Chapter 9

To derive an upward flow in a porous medium, it is clear that pressure must increase more rapidly with depth $y$ than it does when the fluid is motionless. Use this idea to justify writing Darcy's law for vertical flow in a porous medium in the form $$v=-\frac{k}{\mu }(\frac{dp}{dy}-\rho g)$$ where $v$ is the vertical Darcy velocity (positive in the direction of increasing depth), $\rho$ is the fluid density, and $g$ is the acceleration of gravity. Consider a porous medium lying on an impermeable surface inclined at an angle $\theta$ to the horizontal. Show that Darcy's law for the downslope volumetric flow rate per unit area $q$ is $$q=-\frac{k}{\mu }(\frac{dp}{ds}-\rho g\sin \theta )$$ where $s$ is the downslope distance and $q$ is positive in the direction of $s$.

Consider an unconsolidated (uncemented) layer of soil completely saturated with groundwater; the water table is coincident with the surface. Show that of d the upward Darcy velocity $|v|$ required to fluidize the bed is $$|v|=\frac{(1-\varphi )kg({\rho }_s-{\rho }_w)}{\mu }$$ where $\varphi$ is the porosity, ${\rho }_s$ is the density of the soil particles, and ${\rho }_w$ is the water density. The condition of a fluidized bed occurs when the pressure at depth in the soil is sufficient to completely support the weight of the overburden. If the pressure exceeds this critical value, the flow can lift the soil layer.

Assume that a porous medium can be modeled as a cubic matrix with a dimension $b$; the walls of each cube are channels of thickness $\delta$. (a) Determine expressions for the porosity and permeability in terms of $b$ and $\delta .$ (b) What is the permeability if $b=0.1\mathrm{m}$ and $\delta =1\mathrm{mm}?$

Consider one-dimensional flow through a confined porous aquifer of total thickness $b$ and crosssectional area $A$. Suppose the aquifer consists of $N$ layers, each of thickness $b_i(i=1,\dots ,N)$ and permeability $k_i(i=1,\dots ,N).$ Determine the total flow rate through the aquifer if all the layers are subjected to the same driving pressure gradient. What is the uniform permeability of an aquifer of thickness $b$ that delivers the same flow rate as the layered aquifer when the two are subjected to the same pressure gradient?

If fluid is injected along a plane at $x=0$ at a rate of $0.1{\mathrm{m}}^2{\mathrm{s}}^{-1}$, how high is the phreatic surface at the point of injection and how far has the fluid migrated if $\mu ={10}^{-3}\mathrm{Pa}\mathrm{s},\varphi =0.1,k={10}^{-11}{\mathrm{m}}^2$ $\rho =1000\mathrm{kg}{\mathrm{m}}^{-3},$ and $t={10}^5\mathrm{s}$ ? A MATLAB solu- tion to this problem is provided in Appendix $D$.

