Download - 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