04-MScREM2

SYZ1 Petroleum Reservoir Monitoring and Testing

MSc REM

Reservoir Evaluation and Management

Radial Flow and

Well Testing Basics


Principle of fluid flow in porous media

• Potential problemflow due to pressure difference

• 3-D in space, 3-phase fluids, 3-forces(viscous, gravitational and capillary forces)

• Obey three basic lawsThe principle of mass conservationDarcy’s Law- conservation of momentumEquation of State


The Derivation of radial Diffusivity Equation

ZY

X

∆X

∆Y

∆Z ∆X|x

∆Y|y ∆Z|z

∆Z|z+∆z

∆X|x+∆x

∆Y|y+∆y

An 3-D element of homogeneousporous media with dimensions of

x, y, z∆ ∆ ∆

In the x direction, the massflowing into the element is:( )|x xV y z tρ ∆ ∆ ∆

Within time , the massflowing out of the element is:

t∆

( )|x x xV y z tρ +∆ ∆ ∆ ∆

The mass accumulated in theelement within time is:t∆( )| ( )|

( )| ( )|x x x x x

x x x x x

V y z t V y z t

V V y z t

ρ ρ

ρ ρ+∆

+∆

∆ ∆ ∆ − ∆ ∆ ∆

=− − ∆ ∆ ∆


In the y direction:( )| ( )|y y y y yV V x z tρ ρ+∆ − − ∆ ∆ ∆

In the z direction:

( )| ( )|z z z z zV V y x tρ ρ+∆ − − ∆ ∆ ∆ The total mass accumulation in the element within time is:t∆

( )| ( ) |

( )| ( )|t t t

t t t

x y z x y z

x y z

ρφ ρφ

ρφ ρφ+∆

+∆

∆ ∆ ∆ − ∆ ∆ ∆

= − ∆ ∆ ∆


So, we have:( )| ( )| ( )| ( )|

( )| ( )| ( )| ( )|

x x x x x y y y y y

z z z z z t t t

V V y z t V V x z t

V V y x t x y z

ρ ρ ρ ρ

ρ ρ ρφ ρφ

+∆ +∆

+∆ +∆

− − ∆ ∆ ∆ − − ∆ ∆ ∆ − − ∆ ∆ ∆ = − ∆ ∆ ∆

Dividing each terms on both sides of the above equation by :x y z t∆ ∆ ∆ ∆

( )| ( )| ( )| ( )|

( )| ( )| ( )| ( )|

x x x x x y y y y y

z z z z z t t t

V V V Vx y

V Vz t

ρ ρ ρ ρ

ρ ρ ρφ ρφ

+∆ +∆

+∆ +∆

− −− −

∆ ∆− −

− =∆ ∆


Taking the limit of the above equation, i.e., let:, , , 0x∆ → 0y∆ → 0z∆ → 0t∆ →

( ) ( ) ( ) ( )x y zV V Vx y z t∂ ∂ ∂ ∂ρ ρ ρ ρφ∂ ∂ ∂ ∂

− − − =

This is the continuation equation of a single phase, compressible fluid flowing in a 3-D porous media.According to Darcy’s law:

xk P DV g

x x∂ ∂ρ

µ ∂ ∂

=− −

yk P DV g

y y∂ ∂ρ

µ ∂ ∂

=− −


zk P DV g

z z∂ ∂ρ

µ ∂ ∂

=− −

Replacing these in the continuation equation:

( )

k P D k P Dg gx x x y y y

k P Dgz z z t

∂ ∂ ∂ ∂ ∂ ∂ρ ρ ρ ρ∂ µ ∂ ∂ ∂ µ ∂ ∂

∂ ∂ ∂ ∂ρ ρ ρφ∂ µ ∂ ∂ ∂

− + − +

− =

Considering:

• The viscous dominated laminar flow


• The small, constant rock compressibility and Equation of State :

