Hydrogeology Journal

, Volume 18, Issue 1, pp 5–23

Three-dimensional benchmark for variable-density flow and transport simulation: matching semi-analytic stability modes for steady unstable convection in an inclined porous box

Authors

    • US Geological Survey
  • Craig T. Simmons
    • Flinders University
  • Neville I. Robinson
    • Flinders University
Paper

DOI: 10.1007/s10040-009-0556-6

Cite this article as:
Voss, C.I., Simmons, C.T. & Robinson, N.I. Hydrogeol J (2010) 18: 5. doi:10.1007/s10040-009-0556-6

Abstract

This benchmark for three-dimensional (3D) numerical simulators of variable-density groundwater flow and solute or energy transport consists of matching simulation results with the semi-analytical solution for the transition from one steady-state convective mode to another in a porous box. Previous experimental and analytical studies of natural convective flow in an inclined porous layer have shown that there are a variety of convective modes possible depending on system parameters, geometry and inclination. In particular, there is a well-defined transition from the helicoidal mode consisting of downslope longitudinal rolls superimposed upon an upslope unicellular roll to a mode consisting of purely an upslope unicellular roll. Three-dimensional benchmarks for variable-density simulators are currently (2009) lacking and comparison of simulation results with this transition locus provides an unambiguous means to test the ability of such simulators to represent steady-state unstable 3D variable-density physics.

Keywords

Analytical solutionsGroundwater density/viscosityNumerical modelingBenchmarkVariable-density groundwater

Banc d′essai tri-dimensionnel pour la simulation d′écoulement d′un flux de densité variable: comparaison des modes de convexion uniforme instable dans une boîte poreuse inclinée

Résumé

Ce banc d′essai pour simulation numérique tridimensionnelle (3D) d′un flot d′écoulement souterrain de densité ou d′énergie variable permet de comparer les résultats semi-analytiques de transition d′un mode convectif en régime permanent à un autre, dans une boîte poreuse. Des études expérimentales et analytiques antérieures de flux convectif libre dans un milieu poreux incliné ont montré qu′il existe différents modes de convection possibles dépendant des paramètres du système, géométrie et inclinaison. En particulier, il existe une transition nette entre le mode hélicoïdal, consistant en écoulements longitudinaux descendants surimposés à un flux unicellulaire ascendant et un mode d′écoulement unicellulaire purement ascendant. Des bancs d′essai tri-dimensionnels pour simulations d′écoulements de densité variable manquent actuellement (2009) et la comparaison de simulations avec ce dispositif de transition montre clairement la capacité de tels simulateurs à représenter en 3 dimensions la physique des phénomènes instables en régime permanent.

Estándar de comparación tridimensional para la simulación de flujo de densidad variable y transporte: coincidencia de modo de estabilidad semianalítica para una convección estacionaria inestable en una caja porosa inclinada

Resumen

Este estándar de comparación para simuladores numéricos tridimensionales (3D) de flujo de agua subterránea de densidad variable y transporte de solutos o energía consiste en comparar los resultados de la simulación con la solución semianalítica para la transición de un modo convectivo de estado estacionario a otro de una capa porosa. Experimentos previos y estudios analíticos de flujo convectivo natural en una capa porosa inclinada han demostrado que hay una variedad de posibles modos convectivos dependiendo en los parámetros del sistema, la geometría y la inclinación. En particular, existe una transición bien definida desde el modo helicoidal que consiste en rollos inclinados pendiente abajo superpuestos por sobre un rollo unicelular pendiente arriba a un modo que consiste en un rollo unicelular puro y pendiente arriba. Se carece actualmente (2009) de estándar de comparación tridimensionales para simuladores de densidad variable y la comparación de los resultados de simulaciones con este lugar de transición proporciona un medio inambiguo para testear la habilidad de tales simuladores para representar la física del estado estacionario inestable de densidad variable en 3D.

变密度流和运移模拟的三维基准: 半解析稳定性模式拟合倾斜多孔介质箱中稳态非稳定对流

摘要

这一变密度地下水流和溶质或能量运移的三维数值模拟基准由多孔介质自一个稳态对流模式向另一个稳态对流模式转变的半解析解匹配仿真结果 组成。已有对天然条件下倾斜多孔介质层中对流的实验和解析研究表明, 有很多取决于系统参数、几何形状和倾角的对流模式。特别是由下斜的纵向卷叠加上斜单卷的螺旋式模型到仅仅包括上斜单卷模式的转换, 研究较为清楚。目前 (2009) 缺少变密度流模拟的三维基准, 且模拟结果与这种转换点的比较为检验这种模拟器代表稳态的非稳定3D变密度物理机制的能力提供了确定的方法。

Testes de referência tridimensionais para a simulação de fluxo de densidade variável e de transporte: ajustando modos de estabilidade semi-analíticos para convecção instável e estacionária numa caixa porosa inclinada

Resumo

Este teste de referência (benchmark) para simuladores numéricos tridimensionais (3D) de fluxo de água subterrânea de densidade variável e transporte de soluto ou energia consiste em ajustar os resultados da simulação com a solução semi-analítica para a transição de um modo convectivo estacionário para um outro numa caixa porosa. Os estudos experimentais e analíticos anteriores do fluxo convectivo natural numa camada porosa inclinada mostraram que existe uma variedade de modos convectivos possíveis dependendo dos parâmetros do sistema, da geometria e da inclinação. Em particular, há uma transição bem definida do modo helicoidal consistindo de cilindros longitudinais descendentes sobrepostos a um cilindro unicelular ascendente em relação a um modo consistindo num cilindro unicelular ascendente. Actualmente (2009) há uma falta de testes de referência tridimensionais para simuladores de densidade variável e a comparação dos resultados da simulação com este ponto de transição dá um meio inequívoco para testar a capacidade de tais simuladores representarem a física de densidade variável a 3D, instável e estacionária.

