Chapter 1: Flow in Soil
Table of Contents 1.1 Capillary in soil, soil shrinkage & soil expansion (Text book 9.9 & 9.10) 1.2 Head and flow of one and two dimensional (Text book 8.2 & 8.3) 1.3 Seepage analyses (Text book 8.4 – 8.10) 1.4 Filter design (Text book 8.11)
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Adapted from Dr. Lulie’s presentation slide
Adapted from Dr. Lulie’s Lulie’s presentation slide
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Adapted from Dr. Lulie’s presentation slide
Adapted from Dr. Lulie’s Lulie’s presentation slide
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1.1 Capillarity (continued…)
Groundwater table (or phreatic surface) – – the level which underground water will rise in an observation well, pit or other open excavation in the earth Soil beneath groundwater table – filled with water Soil moisture – – any water in soil located above the water table Capillary rise – – phenomenon which water rises above the groundwater table against the pull of gravity but is in contact with the water table as its source Capillary moisture – – the water associated with capillary rise Vadose zone Vadose zone – – the soil region directly above the water table and wetted by capillary moisture
Water in Capillary Tubes
Basic principles of capillary rise in soils related to the rise of water in glass capillary tubes
The rise – rise – attraction between the water and the glass and to a surface tension, which develops at the airair-water interface at the top of the water column in the capillary tube
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Water in Capillary Tubes (continued (continued…)
Value of Ts for water varies according to temperature As temperature increases, the value of T s decreases, indicating a lessening height of capillary rise under warm conditions At room temperature, Ts for water = 0.064 N/m At freezing, Ts for water = 0.067 N/m In applying the development of capillary rise in tubes to capillary rise in soils: hc 31/d mm (McCarthy, D. F., 2002) ≈ ≈
(*provided that d is in millimetres) millimetres)
Water in Capillary Tubes (continued (continued…)
Question: Compute the height of capillary rise for water in i n a tube in having a diameter of 0.05 mm (in SI units)
Solution: hc =
4T s d γ w
=
( 4)(0.064 N / m ) (5 x10−5 m)(9.81kN / m 3 )
= 0.52m
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Water in Capillary Tubes (continued (continued…)
The height of capillary rise is not affected by a slope or inclination in the direction of the capillary tube, or by variations in the shape and size of the tube at at level below the meniscus
Water in Capillary Tubes (continued (continued…)
Capillary rise is not limited to tube, or enclosed, shapes. If two vertical glass plates are placed so that they they touch along one end and, form a V, a wedge of water will rise up in the V because of the capillary phenomenon
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Capillary rise in soil
Shapes of void spaces between solid particles are unlike those in capillary tubes Voids are of irregular and varying shape and size, and interconnect in all directions, not only the vertical The features of capillary rise in tubes are applicable to soils in so far as they facilitate an understanding of factors affecting capillarity, and help to establish an order of magnitude for capillary rise in the different types of soils
Capillary rise in soil (continued ….) Question: Limited laboratory studies indicate that for a certain silt soil, the effective pore size for height of capillary rise is 1/5 of D 10, where D10 is the 10 percent particle size from the grain grain--size distribution curve. If the D10 size for such a soil is 0.02mm, estimate the height of capillary rise. Solution: d = effective capillary diameter = 1/5D10 = 1/5 (0.02 mm) = 0.004 mm hc ≅ 31/d ≅ 31/0.004 mm ≅ 7750 mm ≅ 7.75m
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Capillary rise in soil (continued ….)
