Generalized hydraulic conductivity model for capillary and adsorbed film flow

Abstract

Commonly used transport models of unsaturated flow assume that the movement of pore water is dominated mainly by capillary flow and they neglect adsorbed film flow. These models have been proven to be successful at high and intermediate saturations but typically underestimate the hydraulic conductivity in the dry range, where water movement in equilibrium conditions is dominated by adsorbed film flow. Given these considerations, this paper proposes a simplified configuration of pore water that accounts for the transport processes of both capillary and film flow. Based on the mechanisms of soil water retention, a conception of the specific thickness of the adsorbed film is defined to describe the adsorption strength and adsorption capacity of porous media. Furthermore, a statistical physically based model of relative hydraulic conductivity in the full range of suction is derived. Fractal and Monte Carlo methods are used to determine the pore size distribution of porous media and then the corresponding specific model of relative hydraulic conductivity is derived. The results show that the proposed model agrees well with the experimental data in the entire suction range. It is also found that the pore size distribution of porous media controls the transport characteristic of capillary water but not adsorption film flow which is only related to the mineral content, mineral species, and specific surface area. Additionally, the influences of the model parameters on the transport of porous media are also addressed.

Résumé

Les modèles de transport couramment utilisés d’écoulement en zon non saturée supposent que le mouvement de l’eau porale est dominé principalement par le flux capillaire et ils négligent l’écoulement associé au film adsorbé. Ces modèles se sont avérés être couronnés de succès à des saturations élevées et intermédiaires, mais sous-estiment généralement la conductivité hydraulique dans la gamme sèche, où le mouvement de l’eau dans des conditions d’équilibre est dominée par l’écoulement associé au film adsorbé. Compte tenu de ces considérations, cet article propose une configuration simplifiée de l’eau porale qui tient compte des processus de transport aussi bien de l’écoulement capillaire que celui associé au film. A partir des mécanismes de rétention d’eau du sol, une conception de l’épaisseur spécifique du film adsorbé est définie pour décrire la force d’adsorption et la capacité d’adsorption des médias poreux. En outre, un modèle statistique à base physique de conductivité hydraulique relative pour toute la gamme de succion est dérivé. Les méthodes des fractales et de Monte Carlo sont utilisées pour déterminer la répartition de la taille des pores du milieu poreux, puis le modèle spécifique correspondant de conductivité hydraulique relative en est déduit. Les résultats montrent que le modèle proposé est cohérent avec les données expérimentales pour toute la gamme de succion. On constate également que la répartition de la taille des pores du milieu poreux contrôle la caractéristique de transport de l’eau capillaire, mais pas l’écoulement associé au film d’adsorption qui est seulement lié à la teneur minérale, aux espèces minérales et à la surface spécifique. En outre, les influences des paramètres du modèle sur le transport en milieu poreux sont également abordées.

Resumen

Los modelos de transporte de flujo no saturado comúnmente utilizados asumen que el movimiento del agua en los poros está dominado principalmente por el flujo capilar y descuidan el flujo pelicular por absorción. Estos modelos han demostrado ser exitosos para saturaciones altas e intermedias, pero típicamente subestiman la conductividad hidráulica en el rango seco, donde el movimiento del agua en condiciones de equilibrio está dominado por el flujo pelicular por adsorción. Teniendo en cuenta estas consideraciones, en el presente documento se propone una configuración simplificada del agua en los poros que da cuenta de los procesos de transporte del flujo capilar y pelicular. Basándose en los mecanismos de retención de agua del suelo, se define una noción del espesor específico de la película adsorbida para describir la fuerza de adsorción y la capacidad de adsorción de los medios porosos. Además, se obtiene un modelo estadístico basado físicamente en la conductividad hidráulica relativa en todo el rango de la succión. Se utilizan los métodos Fractal y Monte Carlo para determinar la distribución del tamaño de los poros de un medio poroso y luego se deriva el correspondiente modelo específico de conductividad hidráulica relativa. Los resultados muestran que el modelo propuesto se corresponde con los datos experimentales en todo el rango de la succión. También se ha comprobado que la distribución del tamaño de los poros de los medios porosos controla la característica de transporte del agua capilar pero no el flujo pelicular por adsorción, que sólo está relacionado con el contenido mineral, las especies minerales y la superficie específica. Además, también se abordan las influencias de los parámetros del modelo en el transporte de medios porosos.

