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Well Testing GeomechanicsWell Testing Geomechanics

Maurice Dusseault

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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

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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

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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…

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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...

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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

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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.

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-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

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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

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csCompressibilityCompressibility

Com

pres

sibi

lity

Effective confining stress - σ′

Cc = constant

Hertzian contact model

Brick contact model

Reality

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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

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csPermeability and Permeability and σ′σ′ EffectsEffects

� Negligible for most sandstones (

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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

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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

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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)

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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.

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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

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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!)

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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

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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

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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

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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

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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

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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

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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.

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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