Introduction

Three-dimensional (3D) numerical simulators of variable-density groundwater flow and solute or energy transport have come into practical use in recent years—e.g. SUTRA (Voss and Provost 2002), FEFLOW (Diersch 2002), SEAWAT(Langevin et al. 2007)—for a variety of analyses, usually focusing on coastal seawater intrusion in aquifers, but also on, for example, study of contaminant plume migration, sabkha and saline lake hydrology, brine migration in sedimentary basins, nuclear waste disposal with variable density due to both temperature and concentration distributions, and subsurface thermal energy storage. Benchmarks are tests of simulators that should confirm that the quantitative simulation results are correct—and that the simulator is correctly solving the underlying governing equations of variable-density groundwater flow and solute transport. However, unambiguous 3D benchmarks for confirming the correctness of variable-density simulators are currently lacking. Most of the simulators in use have been tested using two-dimensional (2D) benchmarks. Existing benchmarks consist of matching simulation results with either laboratory experiments or with analytical solutions for variable-density flow physics. Due to the lack of unambiguous tests, further confirmation of simulator correctness has been obtained by comparison among various simulators, based on a variety of numerical methods, of results for some standard problems that lack laboratory or analytical basis.

Although existing laboratory and inter-simulator comparison benchmarks provide some support that a particular simulator works properly, these tests are not unequivocal. When results from simulators match, this indicates that either all or none are functioning properly, because the correct result to which these should be compared for the tested problem (from laboratory or analytical studies) is not available.

Matching simulator results with measurements made in laboratory experiments would provide reliable benchmarks, should it be possible to set up the structure of the simulator (e.g. boundary conditions, initial conditions, internal properties, etc.) exactly equivalent to the experimental conditions. This is often made difficult due to a lack of knowledge of the exact conditions in the laboratory, and so the simulation includes errors in conforming to the precise physical conditions of the laboratory experiment. This requires compromises when configuring the simulation of the experiment. Least ambiguous are benchmarks that require a match with analytical solutions for variable-density physics, but analytical solutions are not easy to obtain for such nonlinear systems.

Stable and unstable variable-density physics must be distinguished because these types represent quite different fluid behaviors that require different benchmarks for confirming the ability of a simulator to correctly reproduce the behaviors. Stable density situations are defined as groundwater systems in which the fluid density is constant or increases with increasing depth. This includes the case of classical seawater intrusion in which, although the fluid density varies laterally in an aquifer, it always increases with depth. Unstable density situations are defined as groundwater systems in which the density decreases with depth, at least in some part of the system. In the stable situation, fluid of a similar density tends to remain contiguous in space, whereas in the unstable situation, fluid of one density passes through fluid of another density in patterns that are difficult to predict. In contrast with constant-density flow fields, which normally do not circulate, stable-density-driven flow fields exhibit circulation, tending to have uniform unimodal patterns. Unstable-density-driven flow fields are complex spatially, often exhibiting several spatial scales of circulation, and these patterns can be chaotic.

The following passages briefly review existing 2D and 3D benchmarks, before presenting a new 3D benchmark based on mathematical stability analysis and semi-analytical solution. Existing benchmarks were reviewed by Diersch and Kolditz (2002), and some of these are considered again in the following, from a different viewpoint. Each benchmark is classified as to whether it is a ‘necessary’ benchmark (meaning that a simulator must satisfy this test if it is functioning properly, but this does not prove that the simulator correctly represents the tested variable-density problem class), whether it is a ‘sufficient’ benchmark (meaning that if a simulator passes this test, it is definitely functioning correctly for the tested variable-density problem class), or whether it is both.

Existing 2D benchmarks

Static sharp horizontal interface in a box (Voss and Souza 1987)

(This is a necessary but not sufficient test for a simulator to represent 2D and 3D stable variable-density physics.) It confirms that flow does not occur when the density is constant in the lateral direction but increases uniformly with depth in a 2D cross-sectional model. This may also be directly employed for testing a 3D model with the same type of density distribution.

Henry steady-state seawater intrusion semi-analytical solution (Voss and Souza 1987)

(This is a necessary but not sufficient test for a simulator to represent 2D stable variable-density physics.) The Henry problem is an appealing and widely used benchmark because it encompasses the fundamental physics of coastal seawater intrusion via a semi-analytical solution (Henry 1964, corrected by Segol 1994). However, Voss and Souza (1987) pointed out that even for a simulator that fails the necessary test above, Static sharp horizontal interface in a box, it is possible for the simulator to correctly match this solution. Insensitivity of the Henry problem to variable-density flow physics is due to the high level of diffusion required in the problem setup to allow closure of the series semi-analytical solution. Other variations on the Henry problem have been proposed (e.g. Simpson and Clement 2004) but these also are not sufficient tests of a simulator.

Elder transient unstable convection simulator comparison problem (Voss and Souza 1987)

(This is a simulator inter-comparison for 2D unstable variable-density physics, neither a necessary nor sufficient test for a simulator; however, should several simulators that satisfy all necessary tests provide differing results for this problem, not all of these simulators can be functioning correctly.) The Elder problem is an adaptation of an experiment and early numerical solution (Elder 1967) created to test simulators. The original experiment and numerical solution was for temperature-dependent unstable convection. Elder managed to generate a simulated pattern that had rough similarity to what was observed in the experiment but there were noted discrepancies in terms of the number of fingers and convective circulation cells. The setup was modified to represent concentration-dependent convection as part of the international HYDROCOIN efforts to benchmark simulators studying nuclear waste disposal safety near salt domes and within other fluid-density-varying systems (as described in Holzbecher (1998)). Several simulators have been able to match Elder′s complex numerical convection patterns and each other (e.g. see Prasad and Simmons 2005). It has been found that the numerical solution for the central convecting finger provides a different direction of movement (upward or downward) depending on the fineness of spatial discretization and that this direction does not converge to a single result with finer discretizations, although the rest of the flow pattern remains generally the same. Thus, the Elder problem is a good problem for comparing a simulator to the general complex pattern of unstable convection generated by other simulators. However, this test also indicates that there may be no unique solution for a transient unstable convection problem because instabilities and fingers are initially generated by otherwise-inconsequential numerical factors that are not possible to control and that have no impacts on correct simulation of stable variable-density and constant-density problems.

