Прикладная Гидрогеология Tomsk Polytechnic University Tomsk, Russian Federation Spring Semester 2014 Yoram Eckstein, Ph.D. Fulbright Professor 2013/2014
Useful links
V. Principles of GroundwaterFlow
Forms of energy endowed in ground-water Mechanical Thermal Chemical Ground water moves from one region to another to eliminate energy differentials The flow of ground water is controlled by the law of physics and thermodynamics
Forces Acting on Ground- Water Gravity –pulls ground water downward External pressure Atmospheric pressure above the zone of saturation Molecular attraction – Cause water to adhere to solid surfaces Creates surface tension in water when the water is exposed to air This is the cause of the capillary phenomenon
Resistive Forces Forces resisting the fluid movement when ground water is flowing through a porous media: Shear stresses –acting tangentially to the surface of solid Normal stresses acting perpendicularly to the surface These forces can be thought of as “friction”
Mechanical Energy Bernoulli equation h - hydraulic head (L, J/N) First term –velocity head (ignored in ground water flow) Second term –elevation head Third term –pressure head
Heads in Water with Various Densities
Darcy’s Law
The limits of validity of Darcy’s Law
Specific Discharge Average Linear Velocity
R x y yy xx zz 1.Consider flux (q) through REV 2.OUT – IN = - Storage 3.Combine with: q = -K grad h q Derivation of the Governing Equation
Confined Aquifers
vertical leakage downward to the confined aquifer
Unconfined Aquifers
Solution of Flow Equations If aquifer is homogeneous and isotropic, and the boundaries can be described with algebraic equations analytical solutions Complex conditions with boundaries that cannot be described with algebraic equations numerical solutions numerical solutions
Gradient of Hydraulic Head
To obtain the potential energy: measure the heads in an aquifer with piezometers and multiply the results by g If the value of h is variable in an aquifer, a contour map may be made showing the lines of equal value of h (equipotential surfaces)
Gradient of Hydraulic Head Equipotential lines in a three-dimensional flow field and the gradient of h The diagram above shows the equipotential surfaces of a two-dimensional uniform flow field Uniform the horizontal distance between each equipotential surface is the same The gradient of h: a vector roughly analogous to the maximum slope of the equipotential field.
Gradient of Hydraulic Head s is the distance parallel to grad h Grad h has a direction perpendicular to the equipotential lines If the potential is the same everywhere in an aquifer, there will be no ground-water flow
Relationship of Ground-Water-Flow Direction to Grad h The direction of ground-water flow is a function of the potential field and the degree of anisotropy of the hydraulic conductivity and the orientation of axes of permeability with respect to grad h In isotropic aquifers, the direction of fluid flow will be parallel to grad h and will also be perpendicular to the equipotential lines
Relationship of Ground-Water-Flow Direction to Grad h For anisotropic aquifers, the direction of ground-water flow will be dependent upon the relative directions of grad h and principal axes of hydraulic conductivity The direction of flow will incline towards the direction with larger K
Flow-Lines and Flow-Nets A flow line is an imaginary line that traces the path that a particle of ground water would follow as it flows through an aquifer In an isotropic aquifer, flow lines will cross equipotential lines at right angles If there is anisotropy in the plane of flow, then the flow lines will cross the equipotential lines at an angle dictated by: the degree of anisotropy and; the orientation of grad h to the hydraulic conductivity tensor ellipsoid
Flow-Lines and Flow-Nets Relationship of flow lines to equipotential field and grad h. A. Isotropic aquifer. B. Anisotropic aquifer
Flow-Nets The two-dimensional Laplace equation for steady-flow conditions may be solved by graphical construction of a flow net Flow net is a network of equipotential lines and associated flow lines A flow net is especially useful in isotropic media
Assumptions for Constructing Flow Nets The aquifer is homogeneous The aquifer is fully saturated The aquifer is isotropic (or else it needs transformation) There is no change in the potential field with time The soil and water are incompressible Flow is laminar, and Darcy’s law is valid All boundary conditions are known
Boundary Conditions No-flow boundary: Ground water cannot pass a no-flow boundary Adjacent flow lines will be parallel to a no- flow boundary Equipotentiallines will intersect it at right angles Boundaries such as impermeable formation, engineering cut off structures etc.
Boundary Conditions Constant-head boundary: The head is the same everywhere on the boundary It represents an equipotential line Flow lines will intersect it at right angles Adjacent equipotential lines will be parallel Recharging or discharge surface water body
Boundary Conditions Water-table boundary: In unconfined aquifers The water table is neither a flow line nor an equipotential line; rather it is line where head is known If there is recharge or discharge across the water table, flow lines will be at an oblique angle to the water table If there is no recharge across the water table, flow lines can be parallel to it
Flow Net A flow net is a family of equipotential lines with sufficient orthogonal flow lines drawn so that a pattern of “squares” figures results Except in cases of the most simple geometry, the figures will not truly be squares
Procedure for Constructing a Flow Net 1.Identify the boundary conditions 2.Make a sketch of the boundaries to scale with the two axes of the drawing having the same scale 3.Identify the position of known equipotential and flow-line conditions
Procedure for Constructing a Flow Net
4. Draw a trial set of flow lines. a.The outer flow lines will be parallel to no-flow boundaries. b.The distance between adjacent flow lines should be the same at all sections of the flow field
Procedure for Constructing a Flow Net
5. Draw a trial set of equipotential lines. a.The equipotential lines should be perpendicular to flow lines. b.They will be parallel to constant-head boundaries and at right angles to no-flow boundaries. c.If there is a water-table boundary, the position of the equipotential line at the water table is base on the elevation of the water table d.They should be spaced to form areas that are equidimensional, be as square as possible
Procedure for Constructing a Flow Net 6. Erase and redraw the trial flow lines and equipotential lines until the desired flow net of orthogonal equipotential lines and flow lines is obtained, e.g.
Procedure for Constructing a Flow Net Flow net beneath an impermeable dam
Example of Flow Net Between Two Streams The water table is neither a flow line nor an equipotential line; rather it is line where head is known
Computing Flow Rate from a Flow Net
Refraction of Flow Lines When water passes from one stratum to another stratum with a different hydraulic conductivity, the direction of the flow path will change The flow rate through each stream tube (flow path) in the two strata is the same (continuity)
Refraction of Flow Lines Stream tube crossing a hydraulic conductivity boundary
Refraction of Flow Lines
B. From low to high conductivity. C. From high to low conductivity
Refraction of Flow Lines
When drawing flow nets with different layers, a very helpful question to ask is “What layer allows water to go from the entrance point to the exit point the easiest?” Or, in other words, “What is the easiest (frictionally speaking) way for water to go from here to there?”
Flow Nets and Lakes
Steady Flow in a Confined Aquifer Steady flow through a confined aquifer of uniform thickness
Steady Flow in an Unconfined Aquifer Steady flow through an unconfined aquifer resting on a horizontal impervious surface L is the flow length
Steady Flow in an Unconfined Aquifer