Geo-E2010 Advanced Soil Mechanics L Wojciech Sołowski. 26 February 2017

Transcription

1 Geo-E2010 Advanced Soil Mechanics L Wojciech Sołowski 26 February 2017

2 Permeability, consolidation and seepage Department of Civil Engineering Advanced Soil Mechanics W. Sołowski 2

3 To learn 1. What is seepage. Why it happen? 2. What is head? Why there is head? Bernoulli equation? 3. Darcy s law and when it is valid? 4. What affect soil permeability? 5. What is consolidation? 6. What is the coefficient of consolidation? How is it linked to consolidation / settlement? 7. How effective stress is affected by flow 8. Laplace equation for seepage 9. Flow nets graphical solution of Laplace equation Department of Civil Engineering Advanced Soil Mechanics W. Sołowski 3

4 Seepage Department of Civil and Environmental Engineering EMC: Soil Mechanics. W. Sołowski 4

5 Seepage & Permeability Seepage is flow of water [through soils] when there is a difference in water levels (known as head ) [on either sides of a structure]. Point of higher energy Point of lower energy Fig. Seepage beneath (a) a concrete dam (b) a sheet pile Permeability is a measure of how easily a fluid (water) can flow through a porous medium (soils). e.g. how many metres in a second 5

6 Bernoulli's equation Based on the conservation of energy law, for steady flow of a non-viscous, incompressible fluid, the total head at a point can be expressed as the summation of three independent components; Pressure head, velocity head & elevation head Elevation Head Velocity Head Total Head h z u w 2 2 v g Pressure Head Heads are in length (unit - m) 6

7 Bernoulli's equation for Flow through soils Seepage velocity through a porous soil can be ignored, i.e. the velocity head is zero 0 h z u w 2 2 v g h z u w Total Head Elevation Head Pressure Head Velocity Head

8 The Head loss (h L ) When water flows from upstream (A) to downstream (B), energy/head is lost because the soil grains opposes the flow. Loss of energy, Total head loss (h L ) = Difference in water levels (Δh) Pore water pressure (U) at any point in the flow region can be written as: h A u A U = Pressure head x ρ w w Z A x g A s Datum h B ub w Z B h B

9 The Head Loss h When water flows from point A to point B, the total pressure at point A is greater than at point B Energy / Head is dissipated in overcoming the soil resistance and hence is the head loss water u A w z A u B w z B h A B where h = pressure drop across points A and B

10 Head loss & Hydraulic gradient (i) Hydraulic gradient (i) is the total head loss per unit length i = Total head at A Total head at B Length AB h A u A w A h B ub w h B Therefore i is: Z A s Datum Z B i h s

11 Permeability Department of Civil and Environmental Engineering EMC: Soil Mechanics. W. Sołowski 11

12 Darcy s law: Darcy s experiment Soil sample: A -> Cross-sectional Area L -> Length Soil sample Head in T: h T = H 1 T Head in B: h B = H 2 Steady state flow conditions H 1 A L H 1 and H 2 constants during the test B Q is measured (volume of water) Datum Q H 2 Department of Civil Engineering Advanced Soil Mechanics. W. Sołowski 12

13 Darcy s law: Soil sample Darcy carried out several tests with different kind of soils considering different values for the following parameters: A, L, H 1 and H 2 H 1 T S L Darcy observed that Q is directly proportional to: B Head difference H 1 H 2 Cross-sectional Area A and inversely proportional to: Sample length L Q Datum Q H 2 That is: H1 2 Q H L At i= (H 1 H 2 )/L hydraulic gradient The constant of proportionality is a soil parameter known as coefficient of permeability (k). Thus Q=Atki i Department of Civil Engineering Advanced Soil Mechanics. W. Sołowski 13

14 Seepage velocity (v s ) At particle level the water seeping through a soil follows a very tortuous path. The average velocity at which the water flows through the soil pores is obtained as follows: v q = v A = v s v s = A/A v v v n= v v v v s = n A v n sup ki n A v A ~ n v: average flow velocity[m/s] v s : seepage velocity [m/s] n: porosity[-] A Av: area of void in the cross section of the specimen q A v v s A v 14

