Sphinx Document for terzaghi updated

Author: JananiPSrinivasan

doc/sphinx/source/examples/Terzaghi/1D/Terzaghi1D-BC.md

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+### Terzaghi One-Dimensional Consolidation
+
+Simulates **Terzaghi's 1D consolidation** using mimetic finite difference operators from the MOLE library. The system models **transient flow and deformation** in a saturated soil column under a **constant compressive load** at one end with **drainage permitted only at the loaded boundary**.
+
+The governing pressure equation is:
+
+$$
+\frac{\partial p}{\partial t} = c_f \nabla^2 p
+$$
+
+with $x\in[0,25]$ meters. Displacement and strain are derived from the pressure, and Darcy’s law is used to compute fluid flux.
+
+#### Boundary conditions:
+
+- **Dirichlet** at \( x = 0 \): \( p(0,t) = 0 \)  
+- **Neumann** at \( x = L \): \( \frac{dp}{dx}(L,t) = 0 \)
+
+This corresponds to a domain with **impermeable backing** and **open drainage at the loaded end**.
+
+---
+
+#### Numerical Strategy
+
+- Pressure is initialized to a uniform value \( P_0 = 10 \text{ MPa} \)
+- Integration is performed using **Forward Euler**
+- Mimetic MOLE operators:
+  - `lap()` for pressure diffusion
+  - `grad()` for Darcy flux
+  - `div()` for residual calculations
+- Spatial discretization uses a **staggered grid** with ghost cells to enforce boundary conditions
+
+---
+
+#### Analytical Benchmark
+
+An analytical solution is computed using a **Fourier series expansion**:
+
+$$
+p(x,t) = \sum_{n=0}^{\infty} \left( \frac{4 P_0}{n \pi} \sin\left(\frac{n \pi x}{2L}\right) e^{- \frac{n^2 \pi^2 c_f t}{4L^2}} \right), \quad n = 2k + 1
+$$
+
+The benchmark solution includes:
+- Pressure field
+- Flux via Darcy’s law
+- Strain and displacement
+- Mass conservation residual
+
+---
+
+#### Outputs
+
+At selected time snapshots (1, 10, 40, 70 hours), the following are printed and plotted:
+
+- **Numerical and analytical pressure** profiles
+- **Darcy flux** from numerical and analytical solutions
+- **Displacement fields**
+- **Mass balance residuals**
+- **Relative L2 error** tables
+- **3D surface plots** for pressure, displacement, and residual evolution
+
+---
+
+#### Physical Parameters
+
+| Parameter        | Value                         | Description                             |
+|------------------|-------------------------------|-----------------------------------------|
+| \( P_0 \)         | 10 MPa                        | Face load                               |
+| \( c_f \)         | \(1 \times 10^{-4}\)          | Diffusivity                             |
+| \( K \)           | \(1 \times 10^{-12} \, \text{m}^2\) | Permeability                      |
+| \( \mu \)         | \(1 \times 10^{-3} \, \text{Pa·s}\) | Dynamic viscosity                 |
+| \( K_s \)         | \(1 \times 10^8 \, \text{Pa}\) | Bulk modulus                            |
+| \( \alpha \)      | 1.0                           | Biot coefficient                        |
+| \( S_s \)         | \(1 \times 10^{-5} \, \text{Pa}^{-1}\) | Specific storage coefficient     |
+| \( \rho \)        | 1000 kg/m³                    | Fluid density                           |
+| \( g \)           | 9.81 m/s²                     | Gravitational acceleration              |
+
+---
+
+#### Code Location
+
+This example is implemented in:  
+- [MATLAB/ OCTAVE (terzaghi1D.m)](https://github.com/csrc-sdsu/mole/blob/master/examples/matlab/terzaghi1D.m)
+
+---