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What Is A Well Test?
• A tool for reservoir evaluation andcharacterization
– Investigates a much larger volume of thereservoir than cores or logs
– Provides estimate of permeability under in-situ conditions
– Provides estimates of near-wellborecondition
– Provides estimates of distances toboundaries
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How Is A Well Test Conducted?qq = 0
t
q
t
p
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Well Test Applications
• Exploration
• Reservoir engineering
• Production engineering
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Well Test Objectives
• Define reservoir limits
• Estimate average drainage area
pressure• Characterize reservoir
• Diagnose productivity problems
• Evaluate stimulation treatmenteffectiveness
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Types of Well Tests
q
Single-Well Multi-WellInternal
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Development of the
Diffusivity Equation
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The Diffusivity Equation
• Describes the flow of
–
A slightly compressible fluid – Having constant viscosity
– In a porous medium
– At constant temperature
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The Continuity Equation
21
vAvAm
(A v)1 (A v)2
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Equation of State for a Slightly
Compressible Liquid
oppcoe
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Flow Equation - Darcy’s Law
or, in differential form,
L
pkAq
xpku xx
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The Diffusivity Equation
t
p
k
c
r
pr r r
1t
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Data Used in Well Testing
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Formation Volume Factor
surf
resoV
VB
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Viscosity
dyv
v + dv
dy
dv A
F
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Fluid Compressibility
p
Vln
p
V
V
1
c oo
oo
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Porosity
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Permeability
pALqk
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Pore Compressibility
p
ln
p
1
cf
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Net Pay Thickness
h = h1 + h2 + h3
Shale
Sand
h3
h2
h1
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Net Pay Thickness
Case 1 Case 3
Case 4Case 2Internal
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SaturationsH2O
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Wellbore Radius
r w
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Total Compressibility
ggwwoof t cScScScc
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Radial Flow andRadius of Investigation
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Radial Flow and
Radius of InvestigationInstructional Objectives
1. Calculate radius of investigation at a
given time or time required to reach a
given radius of investigation.
2. Describe the effects of reservoirproperties on radius of investigation.
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Radial Flow Reservoir Model
Bulkformation
hr w
r Internal
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Ei-Function Solution
kt
r c948
Eikh
qB
6.70pp
2t
i
x
u
duu
exEi
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Ei-Function Graph
0
1
2
3
4
5
6
7
0.001 0.01 0.1 1 10 100
-x
- E i (
- x
)
Ei-function
Log approximation
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Short-Time Approximation for
Ei-Function Solution
i p p
10948 2
kt
r ct Applies when
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Long-Time Approximation to
Ei-Function Solution
kt r c
kh Bq p p t i
2
101688log6.162
Applies when 01.0948 2
kt
r ct
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Pressure Profile During Drawdown
t = 0
Distance from center of wellbore, ft
P r e s s u r e ,
p s i
2000
1000
1200
1600
1800
1400
1 10 1000100 10000
t = 10000 hrs
t = 0.01 hrs
t = 1 hr
t = 100 hrs
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Pressure Profile During Buildup
Distance from center of wellbore, ft
P r e s s u r e , p s i
2000
1000
1200
1600
1800
1400
1 10 1000100 10000
t = 0.01 hrs
t = 1 hr
t = 100 hrs
t = 10000 hrs
t = 0
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Radius of Investigation
Equations• Radius of investigation for a given time
t:
• Time required to reach a given radius of
investigation r i:
ti
c948ktr
k
r c948t
2it
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Wellbore Storage
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Wellbore Storage
Instructional Objectives• Define wellbore unloading
•
Define afterflow• Calculate wellbore storage (WBS)
Coefficient for wellbore filled with A
single-phase fluid
• Calculate WBS coefficient for rising
liquid level
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Fluid-Filled Wellbore -
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Fluid-Filled Wellbore -
Unloading
R a t e
Time
Bottomhole
Rate
Surface Rate
wbwb
sf w
cV
Bqq
dt
dp
24
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Fluid-Filled Wellbore - Afterflow
R
a t e
Time
Bottomhole
Rate
t=0
Surface Rate
wbwb
sf w
cV
Bqq
dt
dp
24
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Rising Liquid Level
R a t e
Time
Bottomhole
Rate
t=0
Surface Rate
cwb
wb sf w
g
g
A
Bqq
dt
dp
144
615.5
24
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