15 Darcy s law: range of permeability k The temperature also affect the permeability of water (usually +20C) Darcy's Law is valid when: 1) the flow is laminar. 2) the ground is fully saturated 3) the ground is homogeneous (consolidation during the specification changes) 4) the country is isotropic (in different directions) Department of Civil Engineering Advanced Soil Mechanics. W. Sołowski 15

16 Influence of chemicals: The permeability on typical clay liners when permeated with leachate with high concentration of Ca +2 cations. Permeability of these liners when permeated with water is 10-9 to cm/s. After Ruhl & Daniel 1997, J. Geotech. Geoenv. Eng. Vol. 123, Department of Civil Engineering Advanced Soil Mechanics. W. Sołowski 16

17 Influence of chemicals The permeability on typical clay liners when permeated with strong acid solution. Permeability of these liners when permeated with water is 10-9 to cm/s. Note that introduction of acid to the wet liner affect the permeability little in short time (as the acidity of the pore liquid reaches the acidity of the leachate very slowly). After Ruhl & Daniel 1997, J. Geotech. Geoenv. Eng. Vol. 123, Department of Civil Engineering Advanced Soil Mechanics. W. Sołowski 17

18 Permeability determination Department of Civil and Environmental Engineering EMC: Soil Mechanics. W. Sołowski 18

19 Determination of Coefficient of Permeability OR Hydraulic conductivity (k) Estimation from empirical equations Laboratory tests In-situ OR field measurements 19

20 Determination of permeability from empirical equations (horrible)

21 Determination of permeability Two main tests: Laboratory Tests (bad) The Constant Head permeability: - suitable for highly permeable materials (e.g. granular soils) - Not suitable for fine-grained soils where the flow rates are very small, evaporation could lead to significant error. The Falling Head permeability: - suitable for fine-grained soils Coefficient of permeability (k) can also be obtained in the odometer and triaxial tests.

22 Determination of permeability: Field (in-situ) Measurement best if done well Laboratory methods give a realistic estimate of the in-situ permeability of a uniform and isotropic soil. Samples must be selected and taken with care to replicate the field conditions. Sometimes difficult/ impossible to obtain undisturbed soil specimens from the field (i.e. soil structure and fabric are difficult to replicated at laboratory) Therefore, some times it is advisable to conduct in-situ permeability tests. Several techniques are available, such as: Pumping from wells: Confined aquifer Unconfined aquifer Borehole tests: Open-end test. Packer test.

23 Consolidation Department of Civil Engineering Advanced Soil Mechanics. W. Sołowski 23

24 Consolidation process: string and piston analogy C 0 P P ' t u t C t t = 0 P P u u 0 S ' C t P t ut S u u 0 t t t = u t P S 0 t u t 24 t

25 Change of volume in one dimensional consolidation (in time dt) d d dh=de z *dz dh=m v *dz*d m v =de z /d dz dh de z d m v element height change of height change in vertical strain change in effective vertical stress compressibility m v is non-linear, so equation only valid for small stress increments

26 Change of volume in oedometer (in time dt) d d dv=adh= -m v *dz*d-dqdt Am v *dz*ddqdt Am v * ddtdq/dz dz dh de z d m v dq dt element height change of height change in vertical strain change in effective vertical stress compressibility rate of flow time increment Volume change dv is equal to the amount of water outflow dq

27 Change of volume in oedometer (in time dt) d Am v * d dtdq/dz i=-du/( w *dz) dq=aki=-akdu/( w *dz) m v * t - k/ w 2 u/z 2 dz dh d m v dq dt du k d element height change of height change in effective vertical stress compressibility rate of flow time increment pressure difference permeability coefficient

28 Change of volume in oedometer (in time dt) d c v = k/m v w m v * t - k/ w 2 u/z 2 - t (k/m v w ) 2 u/z 2 - t c v 2 u/z 2 -ut c v 2 u/z 2 t u z k m v w c v d change in effective vertical stress time increment pressure difference element height permeability coefficient compressibility water unit weight coefficient of consolidation

29 Basic equation for 1 dimensional consolidation d ut c v 2 u/z 2 d t u z c v time increment pressure difference element height coefficient of consolidation

