Reservoir Simulation: From Upscaling to Multiscale Methods Knut–Andreas Lie SINTEF ICT, Dept. Applied Mathematics http://www.math.sintef.no/GeoScale Multiscale Computational Science and Engineering, September 19–21, Trondheim, Norway Applied Mathematics 21/09/2007 1/47 Reservoir Simulation What and why? Reservoir simulation is the means by which a numerical model of the petrophysical characteristics of a hydrocarbon reservoir is used to analyze and predict fluid behavior in the reservoir over time. Reservoir simulation is used as a basis for decisions regarding development of reservoirs and management during production. To this end, one needs to predict reservoir performance from geological descriptions and constraints, fit geological descriptions to static and dynamic data, assess uncertainty in predictions, optimize production strategies, . . . Applied Mathematics 21/09/2007 2/47
Reservoir Simulation What are the challenges today? Reservoir modelling is a true multiscale discipline: Measurements and models on a large number of scales Large number of models Complex grids with a large number of parameters High degree of uncertainty . . . There is always a need for faster and more accurate simulators that use all available geological information Applied Mathematics 21/09/2007 3/47 Physical Scales in Porous Media Flow One cannot resolve them all at once The scales that impact fluid flow in oil reservoirs range from the micrometer scale of pores and pore channels via dm–m scale of well bores and laminae sediments to sedimentary structures that stretch across entire reservoirs. − → Applied Mathematics 21/09/2007 4/47
Physical Scales in Porous Media Flow Microscopic: the scale of individual sand grains Flow in individual pores between sand grains Applied Mathematics 21/09/2007 5/47 Physical Scales in Porous Media Flow Geological: the meter scale of layers, depositional beds, etc Porous sandstones often have repetitive layered structures, but faults and fractures caused by stresses in the rock disrupt flow patterns Applied Mathematics 21/09/2007 6/47
Physical Scales in Porous Media Flow Reservoir: the kilometer scale of sedimentary structures Applied Mathematics 21/09/2007 7/47 Physical Scales in Porous Media Flow Choosing a scale for modelling Applied Mathematics 21/09/2007 8/47
Geological Models The knowledge database in the oil company Geomodels: are articulations of the experts’ perception of the reservoir describe the reservoir geometry (horizons, faults, etc) give rock parameters (e.g., permeability K and porosity φ ) that determine the flow In the following: the term “geomodel” will designate a grid model where rock properties have been assigned to each cell Applied Mathematics 21/09/2007 9/47 Flow Simulation Model problem: incompressible, single phase Consider the following model problem Darcy’s law: v = − K ( ∇ p − ρg ∇ D ) , Mass balance: ∇ · v = q in Ω , Boundary conditions: v · n = 0 on ∂ Ω . The multiscale structure of porous media enters the equations through the absolute permeability K , which is a symmetric and positive definite tensor with uniform upper and lower bounds. We will refer to p as pressure and v as velocity. Applied Mathematics 21/09/2007 10/47
Flow Simulation The impact of rock properties Ex: Brent sequence Rock properties are used as parameters in flow models Permeability K spans many length scales and have multiscale structure max K / min K ∼ 10 3 –10 10 Details on all scales impact flow Tarbert Upper Ness Challenges: How much details should one use? Need for good linear solvers, preconditioners, etc. Applied Mathematics 21/09/2007 11/47 Flow Simulation Gap in resolution and model sizes Gap in resolution: High-resolution geomodels may have 10 6 − 10 10 cells Conventional simulators are capable of about 10 5 − 10 6 cells Traditional solution: upscaling of parameters Assume that u satisfies the elliptic PDE: � � −∇ K ( x ) ∇ u = f. ⇓ Upscaling amounts to finding a new field K ∗ (¯ x ) on a coarser grid such that K ∗ (¯ x ) ∇ u ∗ � = ¯ � −∇ f, u ∗ ∼ ¯ q ∗ ∼ ¯ u, q . Applied Mathematics 21/09/2007 12/47
