Transcript Flow equations in various cases

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Petroleum Geomechanics Design
Maurice Dusseault
Intro to Petroleum Geomechanics
Geomechanics…
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The petroleum geomechanics design approach
Uncertainty in geomechanics and petroleum
engineering applications
Stress, pore pressure and effective stress
Rock strength and rock stiffness
Jointed and intact rock mass behavior
Geological history and rock properties
Coring, preparing, and testing rocks
Intro to Petroleum Geomechanics
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Stress, Strength, Joints, etc…
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Rock properties depend
on geometry, materials,
history, and so on
Rock masses have
discontinuities…
Granular systems have
contact forces
Frictional strength (μ)
is a vital petroleum
geomechanics concept
Also, pore pressures
Intro to Petroleum Geomechanics
μ·F > W
F
F
W
©MBDCI
Major Issues in Design
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We live with massive uncertainty
 Rocks
are deep, often inaccessible, no cores…
 Properties may vary widely from bed to bed
 Processes are very complex (′, p, T…)
 Direct monitoring is usually impossible
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This presents great challenges to the engineer
who is doing design or predictive work
Petroleum geomechanics design is an iterative
process based on monitoring and analysis
Intro to Petroleum Geomechanics
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…UNCERTAINTY…
Reservoirs are heterogeneous & anisotropic at all scales (microns to kilometers)
Intro to Petroleum Geomechanics
70 m of Athabasca Oilsands,
 = 30%, So = 0.8,  > 1,000,000 cP
North of Fort McMurray, Alta
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Geomechanics Design Elements
1. Geometry, lithostratigraphy, classification of
strata (shape, material types, extent, ...)
2. History and present state (T, [], p, previous
history of loading, ....)
3. Appropriate behavioral law (--T law;
diffusion law for transport; Δp, ΔC effects…)
4. Method of analysis (empirical, numerical,
stochastic, analytic, similarity, ...)
5. Verification (monitoring, autopsies…)
Intro to Petroleum Geomechanics
©MBDCI
The Design Loop
Project Definition
Preliminary Model
Project
Geometry
goals
Technology drivers
Cost-benefit
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of wells…
Stratification
GMU choice
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Design Verification
Rock Behavior
Monitoring
--Y
strategy
Back-analysis
Modification of design
…
Project Analysis
Empirical
assessment
Numerical simulation
Model testing, pilots
Predictions
Intro to Petroleum Geomechanics …
behavior
Correlations, logs
Literature
Lab tests
…
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Preliminary Model
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Geometry of the proposed structure (eg: layout of wells, structural units…)
Classification of the strata into geomechanical units (behaviourally similar rock unit)
Location and geometry of the GMUs (eg:
reservoir, overburden, ...) Geology!!
Changes in geometry which are likely (new
wells, further reservoir development)
GMU = geomechanical unit (a behaviorally similar unit)
Intro to Petroleum Geomechanics
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Geological Models: Logs vs. Rocks
REG. TIPO
ER-EO
ER-EO
C-4
C-5
ER-EO
ER-EO
REG. TIPO
B-SUP
B-SUP
B-SUP
B-6/9
C-3
B-6/9
C-6
C-4
C-5
C-7
B-6/9
B-6/9
SMI
C-1
C-6
C-7
GUAS
C-2
C-6
C-7
GUASARE
C-3
C-4
C-5
C-6
C-7
GUASARE
FALLA ICOTEA GUASARE
SVS-30
Intro to Petroleum Geomechanics
SVS-337
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What is a GMU?
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Geo-Mechanics Unit
Nature is too complex to
“fully” model
Simplification needed
A GMU is a “single
unit” for design and
modelling purposes
1 GMU = 1 set of
mechanical properties
GMU selected from
logs, cores, judgment
Intro to Petroleum Geomechanics
Log data
Core data
GMU 1
GMU 2
GMU 3
GMU 4
GMU 5
GMU 6
GMU 7
GMU 8
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GMU’s and Rock Mechanics
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Rocks are heterogeneous, anisotropic, etc…
For analysis, divide system into GMU’s…
 Includes
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Too many subdivisions are pointless
 Can’t
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critical strata, overburden, underburden…
afford to test all of them
Too few subdivisions is risky
TOO MANY?