HYDROCOIN level 1, case 5: groundwater above a salt dome simulator comparison problem (NEA 1988)

(This is a simulator inter-comparison for 2D stable variable-density physics, neither a necessary nor sufficient test for a simulator; however, should several simulators that satisfy all necessary tests provide differing results for this problem, not all of these simulators can be functioning correctly.) This benchmark was also developed as part of the international HYDROCOIN project as described by Holzbecher (1998) in which several simulators produced the same results.

Salt-lake transient unstable convection simulator comparison problem (Simmons et al. 1999)

(This is a simulator inter-comparison for 2D unstable variable-density physics, neither a necessary nor sufficient test for a simulator; however, should several simulators that satisfy all necessary tests provide differing results for this problem, not all of these simulators can be functioning correctly.) The salt-lake problem is a representation of an experiment (Wooding et al. 1997a, b) in a Hele-Shaw cell that demonstrated the dynamics of dense salt plumes sinking from evaporating saline lakes. It was simulated with a generally good match to the complex evolving pattern of sinking brine fingers by Simmons et al. (1999), although the pattern of fingers generated numerically was impacted by the means selected to initiate fingers at the base of the simulated lake. Discussion in the subsequent literature (e.g. Diersch and Kolditz 2002; Mazzia et al. 2001) focuses on this aspect and the many solutions that appear possible depending on numerical controls, rather than on the remarkable coincidence of simulated with experimental results that makes this an excellent problem for simulator inter-comparison. Standard concepts regarding grid convergence may not apply in these types of highly unstable phenomena. Despite better experimental control than the Elder (1967) experiment, and a simulation setup with less-ambiguous representation of the experiment, the problem still suffers from the fact that there may be no unique solution for a transient unstable convection problem, and the numerical solution obtained may depend on arbitrary numerical aspects of the simulator code or problem setup.

Two-dimensional steady-state unstable convection in layers and boxes (Weatherill et al. 2004)

(This is a necessary and sufficient test proving that a simulator correctly reproduces the physics of 2D steady-state unstable convection.) Bories and Combarnous (1973) and Caltagirone (1982) presented semi-analytic and stability analyses for porous media closely related to the classic infinite-extent parallel plate analyses of Horton and Rogers (1945) and Lapwood (1948) for unstable convection in a horizontal layer. These consist of approximate analytical stability criteria for steady unstable convection in a finite 2D box with top and bottom boundary conditions that maintain an unstable density configuration (dense fluid above less-dense fluid). A family of benchmark tests for 2D variable-density groundwater flow and solute transport simulators, based on comparison of numerical model results with these stability analyses of natural convection in infinite, finite and inclined layers (Weatherill et al. 2004), test 2D models via the following criteria:
  1. 1.

    Matching numerical results with analytical results for convective wavelength and critical Rayleigh number (Rac) for very wide horizontal boxes

     
  2. 2.

    Matching simulated Rac as a function of horizontal box geometrical aspect ratio with an analytical expression

     
  3. 3.

    Matching simulated Rac as a function of box inclination with an analytical expression

     
The definition of Ra is:
$$ Ra = \frac{{{\rho_0}\,g\,k\,\beta \,\left( {{C_{\max }} - {C_{\min }}} \right)\,H}}{{\theta \,\,{\mu_0}\,{D_m}}} = \frac{\text{buoyancy\,\, forces}}{{\text{dispersive\,\, and\,\, viscous\,\, forces}}} $$
(1)
where:
  • H = length scale (L) (usually set to the layer thickness when considering convective cells that extend from top to bottom of the layer)

  • Dm = molecular diffusion coefficient of solute in water (L2/T)

  • ρ0 = freshwater density (M/L3)

  • g = gravitational acceleration (L/T2)

  • k = permeability (L2)

  • β = ρ0−1 (∆ρ/∆C) = linear coefficient of density change, as function of concentration C (−)

  • Cmax = maximum concentration (M/M) expressed as mass fraction

  • Cmin = minimum concentration (M/M) expressed as mass fraction

  • θ  = porosity (−)

  • μ0 = fluid dynamic viscosity (M/LT)

(wherein units are indicated as: L for length, T for time, M for mass) and Rac is a single value or function of box aspect ratio and/or inclination angle given by theory.

The first criterion regarding convective wavelength is not unambiguous because, for horizontal boxes, the wavelength produced by simulators (equals the width of two adjacent convection cells in the box) is sensitive to initial seeding. Seeding can consist of any heterogeneity in boundary conditions, initial concentration field, spatial discretization, or other features and properties of the simulation, which generates the initial flow field. Seeding can be intentional, for example, placing anomalous concentration values within an otherwise homogeneous or smoothly varying initial concentration field, or unintentional, for example, numerical artifacts related to mesh construction. Following indications of sensitivity of the number of cells generated in steady state to spatial and temporal discretization for the infinite horizontal box case (Ataie-Ashtiani and Aghayi 2006), later studies by the present authors (unpublished) confirmed that non-dominant-mode seeding could result in various numbers of non-square cells, whereas seeding of initial concentration values at dominant modes results in the correct wavelength and number of square 2D cells. Thus, the first criterion regarding wavelength is a weak one, only strictly enforced when initial seeding is at the dominant expected wavelength. The critical Rayleigh number stability of the first criterion and the correct convective wavelength when seeded at dominant modes, together with the second and third criteria, provide unambiguous confirmation that a simulator is representing this 2D variable-density problem correctly.