Capillary fringe
Capillary rise in soil (continued ….) Table : Representative heights of capillary rise: water in soil
Soil Type
Meter (m)
Small gravel Coarse sand
0.02-0.1 0.15
Fine sand
0.3 to 1
Silt Clay
1 to 10 10 to 30
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Soil shrinkage & soil swelling
1.2 Head and flow of one and two dimensional
One- dimensional flow – the velocity at all points has the same direction and (for an incompressible fluid) the same magnitude Two-dimensional flow – all streamlines in the flow are plane curves and are identical in a series of parallel planes
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Lecture 3 Notes – Two Dimensional Flow In the previous section the seepage problems discussed were all lab models consisting of one-dimensional flow. In field construction, structures used for water barriers generally involve two- or three-dimensional seepage flow, such as: (1) Cofferdam cells (sheet pile wall) and Concrete dams are confined flow; all the boundary conditions are well defined before the construction of flow nets. (2) Earth dams and levees are unconfined flow; the top flow line is not defined in advance of constructing the flow nets. The purposes of studying the seepage conditions under or within these structures are: 1. to estimate the rate of flow (reservoirs for keeping water cut-off ability 2. seepage force (γ wi) (uplift force) (erosion) 3. pore pressure distribution for effective stress analysis In this section we will concentrate on studying two-dimensional steady flow through soil media since most three-dimensional cases can be t reated as two–dimensional cases when the size-dimension in one of the dimensions is much greater than the other two dimensions. The Laplace Equation, a second order partial differential equation, is the theory behind the flow net. This equation is a common mathematical representation of the energy loss through any resistive media (See textbook for its derivation and details). Methods generally used for solving the Laplace’s equation are: 1. 2. 3. 4.
Direct mathematical solution (different boundary condition for different answer) Numerical solution (approximation, finite difference method) Electrical analogy solution (build electric model) Graphical solution (flow nets, a trial-and-error method)
Flow Net Construction
Flow net consists of Flow Lines (velocity line) and Equal Head Lines (equal potential or equal total head). The characteristics or rules to construct a flow net for isotropic permeability are listed below, 1. satisfy the boundary conditions 2. make flow lines intersect constant head lines at 90° 3. draw curvilinear squares
Although Flow Net is a trial-and-error graphical method, an unique solution will always be achieved when all these three construction requirements are met. A detailed explanation of these three characteristics is shown below: 1. Boundary conditions for two dimensional flow a. Soil water interface = constant head line b. Impermeable boundary = flow line 2. flow lines are perpendicular to constant head lines for isotropic permeability (k x = k z)
From Darcy’s law:
x z
V =
V =
V=
− x ∂∂ − z ∂∂ k
k
h x
h z
− x ∂∂ 2 − z ∂∂ 2 h ) + ( k x
( k
h z
)
For isotropic permeability, K z = K z = k, giving
V=
∂∂ 2 ∂∂ 2 (
h h ) +( ) x z
But
� hx 2
hz 2 hn
( ) +( ) =
= normal derivative or normal gradient
The direction of the normal derivative is no rmal to the constant headlines. The velocity, V is in the direction of this gradient and is therefore normal to the constant headlines.
3.
Consider the condition necessary to have Equal Quantities of Flow between Flow Lines, when the head drops between constant headlines are equal.
If q 1 = q 2 = ∆q; V1A1 = V2A2; ki1A1 = ki2A2; k(∆ h/a)(b)(1) = k (∆ h/c)(d)(1)
Giving b/a = d/c = curvilinear rectangles with the same side ratios Use b/a = d/c = 1 = curvilinear squares) easiest to draw) Computation of quantity of flow through a n et Let nf = number of flow channels in the net nh = number of equal head drops across the net Then q = nf ∆ q, since ∆q = k ∆ h (b/a) = k ∆ h; therefore q = n f k ∆ h If H = total head from across the net ∆ h = H/nh; q = nf k (H/nh)
q = kH (nf /nh) nf /nh = Shape factor
Construction Procedure 1. Observe the general pattern of flow. Where does the water enter and exit the soil? What general path does it follow? 2. Located the boundary flow lines (longest and shortest ) and constant head lines (water entrance and exit) 3. Sketch two intermediate flow lines. 4. Begin at the constant head line where the water enters (or exits) the soil and sketch as many constant head lines as required to reach the exit (or entrance) constant head line by drawing curvilinear squares and 90° intersections with flow lines. 5. Adjust these flow lines and constant headlines as necessary to produce curvilinear squares and 90° intersections. 6. If desired, sketch additional flow lines and constant headlines to reduce the size of the curvilinear squares.