TOO FEW?
Intro to Petroleum Geomechanics
©MBDCI
History and Current State
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Geological and tectonic loading history
History of project to the time of analysis
(injection/production history, seismicity, ...)
Current state of extrinsic parameters:
Temperatures, Stresses, Pressures
Future history of what is proposed: changes in
T, , p, V, ....
The Geological and Stress History of
rocks is vital geomechanics knowledge
Intro to Petroleum Geomechanics
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Stress History and Rock Response
′v
diagenesis
History is vital! In this example, deep
burial and erosion have led to the
following conditions:
•The rock is much stronger
•The rock is much stiffer
•Compaction is unlikely
•Sanding is less likely
•Now, v is 3
•Fracturing now “horizontal”
Current state
Intro to Petroleum Geomechanics
′h
Before any systematic reservoir
geomechanics, the reservoir history
should be studied by a structural
geologist who understands stresses,
diagenesis and rock properties
©MBDCI
Sufficient Behavioral Law (I)
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For each GMU, we need a “sufficient”
behavioral law to apply to the entire GMU
For p, T, and C diffusion (transport) processes:
[kij], [ij], [Dij], [ij]
For - processes: strength model, stiffness,
viscosity (creep), yield behavior, ...
For seismic analysis: [vij]P, [vij]S, [Qij], ...
Clearly, the number of parameters increases
dramatically with anisotropy and complexity
Intro to Petroleum Geomechanics
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Simplified Rock Strength “Law”
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“True” strength criteria can be complex; however,
we often fit straight lines to the data to make
analysis simpler.


Y
cohesion
c′
n
3
T
o
Intro to Petroleum
Geomechanics
1
©MBDCI
Sufficient Behavioural Law (II)
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Choose a behavior model which adequately
describes the behaviour (- example)
Linear-elastic
model (A: no rupture; B: brittle rupture)
Non-linear elastic model E = f(σ3′, εv ...)
Elastic, perfectly plastic model
Elastic with strain-weakening, then plastic
Viscoelastic (shales, some rocks at high T)
Viscoplastic (salt and other halides)
Thermoelastic, thermoelastoplastic models, and so on…
The model must fit the problem: too much
complexity confuses and discredits analysis
Intro to Petroleum Geomechanics
©MBDCI
V (Deviatoric
component)
Deviatoric stress
σ1 – σ3
What Type of Stress-Strain Law?
Constitutive models:
A
A: Linear elastic, no
deviatoric dilation
B
B: Perfect plasticity, no
deviatoric dilation
E
C D
C: Instantaneously
strain-weakening, postfailure dilation angle
D: Gradual weakening,
post-failure dilation
E
+ve
A
C, D
B
-ve
Strain (%)
Intro to Petroleum Geomechanics
E: Damage mechanics
emulation of a real
geomaterial
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Sufficient Behavioral Law (III)
Data may be found in the literature, in data
bases, from geological inference, ....
 Experienced persons can give estimates
 A few simple tests may suffice, allowing
comparisons to existing data bases
 A laboratory test program may be used
 Post-analysis may help refine the behavioral
laws used, improving analysis
Don’t undertake complex testing programs unless potential benefits are large
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Intro to Petroleum Geomechanics
©MBDCI
Wilmington. California
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Bowl shaped
Casings sheared on
the shoulders of the
subsidence bowl
Few shears in middle,
where z greatest
Few on flanks
+earthquakes
Data analysis led to
proposed solutions…
Intro to Petroleum Geomechanics
©MBDCI
Methods of Analysis (I)
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Analysis must be founded on a “conceptual”
model which is correct (get the physics right!!)