Existing 3D benchmark

SaltPool transient stable convection simulator comparison problem (Johannsen et al. 2002; Oswald and Kinzelbach 2004)

(This is a necessary and sufficient test proving that a simulator correctly reproduces the physics of 3D steady-state stable convection.) The SaltPool benchmark requires simulation of an experiment (Oswald and Kinzelbach 2004) in a 3D laboratory tank with a porous medium consisting of glass spheres and with less-dense fluid overlying denser fluid. In one phase of the experiment, freshwater enters one upper corner of the box and fluid leaves from the opposite upper corner, causing the interface to tip and the salt concentration at the outflow point to change with time. This experiment is repeated for fluids with ten-times greater density contrast. The experiment was well-controlled and the experimental conditions carefully measured. Matching the evolution of the experiment with a simulator requires calibration of transverse dispersivity and extremely fine vertical discretization to capture the transverse dispersion process sufficiently well. The SaltPool experiment is the only currently available benchmark for 3D stable variable-density physics. The same authors also began development of an experiment-based benchmark for unstable variable-density physics (Oswald et al. 2007) and sought to develop benchmark-like criteria for comparison of simulation results with the measured behavior (Johannsen et al. 2006).

Semi-analytical 3D benchmark

Much understanding was developed in the 1960s through 1980s in fluid mechanics regarding 3D unstable convection as reviewed by Nield and Bejan (2006). Some of these results lend themselves to a benchmark for 3D groundwater simulators. The proposed 3D benchmark consists of matching simulator results with analytically derived stability criteria for steady-state unstable convection in a finite 3D box containing a porous medium, as a function of Ra and box inclination. This benchmark is analogous to that for 2D described above (Weatherill et al. 2004), but refers to the modes of convection that occur in 3D. The loci of 3D stability transitions, Rac, are corrected by co-author Neville Robinson from the classic analyses of 3D convection in porous media by Caltagirone and Bories (1985) that were reviewed by Caltagirone (1982) and Nield and Bejan (2006). The mathematical development is presented later (see Appendix). For the benchmark, only the region Ra ≤ 200 is considered to avoid chaotic-unsteady convection modes that occur at higher Ra. (This is a necessary and sufficient test proving that a simulator correctly reproduces the physics of 3D steady-state unstable convection.)

Stability diagram and modes of 3D convection

The dimensionless benchmark geometry is shown in Fig. 1. The box, of dimensional thickness H (the length scale in Eq. 1), has aspect ratios (= lateral extent/thickness H) of A and B, but in the present work, only the square case A = B is considered; thus, aspect ratio is given by the value of A. The angle of the box, φ, is measured upward from the horizontal plane. The gravity vector is perpendicular to the horizontal plane and oriented downwards. By setting specified concentration boundary conditions on the top and bottom of the box, the density is held higher at the top than at the bottom (the unstable situation), and the other sides of the box are closed to solute transport. All sides of the box are closed to fluid flow. The benchmark prescribes the allowed convective modes of steady-state flow and solute transport within the box according to the stability analysis for all states defined by the pair of values (φ, Ra), Ra ≤ 200.
https://static-content.springer.com/image/art%3A10.1007%2Fs10040-009-0556-6/MediaObjects/10040_2009_556_Fig1_HTML.gif
Fig. 1

The inclined porous box. All sides are scaled by H so that the thickness is 1 and A and B are lateral aspect ratios; φ is the angle that the box makes with the horizontal plane (indicated by shadow). Coordinate direction x is in horizontal plane pointing upslope, y is in horizontal plane pointing across slope, z is perpendicular to the horizontal plane, pointing upwards

The theoretical stability diagram of Caltigirone and Bories (1985; Fig. 2), as corrected at the end of this article (see Appendix), is shown in Fig. 2. Steady-state unstable convection occurs in several modes and the stability region for each mode is shown in Fig. 2. Caltagirone and Bories (1985) also found rough experimental verification of the predicted modes and regions. Polyhedral cells are predicted to occur for low angles when Ra > Rac, upslope unicellular convection occurs for all angles greater than zero when RaRac, and helicoidal convection consisting of downslope longitudinal rolls superimposed upon an upslope unicellular roll occurs for high angles when Ra > Rac. Images of each of these modes were generated via simulation (described later) and these are as follows.

Unicellular convection is illustrated in Fig. 3. There is a single convection cell with a flow direction upwards along the entire bottom of the box and downwards along the entire top of the box. There is no lateral variability (across slope) in the box of either flow or concentration.

Helicoidal convection is illustrated in Fig. 4. This mode is the superposition of two patterns, the unicellular cell and longitudinal rolls. The unicellular cell is the same as that shown in Fig. 3. The axes of the longitudinal rolls extend along the box from the top to the bottom edge, and their configuration is similar to the pattern of roll cells in a horizontal 2D box (see Weatherill et al. 2004). Lateral and transverse sections through the box of Fig. 4 that illustrate these patterns are shown in Fig. 5.

Polyhedric convection patterns are illustrated in Fig. 6a,b. At lower inclinations, the convective modes are not unique inasmuch as they are dependent upon initial conditions. As pointed out by Caltagirone and Bories (1985), there are a variety of convective flow structures possible including polyhedral cells, transverse rolls, longitudinal rolls and combinations of these types. These patterns are dependent on initial seeding of the concentration field and a rich variety of interesting configurations is possible. One of the more-regular polyhedric patterns consists of transverse rolls, as illustrated in Fig. 7, in this case, superposed on the upslope unicellular convection. Transverse rolls have axes that extend across the direction of slope.