( ) PCt t∂ ∂ρ φ φ ρ∂ ∂

=

2 2 2

2 2 2x y z tP P P Pk k k C

x y z t∂ ∂ ∂ ∂φ µ∂ ∂ ∂ ∂

+ + =

It is derived:

• For an isotropic, homogeneous reservoir:

x y zk k k k= = =


2 2 2

2 2 2tCP P P P

x y z k tφ µ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂+ + =

Replace this into the above equation:

Using Cylindrical Co-ordinate system:2 2

2 2 2

1 1 tCP P P Prr r r r z k t

φ µ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ θ ∂ ∂

+ + =

Considering 1-D radial flow:

1 1P Prr r r t∂ ∂ ∂∂ ∂ η ∂

=


Where:

t

kC

ηφµ

=

Is called, “The Hydraulic Diffusivity”.

This is the basic radial diffusivity equation for transient pressure analysis.


Assumptions for the Derivation of the Diffusivity Equation

• The reservoir formation is homogeneous and isotropic with uniform thickness: h = constant,

• The rock and fluid properties are independent of pressure (constant compressibility and fluid viscosity).

• The pressure gradient in the formation is small, so the terms: , , can be ignored.

x y zk k k k= = =

2Px

∂∂

2Py

∂∂

2Pz

∂∂


• The flow to the well is radial (laminar) flow, so the Darcy’s Law can be applied.

• The gravity effect (force) is ignored ( flow is viscous dominated).


Constant Terminal RatePressure Draw Down Solution


∂∂ φµ

∂ ∂∂

∂pt

kc r

r pr

r=

1

Second order , Linear Parabolic PDE

Initial Conditionα φµ= kc

HydraulicDiffusivity

p r p all r ri w,0 − = >e j

Diffusivity Equation


The Constant Terminal Rate DD Solution

Initial condition:, for all r

Inner boundary condition:

, for t > 0

Outer boundary condation:, for all t

0|t iP P= =

0lim

2r

P qrr kh

∂ µ∂ π→

=

|r iP P→∞=


Constant Rate Well

u qr h

k prr

w= =2π µ

∂∂

∂∂

µπ

pr

qkh r

r r ww=

= 21

i.e.

Finite Wellbore Radius Inner B.C.

Line SourceApproximation

Limr → 0

r pr

qkh

∂∂

µπ

= 2

-Inner Boundary Condition


( )p p p r tq

khD

i= − ,µ

π2

DimensionlessPressure

drop

t k tc rD

t w= φµ 2

Dimensionless

Time

r rrDw

=Dimensionless

Radius(position)

Dimensionless Variables


∂∂

∂ ∂∂

∂pt r

r pr

rD

D D

DD

D

D=

1

I.C. p = 0 , all r t < 0D D D

B.C. 1

B.C. 2

∂∂pr at r tD

DD D= − = >1 1 0

p = 0 as r D D ∞

Dimensionless Form of the Diffusivity Equation


270.6( , )0.00105

ti i

C rq BP r t P Ekh kt

φµµ = − − −

“Line Source”, constant terminal rate solutionin an infinite acting reservoir, in Field units due to Matthews and Russell (1967):

Where, is termed “The Ei Function”.( )iE x−2 3

( ) ln ......1! 2(2!) 3(3!)

u

ix

e du x x xE x xu

∞ − − = − = − + −

∫

is the expression of “The Exponential Integral”.


Log Approximation to the Ei Function


For a small x value, e.g., , this term can be further simplified as:

Where, the number 0.5772 is Euler’s constant, so:

0.01x <

( ) ln( ) 0.5772iE x x− = +

270.6( , ) ln 0.57720.00105

t ww wf i

C rq BP r t P Pkh kt

φµµ = = − − −

Substituting the log base 10 into this equation for the “ln” term:

2162.6( , ) log 3.23w wf i

t w

q B ktP r t P Pkh C r

µφµ

= = − −


Re-arranging for DD analysis:

( ) log( )wf p i pP t P m t= −Where:

162.6 qBmkhµ

=

This is the basic equation for transient well test analysis.

