1PET212EGlobal Flow Regimes in

Reservoir Flow

M. OnurSpring 2007

Global Flow Regimes At any given time in the producing life of a

reservoir, the fluid flow condition existing may be characterized as either

a) transient, b) pseudosteady-state or c) steady-state.

What do these terms mean?

Reservoir Flow

Initially, in a virgin reservoir, the pressure at any fixed depth is constant.

As production begins, the pressure near the wellbore drops significantly as near-wellbore fluids expand to satisfy the imposed production condition.

Far away from the well, no measurable pressure drop can be observed at early times locations far away from the well are not aware

that the reservoir is being produced.

Transient As time progresses, pressure drops can be

measured further and further away from the well, an increasing volume of the reservoir fluids expand to contribute to the well's production.

During this period, the reservoir is said to be infinite acting and the flow is transient; pressure drop at outer reservoir boundary is negligible.

The pressure versus time behavior at the producing wellbore contains information about the reservoir permeability.

2Pseudosteady State After a long time, pressure drops can be

measured at all reservoir locations the entire reservoir is contributing to the well's

production. At this time, the pressure changes at the

same rate at every location in the reservoir, i.e., dp(x,t)/dt = constant; pseudosteadystate flow.

The pressure versus time behavior at the wellbore reflects the volume of fluid (or the reservoir pore volume) contributing to production.

Steady State

To see steady-state flow in a reservoir, we must replace reservoir fluids at the same rate that we remove them.

This situation may occur if we have a recharge to thesystem (an assocaited water aquifer) or may alsooccur in secondary and enhanced oil recovery operations - e.g., waterflooding, gas injection, etc.

During steady-state single-phase flow, nothing is changing in the reservoir, i.e., dp(x,t)/dt = 0. Note there is a pressure drop in the reservoir, andpressure data contains information about rechargingsystem parameters.

Comparisons of Flow Regimes

constantdDdt

=

ViscousLiquid

PumpAir

ViscousLiquid

t = 1

t = 100

ViscousLiquid

q

qq

q

Initial, t = 0 Transient

Pseudosteady-state Steady-state

qD

0dDdt

=

Comparisons of Flow RegimesTransient Pseudosteady-state

Steady-state

t tss

t tss

q

p

3Comparisons of Flow Regimes

Pressure responses at the well

Steady-State Radial Flow

Simplified Reservoir ModelReservoir

radius DamageradiusWell

radius

h

Cylindrical horizontal Reservoir; constant rate at every radius;Reservoir permeability, k; damaged zone permeability, ks;No gravity effects; constant viscosity, formation volume factor.

rs re, pe

Darcys Law

Rate at any radius:

Pressure distribution Integrate Darcys law over radius Outer (undamaged zone)

( )rprhrkqBo

=31008.7

( ) =

ee r

r

p

rpo rdrdp

qBkh31008.7

4Pressure Distribution Outer Zone

Perform integration

Note: Even if there is no damaged zone, pressure drop is greatest close to the wellbore radius. Why is this so?

( )

=rr

khqBrpp eoe ln

2.141

Graphical Pressure Distribution

Pressure versus radius

19401950196019701980199020002010

0 200 400 600 800 1000 1200 1400

Radius, ft

Pres

sure

, psi

a

Note

Most of the pressure drop occurs within the first few inches of the wellbore.

As fluids approach the wellbore, the area available to flow is decreasing (2rh)

Pressure losses increase as fluids approach wellbore.

Reservoir drawdown

Integrate Darcys law over entire reservoir

( ) = e

w

e

wf

r

r

p

po rrkdrdp

qBh31008.7

+= e

s

s

w

e

wf

r

r

r

r s

p

po krdr

rkdrdp

qBh31008.7

+=e

e

w

s

s

owfe r

rkr

rkh

qBpp ln1ln12.141

5Simplify

For undamaged reservoir (k = ks)

++=e

e

w

s

w

s

w

s

s

owfe r

rkr

rkr

rkr

rkh

qBpp ln1ln1ln1ln12.141

+=

w

s

sw

eowfe r

rkk

rr

khqBpp ln1ln2.141

=w

eowfe r

rkh

qBpp ln2.141

Skin Factor

Define skin factor, s, as

Skin factor is A dimensionless number = 0 if there is no damage > 0 if there is damage < 0 if the near-wellbore region is stimulated

w

s

s rr

kks ln1

=

Productivity Index

For steady state flow

A positive skin factor will reduce the wells productivity

A negative skin will increase it Are there any other factors that influence a wells

productivity?

