+= = =

Pwf 70.6qkh

----------0.445rw

2

t-------------------- 2S

ln=

Pwf Pi 70.6qkh

----------0.445rw

2

t--------------------

2Sln 2.637410 k

ct

---------------------------------=;+=

b 0


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Wellbore Storage (WBS) Page 17


General definitions

Vwb = volume of liquid in well (ft3)

Awb = cross-sectional area of well (ft2)

l = density of wellbore fluid (lbm/ft3)

h = height of liquid column inwellbore (ft)

Gas-liquid interface

pumping wells, gas lift wells

injection wells (on vacuum)

an approximation for most natu-

rally flowing oil wells (excepthighly undersaturated oils, P>Pb)

Wellbore mass balance

[Mass inflow] - [Mass outflow] = Accumulation of Mass

Assume constant density, l

Note:

Vwb

Awbhliquid

dh

Pt

q ((((RB/D)

qSF ((((RB/D) Pw + Pt + lgh144

Fig. 21. Wellbore storage definitions

qSF q( )24

5.615---------------

td

d vWB( )=

bbl

D

--------lbm

ft3

---------- ft

3

bbl

--------24

5.615

---------------lbm

ft3

---------- ft3

=

qSF q( )24

5.615---------------

td

dvWB where vWB AWBh td

dvWB AWB td

dh=;=;=

qSF q( )24

5.615---------------AWB td

dh=

h144 Pw Pt( )

g----------------------------------=

td

dh 144

g----------

td

dPwassume

td

dPt=;=

qSF q( ) )24

5.615---------------

144AWBg

----------------------td

dPw=


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Definition: Wellbore storage coefficient for a gas-liquid interface

Example:

3.5 tubing, AWB = 0.041 ft2

o = 50 lbm/ft3vwb = 100 bbl

depth = 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

= 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 plot

A - straight line on cartesian plot

Why?A - 100% WBS, q=0 (PBU)

Therefore, cs can be calcu-

lated from the slope of a straightline (intercept must be zero!)

B - Log-log plot, 100% WBS,q=0 @surface (PBU)

cs144ABW5.615l----------------------

25.65ABWl

---------------------------= =bbl

psi--------

cs 25.65AWB

l------------ 25.65

0.041

50---------------

0.02 bblps i--------= = =

qSF q( ) 24cs

----- tddPw

=

td

dPw

t

P qSF24cs------------- m=

Fig. 22. cs from cartesian plot

qSF 24cs-----

td

dPw=


19/68




Estimate cs from any ( )

pair on unit slope line

Completely liquid filled wellbore

Wellbore mass balance

[Mass inflow] - [Mass outflow] = Accumulation of Mass

log t

log Pwm = 1

qSF24cs-------------

Fig. 23. cs from log-log plot. Estimate cs from any (Pw, t) pair on unit slopeline

qSF 24cs-----

Pt--------=

PwqSF24cs-------------t=

Pw( )log qSF

24cs-------------tlog=

Pw( )log m t( )logqSF24cs-------------

log+=

Pwt,

tlndd

x( )

td

dx( )

td

dt( )ln

t( )lndd

x( ) tt( )lnd

dx( )= =

td

d x( ) tt( )lnd

d x( )=

PW tdd PW( ) t

qSF24cS-------------= =

Take log of both sides[ ]

td

d PW( )log t( )logqSF24cS-------------

log+=

m 1 interceptqSF24cS------------- for P==


20/68




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

qSF q( )24

5.615---------------

td

d vWB( )=

Note vWB AWBh=[ ]

qSF q( )24

5.615

---------------vWBtd

d

=

by chain rule c1

---

P

td

dPPwd

d

td

dPw c

td

dPw= =

qSF q( )24

5.615---------------vWBc td

dPw=

qSF q( )24

5.615---------------vWBc td

dPw=

csvWBc

5.615--------------- bb l

ps i-------- where c = average fluid compressibility

csvWBc

5.615---------------

100 1510( )

