GEOTHERMAL RESERVOIR MANAGEMENT USING

STATISTICAL METHODS

Halldora Gudmundsdottir

Roland N. Horne

G E O R G G E O T H E R M A L W O R K S H O P

N O V. 1 4 - 1 5 , 2 0 1 8

I NTRODUCTION

• For sustainable geothermal exploitation we need to understand and predict the reservoir’s response to stimulation.

• Characterizing fractures and faults is a challening task.

• Statistical methods are becoming increasingly popular for predictive analysis in exploration, production and delivery phases.

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Reservoir characterization

Thermal breakthrough predictions

Cool down of heat source

Insufficient heating of fluid

M AIN O BJECTIVES

• Explore statistical methods for reservoir characterization and prediction analysis:

1) Use nonparametric regression to infer well-to-well connectivity using tracer and temperature data.

2) Use clustering for reservoir characterization and predictions.

3) Apply Bayesian Evidential Learning (BEA) to directly predict production temperature and interwell connectivity of the reservoir.

4) Develop neural networks to infer well-to-well connectivity using injection rates and tracer data.

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W ELL -TO-W ELL CONNECTIVITY W ITH ACE

• The ACE algorithm is used to obtain well-to-well connectivity indices.

• The general form of the nonparametric regression:

𝜃 𝑌 =

𝑖=1

𝑝

𝜑𝑖 𝑋𝑖 + 𝜖

• The correlation is maximized by minimizing the error:

𝜖2 𝜃, 𝜑1, … , 𝜑𝑝 = 𝐸[(𝜃 𝑌 −

𝑖=1

𝑝

𝜑𝑖 𝑋𝑖 )2]

• The minimization is an iterative process:

𝜑𝑖 𝑋𝑖 = 𝐸[𝜃 𝑌 −

𝑖=1

𝑝

𝜑𝑖 𝑋𝑖 |𝑋𝑖]

𝜃 𝑌 = 𝐸[

𝑖=1

𝑝

𝜑𝑖 𝑋𝑖 |𝑌]/||𝐸[

𝑖=1

𝑝

𝜑𝑖 𝑋𝑖 |𝑌]||

• The well-to-well connectivity:

𝐼𝑖 =1

𝑛

𝑗=1

𝑛

𝜑𝑖 𝑋𝑖(𝑡𝑗)4

Y: target variableX1,..,Xp: predictor variablesθ, ϕ : transformation functionsϵ: regression errorn: number of time steps

A library of 800 DFNs*:

* The library of networks is modified from Magnusdottir, L. (2013).

W ELL -TO-W ELL CONNECTIVITY

Using tracer responses at producers as predictors in ACE

For 80.5% of the networks, the ACE connectivity was within ±0.05 of the tracer transit time connectivity.

5

P1 P2

P3

BAYESIAN EVIDENTIAL LEARNING (BEL)

• Instead of applying inverse analysis, a method for directly predicting subsurface flow responses is suggested (Satija et al. (2015), Li (2017), Scheidt et al. (2018)).

• The idea is to replace the iterative history matching process with Monte Carlo sampling of the prior parameter distribution.

• These parameters are used in forward simulations to obtain production data at wells.

• A statistical relationship between data and prediction variables is developed in combination with true observed data to make predictions.

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THE EVIDENTIAL FRAMEWORK

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Prior definition

Monte Carlo study

Multiple reservoir models

Flow simulation → prior data variables Flow simulation → prior prediction variables

Dimension reduction Dimension reduction

Observed production data

Prediction and uncertainty quantification

Exploration data, production data

Decision

*Modified from Li, L. (2017)

THE EVIDENTIAL FRAMEWORK

8

Prior definition

Monte Carlo study

Multiple reservoir models

Flow simulation → prior data variables Flow simulation → prior prediction variables

Dimension reduction Dimension reduction

Observed production data

Prediction and uncertainty quantification

Decision

Exploration data, production data

*Modified from Li, L. (2017), Hermans, T. et al. (2018)

THERMAL PREDICTIONS WITH BEL

CASE 1:

• Data variables: Tracer return at P1up to 500 days

• Prediction variables: Temperature response at P1 from 500-1000 days

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P1

4 20

THERMAL PREDICTIONS WITH BEL

10

P2

4 17

P3

534

THERMAL PREDICTIONS WITH BEL

11

P2

CASE 2:

• Data variables: Tracer at P2 & P3 up to 500 days

• Prediction variables: Temperature response at P1from 500-1000 days

P3

P1

9 20

CONNECTIVITY PREDICTIONS WITH BEL

12

P1 P2

P3

CASE 3:

• Data variables: Tracer at P1, P2 & P3 up to 500 days

• Prediction variable: Interwell connectivities

W ELL - TO-W ELL CONNECTIVITY WITH NEURAL

NETWORKS

• Sensitivity analysis of model structure to obtain interwell connectivities:

