Optimisation under Uncertainty with Stochastic PDEs for the History Matching Problem in Reservoir Engineering

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Optimisation under Uncertainty with Stochastic PDEs for the History Matching Problem in Reservoir Engineering Hermann G. Matthies Technische Universität Braunschweig wire@tu-bs.de http://www.wire.tu-bs.de

Overall Problem 2 Motivation: Computational tools for uncertainty quantification Situation: Geostatistical model with stochastic uncertainty and physical/ mathematical flow model Identification: History matching of well production data Goal: Predict well production and uncertainty Reservoir Long Term Goal: Optimal control of well production under uncertainty

Physical / Mathematical / Stochastic Model 3 Physical: multiphase flow in porous media (Darcy, Buckley-Leverett,...) Mathematical: coupled PDEs, conservation and transport/constitutive laws C t w + A(w) = g Stochastic: Prior geostatistic model of data and stochastic sources/sinks uncertainty. Simple Example: Darcy flow with stochastic coefficients c t u (κ u) = f Goal is the prediction of functional of future state u and its uncertainty. Explicit modelling and quantificaton of uncertainty. System identification under prior uncertainty and observed data. Predict future state and quantify uncertainty given posteriori model. Long term goal: Optimal Control of well yields.

Partly Open Problems 4 General Formulation: Coherent stochastic forward / inverse problem formulation, adequate type of identfication, geostatistical (non-gaussian) models. Forward Problem: Fast SPDE solvers based on stochastic Galerkin methods, much faster than Monte Carlo. Inverse Problem: Efficient solvers for nonlinear stochastic problem. Error Control: combined ( spatial & stochastic) error estimates and adaptivity. Uncertainty: Efficient quantification, high-dimensional integration. Software: Modular, independent, software management conformity. Long Term: Optimal learning / adaptive control.

Novel SPDE Techniques 5 General: Functional analytic formulation of stochastic problem, white noise analysis, Malliavin calculus. Solver: Stochastic Galerkin via polynomial chaos, tensor product solver, combination with fast deterministic solvers, low-rank approximation, Karhunen- Loève (POD, SVD), model reduction, efficient parallelisation. Error Control: Goal oriented, dual weighted residual (DWR), adjoint methods. Uncertainty: High-dimensional integration, sparse grids Smolyak methods. Other: Standard state of the art techniques, FEM/FVM & FDM spatial/ temporal discretisation, fast (multilevel) deterministic solvers. 4 3 2 1 1.5.5 x Mean.4.3.2.1 1.5.5 x 1.5.5 y 1 1 1.5.5 y 1 1 Standard Deviation.8.6.4.2 1.5.5 x 1.5.5 y 1 1 Pr{u(x) > 8}

System Identification Approaches 6 Approaches for system identification Maximum Likelihood: find most likely point. Bayesian like update of prior information based on observation stochastic posterior. Least Squares in equilibrium equation. Constitutive relation error CRE Separate certain (conservation: c u t + q = f) from uncertain assumptions (CR: q = κ u). Realisation Kalman filter EnKF Ensemble (via Monte Carlo) Kalman filtering.

Problem Sequence 7 Sequence of problems with increasing difficulty: linear diffusion: Darcy flow with stochastic conductivity stochastic non-gaussian velocity field. diffusion / advection: Solute transport (single phase) with stochastic velocity and (anomalous) dispersion. non-linear multi-phase flow, water air. Simulation NAPL multi-phase flow oil gas water...

Software Framework 8 Loose software coupling: Modular, independent parts. Software engineering: Components communicate via middleware. Tight numerical coupling: Integration of components into overall numerical scheme. Middleware: Component Template Library (CTL), allows distributed / coarse grain parallelisation. Optimiser Platon FEM ParaFep Eigenvalue (P)Arpack Communication Middleware CTL Component Template Library (MPI, PVM, TCP IP, etc...) Software KLE PCE StoFEL KLE StoFEL PCE Tensor Product Solver StoFEL PS

Related Collaboration / Work 9 Numerical / stochastic upscaling: Helmig IWS Uni Stuttgart; Cirpka ETH Zürich; Kulasiri Lincoln University, Christchurch. Stochastic material properties, error estimates: Ibrahimbegović, Ladevèze LMT ENS Cachan/ Paris; Villon UTC Compiegne; Wriggers IBNM Uni Hannover; Carstensen HU Berlin. Numerics for SPDEs: Schwab ETH Zürich; Tempone Florida State University; Babuška U of Texas, Austin; Burrage U of Queensland, Brisbane. System Identification: Faßbender. Coupled Problems: Ohayon CNAM, Paris. Software framework for history matching: Scandpower, Hamburg. Please visit poster early, I have to teach tomorrow morning.

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