Saltwater intrusion modeling in heterogeneous confined aquifers using two-relaxation-time lattice Boltzmann method

Serván-Camas, Borja; Tsai, Frank T.-C.

doi:10.1016/j.advwatres.2009.02.001

Cited by 40 publications

(21 citation statements)

References 27 publications

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“…conditions in Table 2 and the n-lines in Appendix B.2 for g (u) = 1, plotted in Figs. [12][13][14][15][16]. Hence, the OTRT subclass possesses much more advantageous sufficient advection stability criteria than the BGK model, especially in the limit Λ bgk → 0 where the non-negativity conditions become necessary for the BGK model.…”

Section: Tablementioning

confidence: 96%

The role of the kinetic parameter in the stability of two-relaxation-time advection–diffusion lattice Boltzmann schemes

Kuzmin

Ginzburg²,

Mohamad³

2011

Computers & Mathematics with Applications

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a b s t r a c tIn general, explicit numerical schemes are only conditionally stable. A particularity of lattice Boltzmann multiple-relaxation-time (MRT) schemes is the presence of free (''kinetic'') relaxation parameters. They do not appear in the transport coefficients of the modelled second-order (macroscopic) equations but they have an impact on the effective accuracy and stability of the algorithm. The simplest uniform choice (the well known BGK/SRT model) is often inadequate, and therefore a compromise in the complexity of the model is sought. For this purpose, the von Neumann stability analysis is performed for the d1Q3 two-relaxation-time (TRT) advection-diffusion model. The extended optimal (EOTRT) model, which relates the two collision times such that the most stable scheme is set by a suitable choice of the equilibrium parameters, equal for any Peclet number, is then developed. This extends the very recently derived optimal subclass (OTRT) to larger combinations of ''physical'' and ''kinetic'' collision rates. Next, we provide the necessary and/or sufficient stability limits on the EOTRT subclass for a wide range of velocity sets, with and without numerical diffusion, and delineate the interesting choices of free equilibrium weights for the d2Q9 and d3Q15 models. The BGK/SRT model is without advanced advection properties; we prove (for minimal stencil schemes d1Q3, d2Q5 and d3Q7) that the non-negativity of the equilibrium distribution is necessary for its stability in the advection-dominated limit. Beyond the EOTRT and BGK/SRT subclasses of the TRT model, blind choices of the ''ghost'' collision number may result in quite unstable schemes, even for positive equilibrium. However, we find that the d1Q3 stability curves govern the advection properties of the multi-dimensional models and a fuller picture of the TRT stability properties begins to emerge.

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Section: Tablementioning

confidence: 96%

The role of the kinetic parameter in the stability of two-relaxation-time advection–diffusion lattice Boltzmann schemes

Kuzmin

Ginzburg²,

Mohamad³

2011

Computers & Mathematics with Applications

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“…The DDF in porous media is often investigated numerically, using a large variety of numerical models [Voss and Provost, 2002;Diersch, 2002;Langevin et al, 2007;Pruess, 2004;Zhang et al, 2008]. The development of efficient and robust numerical techniques for solving a such problem is still of interest to the scientific community [Huyakorn et al, 1987;Putti and Paniconi, 1995;Xue et al, 1995;Knabner and Frolkovic, 1996;Frolkovic et al, 1997;Ibaraki, 1998;Frolkovic and De Schepper, 2001;Ackerer et al, 2004;Soto Meca et al, 2007;Ackerer and Younes, 2008;Lunati and Patrick, 2008;Camas and Tsai, 2009;Younes et al, 2009;Van Reeuwijk et al, 2009;Hirthe and Graf, 2012;Albets-Chico and Kassinos, 2013;Povich et al, 2013;Lamine and Edwards, 2015] and is even reinforced by the large sets of simulations required for the uncertainty and/or sensitivity analysis [ As the number of users and developers increases, benchmarking of the numerical models for DDF simulation in porous media has become a challenging task [Voss and Souza, 1987;Simmons et al, 1999;Johannsen et al, 2002;Simpson and Clement, 2003;Weatherill et al, 2004;Oswald and Kinzelbach, 2004;Ataie-Ashtiani and Aghayi, 2006;Van Reeuwijk et al, 2009;Voss et al, 2010Stoeckl et al, 2016]. In this context, comparison against analytical or semianalytical solutions (when they exist) is considered as the best way for numerical models validation because these solutions allow for confirming that the gover...…”