Considering skin and using field unit:

2ln ln 0.80908 24

swf i

t w

q B kp p t Skh C rµ

π φµ

= − + + +


Pressure Draw Down Analysis and field Examples


TIME, t

Pressure Drawdown Testing

RATEq

0

0

SHUT-IN

PRODUCING

TIME, t

p = pws i

0 Fig2.3.1

Bottom Hole Pressure Pwf


For an infinite-acting reservoir with an altered Zone

[ ]p t SwD D= + +12 0 80908 2ln .

p p q Bkh t k

c r Swf is

t w= − + + +

µπ φµ4 0 80908 22ln ln .i.e.

Hence plot p versus twf l n

Giving p m t pwf t= + =ln 1

s lope intercept


0

pt=1

ln tNOTE : ln t = 0 corresponds to t = 1

Fig 2.3.2

Deviation from straight linecaused by damage andwellbore storage effects

slope, m = − 4 khπq Bs µ

Drawdown Semilog Plot

Bottom Hole

Pressure Pwf


m q Bkh

s= µπ4 k h k

p p m kc r St i

t w= = + + +

1 2 0 80908 2ln .φµ

S p pm

kc r

t i

t w= − − −

=12 0 809081

2ln .φµi.e.

Intercept of Semilog Straight Line

Slope of Semilog Straight Line

Drawdown Interpretation


q : STB/D

h : ft

k : md

s r : ft

t : hr

c : psit-1

t k tc rD

t w= ×0 000263679

2.

φµ

p pq B

kh

pq B

khD

s s= × = ×

∆ ∆887217

21412. .µ

πµ

µ

φ

: cp

: fraction

p : psi

SPE Field Units


p t SwDD= +1

24ln γ

p t SwDD= +

230262

4 08685910. log .γi.e.

i.e.

∴ = − × × +

p p q Bk h

k tc r Swf i

s

t w

887 22

230262

00002637 4 08685910 2. . log . .µπ γ φµ

p p q Bk h t k

c r Swf is

t w= − + − +

162 6 32275 0868592. log log . .µ

φµ

or

m q Bk h kh ks= − → →162 6. µ

Slope

Intercept

p p m kc r St i

t w= = + − +

1 2 32275 086859log . .φµ

S p pm

kc r

t i

t w= − − +

=11513 3227512. log .φµ

Field Units - Log10 or semilog graph paper


Radius of Investigation Concept


103 5x103 t =10D4

1 100 200rD

PRESSUREDISTURBANCE

FRONT Fig 2.2.8

Radius of Influence

p = 0.1D


q

ACTIVEWELL

OBSERVATIONWELL

MINIMUM OBSERVABLE pDEPENDS ON GAUGE RESOLUTION

∆

rD

"ARBITRARY"CRITERION

Ei SOLUTION

pi

pwo

OBSWELL

PRESSURE

0 t Fig 2.2.9

p Ei rtDD

D

=FHGIKJ

12 4

2

p p khqD = =

∆ 2 0 1πµ

.

Radius of Investigation


The radius of investigation:

0.0325invt

k trCφ µ

=

Note:• This parameter is

independent of flowing rate• Its accuracy depends on

pressure gauge


Principles of Superposition


Superposition in space


A “no-flow boundary”resulted from two wells producing at equal rate


A “constant pressure boundary”resulted from an injection and a production well

producing at equal rate


Superposition in time


Constant Terminal RatePressure Build Up Solution


q

0tp t

Well producing Well shut-in

t

P

0tp t

Draw-down Build-up

t

DD time of t = tp+

+

Injectiontime of t∆

t∆


2

2

( )162.6 log 3.23

162.6 ( ) log 3.23

pi ws

t w

t w

k t tq BP Pkh C r

q B k tkh C r

µφµ

µφµ

+ ∆− = −

− ∆

+ −

Superposition of DD with rate +q for the time tp+ and injection with rate -q for the time