5.615--------------------------------- 0.0002= = =

bb l

ps i--------

qSF q 24cs-----

td

dPw=

qSF 0 initially as open to rate q=

q 24cs-----

td

dPw=

q 0 initially as the well is shut in=

qSF fixed=

qSF 24cs-----

td

dPw=

WBS is over when 24

cs

----- td

dPw

0.01q


21/68


Radius of Investigation (ROI) Page 21


= 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 cp

c psi-

q1

q2

q3

t

PWF

t

P

rw re

t3

t2

t1

Pi

r1 r3r2

Fig. 24. Illustration of ROI

Ri 4t= 2.637 410 k

ct---------------------------------=;

Radius of investigation is INDEPENDENT of q


22/68


Pseudo Steady-State Page 22


Pseudo Steady-StateDepletion of a closed system

Pseudo steady-state occurs when the pressure transient has reached all boundaries in aclosed 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 BCs).

Therefore, linear combinations of solutions are also solutions. The combined linearsolution eliminates the following restrictions:

Single well

Reservoir boundaries

Constant rate

PWF Pi141.2q

kh--------------------------

2t

re2

---------rerw-----

ln 0.75+ for= tre2

4-------

tPWF 141.2q

kh--------------------------

2

re2

-------= 2.637410 k

ct---------------------------------=;

tPWF 141.2q

kh--------------------------

2 2.637410( )k

re

2

c

t

-----------------------------------------0.0744q

c

thr

e

2---------------------------- Note:Vp re

2h reservoir volume== =

tPWF 0.234q

ctVp----------------------

Pt--------= =

1

r---

r

rr

P 1

---

tP

=


23/68


Principle of Superposition Page 23


Multi-well solution

Determine

Check for if

B

A

C

rAB

rAC

q

t

qA

qB

qC

PAPTOTAL

APA PB PC+ +=

P r t( , ) Pi 70.6q

kh----------

0.445r2

t--------------------ln 2S +=

P Pi P r t( , ) 70.6q

kh----------

0.445r2

t--------------------

ln 2S = =

PTOTALA

70.6qA

kh--------------

0.445r2

t--------------------

ln 2SA 70.6qB

kh-------------- Ei

rAB2

4t------------

70.6qC

kh-------------- Ei

rAC2

4t------------

=

1.781x( )ln r2

4t--------- 0.02


24/68




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 70.6qkh

----------0.445rw

2

t--------------------

ln 2S 70.6qkh

---------- Ei2L( )24t

----------------- ==

Ei4L

2

4t---------

0.445 2L( )2

t------------------------------

ln


25/68




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

q

image

qimage

L

L

L

L

no flow boundary

Fig. 28. 90 degree intersecting fault mathematical model


26/68




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


27/68




Variable rate

Single 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)


28/68


Horners Approximation Page 28


General Solution

Horners 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

70.6

kh------- q

i

qi 1

( )

i 1=

m

0.445rw2

t ti 1( )--------------------------

ln 2S= =

Can incorporate dozens of rates

Ei xln( )

Ei

tP 24Production from wellMost recent rate

-------------------------------------------------------------NPqlast----------= =

P Pi

PWF

70.6qlast

kh------------------

0.445rw2

tP--------------------

ln 2S= =

Note: tlast 2 tnext to last>

qlast

qnext PBU

q=0


29/68


Buildup Test Solutions Page 29


Use superposition to model variable rates

q is the rate prior to PBU. Use Horners approximation with multiple rates

P* is always taken as the extrapolation from the MTR irregardless of whether boundariesor 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+