摘要

常用的非饱和流传输模型假设孔隙水的运动主要由毛细水控制,而忽略了吸附膜的水。这些模型已被证明在高和中等饱和度条件下是适用的,但通常会低估干燥范围内的渗透系数,而在干燥条件下,平衡条件下水运动主要由吸附膜水决定。考虑到这些考虑因素,本文提出了一种简化的孔隙水构型,该构型考虑了毛细水和薄膜水的传输过程。基于土壤持水机理,定义了吸附膜比厚度的概念来描述多孔介质的吸附强度和吸附能力。此外,推导了在整个吸力范围内的基于物理统计的相对渗透系数模型。分形和蒙特卡罗方法用于确定多孔介质的孔径分布,然后推导相应的相对渗透系数的特定模型。结果表明,所提出的模型与整个吸力范围内的实验数据吻合良好。还发现,多孔介质的孔径分布控制着毛细水的传输特性,而不控制吸附膜的流动,而吸附膜的流动仅与矿物含量,矿物种类和比表面积有关。另外,还讨论了模型参数对多孔介质传输的影响。

Resumo

Modelos de transporte comumente utilizados no fluxo em zona não saturada presumem que o movimento da água nos poros é dominado principalmente pelo fluxo de capilaridade e eles negligenciam o fluxo em filme-adsorvente. Esses modelos provaram ser bem sucedidos em saturações altas e intermediárias, mas tipicamente subestimam a condutividade hidráulica em alcance seco, onde o movimento da água em condições equilibradas é dominado pelo fluxo do filme-adsorvente. Dadas essas condições, esse trabalho propõe uma configuração simplificada da água nos poros que é considerada para os processos de transporte para ambos capilaridade e fluxo de filme. Baseado nos mecanismos de retenção de água no solo, uma conceitualização da espessura especifica do filme-adsorvente é definida para descrever a força e a capacidade de adsorção para o gradiente poroso. Além disso, um modelo estatístico fisicamente embasado da condutividade hidráulica relativa em um alcance total de sucção é derivado. Métodos de fractal e Monte Carlo são usados para determinar a distribuição do tamanho do poro do gradiente poroso e então o modelo específico correspondente da condutividade hidráulica relativa é derivada. Os resultados mostram que o modelo proposto concorda bem com os dados experimentais no alcance inteiro de sucção. Também é encontrado que a distribuição do tamanho do poro do gradiente poroso controla das características do transporte da água na capilaridade, mas não no fluxo de filme-adsorvente que está apenas relacionado com o conteúdo mineral, espécies minerais, e área de superfície específica. Adicionalmente, as influências dos parâmetros do modelo no transporte do gradiente poroso também são endereçadas.

Appendices

Appendix A: Monte Carlo simulation of fractal basis

According to the fractal scaling law, the total pore number from r_min to r_max can be determined by Eq. (26):

$$ {N}_{\mathrm{t}}={\left(\frac{r_{\mathrm{min}}}{r_{\mathrm{max}}}\right)}^{-{D}_{\mathrm{f}}} $$

(38)

Combining Eqs. (27) and (38), it can be obtained that

$$ -\frac{\mathrm{d}N(r)}{N_{\mathrm{t}}}={D}_{\mathrm{f}}{r}_{\mathrm{min}}^{D_{\mathrm{f}}}{r}^{-{D}_{\mathrm{f}}-1}\mathrm{d}r=f(r)\mathrm{d}r $$

(39)

where $f(r)={D}_{\mathrm{f}}{r}_{\mathrm{min}}^{D_{\mathrm{f}}}{r}^{-{D}_{\mathrm{f}}-1}$ is defined as the probability density function (Yu et al. 2005; Xu et al. 2013). Additionally,

$$ {\int}_{r_{\mathrm{min}}}^{r_{\mathrm{max}}}f(r)\mathrm{d}r=1-{\left(\frac{r_{\mathrm{min}}}{r_{\mathrm{max}}}\right)}^{D_{\mathrm{f}}} $$

(40)

From the viewpoint of probability theory, (r_min/r_max)^Df = 0 must be satisfied so that the probability density function f(r) is meaningful. Fortunately, for natural porous media (e.g., soil), r_min/r_max<<10⁻². Hence, (r_min/r_max)^Df = 0 holds approximately.

Then, the cumulative probability R(r) of pores with size from r_min to r can be captured by integrating the probability density function f(r) as follows:

$$ R(r)={\int}_{r_{\mathrm{min}}}^rf(r)\mathrm{d}r=1-{\left(\frac{r_{\mathrm{min}}}{r}\right)}^{D_{\mathrm{f}}} $$

(41)

From Eq. (41), it is clear that R=0 as r=r_min and R=1 as r=r_max. Since the size of pores is randomly distributed in the range of r_min – r_max for natural porous media, R is a set of random numbers in the range of 0–1.

Rearranging Eq. (41) yields

$$ r=\frac{r_{\mathrm{min}}}{{\left(1-R\right)}^{1/{D}_{\mathrm{f}}}} $$

(42)

Equation (42) indicates that the pore size in the range of r_min–r_max can be determined at a given random number R with a range of 0–1. Therefore, Eq. (42) can be seen as a Monte Carlo probability model capturing the pore size.