Empirical models are based on practice and
“qualitative” assessments
 Experience
is a powerful tool, and requires a
strong understanding of the physics
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Analytical (closed-form) and semi-analytical
models are sets of equations which can be
solved directly (e.g. T(t) around a borehole)
Intro to Petroleum Geomechanics
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Analytic Solution Example
The “simplest” borehole stress analysis model
Hollow cylinder model
b
a
pb
pa
q
Elastic stress solution (Lamé)
(Usually, b >> a)
Intro to Petroleum Geomechanics
2 2
2
a (r  b )
2 2
2
b (r - a )
2 2
2
a (r - b )
q 
2 2
p p
2 b
2 2
2 a
r (b - a )
r (b - a )
r
r 
r
2 2
2
b (r  a )
p p
2 2
2 b
2 2
2 a
r (b - a )
r (b - a )
(Another equation is used to
calculate radial displacements)
©MBDCI
Methods of Analysis (II)
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Numerical models are for complex geometries,
varying boundary conditions, non-linear cases,
coupled processes (eg: flow + T + -)
 Finite
difference (FD), Finite element (FEM)
 Boundary discretization methods (BE, DD, BI)
 Discrete element methods (DEM)
 Hybrid approaches (DD + FEM, closed-form
solutions + FD…)
Different approaches may be better for
different problems (FD for T & p; FEM
for --T; FEM + DD for large problems…)
Intro to Petroleum Geomechanics
©MBDCI
Numerical “Discretization”
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Reality is complex…
To solve problems, a
rock mass is “divided”
into many “elements”
This is “discretization”
{}, {T}, {p}, {Q} {f}
(inputs, outputs, loads,
BC’s) are applied where
and as required
σij, εij, T, p… are
computed as required
Intro to Petroleum Geomechanics
discrete element “grain” model
f1
f1
f2
f2
network or FD model
Q1
Q2
Q3
Q2
finite element (FEM) model
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Methods of Analysis (III)
Probabilistic models use “sampling”
techniques for the variables to study outcome
probabilities (e.g. “Monte-Carlo simulation”)
 Stochastic models could mean that properties
are varied according to pre-defined distributions
 In Petroleum Geomechanics, statistical
approaches have been sparingly used to date;
“deterministic” models are widely preferred
Statistical approaches are necessary for
quantitative risk and cost analysis
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Intro to Petroleum Geomechanics
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Monte Carlo Simulation
Many samplings and solutions
are made to explore the overall
probabilities. These are then
related to cost and risk factors.
B, no A
A+B
Cases
A, no B
Random sampling +
problem solving
Intro to Petroleum Geomechanics
A, no B
B, no A
A+B
Parameter 2
Parameter 3
Probability
Parameter 1
Parameter 4
Risk/cost factor
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Monitoring (I)
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Used to verify the assumptions in the analysis
and the behavioral laws
Used to clarify the physical processes and thus
refine the conceptual model and analysis
Used as a means of controlling processes
through “feed-back”
Used to assure that environmental or safety
regulations are being met (e.g.: MS-monitoring)
Intro to Petroleum Geomechanics
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Microseismic Array
fibre-optics or telemetry
workstation
1
local
processors 2
3
4
5
monitoring or future
production wells
zone of
interest
Intro to Petroleum Geomechanics
sensors
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Data from a mine monitoring case in South Africa
Intro to Petroleum Geomechanics
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Monitoring (II)
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There are several different general approaches
PVT + chemical analyses of inputs/outputs
Wellbore methods, generally logs
Seismics, active and passive (microseismic)
Electrical methods
 Magnetotelluric
probing, electrical impedance
tomography, special multiple electrode methods…
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Deformation measurements and gravity
Miscellaneous methods (casing strain, ...)
Intro to Petroleum Geomechanics
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Hydraulic Fracture Mapping
Characteristic deformation pattern makes it
easy to distinguish fracture dip, horizontal and
vertical fractures
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Dip = 0°
Maximum Displacement:
0.0020 inches
Dip =90°
Maximum Δz:
0.00026 inches
Gradual “bulging” of
earth’s surface for
horizontal fractures
Trough along fracture
azimuth for vertical
fractures
Dipping fracture yields
very asymmetrical
bulges
Tiltmeters are used for fracture mapping
Intro to Petroleum Geomechanics
Dip = 80°
Maximum Displacement:
0.00045 inches
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Lessons Learned…
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Uncertainty and complexity dominate
petroleum geomechanics
So, design is an ongoing process based on…
 Use
of existing knowledge
 Lithostratigraphy,
geophysical data, cores…
 Stresses, pressures, temperatures and changes
 Rock behavioral “laws”
 Appropriate
analysis and predictions
 Measurements and refinement of predictions
 Additions to the knowledge base
Intro to Petroleum Geomechanics