Solutions to the general diffusivity equation continued

1
Solutions to the general diffusivity equation
continued

• (radial continued)

2
The general diffusivity equation

• Inner boundary(well)
• No flow (shut-in)
• Constant rate
• Constant flowing pressure

• Outer boundary
• Infinite acting
• No flow
• Constant pressure (or given influx)

Wellbore storage or not

• Time behavior
• Transient
• Stabilized (time invariant in some respect)
• Steady-state
• Pseudo-steady state
• Boundary dominated flow regime

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Boundary-dominated flow regime

• Inner boundary pwf is given
• Outer boundary no flow
• Time aspect already stabilized
• ALMOST THE SAME as Pseudo-steady state, often
  taken the same

p(r)
pwf
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Boundary-dominated flow regime Field units

• With average pressure (approximately)
• Deliverability

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Example Boundary-dominated flow regime

• Assume s 8
• How much is the production rate?
• What is the flow efficiency?
• What is the damage ratio?

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Boundary dominated flow relation Field units
Production rate in time

Average pressure
ln(q)
q
Exponential decline Semilog plot (y axis is log)
Exponential decline
t
t
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The general diffusivity equation

• Inner boundary(well)
• No flow (shut-in)
• Constant rate
• Constant flowing pressure

• Outer boundary
• Infinite acting
• No flow
• Constant pressure (or given influx)

Wellbore storage or not

• Time behavior
• Transient
• Stabilized (time invariant in some respect)
• Steady-state
• Pseudo-steady state
• Boundary dominated flow regime

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Infinite acting, constant rate, line source

• "Inner boundary" q is given
• Outer boundary Infinite acting
• Time aspect transient

p
r
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Infinite acting, constant rate, line source
"Exponential Integral" Solution
x
If x lt 0.02
"Log Approximation" Solution
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Infinite acting, constant rate, line source
Wellbore pressure relations
Exponential Integral Solution ("Ideal", Field
Units)
"Log Approximation" Solution ("Ideal")
"Log Approximation" Solution ("Real")
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Calculation of pressure beyond the wellbore

• q 20 STB/D (is it important what the skin is?)
• Calculate pressure in the reservoir
• At time 1 hr
• At location r 0.3 ft, 1 ft, 100 ft, 1000 ft

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Calculation of pressure in the wellbore

• q 20 STB/D
• Assume s 5
• Calculate pressure in the wellbore
• At time 1 hr, 10 hr

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End of Infinite Acting Period
In hr
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Example End of Infinite Acting Period

• Calculate end of "Infinite Acting" period (or
  transient solution) if re 2100 ft

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Mathematica
f1,f2, 100 Absf1-f2/f1/.x -gt 0.02