Matematisk modellering av CO 2 -lagring MatMoRA

Transcription

1 Matematisk modellering av CO 2 -lagring MatMoRA H. K. Dahle, J. M. Nordbotten, K.-A. Lie University of Bergen Centre for Integrated Petroleum Research SINTEF ICT CLIMIT-dagene 2010, Oslo oktober

2 Personnel Institution Senior personnel Researchers/PhD/postdocs. UiB Helge K. Dahle Paulo Herrera (postdoc) Jan M. Nordbotten Leonid Vasilyev (PhD) Maria Elenius (PhD) Trine Mykkeltvedt (PhD) Elsa du Plessis (PhD) Uni Research Klaus Johannsen Martha Lien Karsten Pruess (adjoint) Geir T. Eigestad Ivar Aavatsmark Eirik Keilegavlen Jan Tveranger SINTEF ICT Knut-Andreas Lie Halvor M. Nilsen (postdoc) Meisam Ashraf (PhD) Ingeborg S. Ligaarden Stuttgart Princeton Rainer Helmig Holger Class Michael A. Celia Steering committee: Benedicte Kvalheim (Statoil), Martin Iding (Statoil), Jostein Haga (Norske Shell), Roger Bjørgan (SINTEF), Ivar Aavatsmark (Uni Research/UiB), Aage Stangeland (Forskningsrådet/Climit, observatør)

3 Motivation (...) the assessment of hazards and risks related to storage of CO 2 streams in geological formations may include a significant level of uncertainty. This uncertainty should be identified and, wherever possible, quantified (...) OSPAR Guidelines for Risk Assessment and Management of Storage of CO2 Streams in Geological Formations Det er behov for kontinuerlig kunnskapsoppbygging innen basisfagene geologi, geofysikk, geokjemi, geobiokjemi, bergmekanikk og strømning i porøse medier for å bedre forstå hvordan CO2 strømmer og påvirker bergartsystemer kjemisk og mekanisk. Numerisk simulering av CO2-strømning er et spesielt viktig verktøy innen geologisk lagring. På lang sikt er det viktig at slike metoder, foruten å simulere trykk og konsentrasjonsfordelinger, også kan simulere bergmekani- ske og geokjemiske prosesser. Main challenges: Length scales, time scales, and geological uncertainty

4 Geological Storage of CO 2 : Mathematical Modelling and Risk Analysis The aim is to develop quantitative tools to be used in risk assessment analysis of geological storage of CO 2. 1 Improved theoretical understanding through mathematical analysis 2 Develop fast simulation technology 3 Verification and validation of mathematical models and numerical formulations 4 Build confidence in modeling through benchmark studies 5 Develop methods and tools for risk assessment

5 Enhanced Dissolution through Convective Mixing High resolution simulation of migration and convection Parameters based on Svalbard benchmark Collaboration with Energy Resources Engineering Stanford

6 Effect of a Capillary Zone Velocity of brine: Linear stability analysis: Notice two-phase region. Growth rate σ vs. wave number k.

7 Upscaled Convective Mixing Preuss, et.al Hypothesis: Earlier onset larger dissolution rate.

8 Vertical Equilibrium (VE) Models Assume vertical equilibrium across the aquifer: p α = ρ α g

9 Vertical Equilibrium (VE) Models Assume vertical equilibrium across the aquifer: p α = ρ α g

10 Vertical Equilibrium: Upscaling Variables Fine-scale variables p n p w = p c (s n ), k rα = k rα (s n ), λ α = k rα µ α Coarse-scale variables Φ = 1 H K = 1 H H 0 H 0 K rα = K 1 H φdz, S α = 1 H ΦH k H dz, U α = 1 H H 0 0 H 0 φs α dz u H α dz k rα k H dz, Λ α = 1 µ α K rα

11 Vertical Equilibrium: Upscaling Equations Fine-scale φs α t + u α = q α u α = λ α k ( p α ρ α g)

12 Vertical Equilibrium: Upscaling Equations Assumptions u V α u H α p α = ρ α g p α = P α (x, y) ρ α g cos(θ)z Fine-scale φs α t + u α = q α u α = λ α k ( p α ρ α g)

13 Vertical Equilibrium: Upscaling Equations Assumptions u V α u H α p α = ρ α g p α = P α (x, y) ρ α g cos(θ)z Fine-scale φs α t + u α = q α u α = λ α k ( p α ρ α g) Reconstruction s n (z) = p 1 c [P c + ρg cos(θ)z] S n = S n (P c ) P c = P c (S n )

14 Vertical Equilibrium: Upscaling Equations Assumptions u V α u H α p α = ρ α g p α = P α (x, y) ρ α g cos(θ)z Fine-scale Coarse-scale φs α t + u α = q α u α = λ α k ( p α ρ α g) Reconstruction ΦS α + t U α = Q α U α = Λ α K [ ] P α + ρ α g sin(θ)e G s n (z) = p 1 c [P c + ρg cos(θ)z] S n = S n (P c ) P c = P c (S n )

15 Fine- and Coarse Scale Constitutive Relationships Capillary pressure Relative permeability 2 x Capillary pressure Fine Scale Capillary fringe pseudo Sharp interface pseudo Relative permeability Fine Scale Capillary fringe pseudo Sharp interface pseudo CO 2 Saturation CO 2 Saturation

16 Vertical Equilibrium: Dimensionless Groups Aspect ratio: R 1 L = H L kh Extent of vertical fringe (inverse Bond number): ɛ = k V (p c) ρg cos(θ)h Dimensionless time associated with capillary fringe: T f = T k V φ λ nλ w(p c) λ n + λ /(ɛh) 2 w Dimensionless times associated with vertical segregation and horizontal flow: ( ) T v = T λ nk V ρg cos(θ), T h = T λ nu φh φλ = T λ nk H ρg sin(θ) wl H φl H

17 Vertical equilibrium models are valid if (Yortsos 95): Vertically segregated flow if: Capillary fringe established if: R 1 L 1 T v 1 T f 1, and T h /T f 1 Sharp interface model applicable if: ɛ 1

18 Vertical Equilibrium Models: Effect of Capillary Fringe Nordbotten, Dahle, WRR 2010, CMWR 2010

19 Ex: Johansen Formation (Cross Section) z dim = 5, relative error: 0.74 z dim = 10, relative error: D: nz=5 3D: nz= Brine CO Brine CO z dim = 20, relative error: VE solution D: nz=20 vertikalmidlet Brine CO brine CO

20 Eksempel: Matching Data Using VE-model Reproducing seismic data using VE-simulator Seismiske data (2006) / 3D simulering (tough2) VE simulering Chadwick, Noy, Arts & Eiken: Latest time-lapse seismic data from Sleipner yield new insights into CO 2 -plume development, Energy Procedia (2009),

21 Summary The project has shown the importance of creating confidence in numerical modeling and simulation (and further need for verification and validation through benchmarking) In the project we have shown that Vertical Equilibrium models may be very efficient and capture the nonlinear dynamics of the migration of the CO 2 -plume better than traditional 3D reservoir-models To establish efficient and accurate numerical tools, there is a strong need to understand the relative importance of different physical processes