Pqt DD PBU Pi PWF( ) PWS PWF( ) Pi PWS= = = =

P Pi PWS 70.6qkh

----------0.445rw

2

tp t+( )-------------------------

ln 2S 70.6q( )kh

------------------0.445rw

2

t( )--------------------

ln 2S= =

Pi PWS 70.6qkh

----------0.445rw2

tp t+( )-------------------------

ln 2S

0.445rw2

t( )--------------------

ln 2S+=

PWS Pi 70.6qkh

----------0.445rw

2

tp t+( )-------------------------

ln0.445rw

2

t( )--------------------

ln+=

Pi 70.6qkh

----------t

tp t+( )---------------------

ln+=

Note: xln 2.302 xlog=

P WS Pi 162.6qkh

----------tp t+( )

t---------------------

log=

myx-------

P2 P1

tP t2+t2

-------------------- log

tP t1+t1

-------------------- log

--------------------------------------------------------------------------= =

PWS

1000 100 10 1

tP

t+

t------------------

Pi = P* (infinite shut-in)

m162.6q

kh--------------------------=

P2 P1

10( )log 100( )log-------------------------------------------------

P2 P1

1 2------------------- P1 P2= ==

tP t+

t-----------------

t lim 1=Note:

P


30/68


Derivative Analysis (Drawdown case) Page 30


Derivative Analysis (Drawdown case)Bourdet derivative

Drawdown solution

tlnd

d 1

2.302---------------

tlogd

d=

By chain ruled( )dt

----------td

d tlnd( )d tln---------- 1

t---

d( )d tln----------= =

d( )d tln---------- t

d( )dt

----------=

PWF Pi70.6q

kh----------------------

0.445rw2

t--------------------

ln 2S+=

Pi PWF70.6q

kh----------------------

0.445rw2

t--------------------

ln 2S70.6q

kh---------------------- tln

0.445rw2

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

ln 2S+= =

Take Bourdet Derivative

tlnd

dPi PWF( ) t

70.6qkh

---------------------- 1

t---

70.6qkh

----------------------= =td

dtln

1

t---=;

tlnd

dPi PWF( ) tlnd

d P( ) m 70.6qkh

----------------------= = =

PWF

10001

m

162.6qkh

--------------------------=

tPlog

10001

m70.6q

kh------------------------=

kh 70.6qd P( )d tln

----------------@ stabilization

-----------------------------------------------------------=

tlog

d P( )

d tln

----------------log

MTR


31/68




Skin

a. DD

Pi PWF70.6q

kh----------------------

0.445rw2

t--------------------

ln 2S162.6q

kh--------------------------

0.445rw2

t--------------------

log 0.87S= =

Pi PWF mt

0.445rw2

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

log 0.87S+ m tlog

0.445rw2

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

log 0.87S+ += =

Pi PWF

m----------------------- tlog

0.445rw2

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

log 0.87S+ +=

S 1.151Pi PWF

m-----------------------

2.25

rw2

---------------log tlog=

Take t = 1 hour

SDD 1.151Pi PWF1hr

m----------------------------

2.25

rw

2---------------log=

m162.6q

kh--------------------------=

tPlogtp=1

PWF1hr

Semi-log MTR!

tP

tps

P Pi PWF=

P kh 70.6qd P( )d tln

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

-------------------------=


32/68




b. PBUThe instant a well is shut-in, PWF :

