Improving the worthiness of the Elder problem as a benchmark for buoyancy driven convection models

Transcription

1 Improving the worthiness of the Elder problem as a benhmark for buoyany driven onvetion models Maarten van Reeuwijk, Simon A. Mathias The Department of Civil and Environmental Engineering Imperial College London ( m.vanreeuwijk@imperial.a.uk) Craig T. Simmons, James D. Ward Shool of Chemistry, Physis and Earth Sienes Flinders University, Australia This work was partially funded through the WorleyParsons EoNomisTM initiative. 8 Virtual Conferene on Climate Change and CO Storage Imperial College London and Seond Nature

2 Abstrat An important trapping mehanism assoiated with the geosequestration of CO is that of dissolution into the formation water. Although superritial CO is signifiantly less dense than water, eperimental data reported in the literature show that the density of an aqueous solution of CO ould be slightly greater. Under normal situations, the transfer of gas to solution is largely ontrolled by the relatively slow proess of moleular diffusion. However, the presene of variable densities an trigger off gravity instabilities leading to muh larger-sale onvetion proesses. Suh proesses an potentially enhane rates of dissolution by an order of magnitude. Consequently there is a need for future performane assessment models to inorporate buoyany driven onvetion (BDC). A major issue assoiated with BDC models is that of grid onvergene when benhmarking to the Elder problem. The Elder problem originates from a heat onvetion eperiment whereby a retangular Hele-Shaw ell was heated over the entral half of its base. A quarter of the way through the eperiment, Elder (967) observed si plumes, with four narrow plumes in the enter and two larger plumes at the edges. As the eperiment progressed, only four plumes remained. The issue is that depending on the grid resolution used when seeking to model this problem, modelers have found that different shemes yield steady states with either one, two or three plumes. The aim of this paper is to larify and irumvent the issue of multiple steady state solutions in the Elder problem using a pseudospetral method.

3 Shemati diagram of the problem H H H = = = H z = 4H Elder' s eperiment is haraterized by a Rayleigh number, αρ kg( )H Ra = μd E = α = ρ d'/dρ [ML ] ( ) [ML ] is the onentration differene -3 ρ [ML ] is the desnsity when ' = H [L] is the domain thikness k [L ] is permeability - g [LT ] is gravitational aeleration μ [ML D E [L - T T - - ] is the dynami visosity ] is the effetive diffusion oeffiient

4 The multiple steady states Depending on the numerial model and grid-resolution, the stable steady states S, S, and S 3 are observed. z/h.5 Single plume, S 3 4 /H Double plume, S Do these ambiguities render numerial preditions useless? z/h /H Triple plume, S 3 z/h /H

5 A pseudospetral ode for simulating buoyany driven flow Governing equations: where ψ is the streamfuntion and is the onentration. These equations are solved with a pseudospetral method whih employs sine-and osine-series in the horizontal () diretion and Chebyshev polynomials in the vertial (z) diretion. The use of a pseudospetral method avoids all trunation error assoiated with differentiation. Ra z z z t y + = + = + ψ ψ ψ ψ

6 Bifuration diagram S S S 3 Nu 3 4 Ra

7 Bifuration diagram - observations At Ra = 4 there are three stable steady state solutions; The higher states S and S 3 ome into eistene via a fold-bifuration (Johannsen, 3); Below Ra = 76, there is only one stable steady state. Hene, the ambiguities are physial rather than numerial; at Ra = 4, three stable steady states oeist. If Ra is lowered to 6, there will be only one stable steady state. We all this the Low Rayleigh Number Elder Problem.

8 Low Rayleigh Number Elder Problem (Ra = 6) Ra = 6 and t =.3 Ra = 6 and t =.3 (e) Ra = 6 and t =.3 Ra = 6 and t =.3 (g).5 FEFLOW Ra = 6 and t =. Pseudo spetral FEFLOW 895 (f) Ra = 6 and t =. Ra = 6 and t = Ra = 6 and t = (h) A omparison of the pseudospetral method with the ommerial groundwaterflow simulation software FEFLOW shows eellent agreement, regardless of the initial onditions employed.

9 Conlusions The aim of this paper is to larify and irumvent the issue of multiple steady state solutions in the Elder problem. A pseudospetral method was used to produe a bifuration diagram for < Ra < 4 whih is free of spatial disretization error. The results onfirm that the multiple steady states are indeed an intrinsi harateristi of the Elder problem. The eistene of multiple steady states makes the original Ra = 4 Elder problem unsuitable for benhmarking numerial models. To avoid the multiple steady states, we propose a benhmark at Ra = 6. Look out for: M. van Reeuwijk, S. A. Mathias, C. T. Simmons, J. D. Ward (9) Insights from a Insights from a pseudospetral approah to the Elder problem. Water Resour. Res.