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©MBDCI©MBDCI
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Well Testing GeomechanicsWell Testing Geomechanics
Maurice Dusseault
©MBDCI©MBDCI
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csStressStress--Pressure CouplingPressure Coupling
shalereservoir
p(r)
σv(r) σv(r)
p(r)
Initial conditions Early time - ∆tQ
p – σ′ coupling improves well test analysis, even in linear elastic systems
©MBDCI©MBDCI
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csConventional Well Test AnalysisConventional Well Test Analysis
� Well equations (& software) are based on variants of the original Theis solution – 1935
� The basic equation, using head, is…
� The implicit assumption: ∆σ = 0, so ∆σ′ = ∆p� This is a non-coupled solution� It is therefore an incomplete solution…
∂∂∂∂2h/∂∂∂∂ r2 + (1/r)(∂∂∂∂h/∂∂∂∂r) = (S/T)(∂∂∂∂h/∂∂∂∂t)S is reservoir storativity
T is reservoir transmissivity
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csStress Arching in Well TestingStress Arching in Well Testing
stress archingσv
ri H
W
∆p region
no ∆p yet
Now you see why we call it stress arching!
Arching of stresses begins Large-scale arching
∆p leads to a small ∆V, and a redistribution of σv
©MBDCI©MBDCI
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csStressStress--Pressure CouplingPressure Coupling
� In well-test equations, it is implicitly assumed that σv = constant, but…
� This is obviously not correct
� There are actually large changes in σv …
� These are a function of:�Stiffness of the overlying strata
�Compressibility of the reservoir rock
�The degree and radius of drawdown -∆p(r)
�The geometry – thickness of the reservoir, depth below surface, dip…
©MBDCI©MBDCI
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csSome Practical ConsequencesSome Practical Consequences……
� All well test equations are in error�Flow rate predictions wrong by up to 50%
�This has impact on decision-making
�One axi-symmetric elastic closed-form solution has been developed for this…
� Arching effect changes with time�Compaction can be delayed until the ∆p zone is
large enough for σv re-imposition
�Early time behavior will under-predict compaction/subsidence
�Ekofisk in the North Sea, for example
Full mathematical simulation may be needed in critical
cases...
©MBDCI©MBDCI
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csHorizontal Stresses also ChangeHorizontal Stresses also Change……
zone of -p*
*-∆p causes -∆V
A A’
B
B’
σv
σ lossσ gain
σv, ⊥ section A-A’
σ gain
σh
σh, ⊥B-B’
∆p
p
∫Aσ⊥ must be always constant
©MBDCI©MBDCI
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csWell Test Equation AssumptionsWell Test Equation Assumptions……
� They assume that the compressibility of the rock is a constant (no change in ∆σ′ effect)
� They assume that the overburden stress remains constant on the top of the reservoir
� They assume that permeability remains constant with effective stress
� They assume k and Cc are constant with radius
� They assume zero grain crushing or collapse
� These are sometimes OK, sometimes not.
� Geomechanics plays a vital role here.
©MBDCI©MBDCI
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cs
-1200
-1000
-800
-600
-400
-200
0
2000 3.5 7 10.5 14 17.5 21
Distance - X (m)
Pre
ssur
e (k
Pa)
1min2min3min10min20min30min60min
A Coupled Well-Test Model shows clearly the p-effect arising from time-
dependent stress transfer
Effects on PressuresEffects on Pressures
©MBDCI©MBDCI
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csAlso Also -- NonNon--Linear CompressibilityLinear Compressibility
� Mainly in high porosity and fractured strata� C = ƒ(σ′) = ƒ (confining effective stress)
�Grain contacts stiffenwith higher σ′�Fissures, fractures also show the same behavior
� If compression is a first-order factor in the drive mechanism for the well test:�Large errors can arise in the interpretation�Model with C = ƒ(σ′) should be used�Also, fractured media k variation is important
� Even a simple non-linear compressibility can give very different flow model results
©MBDCI©MBDCI
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csCompressibilityCompressibility
Com
pres
sibi
lity
Effective confining stress - σ′
Cc = constant
Hertzian contact model
Brick contact model
Reality
©MBDCI©MBDCI
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csE.g.: Grain Crushing is NonE.g.: Grain Crushing is Non--linear linear
� Depletion or differential volumetric strain causes high σ′, high f ń on individual grains
� Weak (lithic) or cleavable (felspathic) grains crush and fragment (e.g.: Chalk)
� These strains are irreversible (plastic) and also reduce k substantially
feldspar quartz
feldspar crushed, quartz intact
f’n
Easily cleaved or weak grains crush first
©MBDCI©MBDCI
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csPermeability and Permeability and σ′σ′ EffectsEffects
� Negligible for most sandstones (
©MBDCI©MBDCI
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csFracture Compression NonlinearityFracture Compression Nonlinearity
asperities
p p + ∆peffective aperture
∆σ′n
effective stress - σ′n
frac
ture
ape
rtur
e -
a
frac
ture
flow
rat
e -q
),a(k"k" crack σ ′=
∆V
σ′
Linea
r mod
el
Actual behavior
“soft”
“stiff” All aspects of strain and flow in fractured media are non-linear
©MBDCI©MBDCI
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csCoal Bed Methane ExampleCoal Bed Methane Example
� Coal fissures close with +∆σ′, k drops� Gas cannot flow through fractures (2-phases and
capillary blockages impede gas flux)
� After some dewatering time, gas begins to flow
� After a longer time, coal shrinks from CH4 loss
� Stresses change, -∆σ′h, fissures open, flow ↑� Thus, coal gas flow is both σ′ and ∆p dependent � A stress-coupled flow model with non-linear
rock properties is needed for coal seam behavior
©MBDCI©MBDCI
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csPermeability Reduction and Permeability Reduction and σ′σ′
σθσr ∼σ′ - effective stress*
stresses around wellbore
pressure - p
‘k’ - apparent permeability
σ′οpo
fracture-dominated stratum*∆σ′ ≠ ∆p because of p-induced volume change
Q virgin k
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csCoal Gas ProductionCoal Gas Production
Q - Rate
Time
conventional theory using virgin k
with k = ƒ(σ′)(fracture flow)
Well rates are mis-predicted if only small drawdowns used and equations do not include fracture effects (e.g. Austin Chalk)
©MBDCI©MBDCI
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csLateLate--Time Coal PermeabilityTime Coal Permeability
σ′ - stress pressure - p
‘k’ - apparent permeability
fracture-dominated stratum
po
koσ′ο
The CH4 is depleted near the wellbore, causing a
shrinkage of the coal blocks. This leads to a loss of stress and a permeability increase
as the fractures open up.
