4c radial flow semi steady state

7/31/2019 4c Radial Flow Semi Steady State

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AES1310: Rock Fluid Interactions - Part 1

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Radial Flow & Semi-steady state flows

Susanne [email protected]

Radial flow in porous medium: derivation assumptions Transient conditions

Semi Steady State conditions Steady State conditions


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What will be done next?

General equation will be derived to describe the flow throughporous medium in cartesian and radial coordinates.

Discussion of staedy state, semi-steady state and transition

conditions.General description of steady state, semi-steady state and

transition conditions.

Detailed description of transient conditions.


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Derivation of radial flow equation inporous medium

Derivation in radial form to allow description of flow in porous

medium close to a well.

Equations in radial form create feeling for flow through

porous medium.

Cartesian form commonly used in reservoir simulations.


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Assumptions:

Reservoir is homgenous in all rock properties.

Isotropic behavior of permeability.

Production well is completed over the whole formation

thickness -> radial flow can be assumed.

Formation is fully saturated by a single fluid.


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Radial cell geometry

Volume element for mass balance (accounting for porosity):dV = 2..r.h..r


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Mass balance:

Mass flow rate (in) mass flow rate (out) =rate of change of mass in volume element

The left hand of the equation can be rewritten by:

r r r

A rdmq q

dt t

r r r r r

qq q q r q

rq

rr


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With this the mass balance simplifies to:

Darcys law for radial, horizontal flow is:

Substituting this into the mass balance results in:

2

2

q A r q r h r r rr t r t

qr h

r t

2k p

q r hr

2

2

k pr h

rr h

r t


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Rearranging the equation leads to:

This equation is generally applicable for horizontal, radial 1Dflow.

1 k prr r r t


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The density can be expressed by the mass and the volume:

The isothermal compressibility is in terms of the volume is:

In terms of the density:

ii

m

V

1 ii

i T

VcV p

1

1

i

i i i ii i i

i i T

TT

m

c cm p p p


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This is the general equation to compute the isothermalcompressibility for a component i.

The isothermal compressibility can be incorporated in themass balance. First, the mass balance is rewritten:

1 i

ii T

c p

1

1

T

k pr

r

r r t

k pr

r p

r r p t


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Meaning of product of porosity and the density:

Density describes the mass per volume, here the porevolume:

Porosity is the ratio of the pore volume and the totalvolume (matrix + pore volume):

Thus the product describes the mass of fluid per totalvolume:

?

ii

P

m

V

P

total

V

V

i iPi

P total total

m mV

V V V

i i f di l fl i i


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In order to describe the compressibility we need to describe

the compressibility of the total volume, meaning of the rockand of the fluid.

This can also be described by a isothermal compressibility:

itotal

i

mV

1

1

i

itotal ieffective

total iT

T

i

i T

m

Vc

V p m p

p

D i i f di l fl i i


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With this the rhs of the diffusivity equation can be rewritten:

ceff is the efficient isothermal compressibility. Commonly, it isdescribed as the sum of the isothermal compressibility of thefluid and of the pores.

The compressibility of the pores can be related to thecompressibility of the matrix.

1

eff

k p pr c

r r r t

eff fluid poresc c c

D i ti f di l fl ti i


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If we assume that the total volume does not change with the

pressure:

And determine the total volume as the sum of the volume of

the matrix and of the pores, a relationship between the porevolume and the matrix volume changes can be derived:

eff fluid poresc c c

0total

T

V

p

0total matrix P

total matrix pores

T T T

matrix P

T T

V V VV V V

p p p

V V

p p

D i ti f di l fl ti i


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With this and the definition of the isothermal compressibility,we get:

1

1

1

matrixmatrix

matrix T

P

P matrix matrix P PP T

matrix P

T T

matrix total PP matrix matrix

P P

total

P matrix matriP

Vc

V p

Vc c V c V

V p

V V

p p

V V Vc c c

V V

Vc c c

V

11

x

R di l diff ti l f fl id fl


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Radial differential for fluid flowin porous medium

Single Phase

1

eff

k p pr c

r r r t

Equation is basic, partial differential equation for description of

radial flow of any single phase through porous medium.

Equation is non-linear due to implicit pressure dependence ofthe density, the compressibility and the viscosity.

Analytical solutions can only be found if equation is firstlinearized.



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Linearization

2

2

1

1

eff

eff

k p p

r cr r r t

p k k p rr

r r r r pc

r tk p k pr rr r r

The equation can only be linearized if some crude assumptionsare made!Before the equation can be linearized we extend the equation:



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Introducing the isothermal compressibility for the description ofthe derivative of the density with respect to r gives:

For a constant isothermal compressibility, this equation can berewritten:


Linearization

1 1Vc

V p p

1

1

pc

p r

cp p r r

r

pc

r r



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Assuming that:Viscosity is independent of pressure and thus is constantPressure gradient p/r is small (p/r)2 0We get:

If it is assumed that the compressibility is constant, also thecoefficient at the rhs of the equation is constant.


Linearization

2

2

2

2

1

1

eff

eff

cp p p

rr r r k t

cp p p

r r r k t

Radial diffe ential fo fl id flo


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Linearized equation only valid with made assumptions.

