Diffusivity Equation

8/11/2019 Diffusivity Equation

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FUNDAMENTALS OFFUNDAMENTALS OF

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Diffusivity EquationDiffusivity Equation

Ali F. M. Altaee


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To Derive Diffusivity Equation.

(CTP)

To apply constant terminal rate solution(CTR)


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-Pi Pi

r3 r2 r1 r1 r2 r3

re

Const.

Flow rate

Pi

Pit1 t2

t3

r3

r2

r1

r1

r2

r3

re

Const.

Pwf

i Pit1 t2t3

Pressure disturbance as a function of time

re


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Pressure disturbance moves away fromwellbore at a rate determined by

Permeability

Viscosity

Transient flow is defined as the time period

boundary has no effectduring which theon the pressure behavior in the reservoir

its infiniteand the reservoir will behave as

.in size


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Basic Transient Flow E uation

porous media may not be the same as the flow

The fluid content of the porous medium changes

The variables in unsteady-state flow:

me, t

Porosity, Total compressibility, ct


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Basic Transient Flow Equation

Q

1

Q

2

The flow rate into & out of an element of volume Q = Q The variables in unsteady-state flow:

Time, t Porosity,

Total compressibility, ct

Transient Flow equation

must have these

independent variables +

limits


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on nu y qua on MBE

Transport EquationDarcy

Compressibility Equation Isotherm coeff

u y


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on nu y qua on A material balance equation that accounts for

every pound mass of fluid produced, injected,

or remaining in the reservoir.

Transport Equation

The transport equation is Darcys equation in

its eneralized differential form


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Compressibility Equation

Is used in formulating the unsteady-state

changes in the fluid volume as a function of


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Boundary Conditions:

e orma on pro uces a a cons an ra e n o e

wellbore

the reservoir behaves as if it were infinite in size,

i.e. re =

The reservoir is at a uniform pressure when

=, . .,


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where k = permeability, md

r = radial position, ft

=

ct = total compressibility, psi1

t = time, hrs

= porosity, fraction

= v scos y, cp


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CONTINUITY EQUATION

According to the concept of the material-balance equation

Mass entering

volume element

Mass leaving

volume elementur ng nterva t ur ng nterva t

rate of mass

accumulation

durin interval t

---------------- ((5656))


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Illustration of radial flow


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Mass entering the volume element during

---------------- 5757where = velocity of flowing fluid, ft/day

= fluid density at (r + dr), lb/ft3A = Area at r + dr

t = time interval, days

The area of element at the entering side is:

---------------- ((5858))

Combining Equation (58) with (57) gives:

---------------- ((5959))


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Mass leaving the volume element

---------------- 6060

o a ccumu a on o assThe volume of

some element

with a radius

of r

Differentiating

with respect to r

----------------


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Total mass accumulation during t = dV [()t + t()t]

Substituting for dV

Total mass accumulation = (2rh) dr [()t + t()t ] ----------------

Replacing terms of material balance Equation with calculated relationships

Dividing by (2rh) drt

OR---------------- ((6363)) where = porosity

= densit lb/ft3

= fluid velocity, ft/daycontinuity equation


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The transport equation

arcy s aw s essen a y e as c mo onequation

------------ ((6464))

where k = permeability, md

= velocit ft/da

---------------- ((6565))

Expanding the right-hand side by taking the indicated derivatives eliminates the porosity

from the partial derivative term on the right-hand side:

---------------- ((6666))


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porosity is related to the formation compressibilityCOMPRESSIBILITY EQUATION

---------------- ((6767))

Applying the chain rule of differentiation to /t,

Substituting Equation (67) into this equation

substituting the above relation into Equation (66)and the result into Equation (65), gives:

---------------- ((6868))


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Radial Flow of Slightly

Compressible Fluids the permeability and viscosity are constant over

pressure, time, and distance ranges

---------------- ((6969))

Expanding


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Using the chain rule in the above relationship yields:

Recalling that the compressibility of any fluid is related to its density by:

Combining the two equations


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very small and may be ignored

---------------- ((7070))

Define total compressibility, ct, as:

---------------- ((7171))


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petroleum engineering

where the time t is commonly recorded in hours

---------------- ((7373))

where k = permeability, mdr = radial position, ft

p = pressure, psia

ct = total compressibility, psi1

=

e us v y ons an

= porosity, fraction

= viscosity, cp


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When the reservoir contains more than one fluid, total

---------------- 7474

Co,w,g= compressibility of oil, water and gas

So,w,g = fractional saturation of oil, water and gas

---------------- ((7575))

The diffusivity equation can then be written in a more convenientorm as:

---------------- ((7676))


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Laplaces Equation

for a steady-state flow condition, the pressure at any

with time

Substitute in diffusivity equation

---------------- ((7777))

Laplaces equation

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Diffusivity Equation