DERIVATION OF THE
DIFFUSIVITY EQUATION
WELL TESTING CONCEPT
During a well test, the pressure response
of a reservoir to changing production (or
injection) is monitored.
Time
Output
Reservoir
Pressure
Rate
Input
Time
KEY CONCEPTS TO BE USED
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Conservation of mass: mass balance
Darcy’s law: equation of motion
(relationship between flow rate,
formation and fluid properties and
pressure gradient)
Equation of state: relationship between
compressibility, density and pressure
DERIVATION OF THE DIFFUSIVITY EQUATION
Basic assumptions:
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•
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Single phase radial flow
One dimensional horizontal radial flow
Constant formation properties (k, h, ϕ)
Constant fluid properties (µ, B, co)
Slightly compressible fluid with constant
compressibility
Pore volume compressibility is constant (cf)
Darcy’s law is applicable
Neglect gravity effects
RADIAL SYSTEM
r+∆r
r
RADIAL SYSTEM
r
mass in
mass out
r+∆r
volume
element
CONSERVATION EQUATION
(Mas flow in) r - (Mass flow out) r+∆r =
Mass accumulation
Mass flow rate = Area × velocity × density × time
Mass accumulation = change of mass over time ∆ t
Bulk vo lu m e b etween r and r+ ∆ r = π [(r+ ∆ r) 2 -r 2 ] h
= π [r 2 +2r∆ r+(∆ r) 2 -r 2 ] h 2 π r∆ r h
(Mass flow rate in) r = [2π rhv ρ ]r ∆ t
(Mass flow rate out) r+∆r = (2π rhv ρ ) r + ∆r ∆ t
Mass accumulation = ( 2π r ∆ rhφρ )t + ∆t − ( 2π r ∆ rhφρ )t
CONSERVATION EQUATION
(Mas flow in) r - (Mass flow out) r+ ∆ r =
Mass accumulation
( 2π rhv ρ ) r ∆ t − ( 2π rhv ρ ) r + ∆ r ∆ t =
( 2π r ∆ rhφρ )t + ∆t − ( 2π r ∆ rhφρ )t
Divide both sides by 2 π r∆ rh ∆ t
(φρ )t + ∆t − (φρ )t
1 ( rv ρ ) r - (rv ρ ) r + ∆ r
[
]=
∆t
r
∆r
1 ∂ ( rv ρ ) ∂ (ϕ ρ )
−
=
∂t
r ∂r
The above equation is call the continuity equation.
DARCY’S LAW
k ∂p
µ ∂r
Substitute Darcy's law into the continuity equation
∂p
∂ (ϕρ )
1 k ∂
[r ρ
]=
∂r
∂t
r µ ∂r
Expand the derivat ives of the products of terms
∂
∂p
∂ρ ∂p
∂ρ
∂ϕ
1 k
+ρ
[ρ
(r
) + r ( )( )] = ϕ
∂r ∂r
∂r ∂r
∂t
∂t
r µ
Apply the chain rule to th e derivatives of ρ and ϕ
1 k
∂
∂p
∂ρ ∂p 2
∂ρ
∂ϕ ∂p
[ρ
(r
) + r ( )( ) ] = [ϕ
]
+ρ
r µ
∂r ∂r
∂p ∂r
∂p
∂p ∂t
v=−
EQUATION OF STATE
The relationship between compressibility, density
and pressure is given by:
1 ∂ρ
co =
ρ ∂p
The definition of pore volume compressibility is:
1 ∂ϕ
cf =
ϕ ∂p
Use the above equations to replace the derivatives
of ρ a nd ϕ
1 k
∂
∂p
∂p
∂p
[ρ
(r
) + rco ρ ( ) 2 ] = [ϕ co ρ + ϕ c f ρ ]
r µ
∂r ∂r
∂r
∂t
DIFFUSIVITY EQUATION
Divide both sides of the above equation
by k ρ and multiply by µ to obtain:
1 ∂ ∂p
∂p 2 ϕ µ ct ∂p
(r ) + co ( ) =
∂r
r ∂r ∂r
k ∂t
ct = co + c f
The above equation is called the diffusivity equation.
It is non-linear partial differential equation due to the
∂p 2
term co ( ) . To be able to solve it using analytical
∂r
techniques it needs to be linearized.
LINEARIZATION OF THE DIFFUSIVITY EQUATION
To linearize the diffusivity equation we need to make
the assumption that the pressure gradients are small and
use the assumption that fluid compressibility is small.
∂p 2
Therefore the term co ( ) is small compared to other
∂r
terms and can be neglected.
The linearized diffusivity equation becomes:
1 ∂ ∂p ϕ µ ct ∂p
(r ) =
r ∂r ∂r
k ∂t
OIL FIELD UNITS
k = permeability (md)
h = formation thickness (ft)
ϕ = porosity (fraction)
µ = fluid viscosity (cp)
r = radial distance (ft)
rw = wellbore radius (ft)
t = time (hours)
p = pressure (psig or psia)
pi = initial reservoir pressure (psig or psia)
q = flow rate (STB/D)
B = formation volume factor (res. bbl/STB)
ct = total system compressibility (psi-1)
DIFFUSIVITY EQUATION IN OIL FIELD UNITS
ϕ µ ct
1 ∂ ∂p
∂ 2 p 1 ∂p
∂p
(r ) = 2 +
=
−4
∂r
r ∂r ∂r
r ∂r 2.637 ×10 k ∂t
The above equation is second order (highest
derivative with respect to space is two) linear
partial (it has derivatives with respect to two
independent variable r and t) differential equation.
To solve it we need two boundary conditions and
an initial condition.
BOUNDARY CONDITIONS
Rate
1. The flow rate at the wellbore is given by Darcy's law:
kh
∂p
q=
(r
) r=rw at r = rw for t ; 0 .
141.2 B µ ∂ r
q
∂p
141.2qB µ
(r
) r=rw =
∂r
kh
Time
p = p i as r → ∞ for any value of any time.
Pressure
2. If we consider a system large enough so that it
will behave like an infi nite acting system, then far
away from the wellbore the pressure will remain at p i .
pi
Radius
INITIAL CONDITION
Pressure
Since the promlem under investigation is time
dependant, the initial pressure should be known
at time t = 0.
Using uniform pressure distribution throughout
the system at t = 0.
p(r,0) = pi at t = 0 for all values of r .
pi
Radius
DIMENSIONLESS VARIABLES
T o m ak e th e eq u atio n an d its so lu tio n m o re g en eral
fo r an y flu id an d reservo ir system it, is m o re
co n ve n ien t to ex p ress it in d im en sio n less varialb les.
D im e n sio n less p r essu r e:
kh ( p i − p )
1 4 1 .2 q B µ
D im en sio n less tim e:
pD =
tD
2 .6 3 7 × 1 0 − 4 kt
=
ϕ µ c t rw2
D im en sio n less rad iu s:
r
rD =
rw
DIMENSIONLESS VARIABLES
Differential equation:
∂ 2 pD 1 ∂pD ∂pD
+
=
2
rD ∂rD
∂rD
∂t D
Boundary conditions:
∂p D
(rD
) rD =1 = -1
∂rD
(1)
(2)
p D = 0 as rD → ∞ for any value of t D
(3)
Inital condition:
p D (rD ,0) = 0 at t D = 0 for all values of rD
( 4)