## FRACTAL DERIVATIVE MODEL OF WATER ANOMALOUS ADSORPTION IN SWELLING SOIL1)

TIAN Peibo,2), LIANG Yingjie,3)

College of Mechanics and Materials, Hohai University, Nanjing 211100, China

 基金资助: 1)国家自然科学基金资助项目(11702085)

2)田沛博,硕士研究生,主要研究方向为反常扩散力学建模。E-mail:tpb15951767508@163.com

Abstract

In this paper the fractal derivative model of water adsorption in swelling soil was constructed based on the material coordinates, which correlates the moisture content with the spatial position. The cumulative adsorption of water in swelling soil was also derived. The cumulative adsorption in swelling soil underlying the fractal derivative model is a function of the fractal derivative of time and the diffusion coefficient. The fractal derivative order can be used to classify the adsorption process and to characterize the heterogeneity of soil. The feasibility of the fractal derivative model is verified by analyzing the experimental data of the water cumulative adsorption in the black soil and sand, and the proposed model exhibits higher accuracy than the traditional integer order model.

Keywords： material coordinate; cumulative adsorption; swelling soil; fractal derivative; anomalous diffusion

TIAN Peibo, LIANG Yingjie. FRACTAL DERIVATIVE MODEL OF WATER ANOMALOUS ADSORPTION IN SWELLING SOIL1). Mechanics in Engineering, 2022, 44(2): 317-321 DOI:10.6052/1000-0879-21-391

## 1 分形导数累积吸附模型

### 1.1 物质坐标与分形导数

$m=\int_0^{z_{1}} \left( 1+e \right)^{-1}{\rm d}z$

$I\left( t \right)=\int m {\rm d}\vartheta$

$\frac{{\rm d}\vartheta }{{\rm d}{t}^{{\alpha }}} =\frac{\vartheta \left( t \right)-\vartheta \left(t_{1}\right)}{(t-t_{0})^{\alpha}-(t_{1}-t_{0})^{\alpha}} =\frac{1}{\alpha(t-t_{0})^{\alpha-1}}\frac{{\rm d}\vartheta }{{\rm d}t}$

$\frac{{\rm d}\vartheta }{{\rm d}t^{\alpha }} =\frac{\vartheta\left( t \right)-\vartheta \left( t_{1} \right)}{t^{\alpha }-t_{1}^{\alpha }} =\frac{1}{\alpha t^{\alpha {\rm -1}}}\frac{{\rm d}\vartheta }{{\rm d}t}$

$\frac{{\rm d} m}{{\rm d} x^{\beta}}=\lim _{x \rightarrow x^{\prime}} \frac{m(x)-m(x')}{x^{\beta}-x'^{\beta}}=\frac{1}{\beta x^{\beta-1}} \frac{{\rm d} m}{{\rm d} x}$

$\frac{\partial p\left( x,t \right)}{\partial t^{\alpha}} =\frac{p\left( x,t \right)-p\left(x,t_{1} \right)}{t^{\alpha}-t_{1}^{\alpha }}$
$\frac{p(x, t)}{\partial x^{\beta}}=\lim _{x_{1} \rightarrow x} \frac{p(x, t)-p\left(x_{1}, t\right)}{x^{\beta}-x_{1}^{\beta}}$

$\left. \begin{array}{*{20}c} t^{\alpha } =\hat{t}\\ x^{\beta } =\hat{x}\\ \end{array} \right\}$

$\frac{\partial p\left( x,t \right)}{\partial\hat{t}} =D_{\alpha,\beta }\hat{\nabla}^{2}p\left( x,t \right)$

$p\left( x,t \right) =\frac{1}{\sqrt {4\pi D_{\alpha,\beta }\hat{t}}}{\rm e}^{-\frac{\hat{x}^{2}}{4D_{\alpha,\beta }\hat{t}}}$

$p(x,t)=\frac{1}{\sqrt {4\pi D_{\alpha,\beta }\hat{t}^{\alpha }} }{\rm e}^{-\hat{x}^{2\beta }/(4D_{\alpha,\beta }\hat{t}^{\alpha })}$

$\left\langle x^{2}\left( t \right) \right\rangle \sim t^{\frac{3\alpha -\alpha \beta }{2\beta }}$

### 1.2 膨胀性土壤累积吸附的分形导数模型

$\frac{\partial \vartheta }{\partial t^{\alpha }} =\frac{\partial }{\partial m}\left( D_{m}\frac{\partial \vartheta }{\partial m} \right)$

$\vartheta =\frac{1}{\sqrt {4\pi D_{m}t^{\alpha }} }\exp \left( -\frac{m^{2}}{4D_{m}t^{\alpha }} \right)$

$m =\sqrt {{\rm -4}D_{m}t^{\alpha }\ln {(\sqrt{4\pi D_{m}t^{\alpha }} }\vartheta } )$

$I(t)=\int_{\vartheta_{i}}^{\vartheta_{0}} \sqrt{-4D_{m}t^{\alpha }\ln (\sqrt {4\pi D_{m}t^{\alpha }} \vartheta )} {\rm d}\vartheta$

### 图1

Fig.1   Curves of cumulative adsorption for $\alpha =0.4$,0.6, 0.8,1.0 in the fractal derivative model

$I\left( t \right) =S{\rm \cdot }t^{{\rm 1/2}}$

## 2 分形导数模型应用与验证

### 2.1 膨胀性黑土中累积吸附过程应用实例

$RMSE\left( I,n \right) =\sqrt{\frac{1}{n}\sum\nolimits_i^n{(I_{1}-I_{0})^{2}}}$

### 图2

Fig.2   Plots of cumulative adsorption capacity of water in black soil by usingthe fractal derivative model and integer order model

### 图3

Fig.3   Plots of cumulative adsorption of water in the sand by using the fractal derivative model and integer order model

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Sun HG, Meerschaert MM, Zhang Y, et al.