It should be recognized that all modes of convection may be considered polyhedric. Thus, while any pattern is possible in the polyhedric region of the stability diagram (Fig. 2), only the stated mode is possible in the other regions. Whereas polyhedric convection patterns are usually dependent on initial seeding, the unicellular and helicoidal modes are independent of initial state, and are self-generating (e.g. see Caltagirone 1982).
https://static-content.springer.com/image/art%3A10.1007%2Fs10040-009-0556-6/MediaObjects/10040_2009_556_Fig2_HTML.gif
Fig. 2

Convective modes for an inclined porous box (labels indicate name of mode); solid curves are transition loci Rac(φ) between unicellular and helicoidal modes for different aspect ratios, from A = 5 to A = inf (infinity). The dashed curve is the transition locus Rac(φ) between polyhedric and helicoidal convection modes. A is the lateral to vertical box aspect ratio, φ is the box inclination angle in degrees, Ra is the Rayleigh number (Eq. 1). The stable mode occurs along the vertical axis at φ = 0 for Ra values ≤ 4π2. Numerical values for reproducing these curves are given at the end of the article (see Appendix)

https://static-content.springer.com/image/art%3A10.1007%2Fs10040-009-0556-6/MediaObjects/10040_2009_556_Fig3_HTML.gif
Fig. 3

Unicellular steady-state convective mode for an inclined porous box. Velocity pattern illustrated on side of box (for each vector, the line points in the flow direction, away from the spot marking the base; length is proportional to velocity magnitude). Surfaces of constant concentration are shown for 25% (blue), 50% (green) and 75% (brown) of maximum concentration (set on top surface of box). (φ = 55°, Ra = 70)

https://static-content.springer.com/image/art%3A10.1007%2Fs10040-009-0556-6/MediaObjects/10040_2009_556_Fig4_HTML.gif
Fig. 4

Helicoidal steady-state convective mode for an inclined porous box. Velocities and concentrations as in Fig. 3. (φ = 55°, Ra = 100)

https://static-content.springer.com/image/art%3A10.1007%2Fs10040-009-0556-6/MediaObjects/10040_2009_556_Fig5_HTML.gif
Fig. 5

Cross-sectional illustrations of helicoidal convective mode (shown in Fig. 4). a Shows longitudinal section viewed from side with vectors, b shows transverse section across the middle of the slope. Vectors and concentrations as in Fig. 3

https://static-content.springer.com/image/art%3A10.1007%2Fs10040-009-0556-6/MediaObjects/10040_2009_556_Fig6_HTML.gif
Fig. 6

Examples of polyhedric convective mode for a horizontal porous box. a Shows steady-state polyhedral cells that result for an arbitrary initial seeding spaced at the dominant wavelength (twice the box height) along each row of seeds . b Shows unsteady polyhedral cells that result some time after initial seeding with a random pattern. Vectors as in Fig. 3; surfaces of constant concentration are shown for 25% (blue), 50% (green) and 75% (yellow) of maximum concentration (set on top surface of box). (φ = 0°, Ra = 100)

https://static-content.springer.com/image/art%3A10.1007%2Fs10040-009-0556-6/MediaObjects/10040_2009_556_Fig7_HTML.gif
Fig. 7

Polyhedric steady-state convective mode for an inclined porous box with transverse rolls. Velocities as in Fig. 3. Concentration surface is 50% of maximum concentration. This pattern appears without seeding. (φ = 15°, Ra = 48)

Verification of a simulator

Numerical model design

In order to demonstrate how a variable-density groundwater flow simulator can be verified via this benchmark, the US Geological Survey SUTRA simulator (Voss and Provost 2002) is employed; this work was carried out using the graphical preprocessor, SutraGUI (Winston and Voss 2003) and the Model Viewer post-processor (Hsieh and Winston 2002). Although the stability diagram deals with dimensionless Rayleigh number and box geometry, the simulator works in terms of dimensional parameters. For this demonstration, a box aspect ratio A = B = 10 was selected (see Fig. 1); the actual size simulated was 20 m × 20 m × 2 m. Discretization was set at 50 elements across the lateral A and B directions of the box, and 10 elements in the thickness direction, for a total of 25,000 elements and 28,611 nodes. (Both coarser and finer discretizations were also used and are discussed later.) The dimensional parameter values required by SUTRA that were used to generate the case, Ra  = 100, are listed in Table 1, as a reference state.
Table 1

Dimensional parameters used in the SUTRA simulations to create the case Ra = 100, showing parameter name, symbol (refer to Voss and Provost (2002) for strict definitions), units, and value. Other values of Ra were obtained by modifying Dm according to Eq. (1)

Parameter

Units

Value

Matrix compressibility, β

[kg/(ms2)]−1

4.47 × 10−10

Water compressibility, α

[kg/(ms2)]−1

1.0 × 10−8

Fluid base density, ρ0

[kg/m3]

1000

Fluid density dependence, ∆ρ/∆C

[kg/m3]

200

Fluid dynamic viscosity, μ

[kg/(ms)]

0.001

Gravity magnitude, |g|

[m/s2]

9.81

Porosity, θ

[1]

0.1

Dispersivities, αL, αT

[m]

0

Solute diffusivity in fluid, Dm

[m2/s]

3.565 × 10−6

Permeability, k

[m2]

9.0851172 × 10−12

Boundary conditions along all sides of the box were set to no-flow. A specified pressure value of zero was set at one upper corner of the box to provide a reference value for the steady-state numerical pressure result, though this is not theoretically required to obtain a transient pressure result. Boundary conditions along the lateral sides of the box were set to no solute flux. Along the top and bottom of the box, the solute concentration was specified as a constant value, Cmax = 1 on the top, and Cmin = 0 on the bottom, fixing the fluid density-according to Eq. (2) and Table 1–to equal 1,200 kg/m3 on the top and 1,000 kg/m3 on the bottom. Although true fluid–density dependence on concentration (as a mass fraction) for a large concentration range is slightly nonlinear, the dependence for this benchmark is simulated as linear to exactly match the linear dependence that was necessarily assumed to obtain the analytical solution:
$$ \rho \, = \,{\rho _0}\, + \,\left( {\Delta \rho /\Delta C} \right)C $$
(2)
where ρ is fluid density, ρ0 is fluid base density (at concentration C = 0) and (∆ρ/∆C) is a constant value.