Section: Introductionmentioning

confidence: 99%

The Henry problem: New semianalytical solution for velocity‐dependent dispersion

Fahs

Ataie-Ashtiani

Younès

et al. 2016

Water Resources Research

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A new semianalytical solution is developed for the velocity‐dependent dispersion Henry problem using the Fourier‐Galerkin method (FG). The integral arising from the velocity‐dependent dispersion term is evaluated numerically using an accurate technique based on an adaptive scheme. Numerical integration and nonlinear dependence of the dispersion on the velocity render the semianalytical solution impractical. To alleviate this issue and to obtain the solution at affordable computational cost, a robust implementation for solving the nonlinear system arising from the FG method is developed. It allows for reducing the number of attempts of the iterative procedure and the computational cost by iteration. The accuracy of the semianalytical solution is assessed in terms of the truncation orders of the Fourier series. An appropriate algorithm based on the sensitivity of the solution to the number of Fourier modes is used to obtain the required truncation levels. The resulting Fourier series are used to analytically evaluate the position of the principal isochlors and metrics characterizing the saltwater wedge. They are also used to calculate longitudinal and transverse dispersive fluxes and to provide physical insight into the dispersion mechanisms within the mixing zone. The developed semianalytical solutions are compared against numerical solutions obtained using an in house code based on variant techniques for both space and time discretization. The comparison provides better confidence on the accuracy of both numerical and semianalytical results. It shows that the new solutions are highly sensitive to the approximation techniques used in the numerical code which highlights their benefits for code benchmarking.

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“…The TRT collision operator [30,31,43] is sufficient for solute transport in shallow water flow. The TRT collision operator is a particular form of the MRT collision operator with two unique relaxation rates, where s 1 = s 2 = s 7 = s 8 = 1/ a and s 4 = s 6 = 1/ s in Equation (7).…”

Section: Lbm For Anisotropic Advection-dispersion Equationmentioning

confidence: 99%

“…The zeroth, first, and second moments of the EDFs are Following the techniques in dealing with heterogeneous and anisotropic dispersion coefficients [33,43], the EDFs applied to the AADE are…”

Section: Lbm For Anisotropic Advection-dispersion Equationmentioning

confidence: 99%

GPU accelerated lattice Boltzmann model for shallow water flow and mass transport

Tubbs

Tsai

2010

Numerical Meth Engineering

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SUMMARYA lattice Boltzmann method (LBM) for solving the shallow water equations (SWEs) and the advectiondispersion equation is developed and implemented on graphics processing unit (GPU)-based architectures. A generalized lattice Boltzmann equation (GLBE) with a multiple-relaxation-time (MRT) collision method is used to simulate shallow water flow. A two-relaxation-time (TRT) method with two speed-of-sound techniques is used to solve the advection-dispersion equation. The proposed LBM is implemented to an NVIDIA Computing Processor in a single GPU workstation. GPU computing is performed using the Jacket GPU engine for MATLAB and CUDA. In the numerical examples, the MRT-LBM model and the TRT-LBM model are verified and show excellent agreement to exact solutions. The MRT outperforms the single-relaxation-time (SRT) collision operator in terms of stability and accuracy when the SRT parameter is close to the stability limit of 0.5. Mass transport with velocity-dependent dispersion in shallow water flow is simulated by combining the MRT-LBM model and the TRT-LBM model. GPU performance with CUDA code shows an order of magnitude higher than MATLAB-Jacket code. Moreover, the GPU parallel performance increases as the grid size increases. The results indicate the promise of the GPU-accelerated LBM for modeling mass transport phenomena in shallow water flows.

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Saltwater intrusion modeling in heterogeneous confined aquifers using two-relaxation-time lattice Boltzmann method

Cited by 40 publications

References 27 publications

The role of the kinetic parameter in the stability of two-relaxation-time advection–diffusion lattice Boltzmann schemes

The role of the kinetic parameter in the stability of two-relaxation-time advection–diffusion lattice Boltzmann schemes

The Henry problem: New semianalytical solution for velocity‐dependent dispersion

GPU accelerated lattice Boltzmann model for shallow water flow and mass transport

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