Quantifying Parameter Interaction in a Generalized Sensitivity Analysis

Quantifying Parameter Interaction in a Generalized Sensitivity Analysis Darryl Fenwick, Streamsim Technologies Céline Scheidt, and Jef Caers, Stanford University

SA: New Requirements • Need a method that can account for: 1. 2. 3. 4. Multiple interpretations/scenarios Spatial uncertainty Time-varying responses Parameter interactions • Computationally efficient May 9, 2013 Streamsim/Stanford HM JIP Meeting 2

Reservoir Modeling: ED & RSM • ED/RSM preferred method for SA • Limitations: • ED ignores prior PDF • Single response • Smooth response • Discrete parameters • Interpretations/scenarios • Spatial uncertainty • Interactions symmetric May 9, 2013 Streamsim/Stanford HM JIP Meeting 3

A General SA Approach Spear and Hornberger – study of growth of nuisance alga • Divided output of model into two classes, behavior B, and behavior B’ • Analyzed how model parameters influenced the classification May 9, 2013 Streamsim/Stanford HM JIP Meeting 4

Spear & Hornberger - Summary Monte-Carlo Sampling Classification of response into 2 classes Analyze of distribution between classes _ F(p|B) models Presence (B) of algae F(p|B) _ Absence (B) of algae May 9, 2013 Streamsim/Stanford HM JIP Meeting 5

Basic Idea • Influential parameter will distinguish the models into the separate classes • The distributions will be different between classes • Non-influential parameters will have no impact on the classification • May 9, 2013 The distributions will be similar between classes Streamsim/Stanford HM JIP Meeting 6

Proposed Generalized SA Extension of the ideas of Spear and Hornberger in multiple aspects (DGSA): 1. Construction of more than two classes • Use of distance-based clustering 2. Analysis of parameter distribution using a L 1 norm distance 3. Construction of a bootstrap-based confidence interval to account for small sample size 4. Analysis of two-way interactions May 9, 2013 Streamsim/Stanford HM JIP Meeting 7

SA: New Requirements • Need a method that can account for: 1. 2. 3. 4. Multiple interpretations/scenarios Spatial uncertainty Time-varying responses Parameter interactions • Computationally efficient May 9, 2013 Streamsim/Stanford HM JIP Meeting 8

DGSA: Advantages Model responses are used only for classification Proxy models can be employed • What is important is that the responses correctly classify the models • Accuracy of the response itself is inconsequential • Possible significant advantage for cpu-intensive models when a fast proxy model is available May 9, 2013 Streamsim/Stanford HM JIP Meeting 9

DGSA: Advantages Can account for asymmetric interactions • RSM assumes that all interactions are symmetric • However, interactions can be asymmetric, especially with mix of continuous & discrete parameters, spatial uncertainty May 9, 2013 Streamsim/Stanford HM JIP Meeting 10

Application - WCA field • • Offshore turbidite 20 producers 8 injectors 78 x 59 x 116 ~ 120, 000 active grid blocks • 3 -1/2 years production • Uncertainty • Depositional scenario Scheidt, C. and Caers, J. K. “Uncertainty Quantification in Reservoir Performance Using Distances and Kernel Methods – Application to a West-Africa Deepwater Turbidite May 9, 2013 Streamsim/Stanford HM JIP Meeting Reservoir”, SPEJ 2009 11

One Realization – TI 1 Upper Section May 9, 2013 Lower Section Streamsim/Stanford HM JIP Meeting 12

Application of DGSA • 4 continuous parameters in the flow simulation: • • SOWCR: Residual oil saturation: (U[0. 15, 0. 35]) krw. Max : Maximum water relative permeability value: (U[0. 3, 0. 6]) wat. Exp: Water Corey exponent: (U[2, 4])) Kv. Kh: Kv/Kh ratio: (U[0. 1, 1]) • Single response: final cum. oil production • Goal: compare general SA with RSM • 40 runs created using Latin hypercube sampling May 9, 2013 Streamsim/Stanford HM JIP Meeting 13

Parameter Distributions c 1 c 2 May 9, 2013 Streamsim/Stanford HM JIP Meeting c 2 prior 14

Parameter Distributions L 1 -norm distance illustration c 1 c 2 Bootstrap procedure standardizes distance and used for sensitivity test Streamsim/Stanford HM JIP Meeting May 9, 2013 15

Parameter Sensitivity Standardized measure of sensitivity Standardized L 1 -norm distance May 9, 2013 (average of the standardized L 1 -norm distance per class) Streamsim/Stanford HM JIP Meeting 16

Parameter Interactions Conditional interaction: the distribution of parameter pi is influenced by the value of pj in a class ck Bin dependent parameter pj (low, medium, high) Class ck High wat. Exp Sensitivity measure & hypothesis test follow same procedure as for single-way sensitivities May 9, 2013 Streamsim/Stanford HM JIP Meeting 17

Parameter Interactions • Sensitive interactions • krw. Max|wat. Exp • wat. Exp|krw. Max • SOWCR|wat. Exp • Asymmetric interaction May 9, 2013 Streamsim/Stanford HM JIP Meeting 18

Sensitivity Results – RSM Sensitive for DGSA Only main factors used to compute the model Main factors + interactions used to compute the model RSM does not give same results when interaction are considered May 9, 2013 Streamsim/Stanford HM JIP Meeting 19

Application of DGSA to WCA • 1 discrete parameter: • • 4 continuous parameters in the flow simulation: • • • TI: uncertainty in depositional scenario SOWCR: Residual oil saturation: (U[0. 15, 0. 35]) krw. Max : Maximum water relative permeability value: (U[0. 3, 0. 6]) wat. Exp: Water Corey exponent: (U[2, 4])) Kv. Kh: Kv/Kh ratio: (U[0. 1, 1]) 240 runs created using Latin hypercube sampling May 9, 2013 Streamsim/Stanford HM JIP Meeting 20

6 Depositional Scenarios TI 1 TI 3 TI 8 TI 9 TI 10 TI 13 May 9, 2013 Streamsim/Stanford HM JIP Meeting 21

Traditional SA Methods Challenges: 1. Discrete parameter TI • For 6 training images, would require building 6 response surfaces 2. Multiple responses (oil & water rates) 3. Spatial uncertainty • Seed for geostatistical algorithm changes for each run May 9, 2013 Streamsim/Stanford HM JIP Meeting 22

Parameter Distributions wat. Exp SOWCR May 9, 2013 TI krw. Max Kv. Kh Streamsim/Stanford HM JIP Meeting 23

Parameter Sensitivities Standardized measure of sensitivity Standardized L 1 -norm distance May 9, 2013 (average of the standardized L 1 -norm distance per class) Streamsim/Stanford HM JIP Meeting 24

Sensitivity Results - Interactions Note: • krw. Max|wat. Exp sensitive as in previous case • SOWCR insensitive • But sensitive interactions May 9, 2013 Streamsim/Stanford HM JIP Meeting 25

Sensitivity Results - Interactions Note: • Many sensitive interactions with TI • Many asymmetric interactions May 9, 2013 Streamsim/Stanford HM JIP Meeting 26

Summary General SA approach has been developed • Idea: Use model classification and parameter distributions as basis for SA • Addresses some limitations in traditional approach to SA in reservoir modeling • • • May 9, 2013 Account for any type of distributions (interpretations/scenarios) Spatial uncertainty Time-varying (multiple) responses Interactions Computationally efficient Streamsim/Stanford HM JIP Meeting 27