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Robust model aggregation for production forecasting of oil and gas - - PowerPoint PPT Presentation

Aug 21, 2023 249 likes •460 views

The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO Robust model aggregation for production forecasting of oil and gas Gilles Stoltz CNRS HEC Paris Joint work with Rapha el Deswarte (Ecole Polytechnique), S

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Robust model aggregation

for production forecasting of oil and gas Gilles Stoltz

CNRS — HEC Paris

Joint work with Rapha¨ el Deswarte (Ecole Polytechnique), S´ ebastien Da Veiga (Safran), V´ eronique Gervais (IFPEN)

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Problem: production forecasting of oil and gas

Keywords and objectives: Lightening the computational burden of fluid-flow simulations by performing history-matching on the outputs of fixed models rather than updating candidate models with many parameters

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

The Brugge field (synthetic but realistic data)

Reference: Peters et al. (2010), SPE 119094

Can be decomposed into millions of grid blocks, in which petrophysical properties are unknown (= a model)

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Classical approach: Fluid-flow equations (and simulators) relate – the production characteristics of the field (pressure, oil and water rates, etc.) over time – to the model (to the petrophysical properties) One may thus learn the model based on – estimates of the petrophysical properties (using some past measurements) – constraints of closeness of their associated production characteristics to those actually observed over time This is computationally heavy: At each time step, many fluid-flow simulations must be performed (many models are tested)

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Our approach: The Brugge data set comes with 104 geological models (their petrophysical properties were chosen in some way) We reweigh their production forecasts over time depending on past performance That is, we perform history-matching on the outputs of the models, not on their inputs Advantages and disadvantages – Computationally very efficient – Theoretical guarantees of good accuracy, without any stochastic assumption on the data – No construction of an underlying geological model (= no interpretation)

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Examples of model outputs and observations (1/2)

500 1000 1500 2000 2500 3000 3500 4000 600 800 1000 1200 1400 1600 1800 2000 2200 2400

BHP_P7

500 1000 1500 2000 2500 3000 3500 4000 2000 2100 2200 2300 2400 2500 2600 2700

BHP_I2

500 1000 1500 2000 2500 3000 3500 4000 200 400 600 800 1000 1200 1400 1600

QW_P14

500 1000 1500 2000 2500 3000 3500 4000 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

BHP_P13

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000

QO_P15

500 1000 1500 2000 2500 3000 3500 4000 2000 2100 2200 2300 2400 2500 2600 2700

BHP_I1

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000 2500

QO_P19

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000

QW_P12 BHP = pressure at the bottom of the hole; QW = water flow rate; QO = oil flow rate P = producer well; I = injection well; the numbers index the wells

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Examples of model outputs and observations (2/2)

500 1000 1500 2000 2500 3000 3500 4000 600 800 1000 1200 1400 1600 1800 2000 2200 2400

BHP_P7

500 1000 1500 2000 2500 3000 3500 4000 2000 2100 2200 2300 2400 2500 2600 2700

BHP_I2

500 1000 1500 2000 2500 3000 3500 4000 200 400 600 800 1000 1200 1400 1600

QW_P14

500 1000 1500 2000 2500 3000 3500 4000 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

BHP_P13

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000

QO_P15

500 1000 1500 2000 2500 3000 3500 4000 2000 2100 2200 2300 2400 2500 2600 2700

BHP_I1

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000 2500

QO_P19

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000

QW_P12 BHP = pressure at the bottom of the hole; QW = water flow rate; QO = oil flow rate P = producer well; I = injection well; the numbers index the wells

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

How to combine the outputs

For a given well and a given production characteristic: We denote by mj,s the model forecasts and by ys the observed measurements, s t − 1, that occurred prior to a given step t We pick weights wj,t based on this past and aggregate the forecasts

  • yt =

104

  • j=1

wj,tmj,t which we later compare to the observed measurement yt Algorithmic question: how to pick the weights? Theoretical question: what guarantees of performance?

