Models and Modeling in Geomechanics

1. Models and Modeling in Geomechanics Maurice Dusseault

2. Models 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!

3. To Model is to Understand…

4. 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…

5. Conceptual Models Precede Reality • Horizontal wells in Venezuela for heavy oil development, Conoco-Phillips Real installation - 2003 Conceptual model - 1999

6. Simple Stratigraphic Model (Alberta Oil Sands) Surficial deposits Colorado shales Grand Rapids Formation Clearwater Formation McMurray Fmn. Oil Sands Limestones Casing shear locations ~650 m Sequence of sands, shales, etc.

7. Estuarine Accretion Model A conceptual model to explain estuarine sedimentation processes No scale

8. REG. TIPO REG. TIPO SVS-30 SVS-337 Geological Models: Logs →Rocks FLANCO OESTE FLANCO ESTE ER-EO ER-EO ER-EO C-4 B-SUP ER-EO C-5 B-SUP B-SUP B-6/9 C-3 C-6 B-6/9 C-1 C-2 C-4 C-1 C-3 FALLA VLE-400 B-6/9 B-6/9 C-2 C-7 C-3 SMI C-5 C-4 C-2 C-5 C-3 C-2 C-1 C-1 C-4 C-3 C-5 C-2 C-6 GUAS C-6 C-4 C-2 C-3 C-5 C-7 C-4 C-6 C-3 C-5 C-4 C-7 C-7 C-6 C-5 C-6 C-7 C-7 GUASARE GUASARE GUASARE FALLA ICOTEA PDVSA 2-D structural section model

9. BLQ. XIV BLOQUE IX BLOQUE X Model 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…… PDVSA

10. Geophysical and Geological Models ConocoPhillips Stratigraphy

11. Campos Basin Model, Brasil

12. Numerical Models Sandia Nat’l Labs • This mesh is part of a geomechanics model for numerical σ-εanalysis • The smallest grid block is 5050100 m size • A lab specimen perhaps only 5050100 mm!! • Clearly, the numerical model is a simplification of a complex reality…

13. Mathematical Modeling & Scale 553 km = 75109 m3 81515 m = 2370 m3 115 m = 5 m3 Rock sample = 0.001 m3

14. Model of “Stacked” Solution Caverns • Circular shapes assumed, allowing a 1-D (radial) model. Also, details are ignored… “Reality”, from sonar scan (ignored) Model for creep analysis of gas storage caverns in salt Terralog Technologies Inc.

15. A Reservoir Isopach Model Y coordinate distance – feet X coordinate distance – feet 0.0 2.0 4.0 6.0 8.0 10.0 Terralog Technologies Inc. Net sand thickness in feet

16. Seismic Model Channel structures

17. 3-D “Whole Earth” Model Well path Stratigraphic surfaces Bill Huang, 2003, Chevron Based on geophysical logs + core data + high-resolution surface seismic

18. Casing Deformation Problem Terralog Technologies Inc. tubing Reality Mathematical Model (FEM) Mathematical models are based on concepts that are models of reality casing cement Concept

19. Detail in Representation • Ekofisk geological model • 19185 km model size • Goal was to simulate • Overall reservoir behavior • Seafloor subsidence… • Numerical model had • 674124 elements • Four materials • No discontinuities • Hypo-elastic model Chin et al. 2003 ConocoPhillips

20. Ekofisk Model Results Chin et al. 2003 ConocoPhillips

21. Model 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…

22. Material Behavior Models • Scale issues, representativeness, scatter of data, similarity to stress path in the field, etc., etc., are all relevant issues in material models. sudden yield sudden yield stress – (σ1 – σ3) stress – (σ1 – σ3) Model used in calculations Lab test σ­ε curve strain - εa strain - εa +ve +ve ΔV ΔV strain - εa strain - εa -ve -ve

23. n 100 mm  rock 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? A large core specimen A core “plug” 1 m Machu Picchu, Peru, Inca Stonecraft

24. 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

25. Granular 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

26. Granular 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 better Model Reality 0.10 mm

27. Guidelines 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

28. A Coupled Geomechanics Model Solve the flow problem for new Δt increment Y Error OK? N Apply as source-sink terms to reservoir simulator FD Calculate stresses strains, ΔV Reservoir simulator FEM Geomechanics simulator Output pressures Pressures applied as nodal loads ΔT can be included as well Solve σ′­ε problem

29. Coupled 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

30. Different 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