4.11: Problem set
- Page ID
- 38776
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1.A soil profile is in hydraulic equilibrium with a water table located at a depth of 275 cm below the surface. Draw a sketch representing that soil profile, label the soil surface as point “A” and the water table as point “B”. Create a table showing the gravitational head, pressure head, and total head at points A and B.
2.An initially saturated soil sample is brought into hydraulic contact with a thin, porousplate connected to a hanging water column(see Fig. 3-12a)with a vertical length of 55 cm.
a.When the soil reaches hydraulic equilibrium, what is the pressure head at the base of the soil sample?
b.In one complete sentence explain why this pressure head occurs.
3.A cylindrical soil column of 100. cm2 cross-sectional area (Asoil) and 50.-cm height is filled with homogeneous soil and saturated, and 10. cm of water is kept ponded on the surface. The soil column is open to the atmosphere and freely draining at the bottom. The steady-state volumetric flow rate through the soil (Qsoil)is 1.0 x 103cm3h-1.
a.Draw a sketch of this soil column.
b.Create a table to determine the difference in hydraulic head across the column.
c.Convert the volumetric flow rate (Qsoil) to soil water flux (q) using the fact that q= Qsoil/A.
d.Calculate the saturated hydraulic conductivity of the soil.
4.A 1.0-mm diameter tube is pushed through the column described in problem 3 and hollowed out. Steady-state flow is established, with water flowing through the soil in accordance with Darcy’s Law and through the tube in accordance with Poiseuille’s Law. Assume the viscosity of water is 1.0 x 10-3kg m-1s-1.
a.Convert the difference in total water potential across the soil column (ψtas calculated in problem 3) expressed in cm of water to a difference in pressure expressed in Pascals (Pa) using the fact that 1 kPa = 1,000 Pa = 10.2 cm of water.
b.Calculate the volumetric flow rate (Qtube)through the tube using Poiseuille’s Law as expressed in Equation 4-1.Convert your answer from m3s-1to cm3h-1.
c.Calculate the flux for the combined column-tube system in cm h-1. Assume that the tube takes up a negligible portion of the cross-sectional area of the column, so the flow through the soil matrix itself is unchanged. Therefore qcombined = (Qsoil+ Qtube)/Asoil.
d.Calculate the effective saturated hydraulic conductivity for the combined soil/tube system.
e.Write one sentence explaining the practical significance of this example
5. A saturated soil column contains two soil layers, each 10. cm thick, with sand having Ks= 10. cmh-1 underneath loam having Ks= 5.0 cm h-1, and 10. cm of water is kept ponded on the surface. The bottom of the soil column is open to the atmosphere and drains freely.
a.Draw a sketch of this soil column.
b.Create a table to determine the difference in hydraulic head across the column
c.Calculate the hydraulic resistance for each layer.
d.Calculate the flux of water through the columnusing Darcy’s Law for layered soil.
e.Calculate the pressure head at the sand-loam interface