Solve Laplace Eq. in 1D: Head & Darcy Vel.

[Jgoshorn1]

Homework Statement

Solve the Laplace equation in one dimension (x, i.e. (∂^2h)/(∂x^2)= 0)
Boundary conditions are as follows:
h= 1m @ x=0m
h= 13m @ x=10m
For 0≤x≤5 K1= 6ms^-1
For 5≤x≤10 K2 = 3ms^-1

What is the head at x = 3, x = 5, and x = 8?

What is the Darcy velocity (specific discharge)?


NOTE: There are multiple steps that will need to be done. Realize that system is heterogeneous. In a multiple layer system with steady-state conditions, Darcy velocity in one layer must equal the Darcy velocity in the other layers

Homework Equations

h(x) = h_o - [(h₀ - h_D )/D]*x

The Attempt at a Solution

I tried to use the equation above subbing in 3, 5, and 8 for the x and using 10m as D, 1m as h₀, and 13m for h_D

Then I used the specific specific discharge for the Darcy's velocity (q=K(dh/dL))

That was apparently all wrong. Apparently this needs to be broken into 2 systems, coupled. Each individual system can be treated as homogenous. So it need two separate LaPlace equations? I really don't know what to do with this problem, please help!

 

[LawrenceC]

Since the discharge goes through both portions of the domain, you have continuity of the dependent variable h at the interface. In addition you also have continuity of discharge at the interface.

When you integrate the Laplace equation in 1-D, you have two constants of integration for each section of the domain. The matchings of discharge and h serve to eliminate some of the integration constants.

 

[Jgoshorn1]

So, what I've done is set q(in) = q(out) => -K1(dh/dl)=K2(dh/dl) => 6((h-1)/5) = 3((13-h)/5) => h(@x=5) = 5m

Then I set the used set of boundary Conditions (BC) to interpret the solution to LaPlace in 1D (h=cx+D):
For BC @ x=0 and h=1 =>1=c(0)+D => 1=D
For BC @ x=5 and h=5 => 5=c(5)+1 => c=4/5
the I used h=cx+D again to solve for the head at x=3
=>h(@x=3)=(4/5)(3)+1=3.4

For the second system I used the point slope eq (y1-y)=m(x1-x) to get c
=> (5-13)/(5-10) = m = 8/5 = c
Then I used h=cx+D to solve for D
=> 5=(8/5)(5) + D => D=-3
Then to solve for h at x=8
h=cx+D => h(@x=8) => (8/5)(8)-3 = 9.8To get Darcy's velocity I just used q=K(dh/dl) for each system
System 1 = 6(4/5) = 4.8 = q
System 2 = 4(8/5) = 4.8 = q

Yes?

 

[LawrenceC]

I get the same answers as you but went about the problem a little differently.

I solve the ODE over two separate domains:

h1(x)=c1+c2*x
h2(x)=c3+c4*x

I used the Dirichlet boundary conditions along with continuity of head and continuity of Darcy velocity to solve for the 4 constants of integration. Equations worked out to be:

h(x) = 1 + .8x 0<x<5

h(x) = -3 + 1.6x 5<x<10

The < signs should be weak inequalities. I cannot make them on my computer.

 

[paranormal]

I would approach this problem by first understanding the physical significance of the Laplace equation in this context. The Laplace equation is a mathematical representation of the steady-state flow of a fluid in a homogeneous medium, where there are no sources or sinks of fluid. In this problem, the Laplace equation is being used to model the flow of water in a one-dimensional system with varying hydraulic conductivity values.

To solve this problem, I would first break it into two separate systems, as mentioned in the note. Each system can be treated as homogeneous, with a constant hydraulic conductivity value. This means that the Laplace equation can be applied separately to each system, with appropriate boundary conditions.

For the first system, where 0≤x≤5 and K1= 6ms^-1, the Laplace equation can be written as (∂^2h)/(∂x^2) = 0, with the boundary conditions h(0) = 1m and h(5) = 13m. This can be solved using standard techniques for solving Laplace equations, such as separation of variables or the method of images.

For the second system, where 5≤x≤10 and K2= 3ms^-1, the Laplace equation can be similarly written as (∂^2h)/(∂x^2) = 0, with the boundary conditions h(5) = 13m and h(10) = 13m. Again, this can be solved using standard techniques.

Once the solutions for each system are obtained, they can be combined to obtain the overall solution for the entire system. This combined solution will give the head at any point x within the system.

To find the Darcy velocity, I would use the Darcy's law equation, q = K(dh/dL), where q is the specific discharge, K is the hydraulic conductivity, and (dh/dL) is the hydraulic gradient. The hydraulic gradient can be calculated using the head values obtained from the Laplace equation solutions.

In summary, to solve this problem, I would break it into two separate systems, solve each system using the Laplace equation with appropriate boundary conditions, and then combine the solutions to obtain the overall solution and calculate the Darcy velocity.