A Parallel BOA-PSO Hybrid Algorithm for History Matching Alan Reynolds, Asaad Abdollahzadeh, David Corne, Mike Christie, Brian Davies and Glyn Williams We’ll start by describing the motivation for history matching, and what history matching is.

Development Optimization: An Overview We have access to a reservoir model and a reservoir simulator. We wish to optimize a plan for development of the reservoir. Can we apply a metaheuristic to the task of development optimization, using the simulator and the model to evaluate each solution? Well, the answer is “yes but…”. If the model accurately describes what is going on underground, and if the simulator is also accurate, then development optimization would simply be a fairly typical optimization problem – albeit one with an expensive evaluation function. However, the main problem is that there must be considerable uncertainty in the reservoir model – subsurface geology cannot be directly observed.

History Matching: An Overview We have access to: A reservoir model, with considerable uncertainties; A reservoir simulator; Pressure, production and saturation data collected during the lifetime of the reservoir. We wish to adjust the values of the reservoir model parameters, in order to produce alternative, improved reservoir models. Model quality is determined via simulation and comparison of the results with the collected data. So a model is evaluated according to a misfit measure and a metaheuristic can be used to optimize the model. In this paper we focus solely on minimizing misfit. Note though, that in reality we actually want to find a range of diverse solutions, with misfits providing a measure of the likelihood of each of the models. The set of models can then used not only to make predictions but also to give an idea of the uncertainty surrounding the predictions.

History Matching and Dev. Optimization Production Data (Rates) This is an optimization process History Dataset Other Obsvtns (PLTs, RFTs) Accuracy Criteria History Matching Static Data (Structure, Rocks) Uncertainties Multiple Models Run Model Output Filter Models Selected Models Formal & Informal Uncertainties ? Regenerate This is an optimization process too Economic Model Development Planning Dave Output (per model) w/ uncertainties Multiple Plans Chosen development plan w/uncertainties Development rules and constraints Run Rank & Risk Results Regenerate

PUNQ-S3 A synthetic problem based on a real field example. Pressure, volume, temperature and aquifer data are taken from the real-world model, while porosity and permeability fields for the ‘truth’ case were generated randomly, but in a manner so as to be as consistent as possible with the geological model. Simulation is used to produce the production data (bottom hole pressure, water cut, gas-oil ratio), after which Gaussian noise is added to simulate measurement error. Simulation model contains 2660 (19x28x5) grid blocks, of which 1761 are active, and 6 production wells. This make the model reasonable in size and runtime and suitable for experimentation with algorithm parameters. The field is structurally bounded to the east and south by a fault, while the link to a strong aquifer to the north and west means that no injection wells are required. The field initially has a small gas cap at the centre of the structure, and production wells are located around this gas cap. We optimize the value of 45 parameters – porosity values in 5 layers, each of which are divided into 9 regions. Permeability values are calculated using the known correlation between porosity and permeability.

Koma A real world problem, with real world data. 32,000 active blocks in 57 layers – so simulation takes a lot longer. (1000 simulations in 24 hours rather than 3000 in about 30 minutes, using 20 nodes on a Beowulf cluster). Colours show ratio of net reservoir rock to total rock. 3 production wells (in green) and 3 injectors. We have selected 54 parameters to be optimized - 16 pore volume multipliers, 8 permeability multipliers, 21 inter-region transmissibility multipliers, 8 fault transmissibility multipliers, and 1 aquifer volume multiplier.

Estimation of Distribution Algorithms Success of genetic algorithms is sometimes attributed to the building block hypothesis. However, in practice, genetic operators frequently destroy the building blocks. An alternative approach: explicitly identify high quality building blocks. Estimation of distribution algorithms (EDAs) create an explicit model of the location of good solutions. The model is used to generate new solutions. New good solutions are used to update the model. The population implicitly contains information about what building blocks seem to produce better quality results.

UMDA The Univariate Marginal Distribution Algorithm is one of the most basic EDAs. Consider a problem with solutions consisting of 5 bits. Rather than evolve solutions, UMDA evolves a model consisting of 5 probability values. Each value is the probability that, in a good solution, the corresponding bit is (or should be) set. Initially, each value is set to 0.5. Here we see the operation of one iteration of UMDA.

BOA Bayesian Optimization Algorithm. Note that each iteration requires the optimization of the Bayesian network structure, which can take some time. However, this is easily outweighed by the amount of time required to perform reservoir simulations. Indeed, the cost of reservoir simulations is what led us to use more sophisticated but computation heavy optimization algorithms. Results using BOA showed a improvement over a highly tuned genetic algorithm used in industry. Similar results could be obtained using only two thirds of the number of solution evaluations.

PSO 𝑣𝑖𝑗 ← 𝑤𝑣𝑖𝑗 + 𝛼𝑟1 (𝑝𝑖𝑗 − 𝑥𝑖𝑗) + 𝛽𝑟2 (𝑔𝑗 − 𝑥𝑖𝑗) 𝑣𝑖𝑗 ← min (𝑣𝑖𝑗, 𝑉𝑚𝑎𝑥,𝑗) 𝑣𝑖𝑗 ← max (𝑣𝑖𝑗 ,−𝑉𝑚𝑎𝑥,𝑗) 𝑥𝑖𝑗 ← 𝑥𝑖𝑗 + 𝑣𝑖𝑗 Particle Swarm Optimization. PSO has also been shown to outperform the industry used GA. Indeed, it also outperforms BOA.

BOA-PSO hybrid Initial BOA population generated by: initializing a swarm of, for example, 50 particles at random; running the PSO until the initial population of, for example, 500 solutions has been generated. Subsequent ‘child’ populations produced by: generating, for example, 5 seed solutions using the Bayesian network; injecting the 5 seed solutions into the swarm; continuing PSO until the child population (of say 500 solutions) is full. Why look at hybrid algorithms for history matching? The focus of our project was the application of EDAs to history matching, yet the relatively simple PSO produced better results. We wondered whether the strengths of the two algorithms could be combined. Also, the trade-off between getting a very good solution according to misfit and getting many, not quite so good solutions, suggested combining two algorithms where one provides the diversity while the other provides the drive toward the best solutions. We wanted to combine the two algorithms so that this trade-off could be easily controlled – we wanted to be able to have 100% PSO, 0% BOA, or 0% PSO, 100% BOA, or any point in between. The algorithm is parallelized synchronously for the BOA component, and asynchronously for the PSO.

Results: PUNQ-S3 Population size: 200. Number of parents and number of children: 100. Swarm size: 50 Fifty runs per setting.

Results: Koma Based on 5 runs each, due to the much greater time requirements of the reservoir simulation.

Conclusions and Further Research BOA-PSO hybrid clearly outperforms BOA on its own, on both problems. On PUNQ-S3, results using the hybrid are marginally better than those obtained by PSO. The hybrid algorithm continues to outperform PSO on the larger, real-world instance (Koma). There is plenty of room for experimentation with and improvement to the hybrid algorithm. And since BOA has already been shown to outperform the commercialized GA used in industry, this is a good result. Though not with any statistical significance – there is enough to warrant further exploration, but little more. In particular, a method that is less disruptive to the operation of the PSO would have BOA suggest new or additional ‘targets’ for the PSO, rather than replacing solutions. Alternatively, we could use DE rather than PSO. (DE has already been shown to be a promising approach.)