affected region
However, coal shrinkage is appreciable; as drawdown proceeds, fracture permeability impairment is reversed.
©MBDCI©MBDCI
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csFractures & Coal ShrinkageFractures & Coal Shrinkage
Q - Rate
Time
conventional theory
with k = ƒ(σ′), (fracture flow), followed by shrinkage of coal
Early-time impairmentLate-time improvement
These problems caused consternation when coal bed methane was first considered for development
©MBDCI©MBDCI
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csGeomechanics: Cavity CompletionGeomechanics: Cavity Completion
pressure - p
‘k’ - apparent permeability
cavity
po
koσ′ο
The cavity is generated so that the rock is damaged and a high drainage surface area
is generated, accelerating gas drainage. Parameter values are along cavity
midpoint.
damaged region
σ′v- stress
heavily perforated
well
Low σh, high σv
Geomechanics knowledge led to the idea of a “Cavity Completion”, now widely use in coal bed methane (better wells!)
©MBDCI©MBDCI
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Rate
Time
conventional theory
extra $$
closure of aperture(aperture = ƒ(∆σ′))
Production with Cavity CompletionProduction with Cavity Completion
with the cavity completion approach
CasingCavityDamaged zone
The cavity method makes a better well, with good early production, and which gets better with time
©MBDCI©MBDCI
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csLetLet’’ s Revisit Archings Revisit Arching…………
� Early time: -p gives +∆σ′, rock compresses� Occurs near the wellbore first, and σv drops� Farther out, σv must rise, thus -∆p ≠ +∆σ′� But, all well test equations assume∆p = -∆σ′� Errors can arise (up to 50%), in well tests of
thick compressible liquid reservoirs (Ekofisk)�Compaction potential underestimation from arching
�Underestimates of potential well productivity
� → Use a geomechanically rigorous model
©MBDCI©MBDCI
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csOverburden Arching EffectOverburden Arching Effect
Delay of compression always occurs in early time by the arching effect, but eventually stresses change
earlyintermediate
∆σ′ ≠ ∆pstress arching σv
well test riri H
W
ri ∆p region
no ∆p yet
*aspect ratio is W/H; when W>~3-4H, arching disappearing
Arching effects arise in coal bed methane (high loads help dilation) and, realistically, it enters into all reservoir problems
©MBDCI©MBDCI
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csSimilar Effects for an Injection Well!Similar Effects for an Injection Well!
X (m)
0
200
400
600
800
1000
Y (m)
0
200
400
600
800
1000
X Y
Z
displacement (m)0.0320.030.0280.0260.0240.0220.020.0180.0160.0140.0120.010.0080.0060.0040.0020
Injection wellVertical displacement
Inverse of arching!
In this case, it is an injection well. We can see uplift aroundthe injection point. Vertical stresses there also go up
Vertical displacement - m
©MBDCI©MBDCI
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csStress Redistribution EffectsStress Redistribution Effects……
- 2 50
- 2 0 0
- 150
- 10 0
- 50
0
50
10 0
0 10 2 0 3 0 4 0 50 6 0
Time(minutes)
Pre
ssur
e(kP
a)
∆p/∆t at a point near the well
©MBDCI©MBDCI
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csProduction and GeomechanicsProduction and Geomechanics
Q - Well production rate
Time
conventional theory
Coupled well equations(with reservoir-scalestress redistribution)
with k = ƒ(σ′)(fracture flow)
More physically complete flow models give better results, giving better engineering and economic decisions
©MBDCI©MBDCI
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csBetter Flow Models with Better Flow Models with ∆σ∆σ′′
� The conventional well test equations are incomplete and can give errors up to 50%
� If you are concerned and feel geomechanics coupling is an issue:�Use better models (e.g. Rothenburg developed a
model including stress redistribution)
�Use a Finite Element Model that includes full stress-flow coupling
�Compare to the simpler models and evaluate
� Economic consequences may not be trivial, and it is vital in reservoir analyses.
©MBDCI©MBDCI
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csWell Evaluation GeomechanicsWell Evaluation Geomechanics
� Reservoir compressibility often non-linear� Also, permeability changes with ∆σ′� Overburden response effects are vital
�Overburden stress redistribution�Arching and delayed compaction drive�Mis-estimation of well productivity
� Well test protocols including geomechanics�Better models that include stress effects�Better assessment of compaction potential
� Without geomechanics, well test models must be considered incomplete