t∆t∆


( )162.6 log pws i

t tq BP Pkh t

µ + ∆ = − ∆

The Solution for Pressure BU Analysis - Horner Time Function

Where, is due to Horner, termed Horner Time.( )pt t

t+ ∆∆


Pressure Build Up Analysis and field Examples


RATE

t p ∆

∆

t

t

FLOWING

SHUT-IN

t p

BHP

pws

p ( t=0)wf∆

Figure2.5.1

Schematic Flow-Rate and Pressure Behaviour for an Ideal Buildup


pws

ln t + tp ∆∆ t

p*

Deviation from StraightLine caused by

Afterflow and Skin

0

slope, m = −4 π k hq Bs

µ

Semilog (Horner) Plot for a Buildup

Fig 2.5.1b


The extrapolation to “infinite shut-in” time for initial reservoir pressure.

lim pt tPws Pi

tt

+ ∆ → ∆∆ →∞

0 1log log log(1) 01

pt tt t

tt

∆+ + ∆ ∆ = = = ∆ ∆

lim Dt •


1 10 100 1000

0 1 2 3

Pi

Logarithmic

Linear scale

infinite time

The extrapolation to “infinite shut-in” time for initial reservoir pressure.


∆t

pw f

pw s

p ( t=0)w f∆

Determination of the Skin Factor

B a s e d o n t h eL a s t F l o w i n g

P r e s s u r ep ( t = 0 )

w f∆

At the end of the flow period i.e. t = tp

− Only the pressure prior to shut-in is influenced by the skin effect

For an infinite-acting system replace p by p*,

the MTR straight line extrapolated pressurei

Also make the substitution slope of Horner plot

( )

++

µφπµ

−==∆ S280908.0rc

tkln

kh4qp0tp 2

wt

piwf

mkh4

q=

πµ

−


( )p t p m k tc r Swf

p

t w∆ = = + + +

∗0 0 80908 22ln .φµ∴

i.e. ( )S p t pm

k tc r

wf p

t w= = − − −

∗12

0 0809082∆ ln .φµ

where m = slope of Horner plot (MTR)

p* = straight line intercept (MTR extrapolated pressure)

p ( t=0) = flowing bottom-hole pressure just prior to shut-inwf∆

Note "m" is intrinsically negative

Skin Factor from a Buildup


Natural Log (ln) ( )S p t pm

k tc r

wf p

t w= = − − +

∗12

0 7 431732∆ ln .φµ

Log Base 10( )S

p t pm

k tc r

wf p p

t w=

−− +

∗

11513 3227510 2. log .φµ

Note m is a negative quantity

SPE Field Units


Determine and

very accurately

p ( t=0) t( t=0)wf ∆ ∆

Fig 2.5.8

+

t( t=0)∆p ( t=0)

w f ∆

End ofDrawdown

Buildup

∆t

Stabilise flow-rate before shutin

q

Q = cumulative volume

Flow-Rate

∆t

Shutin

Afterflow

∆ − ∆p = p p ( t=0)BU ws wf

∆ − ∆t = t t( t=0)

t =pQq

Test Precautions


3424

3423

3422

pws

8 7 6 5 4

Horner Plot

Early Piper Well(HP Gauge) slope

m = 0.7465 psi−

kh = 1.067*10 md.ftS = 3.08

6

q = 11750 bbl/d

B = 1.28 = 0.75 cpr = 0.362 ft = 0.237

c = 1.234*10 psi

s

w

t

µ

φ-5 -1

(psia)

Fig 2.5.10 lnt t

tp + ∆

∆


Miller Dyes and Hutchinson (MDH) method (1950)


log pws i

t tP P m

t+ ∆

= − ∆

( )log log log( )pws i i p

t tP P m P m t t m t

t+ ∆

= − = − + ∆ + ∆ ∆

Horner solution for pressure BU analysis:

For , condition for MDH method

Horner solution for pressure BU analysis:

pt t>> ∆

( ) ( )log log tanp pt t t cons t+ ∆ ≈ =

tan log( )wsP cons t m t= + ∆


Matthews Brons and Hazebroek (MBH) method (1953)

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