PWF Pi162.6q

kh--------------------------

0.445rw2

tP--------------------

log 0.87S+=

PWF Pi mtP

0.445rw2

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

log 0.87S+=

PWF Pi m tPlog2.25

rw2

---------------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 m2.25

rw

2---------------

log 0.87S+ + +=

PWS PWF

m-----------------------------

tP t+tPt

----------------- log k

ctrw2

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

log 3.23 0.87S+ +=

SPBU 1.151PWS PWFt 0=

mHorner semi-log MTR-------------------------------------------------

k

ctrw2

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

log 3.23tP t+tPt

----------------- log tlog+ +=

P PWS PWF t 0==

tlogt s

m 70.6q

kh

------------------------=

P

PWSskin

d P( )d

tP t+t

----------------- ln

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

m162.6q

kh--------------------------=

tP t+t

-----------------log

PWS


33/68


Ideal vs. Actual PBU/DD Tests Page 33


Ideal vs. Actual PBU/DD Tests

a. Drawdown case

b. Drawdown: log-log plot

m162.6q

kh--------------------------=

tPlog

PWF

tPlog

ETRWBS

MTRkh, S

Infinite acting

Radial 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


34/68


Flow Regimes & Model Recognition Page 34


Flow Regimes & Model Recognition

Radial flow

homogeneous, infinite acting system

single fault

ETR

MTR

P

P

t

WBS dominates

Pi PWF 162.6qkh

---------- tlog constant+=

d P( )d tln

---------------- 70.6qkh

----------=

kh70.6q

Pstabilized-----------------------------=

Using superposition and image wells

Ptotal Pwell P eimag+=

Pi PWF 70.6qkh

----------0.445rw

2

t--------------------

ln 2S 70.6qkh

----------0.445 2L( )2

t------------------------------ln

==

70.6qkh

---------- 20.445

t---------------

ln rw2

ln 2L( )2ln 2S+ +=

PWF Pi 162.6qkh---------- 2

0.445

t--------------- log rw2

log 2L( )2

log 2S+ ++=

Note:td

dtln( ) 1

t---=

d P( )d tln

---------------- td( )dt

----------=d P( )d tln

---------------- t 70.6qkh

---------- 21

t---=

slope doubles2 faults, slope x4

3 faults, slope x8, etc.

tPlog

ETR MTR LTRP

P P

Pm

2m

PWS

10001

m

2m

ETRMTRLTR

tP t+t

-----------------log


35/68




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 LTRP

P

P

P

(kh)inner

(kh)outer

Concentric model:

inner

outer

Radius for kinner:

t is where slope becomes negative

[For ROIs in outer zone, use k of outer zone! No matter if the k ishigher or lower]

increase

decrease

ROI 4t ; 2.637

410 kict

----------------------------------==

tPlog

ETR MTR LTRP

P

Po

P

w < ow > o

Same kh!

kh

------

okh

------

w

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

70.6qP

o

( )----------------------

70.6qPw( )

---------------------------------------------= same kh!

Pw( )wo------- Po( )=

Pw

variablekh!

Pw( )

kh

------

o

kh

------

w

---------------- Po( )=


36/68




Spherical (Partial Penetration Completions)

P 70.6 kh------- q

0.445rw2

t--------------------

ln q0.445 2L( )2

t------------------------------

ln==

70.6 kh------- q 0.445

t--------------- ln q

0.445t

--------------- ln qrw2

2L( )2--------------ln+=

P 70.6 qkh

----------rw2L-------

2

ln=

PWS

10001

m=0

tP t+t

-----------------log t

reality

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

hTh p


37/68




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 mDft

pressure drop in fracture is not negligible

almost never happens

tPlog

Linear

P

m=0.5

flow

Radial

flow

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!


38/68


Gas Well Testing Page 38


- if you can see the bi-linear region, it can be used to estimate frac-conductivity (if thematrix 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), , z, cg vary as a function of pressure. Gas is considered ahighlycompressible fluid whereas oil is considered a slightlycompressible fluid.

Non-darcy flow, or turbulence, can exist in gas wells which shows up as a skin due toextra pressure drop. Therefore, differentiation between true mechanical skin and skindue to non-darcy flow is important

- non-darcy flow signifies that Darcys law does not properly predict the P due to flowof gas in porous media

- in porous media, non-darcy flow develops when Re > 50 ( )

- low 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: and z may vary considerably as a function of pressure. Therefore, to accountfor this, the pseudo-pressure function was developed.

