Énergies renouvelables | Production éco-responsable | Transports innovants | Procédés éco-efficients | Ressources durables

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Comparison of numerical schemes for multiphase reactive transport

T. Faney, A. Michel, Q. L. Nguyen and L.Rouvray

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I. Introduction Motivation

References

Problem description

II. Formulation and Active Set Algorithm

III. Splitting Algorithm

IV. Results

V. Conclusion and Future Work

Outline

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Applications in Geosciences : CO2 Sequestration

Gas Production and Storage

EOR

Mineral Diagenesis

Other Applications : Batteries

Biological processes

I. Motivation

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I. Problem Description

System Sub-System Phase Species

Multi-Phase Multi-Species System

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I. Problem Description

Advection in Porous media

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I. Problem Description

Equilibrium chemical reactions For each reaction r

Closure Equations

Equilibrium reaction rates are unknown Rates elimination using chemical analysis

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Example System Water (H2O,OH-,H+,HCO3

-,Ca++) , Gas (CO2g), Calcite (CaCO3

s)

H2O = H+ + OH-

CO2g + H2O = HCO3- + H+

HCO3- + Ca++ = CaCO3

s + H+

Formula matrix A

13 equations (4 mass balance equations, 6 closure equations, 3 chemical equilibrium equations)

13 variables (P,𝝓𝒇, 𝝓𝒔,Sw,Sg,Ss,xH2O->CaCO3s)

I. Problem Description

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Discretization Global Implicit Approach (Implicit Euler)

Two-points finite volume scheme

Issue with phase appearance and disappearance Complementarity conditions

Active Set algorithm vs Semi-smooth Newton methods

New context variable I Ik = context of cell k

Equations and variables depend on Ik

Split active and non-active species, components and equations

II. Formulation and Active Set Algorithm

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Example continued Ik = {Gas, Calcite}

9 equations (4 mass balance equations, 5 closure equations, 0 chemical equilibrium equations) for cell k

9 variables (P,𝝓𝒇, 𝝓𝒔,Sg,Ss,xCO2g,xCaCO3s,NIH2O,NI

H+) for cell k

II. Formulation and Active Set Algorithm

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NA

NI

PA PI SI

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II. Formulation and Active Set Algorithm

Global non linear equation solver Each active variable is updated at the end of each Newton

iteration

Context variable also needs to be updated

Negative saturation phase disappearance

Equilibrium flash phase appearance

Equilibrium flash can be any 0D chemistry solver LMA / GEM formulation

External or Internal library

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III. Splitting Algorithm

Splitting occurs at different stages

Flow Reactive

Transport

Porosity

update

P, Sα njα

φ

tn tn+1

Transport Flash N P, Sα nj

α

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IV. Results – Active Set

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SHPCO2

case study

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IV. Results – Active Set

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IV. Results – Active Set

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IV. Results – Active Set

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IV. Results – Comparison

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SES Benchmark

Irina Sin and Jerome Corvisier, Ecole des Mines de Paris, Geosciences research center

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IV. Results – Comparison

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Simple benchmark

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IV. Results – Comparison

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T = 100 000 years

Active Set

Splitting (Non iterative)

Splitting (Iterative)

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IV. Conclusion and Future Work

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Active Set algorithm to model multi-phase multi-species flow in porous media

Comparison with splitting approach

Future work involves Non-Linear solver efficiency

Flash solver improvement

Semi-smooth Newton approach

Extension to kinetics and diffusion fluxes

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Thank you