Abstract
The use of the linear Boltzmann equation is proposed for transport in porous media in a column. By column experiments, we show that the breakthrough curve is reproduced by the linear Boltzmann equation. The advection–diffusion equation is derived from the linear Boltzmann equation in the asymptotic limit of large propagation distance and long time.
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Acknowledgements
The seed of this research was planted on the occasion of the Study Group Workshop (Department of Mathematical Sciences, The University of Tokyo, December 2010), which is greatly appreciated. The research was restarted with the support of the Focusing Collaborative Research by the Interdisciplinary Project on Environmental Transfer of Radionuclides (University of Tsukuba and Hirosaki University). The research was partially supported by the JSPS A3 Foresight Program: Modeling and Computation of Applied Inverse Problems. MM acknowledges support from Grant-in-Aid for Scientific Research (17H02081, 17K05572, 18K03438) of the Japan Society for the Promotion of Science. YH acknowledges support from Grant-in-Aid for Scientific Research (15H05740, 17H01478) of the Japan Society for the Promotion of Science.
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Appendices
Appendix 1: Eigenvalues
Let us write the homogeneous equation as
where I is the identity, \(\Xi _{\pm }\) and W are matrices, and \(\Phi _{\pm }\) are vectors defined as
We obtain
where
We label the eigenvalues as \(\nu _n\) (\(n=1,2,\dots ,2N\)).
By deforming the Bromwich contour, we obtain a contour \((-\infty ,\gamma )\), \(\gamma >0\), on which s is real. In this case, we have
where \(\lambda _0\) is real and \(w_0\) a 2N-dimensional real vector. We note that \(M=\mathfrak {R}{M}\) in this case and moreover we can write
Since M is a symmetric real matrix, by the Cholesky decomposition we can write \(M=U^TU\), where U is a real triangular matrix with positive diagonal entries. Hence,
Since U is nonsingular, S and \(USU^T\) have the same inertia, i.e., the same number of positive, negative, and zero eigenvalues according to Sylvester’s law of inertia. Clearly, S has \(N_{\eta }\) positive eigenvalues. Therefore, there are \(N_{\eta }\) eigenvalues \(\lambda _0\).
Next, we assume that s has a small imaginary part. Then, we can write
We treat the imaginary part as perturbation and express the matrix-vector equation as
By collecting terms of order \(O(\epsilon ^0)\), we have \(S(\mathfrak {R}{M})w_0=\lambda _0w_0\). From terms of \(O(\epsilon ^1)\), we have
Let us multiply \(w_0^T\) on both sides of the above equation. We obtain
Using \(w_0^T(S(\mathfrak {R}{M}))^T=w_0^TS\mathfrak {R}{M}=\lambda _0w_0^T\), we obtain
That is, \(\lambda _1\) is pure imaginary and the number of positive \(\mathfrak {R}{\lambda }\) does not change. This fact implies that there are \(N_{\eta }\) eigenvalues \(\nu _n\) (\(n=1,\dots ,2N\)) such that \(\mathfrak {R}{\nu _n}>0\).
Appendix 2: Advection–Diffusion Equation with Absorption
Let us consider the following advection–diffusion equation with the absorption term:
Let us introduce \(\Gamma (x,t)\) as
We have
Let us consider the Laplace transform:
Then, we obtain
We obtain
Thus,
where
Noting that
we obtain
where the complementary error function is given by \(\mathop {\mathrm {erfc}}(x)=(2/\sqrt{\pi })\int _x^{\infty }\mathrm {e}^{-t^2}\,\mathrm {d}t\). Finally,
The above solution reduces to the Ogata–Banks solution (Ogata and Banks 1961) when \(\sigma _a=0\). In particular, we have \(C(x,t)\rightarrow C_0\) as \(t\rightarrow \infty\) if \(\sigma _a=0\). When \(\sigma _a=0\), we have
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Amagai, K., Yamakawa, M., Machida, M. et al. The Linear Boltzmann Equation in Column Experiments of Porous Media. Transp Porous Med 132, 311–331 (2020). https://doi.org/10.1007/s11242-020-01393-1
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DOI: https://doi.org/10.1007/s11242-020-01393-1