1.2 Head and flow of one and two dimensional (continued…)
(1)
(2)
1.3 Seepage analyses
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1.3 Analysis of flow nets and seepage
Many catastrophic failures in geotechnical engineering result from instability of soil masses due to ground water flow Lives are lost, infrastructures are damaged or destroyed, and major economic losses occurred In this subchapter, you will study the basic principles of two -dimensional flow of water through soils
1.3 Analysis of flow nets and seepage (continued…)
The topics that you will study would help you to avoid pitfalls in the analyses and design of geotechnical systems where flow of ground water can lead to instability The emphasis of this chapter is on gaining an understanding of the forces that provoke failures from flow of ground water
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1.3 Analysis of flow nets and seepage (continued…) Learning Objectives: Understand the basic principles of twodimensional flow Be able to calculate seepage stresses, pore water pressure distribution, uplift forces, hydraulic gradients, critical hydraulic gradient, flow under and within earth structures Be able to determine the stability of geotechnical systems subjected to two-dimensional flow of water
1.3.1 Basic Concepts
The twotwo-dimensional flow of water through soils is governed by Laplace’ Laplace’ s equation. The popular form of Laplace’ s equation for two -dimensional flow of water through soils is k x
∂ 2 H ∂ x 2
+ k z
∂ 2 H ∂ z 2
=0
Where k x and k z are the coefficient of permeability in the x and z directions and H is the head The assumptions in Laplace ’ s equation are: (i) Darcy’ s law is valid (ii) The soil is homogeneous and saturated (iii) The soil and water are incompressible (iv) No volume change occurs
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1.3.1 Basic Concepts (continued…)
If the soil were an isotropic material then k x = k z and Laplace’ s equation becomes: ∂ 2 H ∂ x 2
+
∂ 2 H ∂ z 2
=0
The solution of Laplace ’ s equation requires knowledge of the boundary conditions. Common geotechnical problems have complex boundary conditions from which it is difficult to obtain a closed form solution. Approximate methods such as graphical methods and numerical methods are often employed. In this subchapter, graphical method, called the flow net technique or flow net sketching, that satisfies Laplace ’ s equation is discussed.
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3.56 x 10-4
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No of equipotential drops at point a
Elevation loss
hpγw
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hpγw
hpγw
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Uplift forces
Static Liquefaction, Heaving, Boiling, & Piping
Static liquefaction – – the state which the effective stress becomes zero, the soil loses its strength and behaves like a viscous fluid fluid Boiling, quicksand, piping and heaving are used to describe specific events connected to the static liquefaction state Boiling – – the upward seepage force exceeds exceeds the download force of the soil Piping – pipe--shaped” erosion that initiates near – the subsurface “ subsurface “pipe shaped” erosion the toe of dams and similar structures. High localized hydraulic gradient statically liquefies the soil, which progresses to the water surface in the form of a pipe, and water then rushes beneath the structure through the pipe, leading to instability and and failure Quicksand – – existence of a mass of sand in a state of static liquefaction Liquefaction – Liquefaction – can be produced by dynamic events such as earthquakes
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Piping
Piping (continued…) The risk of piping can occur in several circumstances, such as a cofferdam (a) or the downstream end of a dam (b) In order to increase the factor of safety against piping in these cases two methods can be adopted (1) increase the depth of pile penetration in (a) and inserting a sheet pile at the heel of the dam in (b); in either case there is an increase in the length of the flow path of the water with a resulting drop in the excess pressure at the critical section.
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Piping (continued…) A similar effect is achieved by laying down a blanket of impermeable material for some length along the upstream ground surface (2) To place a surcharge or filter apron on top of the downstream side, the weight of which increases the downward forces
Example:-
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Others phenomenon: Quick sand, Frost heave in soils, liquefaction
Quicksand
Dreaded quicksand condition occurs where a sand or cohesionless silt deposit is subjected to the seepage force caused by upward flow of groundwater The upward gradient of the water is sufficient to hold the soil particles in suspension, in effect creating a material with the properties of a heavy liquid Elimination of seepage pressure will return the soil to a normal condition capable of providing support
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Frost heave in soils When freezing temperatures develop in a soil mass, most of the pore water in the soil is also subject to freezing. As water cystallizes, cystallizes, its volume expands approximately 9 percent In considering void ratios and the degree of saturation for soils, soils, expansion of a soil material as a result of freezing might be expected to be on the order of 3 or 4 percent of the original volume
Frost heave in soils (continued……)
In the normal frost heave occurrence, the source of water is the groundwater table Upward movement from a water table to the freezing zone relates to a potential for migration (capillary rise) Height of capillary rise is quite quite limited in clean, coarsecoarse-grained soil
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Soil liquefaction
1.4 Filter design
Text book sec: 8.11
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