30 Basic equation for 1 dimensional consolidation ut c v 2 u/z 2 Can be solved numerically for given case, like oedometric consolidation In particular, there are Casagrande and Taylor solutions on which the standard proposes calculation of c v

31 Basic equation for 1 dimensional consolidation

32 Effective stress & flow Department of Civil Engineering Advanced Soil Mechanics. W. Sołowski 32

33 Influence of Seepage on Effective Stress In Soil Mechanics compression is considered as positive while tension is negatives. + Compressive stresses are positives Tensile stresses are negatives

34 Influence of Seepage on Effective Stress Effective Stress Terzaghi (1936) proposed the principle of effective stress (*), the most important equation in soils mechanics. The effective stress ( ) is the component of the normal stress taken by the soil skeleton. It is the effective stress which controls the volume and the strength of the soil. Karl Terzaghi ( ) u w efective stress total stress pore water pressure (*) All the measurable effects of a change of stress, such as compression, distortion and a change in the shearing resistance are exclusively due to changes in effective stress every investigation of the stability of a saturated body of earth requires the knowledge of both the total and the neutral stresses. (Terzaghi, 1936)

35 Stresses in Soils Due to Water Flow Influence of Seepage on Effective Stress Effective Stress Saturated soil comprises two phases: the soil particles the pore water = + Saturated soil Solid Skeleton Water The strengths of these two phases are very different: the soil skeleton can resist shears: inter particle friction particles interlocking the shear strength of water is zero

36 Influence of Seepage on Effective Stress Effective Stress. Physical Interpretation σ σ : total stresses externally applied σ : stresses that act through the contacts between particles (A m ) p w : water pressure (A w ) p w A t : total area σ A t = A m + A w A t

37 NOTE: γ = γ Influence of Seepage on Effective Stress Static Situation (No flow) σ v = σ v u At C: Water Pressure Total stress Effective stress u ' v v h w L z C A B w (h w + L) w h w + sat L ' L Depth Depth Depth

38 Influence of Seepage on Effective Stress Downward Flow v = w h w + sat z ; At C: u = w [h w + z- z (H/L) ]= w (h w + z - z i) v ' = w h w + sat z [ w (h w + z - z i)] v ' = z ( + i w ) flow u v v ' h w z(h/l) H A L z C z( + i w ) B w (h w +L-H) w h w + sat L ' L+H w z(h/l) L H Depth i = H/L Depth Depth

39 Influence of Seepage on Effective Stress Upward Flow v = w h w + sat z ; At C: u = w [h w + z+ z (H/L) ]= w (h w + z + z i) flow v ' = w h w + sat z [ w (h w + z - z i)] v ' = z ( - i w ) z(h/l) H u v v ' h w A L z C z( - i w ) B w (h w +L+H) w h w + sat L ' L-H w z(h/l) L H Depth i = H/L Depth Depth

40 Summary of effective stress 40

41 Influence of Seepage on Effective Stress Quick Condition flow H Upward seepage v ' = z ( - i w ) If the rate of seepage and thereby the hydraulic gradient are gradually increased, a limiting condition will be reached: h w L z C A B v ' = z ( - i cr w ) = 0 i cri =critical gradient (for zero effective stress). Under such conditions, soil particles are no longer in contact and soil stability is lost. This situation is generally referred to as boiling or a quick condition. ' icr w For most soils, the values of i cr varies from 0.9 and 1.1

42 Quick sand / Boiling 42

43 Seepage calculations & flow nets Department of Civil Engineering Advanced Soil Mechanics. W. Sołowski 43

44 Seepage Theory In many practical cases, the nature of the flow of water through the soils is such that the velocity and gradient vary throughout the medium. In this section, the aim is to understand the phenomena that controls the seepage of water through a porous medium and to derive the differential equation governing this groundwater flow. The numerical and graphical methods that are used in the solution of groundwater flow problems are based on these equations. Therefore, it is important to develop an understanding of the assumptions and limitations involved. Mathematical Framework Equation of continuity Water mass balance equation (General equation for a saturated soil) Darcy's Law Constitutive law 44

