Using Metric Space Methods to

Analyze Reservoir Uncertainty

Darryl Fenwick Rod Batycky

Streamsim Technologies, Inc.

Outline

• General Modeling workflow

• Classic Sensitivity Analysis

• Metric Spaces for Screening & Sensitivity Analysis

• Applications to Reservoir Modeling

• Conclusions

General Modeling Workflow

Need more runs?

Sensitivity Runs

Screen

mo

dels

build models

Learn

no

yes

Model refinement

Model(s) for forecasting

Do more runs

/

make more models

Model Parameters

models

Sensitivity analysis

Sensitivity Runs & Parameters

• Key aspect to sensitivity runs is parameterization. • Which parameters to choose, parameter values,…

• Types of parameters • Non-functional parameters, directly input to each model.

• Fault trans, fluid contacts, Sorw, …

• Functional parameters, create other properties which are input to each model. • Random seed, variogram angle, histogram,…

• Create porosity, perm, Sinit…

• Discrete vs continuous parameters.

• How do the parameters impact the models?

Classic Sensitivity Analysis

Challenges:

• Multiple responses • Discrete parameters

• Fault interpretations • Facies proportion cubes

• Stochastic “noise” in response

• Spatial uncertainty • Geostatistically-derived properties

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PERMXPORO

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PR

0 0.2 0.4 0.6 0.8 1 1.2 1.4

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Sensitivity of parameters on CumOil

Improved Screening Method

• Need method that can: • Apply to different types of parameters and different

responses.

• Identify important parameters with respect to desired response.

• Extract diverse set of parameters for uncertainty quantification.

• Compare models with respect to each other.

• Identify “best” models for HM.

• Generalized screening diagnostics needed.

Metric Space Methods for Screening

• The “distances” between a set of points defines a Metric Space (MS).

• MS methods used in internet search engines, image comparison, protein classification, etc.

• MS methods applied by Scheidt & Caers to reservoir modeling (2008, 2009).

MS Method - Key Concepts

1. Dissimilarity Distance. Measures dissimilarity between two models based upon a

distance measure.

2. Multi-Dimensional Scaling (MDS)

Translates all distances to a lower-dimensional space, separated by the distance measure.

3. Cluster Analysis Groups similar models in MDS space.

Perform sensitivity analysis and model screening using

metric space information

Dissimilarity Distance

• The distance is a measure of dissimilarity between any two models.

• Dissimilarity in terms of: • Geologic properties

• (facies, φ, K, OOIP, …) • Flow response

• (water cut, pressure, oil saturation, …)

1x

2x

Nx

gbN

k

j

k

i

kij zz1

)KK(

tsN

ts

j

tsw

i

tswij qq1

,, )(

Requirements for Distance

• A good distance is: • Easy to understand. • Fast to calculate. • Designed for the purpose of the study.

• Example: Sensitivity analysis of water production rate, qw

• Distance is the difference between each

simulation.

tsN

ts

j

tsw

i

tswij qq1

,, )(

Distance Matrix

• Model distances are represented by the distance matrix D.

• D is symmetric, with zero diagonal entries.

• Number of models, n

• Number of unique pairs in D is given by n(n-1)/2.

1 2 3 4 ...

1 0 12 13 14 ...

2 21 0 23 24 ...

3 31 32 0 34 ...

4 41 42 43 0 ...

... ... ... ... ... 0

Multi-Dimensional Scaling (MDS)

• D matrix is difficult to visualize and understand.

• MDS transforms the dissimilarity distance into an approximate Euclidean distance. • Uses Eigenvalue decomposition of D.

• Display Euclidean distances in MDS plot, visual and

diagnostic tool. • Visualizes models relative to each other.

• Identifies similar models (screening)

• Visualizes model uncertainty & response sensitivity.

From MS to MDS, Summary

Distance Matrix D

1 2 3 4 ...

1 0 12 13 14 ...

2 21 0 23 24 ...

3 31 32 0 34 ...

4 41 42 43 0 ...

... ... ... ... ... 0

Model 1 Model 2

Model 3 Model 4

12

13 24

34

32

14

2D projection of MDS Plot

MDS

Single reservoir model Defined by

similarity distance

Visualization of model similarity

1 2 3 4 ...

1 11 12 13 14 ...

2 21 22 23 24 ...

3 31 32 33 34 ...

4 41 42 43 44 ...

... ... ... ... ... ...