Gas Tests - Pseudo ((P)) Equation Development

a. Continuity equation

b. Darcys law

c. EOS

Red

----------=

MW

RT-----------

Pz----

= P z RMW-----------T=

P( ) 2 Pz------ Pd

PB

P

=

x ux( ) y

uy( ) z uz( )+ + t

( )=

uk

--- P= ux

kx-----

xP

=; uyky-----

yP

=; uzkz-----

zP

=;


39/68


Gas Tests - Pseudo ((P)) Equation Development Page 39 2000-2001 M. Peter Ferrero, IX

- oil and water: slightly compressible fluids

- For gases

(b) + (c) into (a)

Differentiating (P) wrt x, y, z, and t

Input Darcys law into Continuity equation:

Input EOS:

assume isotropic conditions

oec o( )

=

MWRT-----------

P

z----

=

isotropic kx ky kz= =

cx

P

2

yP

2

zP

2

+ +x

2

2

P

y2

2

P

z2

2

P+ + +

ct

2.637410 k

---------------------------------t

P=

P( ) 2 Pz------ Pd

PB

P

=

x 2P

z-------

xP

=x

P z2P-------

x

=;

y 2P

z-------

yP

=y

P z2P-------

y

=;

z 2P

z-------

zP

=z

P z2P-------

z

=;

t 2P

z-------

tP

=t

P z2P-------

t

=;

x

kx-----

xP

y

ky-----

yP

z

kz-----

zP

+ +

t ( )=

MWRT-----------

P

z----

=

MW

RT-----------

x kx

-----

P

z----

xP

MW

RT-----------

y ky

-----

P

z----

yP

MW

RT-----------

z kz

-----

P

z----

zP

+ + MW

RT-----------

t P

z----

=

k kx ky kz= = =


40/68


41/68


Pseudopressure or Real Gas Potential ((P)) Page 41 2000-2001 M. Peter Ferrero, IX

Pseudopressure or Real Gas Potential ((P))

a. (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

Linear

Constant

z

P

P2

P(P)30002000

Gas

z

P

Liquid

o

ec P P

o( )

=

Liquid: slightly compressible system

0 P 2000

2000 P 3000

3000 P3000 psi,

1. Transient flow equation (DD)

Multi-rate test (say 4 points) - flow times must be equal

Pi( ) Pwf( ) aq bq 2

+=

Pi2

Pwf2

aq bq 2

+=

Pi Pwf aq bq 2

+=

P( ) P

Pi PWF 162.6qkh

----------gtP

0.445rw2

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

log 0.87 Sm D q+( )+=

Pi

PWF 162.6

qkh----------

gtP

0.445rw2

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

log 0.87S

m+ 141.2

kh-------

Dq2

+=

Pi PWF

q----------------------- 162.6

kh-------

gtP

0.445rw2

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

log 0.87Sm+ 141.2kh-------Dq+=

Pi PWF

q----------------------- a t( ) bq+=


47/68


Multiple Rate Testing Page 47


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( ) 162.6kh-------

gtP

0.445rw2

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

log 0.87Sm+=

Sm

a t( ) 162.6kh-------

gtP

0.445rw2

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

log

141.2kh-------

--------------------------------------------------------------------------------------=

Sma t( )kh

141.2---------------------- 1.151

gtP

0.445rw2

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

log=

b 141.2kh-------D=

D bkh141.2----------------------= MSCFD------------------ 1

tPre2

4------->

Pi PWF 141.2qkh

----------rerw-----

ln 0.75 Sm D q+( )+=

Pi PWF

q----------------------- 141.2

kh-------

rerw-----

ln 0.75 Sm+ 141.2kh-------Dq+=

a 141.2kh-------

rerw-----

ln 0.75 Sm+=

b 141.2

kh-------D=


48/68




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 (P) analysis:

P2:

Transient flow

q

Pi Pwf

q--------------------

b

a

Pi PWF aq bq 2+=

P

Pi2

PWF2

1637zT

kh------------------------q

tP

0.445rw2

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

log 0.87Sm+1422zT

kh------------------------Dq

2+=

Pi2

PWF2

q------------------------ a t( ) bq+=

a t( ) 1637zTkh

------------------------qtP

0.445rw2

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

log 0.87Sm+=

b1422zT

kh------------------------D=


49/68




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 Pwf2

q---------------------

b

a(t)

