• Author / Uploaded
• Loh Chun Liang

Citation preview

UNIVERSITI TEKNOLOGI PETRONAS

PAB3053 RESERVOIR MODELLING AND SIMULATION

SEPT 2013 Dr. Mohammed Abdalla Ayoub Diffusivity Equation-Linear Petroleum Engineering Department (GPED)

Lesson’s outcomes At the end of this lesson, you will: • Get to know what is the meaning of diffusivity equation for linear, horizontal, single phase flow. • Know how to derive it under different conditions. • Be able to deal with the right assumptions and to fully understand the reasons.

Introduction The mass accumulation = (increase or decrease) over time t The mass that flows IN over time t

-

flowrate, q x area, A

The mass that flows OUT over time t

x=L x+x porosity, f

X axis

x flowrate, q

in

x=0 isometric view

x flowrate, q

flowrate, q

in

x=0

x

x+x X axis plan view

x=L

out

out

Introduction Assumptions o o o o

Flow along x direction, no flow in y or z directions Flow into cuboid at left, out of cuboid at right Total length, L Rock 100% saturated with one fluid

k = Permeability (in the X direction), (D) ρ = Density, (g/cm3) U = Flow velocity (cm/s) t = Time (s) Φ = Porosity μ = Viscosity, cp P = Pressure, atm x = Distance, (cm)









flowrate, q

Fluid flows in at position x = 0, and flows out at x= L

x area, A

Element from x to position x+x is examined Bulk volume of the element is the product of the area, A and length, dx, i.e. bulk volume = A x

x=L x+x porosity, f

X axis

x flowrate, q

in

x=0 isometric view

The pore volume of the element is the product of the bulk volume and the porosity, Φ, i.e. pore volume = A x Φ

x flowrate, q

flowrate, q

in

x=0

x

x+x X axis plan view

x=L

out

out

Derivation of the Diffusivity Equation In terms of mass flow rate, Mass flow rate through the area, A = qρ ((cm3/s)*(g/cm3) = (g/s) Mass flow rate through the area, A at position x = (qρ)x Mass flow rate through the area, A at position x+x = (qρ)x+x Mass flow rate into a volume element at x minus mass flow rate out of element at x + x =(qρ)x- (qρ)x+ x

Derivation of the Diffusivity Equation ∆𝑥 𝑣𝜌A∆t

∆ ∅𝜌 A ∆x

(g)

(g)

-

𝜕 ρv 𝜕x

𝜕 vρ 𝜕x

∆x A. ∆t

(g)

∆x ∆t= ∆(∅𝜌)∆x ∆ ρv - ∆x

-

vρ +

A

𝜕 ρv 𝜕x

=

=

∆ ∅𝜌 ∆𝑡

𝜕 ∅𝜌 𝜕𝑡

𝐴

𝑐𝑚2

v

𝑐𝑚/𝑠𝑒𝑐

𝜌

g/c𝑚3

v𝜌

mass flux

Derivation of the Diffusivity Equation o

Mass flow rate out of the element is also equal to the rate of change of mass flow in the element

q xdx  q x   q  * dx x



 q  * dx x

o

Therefore, the change in mass flow rate =

o

This must equal the rate change of mass in the element with a volume = A x Φ

o

The rate change of mass is equal to

   Afdx t

Derivation of the Diffusivity Equation  q  1  f x A t

o

Hence

o

Since the flow velocity is equal to q/A then:





 v   f x t

 v    f x t

o

Or:

o

Assuming Darcy’s law is applicable:

  k P      f x   x  t

Derivation of the Diffusivity Equation o

o o

Density can be related to pressure through the isothermal compressibility 1 V c V P m Density is equal to mass over volume, which is:  V Hence, the isothermal compressibility is (using quotient rule):

d  f(x)  g(x) f (x) - f(x) g(x)  dx  g(x)  [g(x)] 2

c

  m  

    1  m P  P

Derivation of the Diffusivity Equation o Since,

  P P   c t P t t o

Then,

  k P  P    fc x   x  t o

This is NON-LINEAR PARTIAL DIFFERENTIAL EQUATION, which means some variables in the equation (inputs) depend on the quantity we are trying to find.