45 Seepage H Continuity equation (mass water balance equation) For steady state flow soil Flow entering = Flow leaving Elementary soil prism is considered to derive the equation of continuity of saturated flow: z y dx dy dz x 45

46 Continuity equation Seepage H The volume of water entering the element per unit time is: from q = A. v v dy dz v dx dz v dy dx x y z A soil v v v x y z The volume of water leaving the element per unit time is: x y z vx dx dy dz vy dy dx dz vz dz dy dx For steady state flow through an incompressible medium: Flow entering = Flow leaving v x B v z vz dz z dy dx dz v y v vy dy y x vx dx x z v y From eq n A, B v v x y vz 0 x y z x y v z

47 Seepage Darcy s Law (Constitutive Law) The flows in the x, y, z directions can be given from generalized Darcy's Law: v x = k x. i x = k x h x v y = k y. i y = k y h y Combining these equations to continuity equation: v v x y vz 0 x y z v x v y v z = k z. i z = k z h z v z h h h kx ky kz x y z This is the differential equation governing the groundwater flow through a rigid soil skeleton. Finally, for isotropic soils k x = k y = k z = k = constant, and the continuity equation simplified to: h h h k x y z h h h x y z Laplace's Equation

48 Seepage Solution for the flow problems in soil The differential equation governing the groundwater flow through a rigid soil skeleton is: h h h x y z For 2D flow, this can be simplified as: 2 h x h z 2 = 0 The above differential equation can be solved using the following methods: Analytical Numerical Reduced scale Analogue Graphical? H soil h = f x, y,z v = f x, y,z

49 Seepage Graphical Method: Plane Flow In Earthworks, such as excavations, cuttings and embankments, which are long in comparisons with their other dimensions, the significant flow occurs (by symmetry) mainly in the plane of the cross-sections. Concrete dam Excavation The problems is therefore reduced to two dimensions, in which Laplace s Equations can be solved graphically using a technique known as flow net sketching.

50 Graphical Solution for Laplace Equation The solution for the Laplace s equation represents two families of orthogonal curves known as equipotential lines and flow lines. 2 h x h z 2 = 0 50

51 φ 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7 Seepage Graphical Method: Flow net The flow net is the network of a) flow lines - which represents the trajectory of individual fluid particle, and b) equipotential lines - Along an equipotential the total head or potential is constant and therefore is no flow. A flow net may be used to calculate the seepage flow rate and pore water pressure at any point of the cross-section. φ 0 1: Flow lines 2: Equipotential lines h 1 h 2 K H z

52 Flownet: Potential function Consider a function f(x,z) called the potential function such that: f = v x = - k x h x ; f = v z = - k z h z f(x,z) satisfy the Laplace s equation: 2 2 f f + = x z Integrating: f = - k h +C ; h = 1 k C -f C = constant (x,z) (x,z) By assigning to f a number of values such as f 1, f 2, f 3 we can get a number of equipotential lines along witch h= h 1 = h 2 =h 3, respectively. h 1 h 2 f f d f = dx + dz x z Hence: Seepage (x,z) Along an equipotential line f is constant, so df = 0 z (x,z) Equipotential line h 3 f f dz f x v dx + dz = 0 = - x z dx f z v f x z 1 f 1 f 2 f 3 x

53 Seepage Flownet: Flow (or stream) function Let y(x,z) be a function called the flow function such that: y h = v x = - k z x ; y = v z = - k x h z The total differential of the function y(x,z) is: y y dy dx + dz = -vzdx +vxdz x z Y(x,z) satisfy the Laplace s equation: z Y 1 Y f f + = x z Assigning to Y a number of values such as Y 1, Y 2, Y 3 we can get a number of flow lines in the (x,z) plane. Along a flow line Y is constant, so dy = 0, and: dz dx Y v = - z vx 2 v v x v z v x v z Equipotentia l line f 1 f 2 Flow line The tangent at any point on a curve (Yi) specifies the direction of the resultant of the discharge velocity, i.e. the curve represent the flow path. Comparing 1 and 2 flow lines and equipotentials intersect each other at right angles x