The general objective of this benchmark is to verify the ability of the simulator to reproduce the correct modes of convection at a variety of points (φ, Ra) on the stability diagram, Fig. 2. However, the most-discriminating verification focuses on points in the vicinity of the transition loci. To accomplish tests near the locus, pairs of nearby (φ, Ra) points were selected directly across the transition locus. The initial state of the simulation (φ, Ra) at one of the two points was always set to the numerical solution for its counterpart across the locus, so that when φ or Ra was changed in the simulation to obtain the new state on the other side of the transition locus, the convection mode was forced to completely transform to achieve a new state. In practice, the less-complex unicellular mode was always easy to simulate; thus, this state was usually simulated first for each test across the transition. The interior of each mode′s region was investigated only partly, mainly where interesting modes occurred (transverse rolls), and to test the helicoidal/polyhedric transition locus. For states not near the unicellular-helicoidal transition locus, the initial condition was set to the simulated unicellular result for the same inclination angle.

Simulation procedure

There are many ways of setting dimensional values that obtain the same value of Ra according to Eq. 1, but only the Ra value controls the physics; thus, when verifying the simulator, individual dimensional values are arbitrary. This was demonstrated for 2D unstable convection by Weatherill et al. (2004).
  • In order to configure the simulator for different values of Ra, when setting up (φ, Ra) on the stability diagram to test, typically either permeability, k, or molecular diffusivity, Dm, should be varied from the basic values of Table 1, using Eq. 1. (In this work, only Dm, was varied.)

  • To set a variety of angles, φ, the direction of the gravity vector in the SUTRA simulation was varied (the x- and z-components were selected appropriately, with the y-component set to zero). In effect, the box did not change direction, rather, the direction of gravity changed when the box was to be simulated at a new inclination angle.

For the special case of a horizontal box, φ = 0, the initial condition employed was a low Rayleigh number simulation result for a non-convective state (Ra < Rac) with manual seeding superposed. As a result of the mathematical expression of the diffusion process that dominates for such low Ra solutions, for unstable solute transport, the initial concentration expressed as a mass fraction of solute is a non-linear function with increasing value in the upward direction (see Weatherill et al. 2004). The easiest means to obtain this steady-state 3D distribution is via transient simulation.

Once this quiescent 3D simulation result was obtained, a seeding was manually superimposed on it. Seeding involves either raising or lowering the concentration by a small amount at various points. For example, when the system is seeded in a hexagonal or square pattern of seeds separated by the dominant wavelength (twice the cell size, 2H) along a plane midway between top and bottom of the box, the simulation produces a steady solution with cells centered on these points (Fig. 6a). Other dominant modes may be obtained by other seeding patterns such as with lateral spacing H.

Seeding completely determines the pattern of convection for the horizontal box (φ = 0, Ra > Rac), and the seeding pattern can be arbitrary, or, in attempts to generate steady-state patterns, can be laterally distributed at spacing that is a multiple of the dominant wavelength or box height H. This test is a weaker check on the correctness of a variable-density simulator, but may be considered as a necessary but not sufficient criterion for correct functioning of the simulator. Random seeding patterns may lead to a complex superposition of various convective modes, resulting in interesting transient evolution, though not necessarily reproducible or useful steady patterns for benchmarking purposes. A transient state is shown in Fig. 6b for a seeding at all nodes in a plane midway between top and bottom of the box, where the seed value at each node is selected randomly in the concentration range zero to one.

Time discretization was selected both to give quick simulation results for the transitions and to maintain solution stability. Typical time steps were of 1 month duration, and simulations were continued in time until the initial steady convection pattern changed (or did not change) to a new pattern and it was clear that the new (or unchanged) pattern would be the new steady-state solution. The length of time depended on the state—with transitions taking as little as a few years for high Ra conditions and as much as hundreds of years for simulations with both low Ra and low φ. Some states near the transition locus also required long times to achieve, irrespective of these parameter values, so proximity to the transition locus may also be a controlling factor. (It was also observed that transitions take longer to occur for more-finely discretized meshes.) Time step size was increased to as much as 6 months at later times in some of the longer simulations. Some patterns emerged quickly and others emerged slowly only after a long period of no apparent changes (while it seemed the wrong convection mode would be obtained), so no consistent total simulation time could be set for all cases and simulations must be observed and controlled on an individual basis. For example, the (φ = 55°, Ra = 100) case appeared to remain stable in the unicellular mode for about 6 years, and then took another 9 years to change to the helicoidal mode, which then became the new steady solution.

Results and discussion

Basic results

The (φ, Ra) points on the stability diagram, Fig. 2, for which the simulator was tested, are listed in Table 2 together with the expected convection mode and simulated mode for each point. These results are also displayed graphically in Fig. 8. Simulated modes for all tested points agree with the expected modes, showing that the SUTRA simulator correctly represents the physics of unstable steady convection.
https://static-content.springer.com/image/art%3A10.1007%2Fs10040-009-0556-6/MediaObjects/10040_2009_556_Fig8_HTML.gif
Fig. 8

Match of simulated and theoretical convective modes for an inclined porous box with aspect ratio A = 10 (labels indicate name of mode); solid curve is transition locus Rac(φ) between unicellular and helicoidal modes for aspect ratio, A = 10. Dashed curve is transition locus Rac(φ) between polyhedric and helicoidal convection modes. Each point is labeled with a single letter and is the result of one simulation, as listed in Table 2. Letters indicate the convection mode produced by the simulator: S stable, P polyhedric, T transverse, H helicoidal, U unicellular

Table 2

Locations of points in Fig. 8 with expected and observed states (S stable, U unicellular, P polyhedric, T transverse, H helicoidal). Note that transverse and helicoidal are subsets of polyhedric

