please help with parts a, b, c, d and e Write the equation for 3-D transient gro
ID: 802848 • Letter: P
Question
please help with parts a, b, c, d and e
Write the equation for 3-D transient groundwater flow, given heterogeneity and anisotropy. Describe all variables and their units. How does this equation change if we assume a steady-steady system? Write this equation. Write the equation of 1-D flow with isotropic and homogeneous K for both steady-state and transient systems. Based on these equations, which would you use to estimate hydraulic conductivity of an aquifer? Why? Write the equation for 2-D unconfined flow (aka the Boussinesq equation-see Fetter). Describe when and why the head can be taken out of the derivative.Explanation / Answer
a. Equation for 3D transient ground water flow for heterogenous and anisotropic confined aquifer
d/dx(kx.b.dh/dx)+d/dy(ky.b.dh/dy)+d/dz(kz.b.dh/dz)=Sdh/dt
d/dx(Tx.dh/dx)+d/dy(Ty.dh/dy)+d/dz(Tz.dh/dz)=Sdh/dt
where Tx, Ty and Tz are transmissivities of aquifer in x,y and z directions and its unit is m2/day
S is the storage coefficient of the aquifer and is a dimensionless quantity and has no units
kx,ky,kz= hydraulic conductivity of aquifer in x, y and z directions respectively and has units of m/day
h= hydraulic head of aquifer in meters
t= time in seconds
b= thickness of aquifer in meter
B. If we assume steady flow conditions in a heterogeneous and anisotropic aquifer then there will be no change in storage of aquifer with time which implies dh/dt=0
then equation becomes
d/dx(kx.dh/dx)+d/dy(ky.dy)+d/dz(kz.dz)=0
C. For homogeneous and isotropic ground water flow k will not vary with space Kx=Ky=Kz=K given by laplace equation
d^2h/dx^2+ d^2h/dy^2+d^2h/dz^2= Ss/K*dh/dt
d^2h/dx^2+ d^2h/dy^2+ d^2h/dz^2=S/K*dh/dt
Since the equation is one dimensional then it reduces to
d2^h/dx=S/K* dh/dt
when the flow is steady which means no change in storage with time and hence dh/dt=0
Therefore d^2h/dx=0
For transient 1 D flow in homogeneous and isotropic aquifer
d^2h/dx=S/T*dh/dt
We would use equation of transient flow to find the hydraulic conductivity because hydraulic conductivity=T/b if T transmissivity and aquifer thickness b is known then hydraulic conductivity can be easily calculated.
D. Boussinesq equation for homogeneous and isotropic unconfined aquifer
Since it is homogneous and isotropic so the value of K will remain constant and hence Kx=Ky=K
d/dx (hdh/dx)+ d/dy(h.dh/dy)=Sy/k*dh/dt
d^2h/dx^2+ d^2h/dy^2=Sy/K h mean* dh/dt
d^2h/dx^2+ d^2h/dy^2= Sy/T* dh/dt
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