Upscaling Geological Models Industry-standard methods How do we represent fine-scale heterogeneities on a coarse scale? Combinations of arithmetic, geometric, harmonic averaging � 1 /p � V a ( x ) p dx 1 Power averaging � | V | Equivalent permeabilities ( a ∗ xx = − Q x L x / ∆ P x ) Lx u=0 p=1 V’ V’ u=0 u=0 V V Ly p=0 p=1 u=0 p=0 Applied Mathematics 21/09/2007 13/47 Upscaling Geological Models Is it necessary and does one want to do it? There are many difficulties associated with upscaling Bottleneck in the workflow 220 22 200 Loss of details 20 180 18 160 16 Lack of robustness 140 14 120 12 Need for resampling for complex 100 10 80 8 grid models 60 6 40 4 Not obvious how to extend the 20 2 10 20 30 40 50 60 2 4 6 8 10 ideas to 3-phase flows Need for fine-scale computations? In the future: need for multiphysics on multiple scales? Applied Mathematics 21/09/2007 14/47
Fluid Simulations Directly on Geomodels Research vision: Direct simulation of complex grid models of highly heterogeneous and fractured porous media - a technology that bypasses the need for upscaling. Applications: Huge models, multiple realizations, prescreening, validation, optimization, data integration, .. To this end, we seek a methodology that incorporates small-scale effects into coarse-scale system; gives a detailed image of the flow pattern on the fine scale, without having to solve the full fine-scale system; is robust, conservative, accurate, and efficient. Applied Mathematics 21/09/2007 15/47 Multiscale Pressure Solvers Efficient flow solution on complex grids – without upscaling Basic idea: Upscaling and downscaling in one step Pressure varies smoothly and can be resolved on coarse grid Velocity with subgrid resolution Applied Mathematics 21/09/2007 16/47
From Upscaling to Multiscale Methods Standard upscaling: Multiscale method: ⇓ ⇓ Coarse grid blocks: Coarse grid blocks: ⇓ ⇓ Flow problems: Flow problems: Applied Mathematics 21/09/2007 17/47 From Upscaling to Multiscale Methods Standard upscaling: Multiscale method: ⇓ ⇑ ⇓ ⇑ Coarse grid blocks: Coarse grid blocks: ⇓ ⇑ ⇓ ⇑ Flow problems: Flow problems: Applied Mathematics 21/09/2007 18/47
The Multiscale Mixed Finite-Element Method Standard finite-element method (FEM): � � Piecewise polynomial approximation to pressure, l ∇ K ∇ p dx = lq dx Mixed finite-element methods (MFEM): Piecewise polynomial approximations to pressure and velocity � � � k − 1 v · u dx − k − 1 ρg ∇ D · u dx p ∇ · u dx = ∀ u ∈ U, Ω Ω Ω � � l ∇ · v dx = ql dx ∀ l ∈ V. Ω Ω Multiscale mixed finite-element method (MsMFEM): Velocity approximated in a (low-dimensional) space V ms designed to embody the impact of fine-scale structures. Applied Mathematics 21/09/2007 19/47 Multiscale Mixed Finite Elements Grids and basis functions Assume we are given a fine grid with permeability and porosity attached to each fine-grid block: T i T j We construct a coarse grid, and choose the discretisation spaces U and V ms such that: For each coarse block T i , there is a basis function φ i ∈ U . For each coarse edge Γ ij , there is a basis function ψ ij ∈ V ms . Applied Mathematics 21/09/2007 20/47
(Multiscale) Mixed Finite Elements Discretisation matrices (without hybridization) Saddle-point problem: � ψ i k − 1 ψ j dx, b ij = � B � � v � � f � Ω C = , � C T 0 p g c ij = φ j ∇ · ψ i dx Ω Basis φ j for pressure: equal one in cell j , zero otherwise Basis ψ i for velocity: 1.order Raviart–Thomas: Multiscale: Applied Mathematics 21/09/2007 21/47 Multiscale Mixed Finite Elements Basis for the velocity field Velocity basis function ψ ij : unit flow through Γ ij defined as � w i ( x ) , for x ∈ T i , ∇ · ψ ij = − w j ( x ) , for x ∈ T j , and no flow ψ ij · n = 0 on ∂ ( T i ∪ T j ). Global velocity: v = � ij v ij ψ ij , where v ij are (coarse-scale) coefficients. Applied Mathematics 21/09/2007 22/47
Multiscale Simulation versus Upscaling 10 th SPE Comparative Solution Project Producer A Injector Producer D t r e b a r T s s e Producer B N r e p p U Producer C Geomodel: 60 × 220 × 85 ≈ 1 , 1 million grid cells, max K x / min K x ≈ 10 7 , max K z / min K z ≈ 10 11 Simulation: 2000 days of production (2-phase flow) Commercial (finite-difference) solvers: incapable of running the whole model Applied Mathematics 21/09/2007 23/47 Multiscale Simulation versus Upscaling 10 th SPE Comparative Solution Project Upscaling results reported by industry 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 Watercut Watercut 0.5 0.5 0.4 0.4 0.3 0.3 Fine Grid TotalFinaElf 0.2 0.2 Geoquest Fine Grid Streamsim Landmark 0.1 0.1 Roxar Phillips Chevron Coats 10x20x10 0 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Time (days) Time (days) single-phase upscaling two-phase upscaling Applied Mathematics 21/09/2007 24/47
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