Derivation of the Diffusivity Equation The mass accumulation The mass that (increase or decrease) = flows IN over over time t time t Massin  t  A v  x

The mass that flows OUT over time t

in g

Massout  t  A v  x  x in g

The mass accumulation (increase or decrease) over time t

 t  A v  x  t  A v  x  x

Derivation of the Diffusivity Equation Change of mass of fluid The mass at over the time from t to = time t+t t+t

-

The mass at time t

 f t  t  f t  x A Where x and A are assumed constants

Derivation of the Diffusivity Equation f t t  f t  x A  t  A v  x  t  A v  x x o

By dividing both sides by the area A:

f t t  f t  x o

By dividing both side by xt

f t t  f t  o

 t  v  x  t  v  x  x



 v  x   v  x  x

t x By taking the limit for both sides lim t 0

f t t  f t  t

 f     v  t x

 lim x0 

 v  x  x   v  x x

Derivation of the Diffusivity Equation If Darcy’s law is applicable: v

k P  x

 f      k P  t x   x  But:

 f    f   P t P t

Therefore:

 f   P     k P  P t x   x 

  k P  P    fc x   x  t

NON-LINEAR PARTIAL DIFFERENTIAL EQUATION

Derivation of the Diffusivity Equation o There is no known analytical solution for the Non-Linear Partial Differential Equation. o It can either by solved numerically or by simplifying the equation by making various assumptions and then solve the equation analytically.

Assumptions Simplifying Assumptions for the Non-Linear PDE 1. 2.

3.

Viscosity, μ, is constant (with x and P) Permeability is constant with x and P, i.e. the system is homogeneous. The pressure gradient,P,is very small such that: x

 P    0  x  2

4.

The fluid has a constant compressibility

cf 

1       CONSTANT   P 

Justifications Simplifying Assumptions for the Non-Linear PDE 1.

Viscosity, μ, is constant (with x and P) Quite reasonable since viscosity does not vary greatly for most oils (or water) over small pressure ranges.

2.

Permeability is constant with x and P, i.e. the system is homogeneous. Quite drastic since it says that permeability (k) is constant through the reservoir i.e. that the system is homogeneous in k. For a real system, this is a very simplifying assumption.

Justifications Simplifying Assumptions for the Non-Linear PDE  P    0  x  2

3.

The pressure gradient,

P , is very small such that: x

it is odd but it is designed to get rid of “difficult” terms with terms like (∂P/∂x)2 in them 4.

The fluid has a constant compressibility. is reasonable for most reservoir oils.

Working assumptions  Linearization of the Partial Differential Flow Equation for Linear Flow  f   P     k P  P t x   x 

 Assuming the viscosity and permeability do not depend on location or pressure:  f   P  k     P  P t  x  x 

 By expanding the RHS of the equation:  2P    P   P      2    x  x  x x  x 

Working assumptions  ρ is a function of pressure, therefore, the RHS of the equation can be expressed as:  2P    P   P P       2    x  x  P x x  x 

 Assuming the pressure gradient is very small, the RHS of the equation can be simplified to:  2P    P       2  x  x   x 

 The diffusivity equation becomes  P k   2 P  f       2  P t   x 

Working assumptions  By expanding the LHS of the equation:  f   P  f      f   P P t  P P  t

 Formation compressibility can be expressed as: cf 

1 f f P

 Therefore the LHS of the equation becomes:  f   P  f      f  c f   P P t  P  t

Cont.,  Liquid compressibility can be expressed as:

c

1   P

 Again the LHS of the equation becomes:





 f   P  f    c    f  c f  P P t t  Which can be simplified to:  f   P  fct  P P t t

 Where

ct  c  c f

Final form  By equalising the expanded RHS and LHS: P k   2 P  fct      2  t   x 

 By simplifying the equation: P k  t fct

 2P    2   x 

 And this is the diffusivity equation for Linear, Horizontal, Single phase fluid. or   2 P  fct P  2    k t  x 

Fluid compressibility • As previously quoted

1 dV m d V m   V dP V dP d  1   1  d c        dP       dP c

d  c dP   c P

d dP  c dx dx

• For slightly compressible fluids where c is small and constant d



 cdP

ln   a  cP

   0e c  P  P  o

For black oil (P>Pb) • Diffusivity Equations for a Black Oil: • Slightly Compressible Liquid: (General Form)

c(p) 2   2 p 

fct p k t

• Slightly Compressible Liquid: (Small  p and c form) 2

 p

fct p k t

Checking dimensions and conversions   2 P  fct P  2    k t  x 

1 Pa  sec Pa  Pa  Pa   2 2 m sec m 

Quantity 1 = Quantity 2

 2P  fct P  2     x 0 . 00633 k  t  

number 1  unit 1  number 2  unit 2      number 2 0.00633  number1

0.00633 

     

1 ft 2 1  cp  psi   1  md   day   

In vector notation P k  t fct

 2P    2   x 

k  P fct     t   P k  2 P t fct

Radial Diffusivity Equation P 1   k P  fct   r   t r r   r  P k 1   P   r  t fct r r  r 

Concluding remarks Conceptually, the diffusivity equation is obtained by applying mass balance over a control volume. Equation of motion (Darcy's Law) and equation of state (PVT relations) are then combined with the mass balance equation to obtain the final form of the diffusivity equation.