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Exponentially weighted averages (EWA): learning parameter η > 0, wj,t = exp

  • −η

t−1

  • s=1

(ys − mj,s)2

  • K
  • k=1

exp

  • −η

t−1

  • s=1

(ys − mk,s)2 . Ridge regression: regularization factor λ > 0, (w1,t, . . . , wK,t) ∈ arg min

(v1,...,vK )∈RK

  λ

K

  • j=1

v2

j + t−1

  • s=1
  • ˆ

ys −

K

  • j=1

vj mj,s

  • 2

  Lasso regression: replace the regularization above by λ

K

  • j=1
  • vj
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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Performance guarantees for EWA and Ridge (not Lasso yet): – No stochastic modeling, guarantees for all individual sequences – Mimic the performance of (at least) the best model For all bounded sequences of forecasts mj,t and observed production characteristics yt, RMSE of algorithm RMSE of best model + small“regret”

  • 1

T

T

  • t=1
  • ˆ

yt − yt 2 min

j=1,...,104

  • 1

T

T

  • t=1

(mj,t − yt)2 + O

  • T −1/4

References: several papers of the 90s and early 2000s; see the monograph by Cesa-Bianchi and Lugosi, 2006

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Aggregated production forecasts with EWA (1/2)

500 1000 1500 2000 2500 3000 3500 4000 600 800 1000 1200 1400 1600 1800 2000 2200 2400

BHP_P7

500 1000 1500 2000 2500 3000 3500 4000 2000 2100 2200 2300 2400 2500 2600 2700

BHP_I2

500 1000 1500 2000 2500 3000 3500 4000 200 400 600 800 1000 1200 1400 1600

QW_P14

500 1000 1500 2000 2500 3000 3500 4000 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

BHP_P13

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000

QO_P15

500 1000 1500 2000 2500 3000 3500 4000 2000 2100 2200 2300 2400 2500 2600 2700

BHP_I1

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000 2500

QO_P19

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000

QW_P12 BHP = pressure at the bottom of the hole; QW = water flow rate; QO = oil flow rate P = producer well; I = injection well; the numbers index the wells

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Aggregated production forecasts with EWA (2/2)

500 1000 1500 2000 2500 3000 3500 4000 600 800 1000 1200 1400 1600 1800 2000 2200 2400

BHP_P7

500 1000 1500 2000 2500 3000 3500 4000 2000 2100 2200 2300 2400 2500 2600 2700

BHP_I2

500 1000 1500 2000 2500 3000 3500 4000 200 400 600 800 1000 1200 1400 1600

QW_P14

500 1000 1500 2000 2500 3000 3500 4000 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

BHP_P13

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000

QO_P15

500 1000 1500 2000 2500 3000 3500 4000 2000 2100 2200 2300 2400 2500 2600 2700

BHP_I1

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000 2500

QO_P19

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000

QW_P12 BHP = pressure at the bottom of the hole; QW = water flow rate; QO = oil flow rate P = producer well; I = injection well; the numbers index the wells

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Aggregated production forecasts with EWA (zooming in)

1800 2000 2200 2400 2600 2800 1100 1200 1300 1400 1500 1600 1700 1800 1900

QO_P19

2000 2200 2400 2600 2800 3000 3200 3400 800 900 1000 1100 1200 1300 1400 1500

QW_P14

500 1000 1500 2000 2500 1700 1750 1800 1850 1900 1950 2000 2050 2100

BHP_P13

200 400 600 800 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

QO_P15 BHP = pressure at the bottom of the hole; QW = water flow rate; QO = oil flow rate P = producer well; I = injection well; the numbers index the wells

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Overview of the performance of EWA (in red or blue) versus the best model for the well–production characteristic pair

2 1 4 3 5 10 7 8 9 6 numero du puits (propriete BHP_I) 5 10 15 20 RMSE

rmse : Online EWA/best expert(vert) BHP_I

9 19 2 14 6 7 1311 8 12 3 1817 4 10 1 2016 5 15 numero du puits (propriete BHP_P) 50 100 150 200 250 RMSE

rmse : Online EWA/best expert(vert) BHP_P

8 3 6 7 1 4 2 101813 9 16111720 5 12141915 numero du puits (propriete QO_P) 20 40 60 80 100 120 RMSE

rmse : Online EWA/best expert(vert) QOP

1 4 3 7 8 6 2 9 101816111317 5 1214192015 numero du puits (propriete QW_P) 100 200 300 400 500 600 RMSE

rmse : Online EWA/best expert(vert) QW_P

BHP = pressure at the bottom of the hole; QW = water flow rate; QO = oil flow rate P = producer well; I = injection well; the numbers index the wells

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Can we provide interval forecasts?

