DERIVATION OF THE DIFFUSIVITY EQUATION

WELL TESTING CONCEPTDuring a well test, the pressure responseof a reservoir to changing production (orinjection) is monitored.

Input OutputReservoir

Time

Rat

e

TimePres

sure

KEY CONCEPTS TO BE USED

• 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 EQUATIONBasic assumptions:• 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+∆r

rmass outmass in

volumeelement

CONSERVATION EQUATION

r+ rr(Mas flow in) - = Mass accumulationMass flow rateMass accumulation

(Mass flow out

= change of mass over time tBulk vo

)

l m b

u e

Area velocity density time

∆

= × × ×∆

( ) ( )r+

2 2

2 2

r

2

r

etween r and r+ r = [(r+ r) -r ] h = [r +2r r+( r) -r ] h 2 r r h(Mass flow (Mass flow rate out) (2 )

rate in) [2 ]

2 2

r

t

r r

t t

rhvrhv t

Mass accumulation r rh r rht

π

π ππ ρ

π ππ

ρ φρ

φ ρ∆

∆

+∆

+

= ∆

∆ ∆

∆ ∆ ∆= ∆

= ∆ − ∆

CONSERVATION EQUATION

( ) ( )

r+ rr(Mas flow in) - = Mass accumulation(

(Mass flow out)

(

Divide both sides by 2 r r

2 )2 2

( ) -

2

( )1 [

h

]

t

) rr

t t t

r r

r

r

rhv tr rh r rh

rv rvr

rhv t

r

π ρπ φρ π φρ

π

ρ

ρ

ρ

π

∆

+

+∆

+∆

∆∆ − =

∆ − ∆

∆

∆

∆

∆

( ) ( )

1 ( ) ( )

The above equation is call the continuity equation.

t t t

r t

trvrρ

ρ

ϕ

φ φρ

ρ

+∆−

=∆

∂ ∂− =

∂ ∂

DARCY’S LAW

Substitute Darcy's law into the continuity equation

Expand the derivat

1 ( )[ ]

1 [

ives of the products of terms

Apply the chain rule to th

( ) ( )( )]

e

k prr r r t

k p pr rr r

k pvr

r r r t t

ϕρρµ

ρ ρ ϕρ ρµ

µ

ϕ

∂ ∂ ∂

∂

=∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂+ =

= −∂

+∂ ∂ ∂ ∂ ∂ ∂

2

derivatives of and 1 [ ( ) ( )( ) ] [ ]k p p pr rr r r p r p p t

ρ ϕρ ρ ϕρ ϕ ρ

µ∂ ∂ ∂ ∂ ∂ ∂ ∂

+ = +∂ ∂ ∂ ∂ ∂ ∂ ∂

EQUATION OF STATEThe relationship between compressibility, densityand pressure is given by:

The definition of pore volume compressibility is:

Use the above equations to replace the derivativesof a

1

nd

1o

f

cp

cp

ρρ

ρ

ϕϕ

∂=

∂

∂=

∂

21 [ ( ) ( ) ] [ ]o o fk p p pr rc c c

r r r r t

ϕ

ρ ρ ϕ ρ ϕ ρµ

∂ ∂ ∂ ∂+ = +

∂ ∂ ∂ ∂

DIFFUSIVITY EQUATION

2

Divide both sides of the above equationby k ρ and multiply by to obtain:1 ( )

The above equation is called the diffusivity equation.It is non-linear partial dif

( )

fere

to

t o f

pcr

c c c

cp prr r r k t

µϕ µ∂

∂=

∂ ∂ ∂+ =

∂ ∂+

∂

2

ntial equation due to the

term To be able to solve it using analytical

techniques it needs to be linearized

.

.

( )opcr∂∂

LINEARIZATION OF THE DIFFUSIVITY EQUATION

2

pressure gradients are sTo linearize the diffusivity equation we need to makethe assumption that the anduse the assumption that fluid compressibility is small.

Therefore the term is smal

mall

( )opcr∂∂

l compared to other

terms and can be neglected.The linearized diffusivity equation becomes:1 ( ) tcp prr r r k t

ϕµ∂ ∂ ∂=

∂ ∂ ∂

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

2

2 4

(highestderivative with respect to space i

1 1( )2.637 10

s two) (it has derivatives with respect to two

in

The above equation is second order linear

partialdependent v

aria

tcp p p prr r r r r r k t

ϕ µ−

∂ ∂ ∂ ∂ ∂= + =

∂ ∂ ∂ ∂ × ∂

differential equation.To solve it we need two boundary conditions andan initia

ble r

l con

a

di

nd t)

tion.

BOUNDARY CONDITIONS

w

w

r=r w

r=r

1. The flow rate at the wellbore is given by Darcy's law:kh pq = (r ) at r = r for t 0

2. If we consider a system large enough so

.141.2 rp

that

141.2qB (r ) = r kh

itwill behave like an infi

Bµµ

∂∂

∂∂

i

i

nite acting system, then faraway from the wellbore the pressure will remain at p .p = p as r for any value of any time.→ ∞

Time

Rat

e

q

pi

RadiusPres

sure

INITIAL CONDITION

i

Since the promlem under investigation is timedependant, the initial pressure should be knownat time t = 0.Using uniform pressure distribution throughoutthe system at t = 0.p(r,0) = p at t = 0 for all values of r .

RadiusPres

sure pi

DIMENSIONLESS VARIABLEST o m ake the equation and its so lu tion m ore generalfo r any flu id and reservo ir system it, is m oreconveD im e

niennsio

t to expn less p r

ress it in d im enessur

sion less varialb les.

D im ensio

e:( )

141 .n2i

Dkh p pp

qB µ−

=

D

4

D 2

D im ension less rad ius:

less tim e:2 .637 1

r

0t

w

t w

ktc r

rr

ϕµ

−×

=

=

DIMENSIONLESS VARIABLES

D

2

2

DD r =1

D

D D D

D D D D

1 (1)

p(r ) = -1 (2)r

p = 0 as r for any value o

Differ

f t (3)

ential equation:

Boundary conditions:

Inip (r ,0) = 0 at t = 0 for all values of r

tal conditi :4

o( )

n

D D D

D D D D

p p pr r r t

∂ ∂ ∂+ =

∂ ∂ ∂

∂∂

→ ∞