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Review of Some Elements of Soil-Moisture Theory John G. Fe.rris Review of Some Elements of Soil-Moisture Theory 2 SH GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-D | XJ e3 i i g> Review of Some Elements of Soil-Moisture Theory By IRWIN REMSON and J. R. RANDOLPH FLUID MOVEMENT IN EARTH MATERIALS GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-D UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1962 UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY Thomas B. Nolan, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. CONTENTS Page Page List of symbols.____________________________________ IV Potentials in soil water Continued Abstract...________________________________________ Dl Osmotic-pressure potential.______________________ D20 Introduction.______________________________________ 1 Adsorption potential-______________------------- 21 Zone of aeration._-_--_--__________-__________-_____ 2 Thermal potential-________-_____--------------- 21 Chemical potential..---------------------------- 22 Background physics __--__---.-____________________ 2 Total potential and its measurement______________ 22 Surface tension.________________________________ 2 Molecular causes of surface tension___________ 2 Equilibrium distribution of potential and water in soil__ 24 Definition of surface tension._________________ 3 Distribution in the zone of saturation _____________ 24 Pressure differential across a liquid-gas interface _ 3 Distribution in the capillary fringe_______________ 24 Angle of contact.___________________________ 5 Distribution in pinching and swelling tubes Rise of water in a capillary tube..____________ 5 without horizontal connections____-____-_-_ 25 Water vapor in unsaturated soils_________________ 6 Distribution in tightly packed spheres-________ 27 Molecular causes of vapor pressure.___________ 6 Distribution above the capillary fringe.----------- 28 Vapor pressure and temperature._____________ 6 Distribution in tightly packed spheres.- - ______ 28 Vapor pressure and amount of dissolved solute _ 7 Distribution in randomly distributed pores____- 29 Vapor pressure and hydrostatic pressure in the Movement of soil moisture.-.------------------------ 29 liquid ---------_---------__-_____________ 7 Physical picture. __________________-----------_- 29 Barometric equation ________________________ 7 Pore underlain by a cavity_______-__--------- 30 Kelvin equation. ___________________________ Pore underlain by pores of similar size ______ 31 Derivation.____________________________ Pore underlain by pores of dissimilar size_ _ __ __ 31 Significance._ __________________________ 10 Randomly distributed pores________-_-------_ 32 Diffusion. _____________________________________ 12 Differential equation for unsaturated flow _ _ _ _ _ _ _ 33 Osmosis. ______________________________________ 13 Historical background.-_________-----------_ 33 Force, work, energy, potential, and gradient______ 14 Equation of continuity. _____-__-_-___--_---- 34 Potentials in soil water______________________________ 17 Darcy's law.____________-_____-_----------- 36 Gravitational potential._________________________ 17 Flow equation_________________-_------_---- 36 Hydrostatic-pressure potential.___________________ 19 References.-__--___-_-___---_-------------__--__-_- 37 ILLUSTRATIONS Page FIGURE 1. Intermolecular force as a function of molecule separation._______________________-_-----_--_-------_---_-_ D2 2. Bubble of air blown at the end of a capillary tube______________________________ 4 3. Contact angle of a liquid-vapor surface near a solid. __________________---_-_-__-- 5 4. Rise of water in a capillary tube._________________________________-----_-----_. 5 5. Column of vapor in a gravitational field-_______-__-____-____-____---__--_-_---- 8 6. A curved liquid-vapor surface in contact with a free flat liquid surface._________--_. 9 7. Position of the meniscus in a capillary tube and adjacent vapor pressures.---------- 12 8. Position of the meniscus in a wedge-shaped pore and adjacent vapor pressures_-__-_- 12 9. Ground-water gradient------_______________________________-__--_--__-----_-- 16 10. Approximate flow net in the vicinity of an irrigation pond_______________________ 16 11. Moisture characteristic of a fine sand__------_______-_-__-_----_-------_----__-- 23 12. Plot of the capillary-rise equation for water. _________-_________----__----_-_---- 25 13. Hypothetical variation of pore radius with height above the water table.__-_--_--_- 26 14. Plot Of the capillary-rise equation on the same scale used in figure 13_-------------- 26 15. Unit rhombohedron of a mass of spheres in the most compact packing........______ 28 16. Unit element of pore space in a mass of spheres packed in the most compact manner. 