# Slower than usual transport in porous media

In the last decade a number of researchers have found that in some materials (such as zeolite or building bricks) the moisture diffusion follows a slower than usual pace and thus its transport becomes subdiffusive. There is a wealth of literature on the experimental side of the subdiffusion, but its crucial nonlinear aspects are described and investigated for example in [1-4]. In those experiments the diffusion is manifestly nonlinear due to the complex geometry of the medium.

The slower than classical evolution is probably the result of the so-called trapping phenomenon. During the imbibition, water parcels are being kept in some regions of the porous medium for more prolonged time periods. This forces the total flux to arrive at a certain point with a delay. Therefore, the resulting dynamical equation contains a nonlocal, rather than classical, temporal derivative. It can be written as (after nondimensionalization)

$\partial_t^\alpha u = \left( D(u) u_x \right)_x, \quad x>0, \; t>0, \quad 0<\alpha<1,$

where $\partial_t^\alpha u$ is the so-called Riemann-Liouville fractional derivative given by

$\partial^\alpha_t u(x,t) = \frac{1}{\Gamma(1-\alpha)}\frac{\partial}{\partial t}\int_0^t (t-s)^{-\alpha} u(x,s) ds.$

Notice that a computation of the derivative at a certain instant requires the knowledge of the whole process history. This is the very reason that makes the above equation successful in describing subdiffusion in porous media (but not only: there is a number of other applications which are summarized for ex. in [5]).

It is convenient to transform the above governing equation into its self-similar form by the use of the substitution $u(x,t)=U(x t^{-\alpha/2})$. Then it becomes

$\left(D(U) U'\right)' = \frac{1}{\Gamma(1-\alpha)}\left[(1-\alpha)-\frac{\alpha}{2}\eta \frac{d}{d\eta}\right] F_\alpha U, \quad 0<\alpha< 1,$

where $F_\alpha$ is the Erdèlyi-Kober (E-K) operator

$F_\alpha U(\eta):=\int_0^1 (1-s)^{-\alpha} U\left(\eta s^{-\frac{\alpha}{2}}\right)ds.$

The transformed equation still looks formidable but, as a matter of fact, it is easier to analyse, solve and approximate [6]. In a series of papers [6-9] we have obtained a number of different results from which we would like to mention only one of them – an efficient way of finding approximate asymptotic solutions.

It appears that the nonlocal E-K operator can be accurately approximated by an infinite series of local operators (see [7,8]). This, in turn, can be used to effectively solve the governing equation or, at least, to obtain its approximate solution. The difficulties arise due to the character of the problem – it has a free boundary which has to be found altogether with the solution. The resulting, first order, approximation to the important case $D(u)=u^m$ for $m>1$ can be written in a very simple form

$U(\eta)\approx\left(1-\frac{\eta}{\eta^*}\right)^\frac{1}{m},$

where the approximate wetting front position $\eta^*$ is known explicitly. From here, we can go back to the dimensional variables and fit the above model with the real data to obtain diffusivity – one of the most important characteristics of the porous medium [9].

The advantage of this method is straightforward. Instead of fitting the numerical solution of the nonlocal and nonlinear governing equation, we can use an elementary power-type function. The computation cost is then hugely reduced but, of course, for the price of the accuracy. It is then a matter of experimental error to judge for how much discrepancies we can allow but still, our method should be sufficiently accurate for the field experiments. Below we present a plot showing the accuracy in fitting.

Fitting the model with the real data obtained from [4]. The subsequent lines represent measurements for increasing time. Fitted parameters are α=0.855 and m=6.98.

We can see that the fit is decent and the sample shows essential subdiffusive character. The nonlinear governing fractional equations described the process adequately.

The future work consists of developing a stable and accurate numerical methods for solving the governing equation (there is a need for rigorous convergence analysis).

## References

[1] C. Hall, Anomalous diffusion in unsaturated flow: Fact or fiction?, Cement and Concrete Research 37 (2007) 378–385.

[2] L. Pel, K. Kopinga, G. Bertram and G. Lang, Water absorption in a fired-clay brick observed by NMR scanning, J. Phys. D.: Appl. Phys. 28 (1995) 675–680.

[3] K. Hazrati, L. Pel, J. Marchand, K. Kopinga and M. Pigeon, Determination of isothermal unsaturated capillary flow in high performance cement mortars by NMR imaging, Mater. Struct. 35 (2002) 614–622.

[4] Abd El-Ghany, El Abd and J.J. Milczarek, Neutron radiography study of water absorption in porous building materials: anomalous diffusion analysis, J. Phys. D.: Appl. Phys. 37 (2004) 2305–2313.

[5] Metzler, Ralf, and Joseph Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics reports 339(1) (2000): 1-77.

[6] Ł.Płociniczak, H.Okrasińska, Approximate self-similar solutions to a nonlinear diffusion equation with time-fractional derivative, Physica D 261 (2013), 85-91.

[7] Ł.Płociniczak, Approximation of the Erdelyi-Kober fractional operator with application to the time-fractional porous medium equation, SIAM Journal of Applied Mathematics 74(4) (2014), 1219–1237,

[8] Ł.Płociniczak, Analytical studies of a time-fractional porous medium equation. Derivation, approximation and applications, Communications in Nonlinear Science and Numerical Simulation 24 (1–3) (2015), 169-183

[9] Ł.Płociniczak, Diffusivity identification in a nonlinear time-fractional diffusion equation, Fractional Calculus and Applied Analysis 19(4) (2016), pp. 883-866,