54 Flow Net Flow net: How to construct a flow net? The flow net is constructed by trial and error in order to satisfy the following conditions: 1) Flow lines cross equipotential lines at right-angles (no flow along an equipotential line). 2) Flow lines cannot cross other flow lines (2 molecules of water cannot occupy the same space at the same time). 3) Equipotential lines cannot intersect other equipotential lines. 4) Impermeable boundaries and lines of symmetry are flow lines 5) Bodies of water such as reservoir are equipotential lines 6) It is convenient to construct the flow net such that each element be a curvilinear square. If this condition is fulfilled, the drop in head between any two consecutives equipotential lines is the same and the flow rate through the flow tubes/channel is the same

55 Flow Net Seepage Calculation A flow channel is the strip located between two adjacent flow line. Q is the flow through flow channel per unit length of the hydraulic structure. According to Darcy s Law: h -h h -h ΔQ =k n 1 1 k n 2 1 s1 s permeability section hydraulic gradient If the flow elements are drawn as curvilinear squares: n=s h i -hi+1 ΔH ΔQ =k n i 1 ; h1 -h 2 =.. =hn-1 -h n = Δh = si N d If there are N f flow channels, the rate of seepage per unit length of the hydraulic structure is: Nf Q =Nf ΔQ =k H N d H 1 Flow channel n 1 h 1 s 1 n 2 H=H 2 -H 1 h 2 s 2 Ψ=Ψ A h 3 Flow channel h = potential drop between two consecutive equipotential lines. H = Total hydraulic head (dif. in water elevation: H 1 -H 2 ) N D = number of potentials drops. N f = number of flow channels. H2 Ψ=Ψ B

56 Plane Flow: Flow Net Flow line is simply the path of a water molecule from upstream to downstream, total head steadily decreases along the flow line. H concrete dam datum impervious strata soil

57 Plane Flow: Flow Net Equipotential line is simply a contour of constant total head. H concrete dam datum TH=0.8 H impervious strata soil

58 Plane Flow: Flow Net A Flow net is a network of selected flow lines and equipotential lines. datum concrete dam TH = H TH =0 90º H 90º impervious strata soil 90º

59 Flow Net Plane Flow: Quantity of Seepage (Q) unit length normal to the plane...per permeability Q head loss from upstream to downstream H # of flow channels N kh N f d # of equipotential drops datum concrete dam TH = H TH = 0 90º 90º 90º impervious strata soil

60 Flow Net Application z 1. Water flow? 2. Head Pressure (A)? h 1 h 2 k H Objectives: To determine the flow rate To determine the head pressure in any point of the domain A Procedure: z A Draw a flow net. Equipotential lines perpendicular to flow lines. Square cells

61 Flow net Application z 1. Water flow? 2. Head Pressure (A)? h 1 h 2 K H A z A

62 Flow net Application z 1. Water flow rate? 2. Head Pressure? h 1 h 2 φ 0 φ 8 K H A z A

63 Flow net Application z 1. Water flow rate? 2. Head Pressure? h 1 h 2 φ 0 K φ 8 H A z A φ 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7

64 Flow net Application N f = 5 N d = 8 Water flow rate Q H 5 N f k k h h N 8 d 1 2 H = h 1 h 2 z H h 1 h 2 φ 0 K φ 8 H φ 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7

65 Flow net Application Pressure (A?) z H h 1 h 2 φ 0 K φ 8 H A z A N d at A = 5.5 φ 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7 Therefore, Pressure at A = γ w h 1 + H z A h 1 h U A = γ w H z A + 5h h 2 16

66 Summary What is seepage. Why it happen? Seepage= flow of water in soil. Water flows due to excess pore pressure (hydraulic gradient) What is hydraulic head? Bernoulli equation? Hydraulic gradient? h z u w i = Total head at A Total head at B Length AB Darcy s law and when it is valid? H H L 1 2 Q k At valid for slow isothermal laminar flows Department of Civil Engineering Advanced Soil Mechanics W. Sołowski 66