Angle φ (degrees)

Ra

State expected

State observed

0

20

S

S

3

35

U

U

10

35

U

U

15

37

U

U

20

38

U

U

25

40

U

U

30

40

U

U

3

45

P

T

10

45

P

T

20

45

H

H

15

48

P

T

25

50

H

H

30

50

H

H

40

50

U

U

20

52

H

H

40

60

H

H

3

70

P

T

10

70

P

T

15

70

P

T

20

70

H

H

49

70

H

H

55

70

U

U

55

85

H

H

60

85

U

U

0

100

P

P

10

100

P

H

20

100

P

H

30

100

P

H

33

100

H

H

50

100

H

H

55

100

H

H

60

100

H

H

65

100

U

U

65

150

H

H

70

150

U

U

67.5

200

H

H

72.5

200

U

U

The basic results may be categorized as follows:
  • For the horizontal box case, φ = 0, a stable non-convective state is simulated for Ra < Rac, and polyhedric convection occurs for Ra > Rac. The pattern of polyhedric convection depends on the initial seeding. Both steady-state and unsteady patterns are shown in Fig. 6.

  • For the case of relatively low inclination, φ ≤ 15°, unicellular convection is simulated for Ra < Rac, and convection with transverse rolls (a form of polyhedric convection) occurs for Ra > Rac where Ra is less than about 80. For higher Ra, a pattern of strong transverse rolls that occurs initially slowly evolves into helicoidal convection after a long time. The transition between the stability regions of transverse rolls and helicoidal convection was not more-closely investigated. A helicoidal result is shown in Figs. 4 and 5 and a transverse roll result is shown in Fig. 7.

  • For the case of higher inclination, φ > 15°, the simulation produces unicellular convection for Ra < Rac, and helicoidal convection for Ra > Rac.

  • At non-zero angles less than the predicted transition angle between polyhedric and helicoidal modes, helicoidal convection is simulated in some cases. Helicoidal convection is one form of polyhedric convection, so this result is correct.

An additional characteristic of the helicoidal convection mode should also be considered to be part of the benchmark. The longitudinal cells must be equally spaced across the slope, and the cell size (the distance between adjacent upwelling and downwelling regions) must be equal to the height of the box, H. The correct wavelength (2H) and cell size (H) were obtained for all simulated instances of helicoidal convection.

Other discretizations

For three pairs of points across the unicellular-helicoidal transition locus, convection modes were also tested for two other spatial discretizations. The coarser finite-element mesh had half as many elements in both lateral and thickness directions (25 × 25 × 5 elements) as the reference mesh, and the finer finite-element mesh had twice as many elements in both lateral and thickness directions (100 × 100 × 20 elements). For these points, (φ = 3°, Ra = 35, 45), (φ = 30°, Ra = 40, 50), and (φ = 67.5°, 72.5°, Ra = 200), both the reference and fine mesh tested give results for convection modes that match the stability diagram, with no effect of finer discretization on the convection modes simulated. However, the expected convection modes are not correctly obtained in two instances using the coarse mesh. For (φ = 30°, Ra = 50), helicoidal mode is expected but the simulator produces unicellular convection. For (φ = 67.5°, Ra = 200), helicoidal mode is expected and the simulator produces helicoidal convection, but with the wrong wavelength of longitudinal rolls (5.5 wavelengths or 11 cells, rather than 5 wavelengths and 10 cells). The reason is likely that there is not an integer number of finite elements within each expected cell width for the coarser mesh. Due to the way fluid velocities are represented spatially (based on bilinear finite-element basis functions and resulting bilinear distribution of pressure within each element), the particular coarse discretization employed does not allow the correct flow pattern to be supported. In order to support perfect convection cells, the box width must be an integral multiple of the box height, and, using the present numerical method, there must be an integral number of elements per expected cell width in the lateral direction of the box.

Near the unicellular-helicoidal transition locus

Simulation results could always be obtained, in relatively short simulation times, in agreement with the expected mode for angles greater than roughly 2.5° from the unicellular-helicoidal transition locus, or Ra values more distant than roughly 2.5 Ra units from the transition, on either side. For some states closer to the unicellular-helicoidal transition locus than reported in Table 2, long-time simulations were required before the correct state could be obtained. Some long simulations were abandoned while the result was either a unicellular mode persisting for states slightly above the locus or mild longitudinal rolls persisting for inclinations slightly below the locus.

Longitudinal rolls seem to exhibit a great deal of ‘inertia’ in the near-locus region on both sides. Once they exist on the helicoidal side, they persist in the simulation for a long time when the state crosses the transition from helicoidal to unicellular near the transition locus. Similarly, it takes a long time for the longitudinal rolls to generate when transitioning just across the locus in the opposite direction. Additional simulation tests indicate that the width of the near-locus zone is not dependent on the fineness of discretization. However, the near-locus ‘inertia’ of longitudinal rolls is apparently greater for more-finely discretized meshes, which require the longest simulation times for transitioning.

Whereas ‘inertia’ of longitudinal rolls may be a plausible qualitative explanation for this behavior, it may alternatively be argued that because linear analysis generated the critical transition locus on the (φ, Ra) diagram, a wider transition region such as that observed, is more realistic than the sharp transition locus. Indeed, the simulator includes convective physical behaviors, contained in full numerical solutions of the governing equations, which are excluded by linear stability analysis. This topic may be of interest for future investigations.

