C HAPTER 2
2.4 Summary
withωD+ωO+ωS=1 and ω ∈[0,1].
Here three weighting factors (ω), subscripted asD,OandS, represent volumetric fraction of dead-end pore (ωD) (saturated or unsaturated), of open-end unsaturated pores (ωO) and of open-end saturated pore (ηS) relative to total void in an averaging volume of soil. Researchers (Coats and Smith,1964;Jackson and Klute,1967;Wirner et al.,2014;Phirani et al.,2018, and others) gave methods to measure the immobile or stagnant or dead-end pore volume and they always distinguished two domains – mobile and immobile zone, but only in the terms of fully saturated soil. As per Eq. (2.29), mobile zone comprises of whole open-end pores (saturated as well as unsaturated ones) and immobile zone is of dead-end pore. If the dead-end pore is only the unsaturated part of the soil, then the term ∂ ψ
∂T
open-endturns into zero and
∂ ψ
∂T
dead-end term only survives and then this term will havesmalleffect on ∂ ψ
∂T η
and thus on the liquid flux. Therefore, the impact of dead-end is relative to physical condition of the soil or any other porous medium and depends on equilibrium contact angle of liquid over the solid matrix, weighting fraction and physical dimension of dead-end pores in that particular porous medium. When whole soil is saturated, the temperature derivative term becomes zero and contribute none to actual liquid flux. However, we limit the present study as described here and suggest the generic cases as future scope along with sophisticated experimentations.
The effect of temperature on liquid-gas meniscus curvature in the open-end capillary tube also showed linear drop of capillary pressure with temperature. However, the magnitudes of capillary pressures are much lower compared to the dead-end pore cases. As both the closed-end and open-end capillary tubes show linear drop in capillary pressures with rise in temperatures, the phenomena is critical in designing the subsurface drainage of irrigated fields.
The drainage of water may increase more than intended during temperature rise as the capillary pressure reduces. Closed-end capillary tubes with very large air column at the top behave approximately like the open-end tubes with respect to capillary pressure-vs.-temperature. The results presented here are consistent with the experimental data qualitatively. In addition, the chapter, for the first time, provides a conceptual insight into the dead-end pores’ significance on temperature effects on capillary pressure or on capillary pressure-vs.-saturation relationship at a particular liquid saturation.
2.4.1 Questions?
Some questions remain to be answered and those are the practicability or validity of the above assumptions. Capillary force (∼σ) are dominant than gravity force (∼(ρl−ρa)gL2) and this assumption is true for capillary tubes (De Gennes et al.,2004). Our main focus being capillary pressure in porous media, this assumption is vital in this thesis. We discuss the plausibility of CLP in the next chapter. Linearity assumption of interfacial tension (γ)vs. temperature (T) is valid, cause of, almost linear trend betweenγ andT. Quadratic relation can be fitted as: γ=a+bT+cT2. The experimental results (seeNeumann et al.,1971;Neumann,1974) show that the quadratic coefficient|c|has a very low value than|b|((|c|/|b|)<<1)and thus, proves to be negligible without losing any loss of generality. Air-diffusivity into liquid is null and this means that there is an equilibrium between liquid and air phases. This is only possible when air is saturated with liquid vapor.However, phase equilibrium always exists at the interface of two phases of a species(fromC¸ engel and Boles,2015, p. 823). Therefore, our assumption is an extension of phase-equilibrium criterion. We assume that within this small close-end capillary tube, this criterion exists. Last but not the least, the shape of the meniscus isassumedbeing of spherical, even when CLP of meniscus exists. This assumption puts limits on temperature range (see Appendix-A2.1). Plausibility of this range may be questioned.
Assumption of meniscus shape is taken mostly as spherical, mainly because of lesser effect of gravity in capillary tubes (viz. raindrop will be of perfect sphere shape under no gravity). We argue and question this assumption in the next chapters. We try to understand how meniscus shape has an effect on changes in contact angle and thus, eventually on capillary pressure.
A2 Appendix
A2.1 Limitation of temperature based on meniscus height
The values of meniscus height (hm) with different temperatures are shown in the following Fig.
A2.1for two example cases (G-W-A and P-N-A systems). It is pretty clear thathmincreases with temperature. Our assumption is that meniscus shape is spherical even if meniscus sticks to same place while temperature changes.
0 10 20 30 40 50
298.12 298.13 298.14 298.15 298.16 298.17 298.18 298.19 298.2 298.21 298.22
Meniscus Height (h m) (µm)
Temperature (K)
0 10 20 30 40 50
298.06 298.08 298.1 298.12 298.14 298.16 298.18 298.2 298.22 298.24 298.26
(b) (a)
Figure A2.1:T-vs.-hmplots for: (a) G-W-A system and (b) P-N-A system fromR=0 to 50µm
As a consequence the contact angle decreases with the increase of hm. Contact angle is calculated using following equation:
θe(T) =tan−1 R2−hm(T)2 2Rhm(T)
! within
h 0,π
2 i
(A2.1)
Equation (A2.1) suggests that contact angle (θe) becomes 0◦whilehmreaches theRi.e. in our case 50µm. Similarly with decrease in temperature, contact angle eventually becomes 90◦whilehmis zero. This puts a limitation onhmand that is:hm∈[0,R]. FigureA2.1shows thatθeandhmbecome zero andRrespectively at same temperature. Now limitation onhmor θegives the highest or lowest temperatures. These temperatures can be obtained from fitting relationship T(hm) athm=0 andR for lowest and highest temperatures respectively (say, Tlowest andThighest). Here, the fitting relationship with T(hm=R) andT(hm=0)provide:
(a) for G-W-A system,Thighest=298.21 K andTlowest=298.12 K and (b) for P-N-A system, Thighest=298.26 K andTlowest=298.1 K, respectively.