According to Dranchuk and Quon only applicable for:ceff.p


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Note: The above equation is called diffusivity equation.

Diffusivity equations are known from physics. For example thedescription of the temperature distribution is described by thefollowing diffusivity equation:

With T: absolute temperature; K thermal diffusivity constant


Linearization

1 1T Tr

r r r K t

1 effcp prr r r k t

Conditions of solution


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Conditions of solution

1

eff

k p p

r cr r r t

Most common solution is the constant terminal rate solution:

Initial condition:At some fixed time at which reservoir is at equilibrium

pressure peqwell is produced at constant flow rate at r = rw

Three most common conditions:

Steady-state, semi-steady state and transient.

Solution of radial diffusivity equation


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Solution of radial diffusivity equationTransient

Only short period after pressure disturbance in reservoir, e.g.,

by changing production rate at r= rw.No influence on the pressure response due to the outerboundary (infinite extension of reservoir).

Solution of radial diffusivity equation:Pressure and its time gradient are functions of position andtime ( , )

( , )

p f r t

pg r t

t



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Solution of radial diffusivity equationSemi-steady state conditions

Semi-steady state condition applicable to reservoirs which

have been producing for sufficiently long time Effect of outer boundary felt on pressure responseOuter boundary described by a brick wall Production with constant flow rate: pressure change with

time constant



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From the chain rule we know that:

The change of the volume with the pressure can bedescribed with the compressibility:

0p

r

q const

For r= re

For r= rwc

y y u V V pqx u x t p t

V p p qV c q V c

p t t V c

p constt

For all r&t



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The volume can be described by:

Giving

Note:

The isothermal compressibility c is not necessarily constantbut changes with the pressure, e.g., for gases.

2p qt c h r

2V h r



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Reservoir depletion under semi-steady state conditions:

Once reservoir is producing under semi-steady stateconditions, each well will drain from within its own no-flowboundary (Matthews, Brons, Hazebroek)


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According to boundary condition that flow rate of each well is

constant, we obtained:

If the compressibility does not change with pressure, then

the volume can be replaced by:

With this the averaged reservoir pressure can be described

by:i i

ires

i

i

p q

pq

p q

t V c

qV const qp

ct



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Solution of radial diffusivity equationSteady-state condition

Steady-state conditions apply after transient period.

Describes the drainage of a cell with open boundaries.Constant production rate.Production rate is balanced by fluid flow via outer boundary Pressure maintenance via water influx or injection of

replacing fluid. 0

e

pt

p p const

for all r&t

for r= re



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Solution technique is given in more detail but is general and

can be applied for variety of radial flow problems.Geometry and pressure distribution for semi-steady stateconditions:



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We know that:

So that at time t with the average pressure we get:or more specific

With V the pore volume of the radial cell, q constant

production rate; t total flowing time.Incorporating

Into the radial distribution equation gives:

p q

t V c

iV c p p q t

2

eff e

p q

t c h r

2

1 1eff

e

cp p p qr r

r r r k t r r r k h r

b eff iV c p p q t



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Integration leads to:

The integration constant C1can be determined with theboundary condition for r= rethat p/r = 0:

2

2

21

2

e

e

p qd r r dr

r k h r

p q rr C

r k h r

1 02

12

qCk h

qC

k h



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So that

Further integration gives

Assuming that rw2

/re2

negligible simplifies the equationfurther:

2

1

2 e

p q r

r k h r r

2

2 2

2

1

2

ln2 2

e

w

r w

e w

q rdp dr

k h r r

r rq rp p

k h r r

2

2ln

2 2r w

e w

q r rp p

k h r r



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At r= re

we obtain the well inflow equation under semi-steady stateconditions.

The production index PI is then:

1

ln2 2

ee w

w

rq

p p k h r

2

1ln

2

e w e

w

q k hPI

p p r

r



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Often the Everdingen skin factor is included in the equation,

accounting for additional pressure drop due to the presenceof a skin:1

ln2 2

ee w

w

rqp p S

k h r

2

1ln

2

e w e

w

q k hPI

p p rS

r



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One disadvantage of this equation is that q and pw can be

determined experimentally, but not pe.Pressure difference (pressure draw down) is expressed interms of the average pressure rather than the pressure at theouter boundary:

Replacing the volume with:

e

w

e

w

r

r

r

r

p dV

p

dV

2dV r h dr



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Gives

The pressure is described by:

Allows the computation of the average pressure.

2 2 2 2

2 2

2

2

e e

e

w w

e

w

w

r r

rr r

r

re w e w

r

p r h dr p r drp p r dr

r r r r r h dr

2

2ln

2 2r w

e w

q r rp p

k h r r



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Derivation of solution analogue for steady state conditions.

For semi-steady state conditions:

For steady-state conditions:

2

1 1eff

e

cp p p qr r

r r r k t r r r k h r

10

pr

r r r



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AES1310 R k Fl id I i P 1

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Solution of radial diffusivity equationSteady state conditions

Radial inflow equations for stabilized flow conditions

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4c radial flow semi steady state

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