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Models and Modeling

Models and Modeling in GeomechanicsModels and Modeling in Geomechanics

Maurice Dusseault

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Models and Modeling

Models and ModelingModels and Modeling

A “Model” is a simplified version of reality Complexity, heterogeneity have been reduced Small-scale details have been omitted Complex behavioral laws have been “linearized” Boundary conditions have been simplified

“The right horse for the right course” The model should not be too complex

Computational time increases (parametric analysis hard) It becomes difficult to understand the results

Also, it should not be too simple!

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Models and Modeling

To Model is to Understand…To Model is to Understand…

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Models and Modeling

Types of Models…Types of Models…

Conceptual model Based on physical understanding of the process

Geological models Lithostratigraphic model Structural geological model Depositional model 3-D “whole-earth” data base model, etc., etc.

Mathematical or Numerical model Material behavior model And so on and so forth…

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Models and Modeling

Conceptual Models Precede RealityConceptual Models Precede Reality

Horizontal wells in Venezuela for heavy oil development, Conoco-Phillips

Conceptual model - 1999

Real installation - 2003

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Models and Modeling

Simple Stratigraphic ModelSimple Stratigraphic Model

~6

50 m

Casing shear

locations

Surficial deposits

Colorado shales

Grand Rapids Formation

Clearwater Formation

McMurray Fmn.Oil Sands

Limestones

Sequence of sands, shales, etc.

(Alberta Oil Sands)(Alberta Oil Sands)

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Models and Modeling

Estuarine Accretion ModelEstuarine Accretion Model

A conceptual model to explain estuarine

sedimentation processes

No scale

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Models and Modeling

Geological Models: Logs Geological Models: Logs →→ Rocks Rocks

REG. TIPO

SVS-30

REG. TIPO

SVS-337

SMI

GUASARE

C-7

C-6

C-5

C-4

C-3

C-1

B-6/9

GUASARE

FALLA ICOTEA

ER-EO

FA

LL

A V

LE

-400

C-4C-5

C-6

C-7

GUAS

GUASARE

C-7

C-6

C-5

C-4

C-3

ER-EO

B-SUP

B-6/9

C-1

C-3C-4

C-5

C-6

C-7

C-2C-2

ER-EO

B-6/9

B-6/9

C-1

C-2

C-3

C-4

C-5C-3

C-5

C-2

C-4

C-6

C-7

C-7

C-6C-5C-4C-3C-2

C-1

C-2

C-3

ER-EO

B-SUP

B-SUP

2-D structural section modelPDVSA

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Models and Modeling

Model of Fault TracesModel of Fault Traces

In the centre of Lake Maracaibo, Venezuela

Only the fault traces at the reservoir depth are shown

Fault-block controlled oil field (Lago Medio)

In fact, faults are far more complex… Splay structures En-echelon faults Gouge zones……

BLOQUE IX

BLQ. XIV

BLOQUE X

PDVSA

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Models and Modeling

Geophysical and Geological ModelsGeophysical and Geological Models

Stratigraphy

ConocoPhillips

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Models and Modeling

Campos Basin Model, BrasilCampos Basin Model, Brasil

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Models and Modeling

Numerical ModelsNumerical Models

This mesh is part of a geomechanics model for numerical σ-ε analysis

The smallest grid block is 5050100 m size

A lab specimen perhaps only 5050100 mm!!

Clearly, the numerical model is a simplification of a complex reality…

Sandia Nat’l Labs

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Models and Modeling

Mathematical Modeling & ScaleMathematical Modeling & Scale

553 km = 75109 m3

81515 m = 2370 m3

115 m = 5 m3

Rock sample = 0.001 m3

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Models and Modeling

Model of “Stacked” Solution CavernsModel of “Stacked” Solution Caverns

Circular shapes assumed, allowing a 1-D (radial) model. Also, details are ignored…

“Reality”, from sonar scan

Model for creep analysis of gas storage caverns in salt

(ignored)

Terralog Technologies Inc.

