1. Energy Resources Engineering Department Stanford University, CA, USA
A multiscale method for large-scale inverse modeling example of inverting single-phase flow data
Jianlin Fu
Jef Caers
Hamdi Tchelepi
Energy Resources Engineering Department
Stanford University, CA, USA

2. Multiscale modeling
Motivation
Engineering of subsurface is dependent on large- and small-scale heterogeneity
Different practical problems are solved at different representation scales
Different data have different scale of information
High-resolution reservoir modeling entails multiscale data integration
Can we integrate all well data, seismic data, and production data into a model at any appropriate scale, including the fine scale?

3. State-of-the-art
Methods & problems
High-resolution modeling to integrate production data is CPU demanding: flow simulation is expensive
Most approaches ignore the important fine-scale details
Upscaling may filter out important small-scale variations
Statistical downscaling ignores physics
Physical methods ignore geology

4. Existing philosophies (upscaling)
High-resolution inverse modeling
Model
updating
(Stochastic)
Optimization:
GDM
PPM
upscaling ?
Flow
simulation
problem
solution 4

5. Existing philosophies (downscaling)
High-resolution inverse modeling
Model
updating
Upscaling
Flow
simulation
Statistical downscaling …
History matching may not be preserved ! 5

6. Summary of challenges
Cannot run fully high-resolution flow simulations at the fine scale
Reconcile two types of data
Production data (ill-posed inverse problem)
Spatial continuity model (interpreted from geological knowledge, e.g., variogram, TI)

7. A new philosophy High-resolution inverse modeling s s p p m Coupled
PDEs m
Gradient
Gradient m
Model
updating
Coupled
PDEs 7

8. What are the difficulties ?
How to run multiscale flow simulations?
How to compute fine-scale gradient for optimization (history matching)?
How to maintain geological consistency? 8

9. The multiscale solution (1)p p m
Coupled
PDEs m
Model
updating
Coupled
PDEs 9

10. Multiscale flow simulation
Fine-scale flow simulation? Too expensive or impossible!
Anxnpnx1=qnx1
Solution: reduce n
Upscaling vs multiscale

11. Multiscale flow simulation
Multiscale flow simulation? How …?
p=? x
Solution:
Fine-scale reconstruction 11

12. Multiscale flow simulation
Fine-scale pressure
Multiscale pressure
Fine-scale saturation
Multiscale saturation
Courtesy of Zhou

13. Multiscale flow simulation
Error and speedup
Courtesy of Wang 13

14. The multiscale solution (2)
Coupled
PDEs
Gradient
Gradient
Model
updating
Coupled
PDEs 14

15. Gradient  Define an objective function:  Minimization of J requires:

16. Basic adjoint method = 0  Forward flow PDE:  The Lagrangian:
Lagrange multiplier
 Forward flow PDE:
Forward simulation
 The Lagrangian:
= 0
 Adjoint PDE:
Adjoint simulation
 Gradient: 16

17. Multiscale adjoint simulation
Run forward simulation
Compute objective function
Solve coarse-scale adjoint equation
Solve fine-scale adjoint equation
Assemble fine-scale gradient 17

18. The multiscale solution (3)
Coupled
PDEs
Gradient m
Model
updating
Coupled
PDEs 18

19. Model updating + . Model updating: Model updating: Gradient is used:
Gradient-based gradual deformation (Hu, 2004) Z1 Z2 Z3
Model updating:
Model updating: +
Z(α)
Optimize α
Gradient is used:
1) Optimization of α
2) Selection of Z2, Z3, … . 19

20. An illustrative example
Experimental configuration (A small 2D case)
Horizontal injector
Horizontal producer
lnK field
Initial pressure field
Constant
Constant
p=15
p=0

21. An illustrative example
History matching to pressure data
Reference field
Initial field

22. An illustrative example
Proposed method
Computationally efficient
Multiscale matched
Fine-scale matched
Almost identical convergence rate as fine-scale 22

23. An illustrative example
Physically accurate
Proposed method
Multiscale simulation
Fine-scale simulation
Absolute error decreases as the iteration proceeds

24. An illustrative example
Physically accurate
Observations
Before matching
Multiscale matching
Fine-scale simulation

25. An illustrative example
Multiscale matched models
Realization 1
Realization 2
Realization 3 25

26. An illustrative example
Experiment: what if we do not enforce geological
Consistency (pure deterministic optimization)?
Matched lnK field
Need for enforcing geological consistency 26

27. Conclusions Computationally efficient Physically accurate
Similar iterations but more efficient
Capable of handling large-scale cases
Physically accurate
Small absolute error btw multi- and fine-scale
Geologically consistent
Spatial structure preserved
Let’s just give a very simple summary on the observations.

28. Thank you !
Questions ?