Hydrology and Fluid Flow GLE/CEE 330 Lecture Notes Soil Mechanics William J. Likos, Ph.D. Department of Civil and Environmental Engineering University of Wisconsin-Madison

The Hydrologic Cycle

Water volume, in cubic miles Where is the Water? Water source Water volume, in cubic miles Percent of total water Oceans 317,000,000 97.24% Icecaps, Glaciers 7,000,000 2.14% Ground water 2,000,000 0.61% Fresh-water lakes 30,000 0.009% Inland seas 25,000 0.008% Soil moisture 16,000 0.005% Atmosphere 3,100 0.001% Rivers 300 0.0001% Total water volume 326,000,000 100% Source: Nace, U.S. Geological Survey, 1967 and The Hydrologic Cycle (Pamphlet), U.S. Geological Survey, 1984

Water Table Fluctuations (USGS)

Saturated and Unsaturated Zones Low hydraulic conductivity; k = f(S)

Aquifers (Water Encyclopedia)

Perched Groundwater Table

Why do Geotechs Care about Water? 1) Predict quantity of flow Earth dam Typical Section

Why do Geotechs Care about Water? 2) Predict rate of flow (e.g., contaminant transport)

Why do Geotechs Care about Water? 3) Predict pore water pressure! Terzaghi’s Effective Stress s’ = effective stress s = total stress uw = pore water pressure sv uw Total stress (self weight, external loads) is carried by the soil skeleton and the pore pressure = s’+uw The stress carried by the soil skeleton governs engineering behavior (strength, volume change). If the pore pressure changes, the effective stress changes, and changes in behavior occur

1D Fluid Flow – Darcy’s Experiment Qin Qout Influent water reservoir Effluent water reservoir Henry Darcy (1803-1858) h1 L h2 datum A Saturated Sand Flow

Influent water reservoir Effluent water reservoir Qin Qout Influent water reservoir Effluent water reservoir h1 L h2 datum A Saturated Sand Flow Q = volumetric flow rate (volume/time), e.g., cm3/sec At steady state, Q = Qin = Qout Darcy found that Dh = total head loss i = Dh/DL = “hydraulic gradient”

Darcy’s Law For steady state saturated flow in porous media k = “hydraulic conductivity” (length/time) Dht = total head loss (length) DL = distance over which head loss occurs (length) A = cross-sectional area of flow (length squared) L A datum h1 h2 Qin Qout

q = v = “Darcy velocity” or “discharge velocity” v k So, k is a property of the soil (and fluid) that quantifies conductivity to water flow. Sand – high k Clay – low k i

Typical Values of Hydraulic Conductivity

Hydraulic Conductivity vs. Intrinsic Permeability Hydraulic conductivity (k), units = length/time (cm/s) depends on both soil and fluid properties soil properties → void ratio (e), grain size distribution (D10) fluid properties → density (r), viscosity (m) Intrinsic permeability (K), units = length2 (m2) depends only on soil properties for water, K is about 5 orders less than k for example, if k = 10-5 cm/s then K = 10-10 cm2

Hydraulic Conductivity vs. Intrinsic Permeability Example:

Discharge Velocity vs. Seepage Velocity Discharge velocity (v), aka “Darcy” velocity gross flow velocity out of a cross-section Seepage velocity (vs) actual fluid velocity around soil grains governs “piping” (internal erosion processes) vs > v For mass conservation at discharge: Q1 = vsAv Q2 = vAt At = Av + As Av As Since n < 1, vs > v

Total Hydraulic Head Bernoulli’s Equation: ht = total hydraulic head: (m) z = elevation head: (m) uw/gw = pressure head: (N/m2)/(N/m3) = (m) v2/2g = velocity head: (m2/s2)(m/s2) = (m) Daniel Bernoulli (1700-1782) We may also write: For most flow processes in soil v is small; so we usually assume:

Total Hydraulic Head, ht Total hydraulic head (“total head”) is the governing variable for fluid flow. There MUST be a difference in ht for flow to occur between any two points. “Head Loss” Dht =ht1 - ht2 ≠ 0 Consider a beaker of water Will there be flow between 1 and 2? (Is there a “head loss” between 1 and 2?) z ht1 = ht2 so Dht = 0 so NO FLOW he 1 ht1 hp z (+) 2 ht2 Datum (z = 0) he or hp or ht

(flow will stop when water level reaches Point 3) Can water flow uphill? YES! – but only if Dht ≠ 0 3 hp1 FLOW hp2 2 he2 z (+) 1 he1 Datum ht1 > ht2 So FLOW from 1 to 2 (flow will stop when water level reaches Point 3)

ksilt= 10-5 cm/s = 2.8 X 10-2 ft/day Qin ksilt= 10-5 cm/s = 2.8 X 10-2 ft/day Qout 3’ 6’ 25’ 2’ 10’ A B C 5’ datum Point he (ft) hp (ft) ht (ft) A 5 20 25 B ? C 10 (Need to find total head at B by interpolation)

ksilt= 10-5 cm/s = 2.8 X 10-2 ft/day Qin ksilt= 10-5 cm/s = 2.8 X 10-2 ft/day Qout 3’ 6’ 25’ 2’ 10’ A B C 5’ datum ht 25’ htb assume head loss is linear in the soil 10’ 3’ 6’ x A B C

ksilt= 10-5 cm/s = 2.8 X 10-2 ft/day Qin Qout 3’ 6’ 25’ 2’ 10’ A B C 5’ datum Point he (ft) hp (ft) ht (ft) A 5 20 25 B 15 C 10

Drawing Head Diagrams General approach (for steady state systems with homogeneous soil):   Establish elevation datum (arbitrary) Compute and draw he profile relative to elevation datum. Compute hp for points located “outside” the soil. Recall that hp is the height from the point to the nearest peizometer (water level). Compute ht from ht = he + hp. Pressure head at any point “inside” the soil can then be calculated graphically from the pressure head profile or analytically using the fact that the head loss is linear within the soil.