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Models and Modeling

A Reservoir Isopach ModelA Reservoir Isopach Model

Terralog Technologies Inc. 0.0 2.0 4.0 6.0 8.0 10.0Net sand thickness in feet

X coordinate distance – feet

Y c

oord

inat

e di

stan

ce –

feet

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Models and Modeling

Seismic ModelSeismic Model

Channel structures

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Models and Modeling

3-D “Whole Earth” Model3-D “Whole Earth” Model

Well path

Stratigraphic surfaces

Based on geophysical logs + core data + high-resolution surface seismic

Bill Huang, 2003,

Chevron

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Models and Modeling

Casing Deformation ProblemCasing Deformation Problem

Ter

ralo

g T

echn

olog

ies

Inc.

tubi

ng

Concept

Reality Mathematical Model (FEM)

Mathematical models are based on concepts that

are models of reality

casing cement

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Models and Modeling

Detail in RepresentationDetail in Representation

Ekofisk geological model 19185 km model size Goal was to simulate

Overall reservoir behavior Seafloor subsidence…

Numerical model had 674124 elements Four materials No discontinuities Hypo-elastic model

Chin et al. 2003 ConocoPhillips

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Models and Modeling

Ekofisk Model ResultsEkofisk Model Results

Chin et al. 2003 ConocoPhillips

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Models and Modeling

Model ComplexityModel Complexity

The Ekofisk model had 66,000 elements, but… Well over 1 million degrees of freedom A highly complex material behavior law (Chalk) Properties that are functions of σ′ and p Full flow-coupling to geomechanics behavior

Using a finite difference flow simulator in parallel Compaction bridging effects (some heterogeneity)

For one solution (i.e. one set of parameters) 30 year simulation, >600 iterations, 1.5 hours CPU

And, the results were quite good…

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Models and Modeling

Material Behavior ModelsMaterial Behavior Models

Scale issues, representativeness, scatter of data, similarity to stress path in the field, etc., etc., are all relevant issues in material models.

strain - εa

stre

ss –

(σ

1 –

σ3)

sudden yield

strain - εa

ΔV+ve

-ve

strain - εa

stre

ss –

(σ

1 –

σ3)

sudden yield

strain - εa

ΔV+ve

-ve

Lab test σ ε curve

Model used in calculations

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Models and Modeling

How Do We “Model” This Rock Mass?How Do We “Model” This Rock Mass?

Joints and fractures often dominate flow and deformation behavior

Representative testing is generally never possible

So, of what value is an extremely complex behavioral law based on small test specimens?

1 m

A large core specimenA core “plug”

Machu Picchu, Peru, Inca Stonecraftrock

n

100 mm

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Models and Modeling

Local, Reservoir and Regional Scales Local, Reservoir and Regional Scales

Regional Scale Stresses Basin scale: 50 km to 1000 km Often called “far-field stresses”

Reservoir Scale Stresses A reservoir, or part of a reservoir Scale from 500 m to several km Salt dome region: 5-20 km affected zone

Local Scale Stresses Borehole region: 1-5 m Drawdown zone (well scale) 100-1000 m

Small Scale Stresses (less than 10-20 cm)

~100 km

~4 km

~400 m

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Models and Modeling

Granular ModelsGranular Models

Rocks are heterogeneous at all scales (microns to kilometers)

In granular media, macroscopic stresses are transmitted through grain contact forces (fn, fs)

fs = shear force

fn = normal force

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Models and Modeling

Granular ModelsGranular Models

Most behavior aspects of granular media can be emulated in a 2-D model with only a few thousand elliptical “grains”!

A 3-D model with 10,000 grains is betterRealityModel

0.10 mm

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Models and Modeling

Guidelines for all Types of ModelsGuidelines for all Types of Models

Think carefully about what you are trying to represent in your model

The data base may be extremely densely populated, but the model much less so…

Avoid all unnecessary complexity Most geomechanics problems can be reasonably

solved with a linear elastic rock model, 6-8 GMUs Keep the geometry and BC’s simple Increase complexity only when it is necessary

E.g.: going to an elastoplastic reservoir rock model

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Models and Modeling

A Coupled Geomechanics ModelA Coupled Geomechanics Model

Reservoirsimulator

Geomechanicssimulator

FD

FEM

Solve the flow problem for new Δt increment

Output pressures

Solve σ′ ε problem

Pressures applied as nodal loads

ΔT can be included as well

T

L

L

1T

Kv

Calculate stressesstrains, ΔV

Apply as source-sink terms

to reservoir simulator

Error OK?N

Y

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Models and Modeling

Coupled Geomechanics ModelCoupled Geomechanics Model

This approaches uses pre-existing powerful reservoir simulators for diffusion problems

The iteration loop converges very rapidly The approach is rigorous The logic is straightforward You can introduce non-linearities easily…

k = ƒ(σ′) Non-linear stiffness - C = ƒ(σ′) And so on

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Models and Modeling

Different Analysis ModelsDifferent Analysis Models

Empirical Model, also a Physical Model Probabilistic or Stochastic Model Closed-Form Solution (Analytic Solution) Semi-Analytic Solution (Integrals…) Mathematical models

Finite Difference methods Finite Element method Boundary Element methods Streamline models for flow problems