Combining Eqs. (25) and (42), the Monte Carlo model of relative hydraulic conductivity in the full range of the matric head can be obtained:

$$ {k}_{\mathrm{r}}=\left\{\begin{array}{c}\left[\sum \limits_{i=1\left({r}_{\mathrm{min}}\right)}^{j\left({r}_{\mathrm{c}}\right)}\frac{1}{{\left(1-{R}_i\right)}^{4/{D}_{\mathrm{f}}}{\tau}_i}+\frac{8}{3}\sum \limits_{i=j\left({r}_{\mathrm{c}}\right)}^{n\left({r}_{\mathrm{max}}\right)}\left(\frac{2{\beta}^3{\left(1-{R}_c\right)}^{-3\alpha /{D}_{\mathrm{f}}}}{{\left(1-{R}_i\right)}^{\left(4-3\alpha \right)/{D}_{\mathrm{f}}}{\tau}_i}-\frac{\beta^4{\left(1-{R}_c\right)}^{-4\alpha /{D}_{\mathrm{f}}}}{{\left(1-{R}_i\right)}^{\left(4-4\alpha \right)/{D}_{\mathrm{f}}}{\tau}_i}\right)\right]/\sum \limits_{i=1\left({r}_{\mathrm{min}}\right)}^{n\left({r}_{\mathrm{max}}\right)}\frac{1}{{\left(1-{R}_i\right)}^{4/{D}_{\mathrm{f}}}{\tau}_i}\kern0.66em ,h\le {h}_{c,\min}\\ {}\\ {}\left(\frac{16}{3}{\beta}^3{\xi}^{3\alpha}\sum \limits_{i=1\left({r}_{\mathrm{min}}\right)}^{n\left({r}_{\mathrm{max}}\right)}\frac{1}{{\left(1-{R}_i\right)}^{\left(4-3\alpha \right)/{D}_{\mathrm{f}}}{\tau}_i}-\frac{8}{3}{\beta}^4{\xi}^{4\alpha}\sum \limits_{i=1\left({r}_{\mathrm{min}}\right)}^{n\left({r}_{\mathrm{max}}\right)}\frac{1}{{\left(1-{R}_i\right)}^{\left(4-4\alpha \right)/{D}_{\mathrm{f}}}{\tau}_i}\right)/\sum \limits_{i=1\left({r}_{\mathrm{min}}\right)}^{n\left({r}_{\mathrm{max}}\right)}\frac{1}{{\left(1-{R}_i\right)}^{4/{D}_{\mathrm{f}}}{\tau}_i}\kern1.32em ,h>{h}_{c,\min}\end{array}\right. $$

(43)

where R_c is a random number corresponding to the critical radius. ξ=r^*/r_min is a parameter indicating that the suction characteristic radius r^* is smaller than r_min, subject to ξ<=1.

Appendix B: notation

A _F :

Cross-section area of film flow for a single pore (L²)

A _F :

Hamaker constant (ML²T⁻²)

A _REV :

Cross-section area of the REV (L²)

C _f :

Fractal factor

D _f :

Area fractal dimension

D _T :

Tortuosity fractal dimension

e :

Electron charge (TI)

g :

The gravity acceleration constant (LT^–2)

h :

Matric head (L)

h _c,max :

Maximum matric head retained by capillarity (L)

h _c,min :

Minimum matric head retained by capillarity (L)

k _B :

Boltzmann constant (ML²T⁻²Θ⁻¹)

k _r :

Relative hydraulic conductivity

K _s :

Saturated hydraulic conductivity (LT⁻¹)

L ₀ :

Representative length of a pore (L

L _t :

Tortuous length of a pore (L)

n :

Porosity

∆P:

Pressure drop (ML⁻¹ T⁻²)

_Q ^C _U,t :

Total volumetric flow rate of capillary water (L³T⁻¹)

_Q ^F _U,t :

Total volumetric flow rate of film flow (L³T⁻¹)

q ^F :

Volumetric flow rate of film flow for a single pore (L³T⁻¹)

Q _S,t :

Total volumetric flow rate of saturated REV (L³T⁻¹)

Q _U,t :

Total volumetric flow rate of unsaturated REV (L³T⁻¹)

r :

Radius of grains or pores (L)

r ^* :

Suction characteristic radius (L)

r _c :

Critical pore radius (L)

r _i :

Radius of ith pore (L)

r _max :

Maximum pore radius (L)

r _min :

Minimum pore radius (L)

S :

Surface area of the grain (L²)

S _specific :

Specific surface area of the grain (L⁻¹)

T :

Kelvin temperature (Θ)

V :

Grain volume (L³)

${\overline{v}}_{\mathrm{p}}$ :

Average liquid velocity (LT⁻¹)

Z :

Ion change

Π :

Disjoining pressure (ML⁻¹ T⁻²)

Π _e :

Ionic-electrostatic component of disjoining pressure (ML⁻¹ T⁻²)

Π _m :

Molecular component of disjoining pressure (ML⁻¹ T⁻²)

Π _s :

Structure component of disjoining pressure (ML⁻¹ T⁻²)

α :

Adsorption strength of materials

β :

Adsorption capacity of materials

δ :

Specific thickness of adsorbed water

ε :

Relative permittivity of water

ε ₀ :

Permittivity of free space (M⁻¹L⁻³T⁴Ι²)

λ :

Shape factor

μ :

Viscosity of pore water (ML⁻¹ T⁻¹)

θ :

Contact angle

θ _a :

Adsorbed water film volume of a grain (L³)

ρ :

Density of pore water (ML⁻³)

σ :

Water surface tension (ML⁻¹ T⁻²)

τ _i :

Tortuosity of ith pore

τ :

Pore tortuosity

ω :

Film thickness (L)