𝑆𝑚𝑛 =𝜕𝑌𝑚𝜕𝑋𝑛

• Sensitivity is evaluated over entire training set:

𝑆𝑚𝑛,𝑎𝑣𝑔 =σ𝑑=1𝐷 𝑆𝑚𝑛

2

𝐷

13

*Artun (2017), Demiryurek et al. (2008)

INJE

CT

OR

S

PR

OD

UC

ER

S

PALINPINON G EOTHERMAL F IELD

OK7 - ACE OK7 - NN

PN-1RD 0.0649 0.0873

PN-2RD 0.1234 0.1228

PN-3RD 0.0195 0.0068

PN-4RD 0.0195 0.0903

PN-5RD 0.1299 0.0837

PN-6RD 0.0649 0.0719

PN-7RD 0.0065 0.0568

PN-8RD 0.0195 0.0864

PN-9RD 0.1039 0.0931

Input & Output Data Multiple initializations of the neural network

Best fit

SYNTHETIC CASE

15

SUMMARY AND NEXT STEPS

• ACE regression• ACE regression produced well-to-well connectivity indices in good agreement with tracer

transit times. • Temperature data performed worse than tracer data.

• K-means clustering:• Clustering showed promise in grouping networks according to reservoir character and

behavior. • Determining optimal number of cluster is challenging.

• Bayesian Evidential Learning• BEL was able to make predictions based on tracer and temperature data. • With real data, the relationship between data and forecasting variables will become less

linear.

• Neural networks:• Show promise in producing well-to-well connectivities.• Can be used to develop proxy models for the reservoir for fast optimization of

injection/production schedules.16

NEXT STEPS

• Generate more realistic realizations of geothermal reservoirs.

• Add number of injection and production wells and increase complexity of injection schedules.

• Consider field scale data.

• Investigate other statistical methods that can be used for predictions and optimization of injection/production operations.

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Acknowledgements

• We would like to thank the Department of Energy Resources Engineering at Stanford University and Landsvirkjun, National Power Company of Iceland, for the financial support that made this research possible.

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References

Artun, E. (2017). Characterizing interwell connectivity in waterflooded reservoirs using data-driven and reduced-physics models: a comparative study. Neural Comput. & Applic, 28:1729-1743.

Demiryurek, U., Banaei-Kashani, F. & Shahabi, C. (2008). Neural-Network based Sensitivity Analysis for Injector-Producer Relationship Identification. In Proceedings of the SPE Intelligent Energy Conference and Exhibition, Amsterdam, The Netherlands, 25-27 February 2008.

Horne, R.N. & Szucs, P. (2007). Inferring well-to-well connectivity using nonparametric regression on well histories. In Proceedings of the 32nd Workshop on Geothermal Reservoir Engineering. Stanford University, 22-24 January 2007.

Li, L. (2017). A Bayesian Approach to Causal and Evidential Analysis for Uncertainty Quantification Throughout the Reservoir Forecasting Process. PhD thesis, Stanford University.

Magnusdottir, L. (2013). Fracture Characterization in Geothermal Reservoirs Using Time-lapse Electric Potential Data. PhD thesis, Stanford University.

Satija, A. & Caers, J. (2015). Direct forecasting of subsurface flow response from non-linear dynamic data by linear least-squares in canonical functional principal component space. Advance in Water Resources, 77:69-81.

Scheidt, C., Li, L. & Caers, J. (2018). Quantifying Uncertainty in Subsurface Systems. New York, NY: American Geophysical Union, Wiley.

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20

W ELL -TO-W ELL CONNECTIVITY

Using thermal responses at producers as predictors in ACE

For 58.3% of the networks, the ACE connectivity was within ±0.05 of the tracer transit time connectivity.

21

P1 P2

P3

RESERVOIR CHARACTERIZATION USING CLUSTER

ANALYSIS

• Exploratory capabilities of unsupervised learning can potentially be helpful for reservoir characterization.

• K-means clustering is used to group fracture networks and hence infer about the character of the reservoir as well as thermal behavior.

• K-means minimizes variation within a cluster as much as possible:

min𝐶1,…,𝐶𝐾

𝑘=1

𝐾

𝑊(𝐶𝑘)

𝑊 𝐶𝑘 =1

𝐶𝑘

𝑖,𝑖′∈𝐶𝑘

𝑗=1

𝑝

(𝑥𝑖𝑗 − 𝑥𝑖′𝑗)2

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K: number of clustersCk: cluster number kW(Ck): distance measurep: number of featuresx: features

CLUSTER ANALYSIS – EXAMPLE 1

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P1 P2

CLUSTER ANALYSIS – EXAMPLE 2

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P1 P2

P3