Step $1$

To calculate the various stress components as requested, we'll need to use the given information and formulas. Let's go step by step:

$($a$)$ Compute the total vertical stress assuming a lithostatic gradient of $23\mathrm{.}8$ MPa$/$km:

Total Vertical Stress $()=$ Lithostatic Gradient $(*$ g $*$ depth$)$

Explanation:

Given:

Lithostatic Gradient

Depth $($depth$)=$

$($to convert to meters$)$

Now, calculate $$:

Step $2$

$($b$)$ Compute the effective vertical stress assuming hydrostatic pore pressure gradient:

Effective Vertical Stress $(\text{'})=-$ Pore Pressure

In this case, since it's hydrostatic, the pore pressure is given by the weight of the overlying fluid, which is $\backslash$rho $*$ g $*$ depth:

Pore Pressure $=($Fluid Density $*$ g $*$ depth$)$

Explanation:

Assuming typical values for fluid density $(1,000$ kg$/$m$^3)$ and gravitational acceleration $(9\mathrm{.}81$ m$/$s$^2)$:

Pore Pressure $=$

$^3$

$^2$

Now, calculate $\text{'}$:

$\text{'}=7,766,950\mathrm{.}55$ Pa $-3,048,665\mathrm{.}26$ Pa $=4,718,285\mathrm{.}29$ Pa

Step $3$

$($c$)$ Compute horizontal effective principal stresses assuming horizontal isotropy:

Use the given expressions to calculate $$ and $$:$=/(1-)*+/(1-)*$

$=/(1-)*+(1-)/(1-)*$

Given:

$=0\mathrm{.}23($Poisson$$s ratio$)$

$==0($both horizontal strains are nearly zero$)$

Now, calculate $$ and $$:

$=0\mathrm{.}23/(1-0\mathrm{.}23)*7,766,950\mathrm{.}55$ Pa $=2,382,717\mathrm{.}81$ Pa

$=0\mathrm{.}23/(1-0\mathrm{.}23)*7,766,950\mathrm{.}55$ Pa $=2,382,717\mathrm{.}81$ Pa

Explanation:

$($d$)$ Write the tensor of effective principal stresses as a matrix:

The tensor of effective principal stresses in two dimensions is represented as a matrix:

$|10|$

$|02|$

In this case, $1=$ and $2=,$ which are both $2,382,717\mathrm{.}81$ Pa$.$

So$,$ the matrix is:

$|2,382,717\mathrm{.}810|$

$|02,382,717\mathrm{.}81|$

Answer

$($e$)$ Compute total horizontal principal stresses:

The total horizontal principal stresses $()$ are equal to the effective horizontal principal stresses in this case since there's no additional stress:

$==2,382,717\mathrm{.}81$ Pa

$($f$)$ Compute the ratio between total horizontal and total vertical principal stresses:

Ratio $=/$

So$,$ the ratio between total horizontal and total vertical principal stresses is approximately $0\mathrm{.}3064$