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Chapter 5

The Basic Differential Equation for Radial Flow in a

Porous Medium§ 5.1 Introduction

To derive and to solve the radial fluid flow in porous medium

)1.5(1

t 

 pc

r 

 pr 

k 

r r   

 

  

 

   

 

  

)20.5(r 

1

)19.5(r 

12

2

t 

 p

k 

c

r 

 pr 

r or 

t 

 p

k 

c

r 

 p

r 

 p

 

  

 

 

 

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§ 5.2 Derivation of the Basic radial differential equation

• Assumptions:• -- The reservoir is homogenous in all rock

• properties and isotropic with respect to permeability

• -- h=const. and hperf =h

• -- Single phase fluid

•

• Why not Cartesian geometry?C.V.

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• Conservation of mass:

• Mass flow rate (in)－mass flow rate (out)

•

  ＝ Rate of change of mass in the volume element

• Using Darcys law for a radial flow

C.V.

 

)2.5(2)(

22

t hr 

r 

qor 

t dr hr dr hr 

t qq r dr r 

 

    

  

            

)3.5(1

22

)2(

t r 

 pr 

k 

r r 

t rh

r 

 prhk 

r 

r 

 pk rhq

 

  

 

 

  

 

   

 

  

    

 

   

 

 

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§ 5.4 The Linearization of Equation 5.1 for Fluids of small and

constant compressibility

)1.5(1 t 

 pcr 

 pr k 

r r   

  

  

   

 

  

t 

 pc

r 

 pr 

k 

r 

 pk 

r 

 pr 

r 

 pk 

r 

 pr 

k 

r   

 

  

 

   

 

  

 

  

   

    2

2

r 

1

r r 

 pc pc

 p  

         

  

  

1c

From Eq.(5.4)

t 

 pc

r 

 pr 

k 

r 

 pk 

r 

 pr 

r 

 pk 

r 

 pr 

k 

r   

 

 

 

 

    

 

  

 

    

   

    2

2

cr 

1

 small r 

 pce

r 

 pr  f  k ce

k 

r 

 

  

 

sin0;)(,sin0    

 Note:

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)20.5(

r 

1

)19.5(r 

12

2

t 

 p

k 

c

r 

 pr 

r 

or 

t 

 p

k 

c

r 

 p

r 

 p

 

 

 

 

 

 

It is for the flow of liquids or for c·p

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§ 5.3 Conditions of Solution

Radial flow equation：

The most common and useful analytical solution is for the

)1.5(1

t 

 pc

r 

 pr 

k 

r r   

 

 

 

 

   

 

  

r at  p pr r at const qconditionsboundary

r all  for  p pconditioninitial 

i

w

i

.:

:

constant terminal rate solution (Chapter 7&8)

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Radial flow equation：

The three most common conditions

(1) Transient --- Early time; no boundary effect

(Infinite acting reservoir)

(2) Semi steady state --- The effect of the outer boundary has been felt.

where

)1.5(1

t 

 pc

r 

 pr 

k 

r r   

 

  

 

   

 

  

),(

),(

t r  f  r  p

t r  g  p

)7.5(.

0

t and r all  for const t 

 p

and 

r r at 

r 

 pe

)8.5(

1

qdt 

dV 

dt 

dpcV 

dV cVdp

dpdV 

V c

)10.5(2

 

    

hr c

q

dt 

dpcV 

q

dt 

dpor 

e

 pressureaverage p p  

.

..

avg rateqi

qi p p

avg vol Vi

Vi p p

i

i

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(3)Steady state

due to natural water influx or the injection of some fluid and

ee   r r at const  p p 

  .

t and r all  for 0

t

 p