Lambe and Whitman (1969)

Lambe and Whitman (1969)

Example Problem: Determine the components of head at A, B, and C What is the hydraulic conductivity if you measure v = 2 cm/hr?

Point he (cm) hp (cm) ht (cm) A 80 20 100 B 10 90 C 40 45 85 D 50 30

Determining Hydraulic Conductivity Varies over ~13 orders of magnitude! tight shale ~ 10-11 cm/s to clean gravel ~ 102 cm/s No one method is ideal for all soil types Laboratory Methods: Constant Head Falling Head Constant Flow Field Methods: Point Infiltrometers (small scale) Pumping Tests (large scale) Challenges: Disturbance! Scale? Anisotropy Advantages: Control of boundary conditions Control of stress state [k = f(e)] Cost effective? (varved clay; kh ≠ kv)

Constant Head Method Qin Dh A = (k)(Dh/DL)(A) Qin = Qout s1 s3 Qout DL Steady-state flow established under constant gradient (constant head) Measure Q for applied gradient to determine k from Darcy’s Law Want to keep i < 30 to avoid changes to soil fabric Qin Requires undisturbed specimen Flexible wall or rigid wall systems Very common Best for relatively high k (e.g., sand) Dh A = (k)(Dh/DL)(A) Qin = Qout s1 s3 Qout DL

Qin = 892 ml in 112 sec Dh = 60cm d=18 cm Qout DL=16.7 cm Constant Head Example Problem Qin = 892 ml in 112 sec Dh = 60cm d=18 cm Qout DL=16.7 cm = (k)(Dh/DL)(A) Qin = Qout

Falling Head Method graduated Dh@ t0 standpipe (a) t0 t1 falling head Hydraulic gradient changes with time Works for relatively low k (e.g., clay) graduated standpipe (a) Dh@ t0 t0 t1 falling head boundary a Dh@ t1 A Qout DL constant head boundary

Constant Flow Method Q Syringe Pump A Dh DL Q = (k)(Dh/DL)(A) Apply Q rather than measure it Measure Dh rather than apply it Q = -0.1 cc/min Q = 0.1 Q = -0.2 Q = 0.2 Q = -0.3 Q = 0.3 Q = (k)(Dh/DL)(A)

Constant Flow Method Can apply extremely small flow rate Q Thus, we can measure extremely small k, and still keep low gradient (i<30) k as low as 10-14 cm/s Very useful for low perm materials (clay, shale, concrete!)

Empirical Correlations “Empirical” = Relying on or derived from observation or experiment Often desirable to estimate k from other (easily measured) properties Hazen’s correlation % Finer k = hydraulic cond. (cm/s) C = empirical coefficient (0.8 to 1.2; commonly 1.0) D10 = 10% finer diameter from sieve analysis (mm) 0.1mm < D10 < 3mm and Cu < 5 D10 Log D Makes sense that the smaller grains govern hydraulic conductivity

Anisotropic Flow k4 H4 kx k3 H3 k2 H2 kz k1 H1 (varved clay; kh ≠ kv) Generally kx > kz

Anisotropic Flow Example Silt (5 mm) Analysis area (repeating structure) kz kx Clay (20 mm) ks = 3 × 10-4 cm/s kc = 6 × 10-7 cm/s

Capillarity Sandstone, Golden Colorado hc ~ 9’ (275 cm)

Capillarity

Height of Capillary Rise (Lu and Likos, 2004)

2D Fluid Flow – Flow Net Analysis What is the distribution of total head and flow velocity in the domain? y x z dz S = 1.0 dx

Conservation of Mass: dz S = 1.0 dx (any velocity change in one direction must be negated by an equal and opposite velocity change in the other direction)

3) From mass conservation 1) From Darcy’s Law: 3) From mass conservation Laplace Equation 2) Define “velocity potential” f Partial diff. equation defining distribution of f (total head) in 2 dimensions at steady state

Solution to Laplace eq. is a “flow net” Equipotential lines – same total head (like contour lines) Streamlines (Flow lines) lines along which flow occurs Equipotential Lines Flow Lines

Procedures for drawing flow nets: Draw 2-D cross-section of problem to scale Draw boundaries in ink Take advantage of symmetry Select integer number of flow “tubes” (Nf ~4 to 6) Sketch flow lines Sketch equipotential lines Follow b/a = 1 rule EL and FL must intersect at right angles No FL or El may intersect Lines are smooth b a EL EL FL ht1 FL ht2

Using Flow Nets – Flow Prediction Nf = 3 Nd = 12 b/a = 1 Dh = 10 m k = 1× 10-3 cm/s 10m

Cut-Off Walls (Terzaghi, Peck, Mesri)

Impervious Blankets, Graded Filters (Terzaghi, Peck, Mesri)