Sm 1.151a t( )kh

1637zT------------------------

tP

0.445rw2--------------------

log=

Dbk h

1422zT------------------------=

(Flow times must be equal)

Pi2

PWF2

1422zT

kh------------------------q

rerw-----

ln 0.75 Sm+1422zT

kh------------------------Dq

2+=

Pi2

PWF2

q

------------------------ a t( ) bq+=

a t( ) 1422zTkh

------------------------qrerw-----

ln 0.75 Sm+=

b1422zT

kh------------------------D=

Pi PWF aq bq 2

+=

q

Pi Pwf

q--------------------

b

ab

a(t)

PSS

Transient


50/68




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 precedinggraph. 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

bDkh

-----------

Pi PWF aq bq

2

+=

PWS 14.7psia

Pi2

PWF2

q c Pi2

PWF2

( )n

=

Pi2

PWF2

( )n qc---=

Pi2

PWF2

( ) q1

n---

1

c---

1

n---

=

Pi2

PWF2

( )log 1n--- qlog

1

n---

1

c---log+= where

1

n---

1

c---log is constant

log (q)

Pi2

Pwf2

( )logslope = 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

1

n--- c

q

Pi2

PWF2

( )n

--------------------------------=

q c Pi2

PWF2

( )n

=


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Multi-flows followed by one PBU

a. Flow after flow (discussed)

Theoretical (equal flow times)

Lees 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 evenone 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 initialor 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 PBUs

Pi Pwf a t( )q bq2

+= - all rates in transient flow

Pi Pwf a t( )q bq2

+= - stabilized deliverability equation (1 rate in PSS)


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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 equationwill be high (optimistic)

c. Modified isochronal

Applicable to any permeability system

q1

q2

q3

t

q

q4

t

P

Pi PWF

q----------------------- 162.6

gkh---------

tP

0.445rw2

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

log 0.87Sm+ 141.2gkh---------D q+=

a t( ) 162.6gkh---------

tP

0.445rw

2--------------------

log 0.87Sm+=

b 141.2gkh---------D=


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Reduced time required to conduct

Well produced at 4 rates, PBU following each rate. Flowing periods/PBUs all sametime 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 PBUs

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+=


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3. Develop deliverability equation:

transient

- kh is calculated from PBUs

- Sm and D are calculated intercept and slope, respectively, from a plot of Sg vs. q

- 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 bq 2

+=

Pi PWF 162.6kh-------

tP

0.445rw2

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

log 0.87Sm+ 141.2gkh---------Dq+=

Pi PWF 141.2qg

kh

-------------re

rw

-----

ln 0.75 Sm+ 141.2

kh

-------Dq2

+=

rerw-----

ln 7.5


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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 requiremultiple PBUs. The disadvantage is that if a reliable kh value cannot be estimated fromthe 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! SG = Sm + Dq only works

for flow-PBU-flow-PBU...


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Odeh-Jones Analysis Page 56


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 162.6kh-------

0.445rw2

t--------------------

log 0.87SG++=

t

q1

q2

t0 t1

Pi PWF( ) P=

162.6kh------- q1

0.445rw2

t--------------------

0.87Sq1 q10.445rw

2

t t1( )--------------------

log 0.87Sq1 q20.445rw

2

t t1( )--------------------

log 0.87Sq2+ +log=

m 162.6kh-------=

Pi PWF( )

m q1 t q1 t t1( ) q2 t t1( )log q10.445rw

2

t--------------------

log q10.445rw

2

t--------------------

log+log+log mq20.445rw

2

t--------------------

log 0.87S=

q2

Pi PWF( )q2

---------------------------mq2------ q1 tlog q2 q1( ) t t1( )log+[ ] m

0.445rw2

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

log 0.87S++=

q1 tlog q2 q1( ) t t1( )log+

Pi PWF( )

q2--------------------------- vs.