( )

+

==s

rrB

khpp

qJ

w

ewfeo

ln2.141

Average Reservoir Pressure

Productivity Index is usually expressed in terms of average reservoir pressure rather than external pressure, pe Average pressure is also used in material

balance calculations Obtainable from analysis of well test data

( )( ) ( )

( )22,2

2

,2

we

r

rr

r

r

r

rr

drtrrp

drrh

drtrprhtp

e

w

e

w

e

w

=

=

6Productivity Index

Steady-State Flow

Productivity Index expression

+

= s

rr

khqBpp

w

eowf 2

1ln2.141

( )

+

==s

rrB

khpp

qJ

w

ewfo

21ln2.141

Notes

Productivity Index for steady state flow is constant Pressure at each point in the reservoir does not

change with time PI strongly influenced by the skin factor

We would like to identify wells where skin factor is large; we can increase production by a stimulation workover.

Skin

In practice, skin may be due to a variety of factors Damage to formation due to invasion of mud filtrate

and mud solids Partial penetration Migration of fines Asphaltines

Treatment of skin will depend on the specific cause.

Productivity Index Steady State Radial Liquid Flow

In terms of external reservoir pressure

In terms of average reservoir pressure

( )

+

==s

rrB

khpp

qJ

w

ewfeo

ln2.141

( )

+

==s

rrB

khpp

qJ

w

ewfo

21ln2.141

7Average Permeability

Radial beds in parallel No skin damage No cross flow between

layers All layers at the same

pressure All layers with same

radial extent Same fluid in each layer

Layer Rates

For this system, total rate is the sum of layer rates

=

=++=n

iiqqqq

121 ...

( ) ( ) ( )...

ln2.141ln2.141ln2.141

2211 +

+

=

w

e

wfe

w

e

wfe

w

e

wfe

rrB

pphk

rrB

pphk

rrB

ppkh

=

=++=n

iiihkhkhkkh

12211 ...

Beds in series

If we put several radial beds in series, we can show that

Note: the derivation of the steady state radial flow with skin damage was a special case of radial beds in series.

( )

=j

jj

w

e

avg

krr

rr

k1ln

ln

Example Consider a radial system comprised of 3

composite zones with the following properties. Find the average permeability.

1507501503

80150.52

1050.251

k (mD)Outer Radius

Inner Radius

Layer

8Solution

For this problem, rw = 0.25, re = 750 Applying our formula

( )md 4.23

150750ln

1501

5150ln

801

25.05ln

101

25.0750ln

ln

ln

1

=

+

+

=

= j

jj

w

e

avg

krr

rr

k

Pseudosteady State RadialLiquid Flow

Accumulation Term

During pseudosteady state flow,

Reservoir flow equation for pseudosteady state

( ) constant , == A

ttrp

Acrpkr

rr t=

00633.0

Boundary Conditions

The previous differential equation is second order; we need two boundary conditions

In addition we have an unknown constant, A Outer boundary sealed:

Inner boundary at constant rate:

0=

errp

( ) Bqrpkrh sc

rw

=

210127.1

3

9Determination of Constant A

Integrate flow equation over reservoir

Inner boundary condition

=

=

200633.0

00633.0

22wet

rr

r

rt

r

r

rrAcrpkr

rpkr

rdrAcdrrpkr

r

we

e

w

e

w

( )hBq

rpkr sc

rw 210127.1 3=

Rate of Change of Pressure with time

Solve for A

During pseudo-steady state flow, pressure is a linear function of time; slope inversely proportional to pore volume.

( )( ) twe sc

scwet

crrhBq

tpA

hBqrrAc

22

3

22

615.5210127.1200633.0

==

=

Productivity Index

Pseudosteady State Flow

If skin were included

( )21ln

2.141

=

w

e

sc

wfe

rr

Bqppkh

( )s

rr

Bqppkh

w

e

sc

wfe +

=

21ln

2.141

( )

+

==

srrB

khpp

qJ

w

ewfe

sc

21ln2.141

Note During Pseudosteady state flow, pressure is

changing with time; however J (or PI) is a constant.