Distance Matrix D

Cluster Points in MDS Plot

MDS

MDS Plot

Applications to Reservoir Modeling

• 43000 active cell, waterflood

• 100+ producers, 20+ injectors, 25years history.

Field oil rate Field wtr rate Field inj rate

Create Multiple Models

Parameter Values

Corr. length Low high

Corr. angle 45 90 135

Sorw 0.2 0.3

Tzmult (k=3) 1 0.001

Kv/Kh ratio 0.1 0.01 0.001

Sensitivity Runs

Learn 72

models Sensitivity analysis

Static Property-based Distance Measure

• A distance based on static gridblock properties is fast to compute. No flow simulation required.

• Useful if there are many models or flow simulation per model is expensive.

• Is a static-based distance a good proxy for flow response?

• Depends on the flow response we are studying.

“Green Field” Uncertainty Quantification

• Quantify uncertainty in cumulative oil production.

• Static-based distance measure is the difference between each model of local gridblock permeability.

• 72 models, 2556 pairs->MDS + clustering -> 5 groups.

gbN

k

j

k

i

kij zz1

)KK(

• Extract centroids of clusters for flow simulation.

• 5 flow simulations.

“Green Field” Uncertainty Quantification

Cum Oil Prod (5 Models) Cum Oil Prod (All Models)

• 5 centroids capture the spread in uncertainty in cumulative oil production.

• Distance based on gridblock Kz is a good proxy to flow simulation response of cumulative oil production.

Flow-Based Distance Measure

• Requires a flow simulation.

• Flow-based distance measures based on:

• Grid properties (So, Sw, Sg, P).

• Total rates, phase (oil, water, gas) rates.

• Inter-well connectivity.

• Flow-based distance data support:

• Field level, well level, time levels

Connectivity-Based Distance

• A benefit of streamline simulation is quantification of well-pairs.

• Connectivity simulations -> fast.

• Quantify connectivity, Q, between well-pairs.

flux between wells, Q

Connectivity-based Distance

wellpairsN

k

j

k

i

kij QQ1

)(

run i run j

• Distance based on difference in flow rate of a well-pair between two models.

MDS Plot based on Connectivity

• MDS gives 5 clusters.

• Grouping is not a function of Kv/Kh

• Grouping is a function of variogram angle.

• Connectivity analysis could be applied at early model building stage.

Kv/Kh Vario Angle

Flowrate-based Distance Measure

• Flow-based distance, field oil rate.

• We can also calculate a distance with respect to historical data.

• We can map “history” in MDS space.

Objective Function

tsN

ts

j

tso

i

tsoij qq1

,, )(

tsN

ts

hist

tso

i

tsoij qq1

,, )(

“Brown Field” Application – Classic Screening

• Objective function with respect to history.

• Only know how models relate to history, but not each other.

Objective Function Field Oil Rate

Run #

“Brown Field” Application – History Matching.

• Compute the difference in field oil rate at each timestep between each model.

• Interested in model differences between each other and history.

MDS Plot

“Brown Field” Application – History Matching

• Extract parameter properties of each cluster (sensitivity analysis).

• Cluster with history has variability in model parameters.

• Additional workflows with the “history” cluster…

• Retain parameter variability in history matches. • Generate new models with most “important” parameters. • Pass to full-physics HM • Well-level HM.

“Brown Field” Application – History Matching.

• Parameter variability for the “history” cluster.

• Forecast all models in the “history” cluster.

• Retain uncertainty in forecasts.

HM Model Diagnostic

• Visualization of models & comparison with history

• Identification of models “close” to history

• Have we “bracketed” our history with our models?

200 models

True earth

L=200

Diagnosing Wrong Priors

True earth

Should we attempt to history match this data?

Field-level vs Well-level

• MDS workflows can be applied to analysis at well-level.

w tsN

w

N

ts

j

tswo

i

tswoij qq1 1

,,,, )(

tsN

ts

j

tso

i

tsoij qq1

,, )( field-level

well-level

field-level MDS well-level MDS

Conclusions

• Introduce Metric Space and MDS as a new method to screen models.

• A general method based on differences that can work with any parameter type any model response.

• Compare models to history and to each other. • Quantify and retain diversity within the HM. • Guide which parameters are important to vary in the HM. • Diagnose wrong priors, ensemble close/far from history.

• Cluster analysis quantifies: • Impact of parameters on each cluster. • Diversity in each cluster or diversity of the centroids. • Uncertainty in forecasts and history matches.

• Software allows easy application of MS methods to reservoir uncertainty workflows.