STF

q2------------


57/68


Odeh-Jones Analysis Page 57


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'

162.6kh------- kh 162.6m'-------=

b m

0.445rw2

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

log 0.87S+=

SG 1.151bm------

0.445rw2--------------------

log=

b

STF/q2

Pi Pwf

q2-------------------- b2'

b1'

q2

q1

S2 1.151b2m--------

0.445rw2

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

log=

S1 1.151 b1m--------

0.445rw2

-------------------- log=

S2 > S1


58/68


Flow Regimes Page 58


Based on relation SG = Sm + Dq (if flow tests are performed followed by PBUs)

Flow Regimes

1. 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

Pcskh, S, Pi

hp hT

m=0.5

early radial: khp, Sm(usually masked by WBS)

kv/kh khT, Sg, Pi


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Flow Regimes Page 59


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- rarelyseen)

- linear - flow from matrix to fractures

- late radial - radial flow in matrix (basically pure radial)

P

P'

m = 0.5 (linear region)

- characteristic of st imulated 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 (l inear, fracture length)

late radial: kh, S

logP,P'

t


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61/68


Pressure level in surrounding reservoir Page 61


Late radial - khh, SGNeed late radial to estimate the wells 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


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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*=PiLTR

P

tpA

--------

tpA

--------


63/68




, 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

------------------------------------- PDMB H= P

P P

70.6qkh

----------

-----------------------2.302 P P( )

m-------------------------------------= 162.6

qkh

---------- 70.6qkh

----------

2.637410( ) 7.65( )

0.039( ) 0.8( ) 1.7 510( )------------------------------------------------------------- 3800

ft2

hr------= =

tPA

--------3800( ) 13630( )

6.97610

---------------------------------------- 7.45= =

PDMB H5.45

2.302 P P( )m

-------------------------------------= =

P 4577 5.45( ) 70( )2.302

---------------------------- 4411psia= =


64/68




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 infollowing 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 118.6qkh

----------0.00388kt

ctre2

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

exp=

P PWS( )log 118.6qkh----------

log 0.00168

kt

ctre2-----------------

=

P PWS( )log A Bt+=

P PWS( )log P

250ctre2

k--------------------------- t

750ctre2

k---------------------------

or

0.51re( )2

4------------------------ t

0.89re( )2

4------------------------

P

P PWS( )log

P

P

P

t

P PWS( )logToo large

Too small

P


65/68


Drill Stem Tests (DST) Page 65


Advantages

requires no knowledge of reservoir properties (A, , 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 notrequired)

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 pressuresprior 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 wellborefluid composition

Memory gauges must be programmed on surface prior to running in hole on SL


66/68


Wellbore Effects Page 66


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 PBUs 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 transmissioncable (pressure, time temperature)

Excellent for remote locations where wire line intervention is difficult. Negates the needto 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 (Bernoullis 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 corrupteduntil 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 possibleto top of perforations or within perforated interval or below interval (within 50 feet shouldbe OK)


67/68


Index Page 67


Index

Buildup Test Solutions 28

Continuity equation (cylindrical coordinates) 7

Darcys Law 8Derivative Analysis (Drawdown case) 30Drawdown test 4Drill Stem Tests 65Drillstem test (DST) 5

Falloff test 5Flow efficiency 13Flow Regimes & Model Recognition 34

Horizontal wells 60Horners Approximation 28

Injection test 4Interference/pulse test 5Isotropic 8

Mathews-Brohs-Hazeroch 62Modified Muskat Method 63Multiple Rate Testing 45Multi-rate Drawdown Test Analysis 56

!

Odeh-Jones Analysis 56

#Pressure buildup test 4

Pseudo (Y(P)) Equation Development 38'

Radius of Investigation 21

)

Skin (drawdown) 31


68/68


Skin and D for Gas Wells 45Skin Development 12Solutions to the Diffusivity Equation 11

Exact solution 11

Infinite reservoir, line source 11Line source solution 11Van Everdingen & Hurst terminal rate solution 11

Superposition 22

1

Wellbore 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

pressure transient analysis - houston university

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