External pressure (or average average) and wellbore pressure are changing at exactly the same rate, so difference between them is constant.

We can also derive a Productivity Index equation in terms of average pressure.

( )

+

==

srrB

khpp

qJ

w

ewf

sc

43ln2.141

10

Non-circular Reservoirs

Productivity Index expressed in terms of reservoir area, A, and Dietz shape factor, CA

( )

+

==

sCr

AB

khpp

qJ

Aw

wf

sc

22458.2ln

212.141

Sample Dietz Shape Factors

Pseudo-Steady State Well PressureResponse

This flow period is observed when no-flowboundaries of reservoir are felt by the well:

hrwf ptmp 0+=

( )

++

+= sCr

Akh

BqthAc

Bqp Aw

sc

t

sc 087351.0loglog6.162234.0 2

thAc

Bqdt

pdtpt

sc

== 234.0 (unit slope on log-log plot)

Reservoir Limit Test Analysis

1E-4 1E-3 1E-2 1E-1 1E+0 1E+1 1E+21E-1

1E+0

1E+1

1E+2

1E+3

Cartesian plotSlope of straightline ise m*

Flow

ing

pres

sure

, psi

0 5 10 15 20 25 30 354840

4880

4920

4960

5000

p0hr

t, hour

pan

dp

' , p

si

11

Determination of Pore Volume

Based on Late time straight line on Cartesian plot:

( )

= hrhrr

rA ppmm

mC 01

303.2exp456.5

hAcBqmslope

t

sc

234.0==

t

scsct cm

BqhAm

BqhAc ==234.0234.0

Dietz Shape Factor CA

CA is called Dietz Shape factor and depends on well/reservoir geometry.

( )

= hrhrr

rA ppmm

mC 01

303.2exp456.5

CA = 31.62 CA = 30.88 CA = 12.99

ykbt

bDA thAckht

=4

,10637.2

ylst

ylsDA thAckht

=4

,10637.2

Dietz Shape Factor CAProductivity Index

Productivity Index is a measure of the well's ability to produce fluids under an imposed reservoir pressure drop.

J Productivity Index, STB/d/psiAverage pressure in drainage area, psia

Bottomhole flowing pressure, psia

Production rate, STB/day

( )wf

sc

pptqJ =

( )tp ( )tpwf( )tq

12

Productivity Index

Function of many parameters Transmissibility, kh/ Storativity, cth Skin damage, s Drainage area of the well, A Reservoir and well geometry

Can we determine individual values of theseparameters?

Productivity Testing

Productivity Test

[ ] scwfwfsc qJppppJq 1==Is Well A or B more productive?

Why is Well C exhibiting a curvedbehavior instead of straight-line?

Oil Well Productivity Index Single-phase flow of oil and partially penetrating

well (hw) in a homogeneous-isotropic system withthickness h:

)(4ln

2.141 2/1

2

wf

wp

wA

sc pp

shhs

rCeAB

khq

++

=

J

What could be the reason(s) for the well producing less?Is it due to skin, limited entry, or low permeability?

13

Gas Well Productivity Index Single-phase flow of gas and limited entry well

(hw) in a homogeneous-isotropic system withthickness h:

What could be the reason(s) for the gas wellproducing less? Is non-Darcy flow significant?

J

( ))(

4ln1422

222/1

2

wf

scw

pwA

sc pp

Dqshhs

rCeAzT

khq

+++

=

Limitations of Productivity Testing Assumes stabilized flow conditions (steady

state or pseudo-steady state flow regimes).

It does not allow us to determine individualvalues of parameters (kh, skin, limited entryskin, non-Darcy term, etc.

So, we will need (actually transient) tests ifwe wish to estimate the individual values of these parameters.

Transient Flow Transient Flow includes a set of transient flow

regimes

Wellbore storage dominated flow Spherical flow Radial Flow Linear Flow etc

These are all subsets of an overall transient flow regime