28 17. Dewatering of a main pore and the reentrants_____________-__._________-___----- 28 18. Pore underlain by a cavity.___________________________________________________ 30 19. Pore underlain by a pore of similar size.________________________________________ 31 20. Pore underlain by a smaller pore__----_-_-__-_-_---_-----_--___-_-------------- 31 21. Pore underlain by a larger pore________________________________________________ 32 22. Unit rectangular parallelepiped-________________-__________-__._--_---_---__--- 34 in LIST OF SYMBOLS a acceleration r radius of a capillary tube A area of a liquid surface r0 critical distance of separation between the molecules b mole fraction of dissolved material of a liquid C a constant R radius of curvature of a hemispherical liquid surface d a digit or group of digits RI and R2 principal radii of curvature of a liquid dS area of an infinitesimal element of capillary surface surface D density of a liquid or gas R the gas constant D(tt} diffusivity as a function of concentration of ions dS area of a small element of capillary surface or molecules T absolute temperature e the exponential base, 2.71828 . t time / a digit u a scalar point function such as potential F force or resultant force v volume of a mass of water Fa force of adhesion Vi specific volume of a liquid Fc force of cohesion vv specific volume of a vapor Fg force of gravity V mass flow of moisture (a vector) g acceleration of gravity Vx, Vv, Vz components of the mass flow of moisture in G gravitational potential the x, y, z directions Gm gravitational potential in ergs per grain of water Vv volume flow of moisture (a vector) G, gravitational potential in ergs per cubic centimeter W work or energy of water x, y, z distances in a rectangular coordinate system h water-table elevation 2 distance above or below a datum AA ^difference in water-table elevation a angle between a force and direction of movement i, j, k a system of unit direction vectors )8 moisture content on a dry-weight basis K coefficient of capillary conductivity ~ specific moisture capacity I length of a capillary tube O2j m mass 7 density of a gas or vapor Al molecular weight A operator indicating change in value of a quantity o a digit e osmotic-pressure potential p (a) pressure; (b) pressure on the concave side of a 77 coefficient of viscosity curved liquid surface; (c) pressure of the vapor in a 6 (1) angle of contact; or (2) angle between the incli­ capillary opening; (d) pressure of the vapor a distance nation of the water table and the horizontal above a free flat water surface; or (e) vapor pressure K diffusivity of a solution X heat of vaporization at constant temperature Po (a) pressure of the vapor immediately above a flat £ adsorption potential water surface; (b) pressure on the convex side of a TT 3.14159 . curved liquid surface; or (c) vapor pressure of a free p density of liquid water pure solvent pa bulk density of a soil p' pressure of the liquid on the convex side of a o- surface tension meniscus I, capillary potential pa pressure just over an element of capillary surface 0 total potential p'a pressure just under an element of capillary surface ^ hydrostatic-pressure potential P hydrostatic pressure ft concentration of ions or molecules Q discharge or flow rate V the vector operator del, or nabla FLUID MOVEMENT IN EARTH MATERIALS REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY By IRWIN REMSON and J. R. RANDOLPH ABSTRACT tial, which occur when temperature differences occur in the This review was assembled from the existing literature to make system; they are relatively unimportant in causing moisture available a convenient introduction to this subject, which is of movement except when the temperature gradients are large; interest to workers in many diverse fields of hydrology. (f) finally, chemical potential, which is due to the osmotic energy Surface tension at the liquid-vapor interfaces largely controls of free ions in the aqueous solution; it is usually unimportant in the occurrence and movement of moisture in unsaturated soils. nonsaline soils because solutes generally move more readily than Surface tension is defined as the amount of work or energy re­ the water with respect to the soil. quired to produce a unit increase in the area of a liquid surface. In a hypothetical distribution of vertically swelling and The pressure on the concave side of a liquid-vapor surface in a pinching horizontally disconnected capillary tubes, there are round capillary tube exceeds the pressure on the convex side by positions of both minimum and maximum capillary rise. These an amount equal to twice the surface tension divided by the tube are the most common positions of the top of the capillary fringe, radius. The rise of liquid in a round capillary
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