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)
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
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)
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)
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)
The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO Examples of model outputs and observations (1/2) BHP_P7 BHP_I2 2400 2700 2200 2600 2000 2500 1800 2400 1600 1400 2300 1200 2200 1000 2100 800 600 2000 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 QW_P14 BHP_P13 1600 2400 2300 1400 2200 1200 2100 1000 2000 800 1900 600 1800 400 1700 200 1600 0 1500 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 QO_P15 BHP_I1 2000 2700 BHP = pressure at the bottom of the hole; QW = water flow rate; QO = oil flow rate 2600 P = producer well; I = injection well; the numbers index the wells 1500 2500 2400 1000 2300 2200 500 2100 0 2000 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 QO_P19 QW_P12 2500 2000 2000 1500 1500 1000 1000 500 500 0 0 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000
BHP_P7 BHP_I2 2400 2700 2200 2600 2000 2500 1800 2400 1600 1400 2300 1200 2200 1000 2100 800 600 2000 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 QW_P14 BHP_P13 1600 2400 2300 1400 2200 1200 2100 1000 2000 800 The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO 1900 600 1800 400 1700 Examples of model outputs and observations (2/2) 200 1600 0 1500 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 QO_P15 BHP_I1 2000 2700 2600 1500 2500 2400 1000 2300 2200 500 2100 0 2000 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 QO_P19 QW_P12 2500 2000 2000 1500 1500 1000 1000 500 500 0 0 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 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
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 m j , s the model forecasts and by y s the observed measurements, s � t − 1, that occurred prior to a given step t We pick weights w j , t based on this past and aggregate the forecasts 104 � � y t = w j , t m j , t j =1 which we later compare to the observed measurement y t Algorithmic question: how to pick the weights? Theoretical question: what guarantees of performance?
The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO Exponentially weighted averages (EWA): learning parameter η > 0, � � t − 1 � ( y s − m j , s ) 2 exp − η s =1 w j , t = � � . t − 1 K � � ( y s − m k , s ) 2 exp − η k =1 s =1 Ridge regression: regularization factor λ > 0, 2 � � K t − 1 K � � � v 2 ( w 1 , t , . . . , w K , t ) ∈ arg min j + ˆ λ y s − v j m j , s ( v 1 ,..., v K ) ∈ R K j =1 s =1 j =1 � K � � � v j � Lasso regression: replace the regularization above by λ j =1
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 m j , t and observed production characteristics y t , RMSE of algorithm � RMSE of best model + small“regret” � � � � T T � � � � � � 2 � � T − 1 / 4 � � 1 � 1 ( m j , t − y t ) 2 + O y t − y t ˆ min T T j =1 ,..., 104 t =1 t =1 References: several papers of the 90s and early 2000s; see the monograph by Cesa-Bianchi and Lugosi, 2006
The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO Aggregated production forecasts with EWA (1/2) BHP_P7 BHP_I2 2400 2700 2200 2600 2000 2500 1800 2400 1600 1400 2300 1200 2200 1000 2100 800 600 2000 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 QW_P14 BHP_P13 1600 2400 2300 1400 2200 1200 2100 1000 2000 800 1900 600 1800 400 1700 200 1600 0 1500 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 QO_P15 BHP_I1 2000 2700 BHP = pressure at the bottom of the hole; QW = water flow rate; QO = oil flow rate 2600 P = producer well; I = injection well; the numbers index the wells 1500 2500 2400 1000 2300 2200 500 2100 0 2000 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 QO_P19 QW_P12 2500 2000 2000 1500 1500 1000 1000 500 500 0 0 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000
BHP_P7 BHP_I2 2400 2700 2200 2600 2000 2500 1800 2400 1600 1400 2300 1200 2200 1000 2100 800 600 2000 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 QW_P14 BHP_P13 1600 2400 2300 1400 2200 1200 2100 1000 2000 800 The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO 1900 600 1800 400 1700 Aggregated production forecasts with EWA (2/2) 200 1600 0 1500 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 QO_P15 BHP_I1 2000 2700 2600 1500 2500 2400 1000 2300 2200 500 2100 0 2000 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 QO_P19 QW_P12 2500 2000 2000 1500 1500 1000 1000 500 500 0 0 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 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
The problem at hand How to combine the outputs Interval forecasts IRSDI-PGMO Aggregated production forecasts with EWA (zooming in) QO_P19 QW_P14 1900 1500 1800 1400 1700 1300 1600 1200 1500 1100 1400 1000 1300 900 1200 1100 800 1800 2000 2200 2400 2600 2800 2000 2200 2400 2600 2800 3000 3200 3400 BHP_P13 QO_P15 2100 2000 1900 2050 1800 2000 1700 1950 1600 1900 1500 1850 1400 1800 1300 1750 1200 1700 1100 500 1000 1500 2000 2500 200 400 600 800 1000 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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