Multiscale Parameterization and History Matching in Structured and Unstructured Grids:Theory and Field Application

E. W. Bhark, A. Rey, A. Datta-Gupta and B. Jafarpour

• Develop structured history matching workflow

• Coarse (regional) scale Novel grid-connectivity-based

parameterization

• Flexible, efficient application for

large models, complex geology

Calibrate multiscale heterogeneity

Avoid traditional regional multipliers

• Local (grid cell) scale Established streamline-based method

• Vasco et al. (1998); Datta-Gupta and King (2007)

Refine prior preferential flow paths

Motivation

2

• Parameterization in history matching

Methods of linear transformation

Grid-connectivity-based parameterization

• Structured history matching workflow

• Field application

Offshore reservoir model (Rey et al. [2009], SPE124950)

Outline of presentation

3

• Reduce redundant model information

Preserve important heterogeneity

• Improves:

Solution non-uniqueness and stability, computational efficiency

Why re-parameterization?

~5,000 Unknowns 100 Unknowns 50 25

Ex., high-resolution

(3D) abs. permeability

4

Parameterization by linear transform

5

v1 + v2 + v3 + … + v50 + … + = vN

M

N

M v

v

v

u

u

u

2

12

1

2

1

u = v

• Required basis properties

Compression power: most

energy in fewest coefficients vi

Amenable to efficient

application for very large grids

for M << NvuΦ NM

N-parameter

high-resolution

model

Grid-connectivity-based transform basis

(1) Model (or prior) independent

Can benefit from prior model information

(2) Applicable to any grid geometry (e.g., CPG, irregular unstructured,

NNCs, faults)

(3) Efficient construction for very large grids

(4) Strong, generic compression performance

(5) Geologic spatial continuity

6

Highlights of new basis

M

N

M v

v

v

u

u

u

2

12

1

2

1

=

Concept: Develop as generalization of discrete Fourier basis

KEY: Perform Fourier transform of function u by (scalar) projection

on eigenvectors of grid Laplacian (2nd difference matrix)

Basis development

• Interior rows Second difference

Periodic operator (circulant matrix)

• Exterior rows Boundary conditions control

eigenvector behavior

7

• Decompose L to construct basis functions (rows of )

Always symmetric, sparse

Efficient (partial) decomposition by restarted Lanczos method

Orthogonal basis functions;

• In general (non-periodic) case

Eigen(Lanczos)vectors vibrational modes of the model grid

Eigenvalues represent modal frequencies

Basis development

vΦvΦuvΦuT 1

5 10 15 20 25 30 35 40 45 50

5

10

15

20

25

30

35

40

45

50

Grid LaplacianCPG Unstructured

2-point connectivity (1/2/3-D)

8

• Modal shape modal frequency

• Constant basis Zero frequency

• Discontinuities honored

Basis vec. 1 Basis vec. 2 Basis vec. 3 Basis vec. 4 Basis vec. 5

Corner-point Grid

(Brugge)

Basis functions: Examples

Basis vec. 9

9

Unstructured grid

Unstructured grid(local refinement)

Basis function 1 Basis function 3 Basis function 5 Basis function 8 Basis function 10

Basis functions: Examples

Multiple subdomains

10

Channel structure

Parameterize

multiplier field

Additional

spatial

detail?

NO

Add higher-

frequency modes to

basis

YES

Calibrated Model

(1) START: Prior model

Prior spatial hydraulic

property model

Update in transform

domain

Back-transform

multiplier field to

spatial domain

Flow and transport

simulation

Mu

ltis

cale

ite

rate

Unit-multiplier field at

grid cell resolutionG

rad

ien

t-b

ased

itera

te

Streamline-,

sensitivity-based

inversion (GTTI)

Structured multiscale workflow

Data misfit

tolerance?

NO

YES

(2) Regional update (3) Local update

FINISH

11

Field application: Offshore reservoir

12

Reservoir

• > 300,000 cells

• Mature waterflood

• 8 years of production history

• 4 producers and 4 water injectors

• Complex depositional sequence of turbidite sand bodies / facies

• Rey et al. (2009), SPE124950

Parameter• Permeability

Data

• Water cut

Prior model facies (5)

Conceptual heterogeneity model

Next objective:

Use parameterization to assist

in heterogeneity identification

and updating

P2I2

P3

I3

P1I1

I4

P4

Initial Kx:

Average of measurements

at wells per facies (5)

Prior geo-model

Facies ID

13

Prior geo-model

Multiplier field

Workflow: Prior model & multiplier field

F6F5

F3

F1

F2

14

Prior geo-model

Multiplier field

Basis functions

Facies 5:

• Multiplier field is linear

combination of basis functions

1 3 6 8 15

v1 …+ v3 …+ v6 …+ v8 …+ v15

F5 multiplier field:

u =

Facies basis functions

15

Adaptive multiscale inversion

Prior geo-model

Multiplier field

Basis functions

Multiscale inversion

• Sequentially refine within-facies heterogeneity

From coarse to finer scales

Adaptive inclusion of basis functions

• End refinement when production data become

insensitive to addition of basis functions

1 5 10

16

Multiscale update

Kx: Adaptive multiscaleNumber of leading basis

functions per facies

10

10

5

1

10

36

17

Adaptive multiscale

Comparison with previous calibration

Manual zonation

Rey et al. (2009)This study

Facies zonation Tx multiplier

18

Tx multiplier

Data misfit: WCTInitial and multiscale

P2

P4

P3

P1

19

Prior geo-model

Multiplier field

Basis functions

Multiscale inversion

Streamline-based inversion

High-resolution permeability model

Streamline-based inversion

• Refine at grid-cell scale

• Streamline paths determined by

heterogeneity, well pattern

20

Streamline-based update

Kx changeFinal Kx match

• Local updates

• Minimal updates along prior preferential flow paths

21

Kx (md)Kx (md)

Final Data misfit: WCTMultiscale and streamline

P2

P4

P3

P1

22

Comparison of data misfit: WCTMultiscale/SL and Business Unit

23

P2

P4

P3

P1

24

Comparison with previous calibration

This study

I4

I3

I2

• Regional

parameterization

more consistent with

model constraints

High perm

(> upper limit

near P3)

Potential

channel

Figure 26: Rey et al. (2009)

SOURCE

TMX: Rey et al. (2009)

TMX mult.

P3

I3 I4

• Multiscale approach to history matching

Builds on well-established ‘structured’ workflow

Regional heterogeneity

Generalized grid-connectivity-based parameterization

Efficient, flexible application to any reservoir model geometry

Refine local heterogeneity

Prior preferential flow paths captured by streamlines

• Field application

Demonstrates practical feasibility

Improvement upon heterogeneity characterization using

standard zonation approaches

25

Summary