67 Summary What affect soil permeability? soil type, fluid viscosity and soil microstructure. Soil microstructure is affected e.g. by chemical additions to the pore water. Fluid viscosity is affected by the type of fluid and temperature What is consolidation? Slow dissipation of pore pressures connected to change of volume. Usually we think 1 dimensionally which is not always correct Department of Civil Engineering Advanced Soil Mechanics W. Sołowski 67

68 Summary What is the coefficient of consolidation? c v = k/m v w c V determines the consolidation characteristics of soil. Note it is not constant. Using it we can compute time in which given soil layer will achieve given consolidation degree How effective stress is affected by flow see slide 40 Department of Civil Engineering Advanced Soil Mechanics W. Sołowski 68

69 Summary Laplace equation for seepage h h h x y z 0 Flow nets graphical solution of Laplace equation For how to use flow nets & construct them, see slides & exercises Department of Civil Engineering Advanced Soil Mechanics W. Sołowski 69

70 Thank you

71 Laboratory tests (extra slides) Department of Civil Engineering Advanced Soil Mechanics. W. Sołowski

72 Laboratory Test (I): The constant head permeability A simple method of permeability measurement. Water flows in one-dimension along the axis of the sample. There are inlet and outlet filters. The hydraulic gradient required to maintain a flow rate Q is determined from the head difference ΔH indicated by manometers inserted at two points at distance L apart along the direction of flow. The flow rate is determined using measuring cylinder and a stopwatch. The hydraulic gradient i is found for a number of different flow rates Q. H 1 Filter ΔH A L Q Datum H 2

73 Laboratory Test (I): The constant head permeability ΔH = constant during the test Darcy's Law may be applied directly: Q = k i A Q A = k i = k H L ΔH Q L k= A ΔH H 1 Filter A L The constant head test is unsuitable for determining the permeability of fine-grained soils (low permeability soils -> flow rate very small). Q Datum H 2 It is a slow test recommendable for clean gravels and sands. For soils with K: cm/s

74 Laboratory Test (II): The Falling Head permeability a For fine-grained soils. Water flows from a small-bore tube of cross sectional area a, through the soil sample which is d h contained within a larger tube of cross-sectional area A. h The water level in the upper tube falls as water flows through the soil sample Filter A L h Q

75 Laboratory Test (II): The Falling Head permeability Initial conditions: t = 0 -> h=h 1 ; At: t = T -> h=h 2 Applying Darcy s Law: a dh h Q = A K ; - a dh = Q dt L h -a dh =K A dt L ; dh A K - = dt h a L Filter A L h h 2 A K ln h = t ; a L T h1 0 h h 2 -ln = T 1 A K a L h Q K= a L ln h h A T 1 2 h = h 1 t = 0 t = T h = h 2

76 Field tests (extra slides) Department of Civil Engineering Advanced Soil Mechanics. W. Sołowski

77 Confined vs Unconfined Aquifer Aquifer is a layer with relatively high permeability Unconfined part of the aquifer Confining unit Confined part of the aquifer

78 Field Measurement of Permeability Pumping well test A. in an unconfined aquifer It is most suitable for use in homegeneuos coarse soil. Continuous pumping at constant rate from the test well (steady-state conditions). Radial seepage resulting in the water table being drawn down to form a cone of depression. Water levels are observed in a number of boreholes spaced on radial lines at various distances from the test well. It is assumed that the hydraulic gradient at any distance r from the centre of the well is constant with depth and is equal to the slope of the water table: dh i= r dr Dupuit s assumption (accurate except points close to the well) Perforated casing 1. It may take several days to achieve steady-state conditions for soils with very low-permeability 2. Typically 4 observation wells are installed, in 2 rows at right angles to each other

79 Field Measurement of Permeability Pumping well test A. in a unconfined aquifer. The test enables the average coefficient of permeability of the soil mass below the cone of depression. Consider two boreholes located on a radial line at distances r 1 and r 2 from the centre of the well, the respective water levels being h 1 and h 2. At a distance r from the well the area through which the seepage take place is 2prh where r and h are variables. Applying Darcy s Law: permeability dh Q =k 2p rh dr area hydraulic gradient r dr 2 h Q =2 p k 2 hdh r1 r h1 r Qln = p k h2 h1 r1 Q = k i A Qln r r k = p h h Surface area of a cylinder = 2πrh