Determine the minimum critical Rayleigh number for the onset of convection in a layer of porous material heated from below with an isothermal and impermeable lower boundary and an isothermal constant pressure upper boundary. This boundary condition corresponds to a permeable boundary between a saturated porous layer and an overlying fluid. What is the horizontal wavelength that corresponds to the minimum value of ${\mathrm{Ra}}_{\mathrm{cr}}$ ? Take the layer thickness to be $b$, and let the upper boundary, $y=0$, have temperature $T=T_0$ and the lower boundary, $y=b$, have temperature $T=T_1$. Assume that at the onset of convection $T^{\prime }$ has the form $$T^{\prime }=T_0^{\prime }\sin \frac{2\pi x}{\lambda }Y(y)$$ and show that $Y(y)$ is a solution of $$\frac{d^{4}Y}{d{\overline{y}}^{-4}}-2a^2\frac{d^{2}Y}{d{y}^{-2}}+Y(a^4-a^{2}Ra)=0$$ where $$a\equiv \frac{2\pi b}{\lambda }\overline{y}\equiv \frac{y}{b}$$ Show that the general solution of Equation (9.135) can be written as $$Y=c_1{e}^{\gamma \overline{y}}+c_2{e}^{-\gamma \overline{y}}+c_3\sin \delta \overline{y}+c_4\cos \delta \overline{y}$$ where $c_1,c_2,c_3,$ and $c_4$ are constants of integration and $$\begin{array}{l}{\gamma }^2=a^2+a\sqrt{\mathrm{Ra}}\\ {\delta }^2=a\sqrt{\mathrm{Ra}}-a^2\end{array}$$ Show that the boundary conditions are $$\begin{array}{rl}Y=0& \text{ on }\overline{y}=0\text{ and }1\\ \frac{d^{2}Y}{d{y}^2}=0& \text{ on }\overline{y}=1\\ \frac{d}{d\overline{y}}(\frac{d^{2}Y}{d{\overline{y}}^2}-a^{2}Y)=0& \text{ on }\overline{y}=0.\end{array}$$ Substitute Equation (9.137) into each of these boundary conditions to obtain four homogeneous equations for the four unknown constants $c_1,c_2,$ $c_3,$ and $c_4.$ Show that a nontrivial solution of these equations requires $$\gamma \tan \delta +\delta \tanh \gamma =0$$ This transcendental equation is an eigenvalue equation that implicitly gives ${\mathrm{Ra}}_{\mathrm{cr}}$ as a function of $a$, since both $\gamma$ and $\delta$ are defined in terms of $\mathrm{Ra}$ and $a$ in Equations (9.138) and (9.139) . The critical Rayleigh number can be found by numerically solving Equations (9.138),(9.139) , and (9.143) . The value of min $\left({\mathrm{Ra}}_{\mathrm{cr}}\right)$ turns out to be $27.1.$ One way of proceeding is to choose a value of $a$ (there exists an ${\mathrm{Ra}}_{\mathrm{cr}}$ for each a). Then try a value of $\delta$. Compute $\gamma$ from ${\gamma }^2=2a^2+$ ${\delta }^2.$ Then compute $\tan \delta /\delta$ and $-\tanh \gamma /\gamma$. Iterate on $\delta$ until these ratios are equal. With $\delta$ determined ${\mathrm{Ra}}_{\mathrm{cr}}$ follows from Equation $(9.139).$ Repeat the process for different values of $a$ until $min\left({\mathrm{Ra}}_{cr}\right)$ is found.

Consider a porous layer saturated with water that is at the boiling temperature at all depths. Show that the temperature-depth profile is given by $$\frac{1}{T_{b0}}-\frac{1}{T}=\frac{R_v}{L}\ln (1+\frac{{\rho }_{l}gy}{p_0})$$ where $T_{b0}$ is the boiling temperature of water at atmospheric pressure $p_0,{\rho }_l$ is the density of liquid water which is assumed constant, and $R_r$ is the gas constant for water vapor. Start with the hydrostatic equation for the pressure and derive an equation for $dT/dy$ by using the formula for the slope of the Clapeyron curve between water and steam $$\frac{dp}{dT}=\frac{L{\rho }_l{\rho }_v}{T({\rho }_l-{\rho }_v)}\approx \frac{L{\rho }_v}{T}$$ where ${\rho }_v$ is the density of water vapor. Assume that steam is a perfect gas so that $${\rho }_v=\frac{p}{R_{r}T}$$ Finally, note that $p=p_0+{\rho }_{l}gy$ if ${\rho }_l$ is assumed constant. What is the temperature at a depth of $1\mathrm{km}$ ? Take $R_v=0.462\mathrm{kJ}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1},L=2500\mathrm{kJ}{\mathrm{kg}}^{-1}$ $$T_{b0}=373\mathrm{K},p_0={10}^5\mathrm{Pa},{\rho }_l=1000\mathrm{kg}{\mathrm{m}}^{-3}$$ $g=10\mathrm{m}{\mathrm{s}}^{-2}$.

Calculate pressure as a function of depth in a vapor-dominated geothermal system consisting of a near-surface liquid layer $400\mathrm{m}$ thick overlying a wet steam reservoir in which the pressure-controlling phase is vapor. Assume that the hydrostatic law is applicable and that the liquid layer is at the boiling temperature throughout. Assume also that the steam reservoir is isothermal.