Helicoidal-polyhedric transition locus

Despite the fact that all modes are allowed in the polyhedric region of the stability diagram, the helicoidal-polyhedric transition locus (at about 31.5° inclination) can be used for simulator verification. Whereas all modes are allowed to the left (at lower angles) of this locus, only the helicoidal mode is allowed in the region between this locus and the helicoidal-unicellular transition locus. Thus, simulators should produce only helicoidal convection in the region between these loci, and the present simulated results in this region are correct. However, because helicoidal convection is also an allowed mode to the left of the helicoidal-polyhedric transition locus, its exact location cannot be distinguished by these simulations. Moreover, Caltagirone and Bories (1985) indicated that laboratory results for this transition occurred at a lower inclination (approximately at 15°), but for the same reasons as just given previously, a transition locus at lower inclination also was not observable in the present simulation results. Caltagirone and Bories (1985) suggested that this lower transition angle may result from the finite aspect ratio of box width to height; however, this was tested here by simulation of a box with A = 50 at inclination of 10°. The result is helicoidal mode (following initial growth of transverse rolls), even when seeded specifically to initially generate both transverse and longitudinal rolls (a pattern that only occurs in the polyhedric mode). Ormond and Genthon (1993) confirmed via linear stability analysis that transverse rolls are not stable for angles above the transition locus and that this transition locus occurs at lower angles when the permeability in the thickness direction is lower than in the lateral box directions.

Conclusions

Most existing tests (or benchmarks) for variable-density groundwater flow and solute transport simulators are based on synthetic problems that serve merely to compare simulators with one another. Although there are no known correct or analytically derived answers to these problems, such tests are useful for establishing some degree of viability and confidence in simulators to represent the physics of these problems. However, such tests do not unambiguously confirm that a simulator is correctly representing variable-density physics.

A few laboratory experiments exist for variable-density flow physics, and these may serve as benchmarks as well. However, the weakness of experiment-based benchmarks is that the exact conditions of the experimental setup are rarely known completely and the simulator must be configured on the basis of reasonable assumptions concerning the experimental setup. Thus, these give confidence that a simulator can approximately reproduce laboratory observations, but these do not unambiguously confirm that a simulator is correctly representing variable-density physics.

Benchmarks based on analytical solutions of the governing equations for variable-density groundwater flow and solute transport provide unambiguous benchmarks because they have exact requirements for simulation setup and quantitative mathematical solutions with which the simulators must be compared. One might criticize analytical-solution-based benchmarks as not being realistic enough to test a simulator′s ability to represent real variable-density problems in groundwater systems. Indeed, analytical benchmarks are neither geometrically complex, nor do they have complex spatial distributions of parameter values. However, it is clear that these benchmarks unambiguously answer the question of whether or not a simulator is able to correctly solve the partial-differential equations governing unstable variable-density physics. This is a prerequisite if a simulator will also be able to correctly simulate variable-density physics in more complex geometries with more complex parameter distributions.

In practice, a 3D variable-density groundwater flow and solute transport simulator must pass several initial benchmark tests before attempting the currently proposed benchmark for steady unstable physics. Normally, it must first be confirmed that this type of simulator correctly represents the physics of constant-density flow and solute transport in 2D and 3D by checking it against analytical solutions for hydraulic head (to test the ‘groundwater flow’ simulator) and solute concentration (to test the ‘solute transport’ simulator). Thereafter, the simulator can be checked for its ability to represent stable variable-density physics (by the 2D and 3D benchmarks listed above). Finally, the simulator can be tested using benchmarks for unstable variable-density physics, for example, the one offered here.

The benchmark proposed in this study consists of analytical stability criteria for steady-state unstable convection in a finite 3D box. This 3D benchmark is analogous to one previously reported for 2D (Weatherill et al. 2004), but refers to the modes of convection that occur in 3D: polyhedral cells (occur for low angles when Ra > Rac), upslope unicellular convection (occurs for all angles greater than zero when Ra < Rac), and helicoidal convection (occurs for high angles when Ra > Rac). The 3D stability criteria are corrected here from classic analyses of 3D convection in porous media by Caltagirone and Bories (1985) that were reviewed by Caltagirone (1982).

Three-dimensional simulators may be tested by:
  1. 1.

    Determining the ability of the simulator to reproduce the unicellular/helicoidal transition locus. (Match simulated steady-state convective modes with the predicted modes near the unicellular/helicoidal transition locus for sets of simulated points (φ, Ra) close to the locus on either side.)

     
  2. 2.

    Confirm simulator ability to produce correct steady-state convection modes within stability regions. (Match simulated convective modes with the predicted modes for sets of simulated points (φ, Ra) in each region of the stability diagram.)

     
  3. 3.

    For the helicoidal mode, the simulated transverse wavelength of longitudinal rolls must match the predicted convective wavelength.

     
  4. 4.

    For horizontal boxes, the simulator must produce quiescent flow for Ra < Rac, and for Ra > Rac, must produce polyhedric convection patterns (with wavelength roughly correct, irrespective of pattern and initial seeding, and with steady polyhedric patterns when seeding is at the dominant wavelength; however, this is a weak test).

     

These tests may be repeated several times for boxes with different aspect ratios.

This 3D benchmark, based on analytically derived stability criteria, provides a necessary and sufficient test to verify the ability of a 3D variable-density groundwater flow and solute transport simulator to correctly represent the physics of steady-state unstable variable-density groundwater flow.

Two additional measures, not evaluated here, may possibly be compared with simulated results as a means of expanding this benchmark: analytical solution for the solute flux through the box as characterized by the Sherwood number, Sh, (the mass transfer analogue of the energy-transfer Nusselt number, Nu) (Caltagirone 1982), and analytical description of the pattern of fluid motion such as characterized by the closed-loop particle paths illustrated by Ormond and Genthon (1993). Unambiguous benchmarks are also required for transient unstable variable-density convection such as for the onset of convection; initial attempts have been made by Wooding et al. (1997a, b), Kooi et al. (2000) and Johannsen et al. (2006).

For the US Geological Survey SUTRA simulator, simulated modes for all tested points agree with the expected modes in the demonstrated verification. Thus, it may be concluded that this simulator correctly represents 3D steady unstable variable-density physics.

Acknowledgements

This research was supported in part by the US Geological Survey. Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the US Government.

Copyright information

© Springer-Verlag 2009