A fractal Richards' equation to capture the non-Boltzmann scaling of water transport in unsaturated media

Advances in Water Resources, 2013, 52:292-295

Lin GX.

An effective phase shift diffusion equation method for analysis of PFG normal and fractional diffusions

Journal of Magnetic Resonance, 2015, 259:232-240

Reyes-Marambio J, Moser F, Gana F, et al.

A fractal time thermal model for predicting the surface temperature of air-cooled cylindrical lion cells based on experimental measurements

Journal of Power Sources, 2016, 306:636-645

Balamlin AS, Boryreyes J, Shapiro M,

Towards a physics on fractals: differential vector calculus in three dimensional continuum with fractal metric

Physica A, 2016, 444:345-359

Hu ZH, Tu XK.

A new discrete economic model involving generalized fractal derivative

Advances in Difference Equations, 2015, 1:1-11

Chen W.

Time-space fabric underlying anomalous diffusion

Chaos, Solitons ＆ Fractals, 2006, 28(4):923-929

Chen Wen, Liang Yingjie, Yang Xu.

Soil infiltration rates and hydrology model classifications based on the Hausdorfffractal derivative Richards' equation

Applied Mathematics and Mechanics, 2018, 39(1):77-82 (in Chinese)

Qu Yi, Sun Hongguang, Li Zhipeng.

The influence of medium permeability on diffusion process in fractal structure

Science Technology and Engineering, 2019, 19(7):15-19 (in Chinese)

Wang Shuhong, Liang Yingjie.

Hausdorff fractal derivative diffusion imaging model for anomalous diffusion in sephadex gel

Chinese Quarterly of Mechanics, 2020, 41(4):666-673 (in Chinese)

Quan Ming.

Classification of expansive soil and construction scheme of foundations in expansive soil area

Cement Technology, 2021(1):73-80 (in Chinese)

Zhang Chunxiao, Xiao Hongbin, Bao Jiamiao, et al.

Stress relaxation model of expansive soils based on fractional calculus

Rock and Soil Mechanics, 2018, 39(5):1747-1760 (in Chinese)

Su N.

Equations of anomalous absorption onto swelling porous media

Materials Letters, 2009, 63(28):2483-2485

Su N.

Theory of infiltration: infiltration into swelling soils in a material coordinate

Journal of Hydrology, 2010, 395(1):103-108

Gan Yongde, Liu Huan, Jia Yangwen, et al.

Swelling soil saturated water movement parameters calculating models

Engineering Science and Technology, 2018, 50(2):77-83 (in Chinese)

Jin Wenting, Xiao Hongbin.

Experimental study on shear creep characteristic and rheological model of Nanning expansive soils

Highway Engineering, 2011, 36(6):64-69, 77 (in Chinese)

Voller VR.

A direct simulation demonstrating the role of special heterogeneity in determining anomalous diffusive transport

Water Resources Research, 2015, 51(4):2119-2127

Li Zhenyu, Xiao Hongbin, Jin Wenting, et al.

Study of nonlinear rheological model of Nanning expansive soils

Rock and Soil Mechanics, 2012, 33(8):2297-2302 (in Chinese)

Diffusive transport of volatile pollutants in nonaqueous-phase liquid contaminated soil: a fractal model

Transport in Porous Media, 1998, 30(2):125-154

Liang YJ, Chen W, Magin RL.

Connecting complexity with spectral entropy using the Laplace transformed solution to the fractional diffusion equation

Physica A, 2016, 453:327-335

Zhang Donghui, Shi Mingheng, Jin Feng, et al.

Diffusion characteristic in fractal porous media

Journal of Engineering Thermophysics, 2004, 25(5):825-827 (in Chinese)

Smiles DE, Rosenthal Margaret J.

The movement of water in swelling materials

Soil Research, 1968, 6:237-248

Karalis TK.

Mechanics of Swelling

Berlin:Springer-Verlag, 1992

Philip JR, Smiles DE.

The steady-state measurement of the relation between hydraulic conductivity and moisture content in soils

Water Resources Research, 1968, 5:1029-1030

Smiles DE.

Hydrology of Swelling Clay Soils. Encyclopedia of Hydrological Sciences

Chichester: Wiley, 2005

Smith RE.

Infiltration Theory for Hydrologic Applications

Washington, DC: American Geophysical Union, 2002

Stockdale A, Banwart S.

Recovery of technologically critical lanthanides from ion adsorption soils

Minerals Engineering, 2021, 168:106921

Bridge BJ.

An experimental study of vertical infiltration into a structurally unstable swelling soil, with particular reference to the infiltration throttle

Soil Research, 1973, 11(2):121-132

/

 〈 〉