80 In-situ OR Field Measurement of Permeability Field measurement: Well pumping in a confined aquifer. In this analysis it is assumed that the well penetrates to the bottom of the aquifer. The water level in the well must not be draw down below the water table. Appling Darcy's Law : dh Q =k 2p rt dr permeability area hydraulic gradient r r dr h 2 p kt = dh r h Q Qln r2 r1 k = 2p T h h 2 1

81 Permeability of stratified soils (extra slides) Department of Civil Engineering Advanced Soil Mechanics. W. Sołowski

82 Permeability of Stratified Soils Flow perpendicular to the bedding plane Where h 1,h 2,h 3..h n h k 1,k 2,k 3..k n v thickness of the each of the n layers total thickness Darcy s coefficients of permeability of the layers respectively Velocity For the continuity of flow, the discharge q, is the same through all the layers.

83 Permeability of Stratified Soils Flow perpendicular to the bedding plane Let the total head lost be Δh and the head lost in each of the layers be Δh 1, Δh 2, Δh 3..Δh n. h = h 1 + h 2 + h h n The hydraulic gradients are: i 1 = h 1 h 1, i 2 = h 2 h 2, i 3 = h 3 h 3.. i n = h n h n If i is the gradient for the deposit, i = h h Since q is the same in all the layers, and area of cross-section of flow is the same, the velocity v is the same in all layers. Let k z be the average permeability perpendicular to the bedding planes. v = k z. i = k 1 i 1 = k 2 i 2 = k 3 i 3 = k n i n

84 Permeability of Stratified Soils Flow perpendicular to the bedding plane k z. h h = k h 1 h 2 h 3 h n 1 = k h 2 = k 1 h 3 = k 2 h n 3 h n = v Substituting the expression for h 1, h 2.. in terms of v in the equation for h, we get: vh k z = vh 1 k 1 + vh 2 k 2 + vh 3 k vh n k n OR h k z = h 1 k 1 + h 2 k 2 + h 3 k h n k n 1 R eq = 1 R R R 3 +

85 Permeability of Stratified Soils Flow parellel to the bedding plane Where h 1,h 2,h 3..h n h k 1,k 2,k 3..k n v 1,v 2,v 3..v n thickness of the each of the n layers total thickness Darcy s coefficients of permeability of the layers respectively Velocity of flow through each layer Hydraulic gradient i will be the same for all the layers as for the entire deposit. Since v = ki, and k is different for different layers, v will be different for the layers, say v 1, v 2, v 3.. v n. Also, v 1 = k 1 i ; v 2 = k 2 i.. and so on. Considering unit dimension perpendicular to the plane of the paper, the areas of flow for each layer will be h 1, h 2, h 3.. h n respectively, and it is h for the entire deposit.

86 Permeability of Stratified Soils Flow parallel to the bedding plane Discharge through the entire deposit = discharge through the individual layers. Assuming k z to be the average permeability of the entire deposit parallel to the bedding planes, and applying the equation: q = q 1 + q q n We have, q = v. A = k i. A k x. ih = k 1 i. h 1 + k 2 i. h k n i. h n k x. h = k 1 h 1 + k 2 h k n h n R eq = R 1 + R 2 + R 3

87 Consolidation (extra slides) Department of Civil Engineering Advanced Soil Mechanics. W. Sołowski 87

88 Terzaghi s consolidation theory u t kz m w v 2 u 2 z c v 2 u 2 z u t k z w m v c v pore water pressure time vertical conductivity unit weight for water compressibility coefficient of consolidation The solution with the help of Fourier series can be written: u T v 4 p p kz m w m v 1 sin 2m 1 t h 2 c h v 2 t 2m 1p z 2m1 2h e p = u and m = positive integer (kokonaisluku) T v 2 2 p T time factor including time and circumstance variables v / 4 88

89 Settlement time t u T v h c 2 v t u h time needed to achieved consolidation degree U (t) (%) the longest seepage distance Nomograms to define T v. Settlement S c (t) in time t is: S c t U t S c Coefficient of consolidation (oedometer test): c v k m w v Mk w M tangent modulus