500 1000 1500 2000 2500 3000 3500 500 1000 1500 2000

Standard request (and offer) with stochastic modelings. Not so clear within the theory of individual sequences...

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Our individual-sequences approach for interval forecasts

  • 1. On the first part of the data set, t = 1, . . . , T,

when one-step ahead aggregated forecasts are provided

– use the algorithms as explained above

  • 2. On the second part of the data set, t = T + 1, T + 2, . . .

when interval forecasts are to be provided

– The models still provide forecasts mj,T+s for s 1 – Consider all possible (bounded) continuations y ′

T+1, y ′ T+2, . . .

  • f the observed characteristics

– Deduce a series of aggregated forecasts y ′

T+1,

y ′

T+2, . . .

– Obtain the intervals as the convex hulls of all these possible aggregated forecasts – Possibly enlarge them to take into account some noise (observed characteristics are measured with noise)

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Interval forecasts with Ridge (1/3)

500 1000 1500 2000 2500 3000 3500 4000 2000 2100 2200 2300 2400 2500 2600 2700

BHP_I2

500 1000 1500 2000 2500 3000 3500 4000 2000 2100 2200 2300 2400 2500 2600 2700

BHP_I5

500 1000 1500 2000 2500 3000 3500 4000 1900 2000 2100 2200 2300 2400 2500 2600 2700

BHP_I10

500 1000 1500 2000 2500 3000 3500 4000 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

BHP_P2

500 1000 1500 2000 2500 3000 3500 4000 1000 1200 1400 1600 1800 2000 2200 2400

BHP_P6

500 1000 1500 2000 2500 3000 3500 4000 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500

BHP_P12

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000 2500

QO_P10

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000 2500

QO_P11

BHP = pressure at the bottom of the hole; QW = water flow rate; QO = oil flow rate P = producer well; I = injection well; the numbers index the wells

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Interval forecasts with Ridge (2/3)

500 1000 1500 2000 2500 3000 3500 4000 2000 2100 2200 2300 2400 2500 2600 2700

BHP_I2

500 1000 1500 2000 2500 3000 3500 4000 2000 2100 2200 2300 2400 2500 2600 2700

BHP_I5

500 1000 1500 2000 2500 3000 3500 4000 1900 2000 2100 2200 2300 2400 2500 2600 2700

BHP_I10

500 1000 1500 2000 2500 3000 3500 4000 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400

BHP_P2

500 1000 1500 2000 2500 3000 3500 4000 1000 1200 1400 1600 1800 2000 2200 2400

BHP_P6

500 1000 1500 2000 2500 3000 3500 4000 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500

BHP_P12

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000 2500

QO_P10

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000 2500

QO_P11

BHP = pressure at the bottom of the hole; QW = water flow rate; QO = oil flow rate P = producer well; I = injection well; the numbers index the wells

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

Interval forecasts with Ridge (3/3)

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000

QO_P15

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000

QO_P9

500 1000 1500 2000 2500 3000 3500 4000 100 200 300 400 500 600 700 800

QW_P10

500 1000 1500 2000 2500 3000 3500 4000 500 1000 1500 2000

QW_P20

BHP = pressure at the bottom of the hole; QW = water flow rate; QO = oil flow rate P = producer well; I = injection well; the numbers index the wells

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The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO

An announcement for those who like real-world machine learning!

PGMO /IRSDI: call for projects in industrial data science Team = academic members + industrial partner Funding = 10-15 kE, for one year Application = only 3-4 pages; deadline at May 14