# IMTECO
## Dynamics of bog-pool and impact of infrastructures

### List of researchers
- [Irene Tierney](mailto:[e-mail])
- [Matteo Icardi](mailto:[e-mail])
- [David Large](mailto:[e-mail])
- [Oliver Ward](mailto:[e-mail])
- [Ivan Sudakow](mailto:[e-mail])
- [Valerie Livina](mailto:[e-mail])
- [Luis Espath](mailto:[e-mail])
- [Alex Trenam](mailto:[e-mail])

### Resources and data available
- [Google Maps bog-pools area -- Kinlochbervie, Lairg](https://maps.app.goo.gl/7MWKYZdRBV7qstNi8)
- [Google Maps bog-pools area -- Flow contry](https://maps.app.goo.gl/6pHLTcrds57ahmN26)
- [Topographic maps](digimap.edina.ac.uk)
- [IMTECO presentation](https://multiform-uon.github.io/mathgeo/dundee_challenges/Bog%20Pool%20Hydrology.pdf)
- [Water level elevation model -- Arctic pools system](https://www.math.utah.edu/~golden/docs/publications/Bowen_et_al_JFG_2018.pdf)
- [Peat degradation on wind farms -- EJ paper](https://www.mdpi.com/2071-1050/16/17/7713)
- [England peat map](https://naturalengland.blog.gov.uk/2025/05/12/a-new-peat-map-for-england/)

### Discussion points
- Peat depth: Kinlochbervie 3mt (1-6mt)
- Infrastructure design normally based only on peat depth and contours. It would be beneficial to support the impact assessment with a quantitative method that takes into account other data (vegatation, geology, hydrologic flow, future development of the pool system, etc...)
- Buffer zones? Observations reveal year-scale direct disturbances (erosion or excessive pooling) of peat in the order of a few meters. Current buffer zones are 10x (~30mt). These do not consider the local peat properties and hydrology.
- What mitigation can be put in place to reduce disturbances?
- Spatial data and availability
- Elevation maps: low-res easily available, dynamic high-res data available in some sites
- Peat depth maps: 1st survey (every 100mt), 2nd survey (50mt), 3d survey (10mt)
- Underlying topography: ?
- Groundwater-dependent vegetation surveys
- Permeability/porosity measurements
- Observation: nearby pools might have different water levels -> almost impermeable medium below pools? deep peat as a hydrogel? air pockets?
- How to measure permeability in peat? (Ehsan) correlation betweeen pore pressure and permeability or more accurate relaxation curves of pore pressure. What's the underlying mathematical model used to interpret this data? are they valid for peat?
- Policy implications?

### Possible approaches
#### Three-dimensional complete models
- Very expensive/slow computationally
- E.g., ChemoHydroMechanical model (Santiago)
- Fenics FEM code

#### Semi-analytical real-time model (Laplace disturbances)
- Assume fully saturated (unconfined?) flow
- For non-saturated flow, could use Richards equation
- 3D permeability map.
- Pools as reservoirs (or alternatively modelled with surface-subsurface models)
- Idealised scenarios with
- semi-infinite domain, or
- finite domain with physical boundaries
- Infrastructure as discontinuities or impermeable barriers
- Compute Green's functions or length-scale of disturbances. Can be done semi-analytically or numerically
- Identify drying or flooding area near the infrastructure
- Second-order effects (erosion followed up by further hydrological changes) not directly included (can be included through iterative scenarios)
- This can become a simple interactive software tool or a simple look-up table/diagram: length-scale of erosion/degradation vs slope angle and peat depth

A simple mathematical formulation is given by the Laplace equation for the hydraulic head $h$:

\begin{align}
\nabla \cdot (\kappa \nabla h) = 0,
\end{align}

where $\kappa(x)$ is the heterogeneous peat permeability. Pools can be treated as boundary conditions (Dirichlet: fixed head) while infrastructures can be idealised as impermeable barriers (Neumann: zero-flux) or zero permeability inclusions.

\begin{align}
\kappa \nabla h \cdot n &= r(t) && \text{(top surface: rainfall/flux input)}, \\
h &= h_{\text{pool}} && \text{(pools: fixed head)}, \\
\kappa \nabla h \cdot n &= 0 && \text{(infrastructure/barriers: no-flux)}, \\
\kappa \nabla h \cdot n &= 0 && \text{(bottom: impermeable base)}.
\end{align}

Extensions to include topography can be obtained by letting $h = w + \tau(x,y)$. This introduces additional source/sink terms proportional to the curvature $\Delta \tau$, allowing accumulation in valleys and drainage from slopes.

#### Large-scale modelling of peat areas using stochastic differential equations (SDEs)
- consider a selected large peat area as the object of interest (rather than a separate pool with detailed modelling)
- derive an SDE to describe the aggregated properties of the peat area, such as susceptibility to landslide, with input variables like precipitation, orography vector field, etc.
- [see preprint](https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1817/egusphere-2025-1817.pdf)

#### Two-dimensional meso-scale model
Let $P$ be the peat depth, and let $w$ be the water table depth. $\tau$ is the topography (altitude) map and $\kappa$ is the permeability of the ground. A first attempt at a simplified model could be

\begin{align}
\partial_t P &= \underbrace{g(w)}_{\text{growth/decay
}} +\underbrace{\varepsilon\Delta P}_{\text{diffusion}} + \underbrace{P \Delta \tau}_{\substack{\text{accumulation/erosion} \\ \text{in valleys/peaks}}} \\
\partial_t w &= \underbrace{r(t)}_{\text{rainfall}} + \underbrace{\operatorname{div}(\kappa\nabla w)}_{\text{diffusion}} + \underbrace{\kappa w \Delta\tau}_{\substack{\text{accumulation/decay} \\ \text{in valleys/peaks}}}
\end{align}
where $\varepsilon > 0$ is a small tuneable parameter introduced for stability of the system.

Construction sites could be included as a modification to the permeability $\kappa$, and then the impact could be assessed by comparing the original profiles to the modified version.

#### Revised 2D meso-scale model

$P$ -- peat depth, $w$ -- water table depth, $\tau$ -- topography, $\kappa$ -- permeability, $D$ -- diffusivity

\begin{align}
\partial_t P &= \underbrace{\operatorname{div}\left(D(P, w)\nabla P\right)}_{\text{diffusion}} + \underbrace{\operatorname{div}\left(P\nabla\tau\right)}_{\substack{\text{topography} \\ \text{transport}}} + \underbrace{g(P, w)}_{\text{growth}} - \underbrace{f_{\rm{erode}}(P, w)}_{\text{erosion}} \\
\partial_t w &= \underbrace{\operatorname{div}\left(\kappa(P, w)\nabla\left(w + \tau\right)\right)}_{\text{diffusion with topography}} + \underbrace{r(t)}_{\text{rainfall}} - \underbrace{f_{\rm{evap}}(P, w) - f_{\rm{trans}}(P, w)}_{\text{evapotranspiration}}
\end{align}

### Next steps
1. Complete the mathematical derivation of the 4 approaches
2. Develop simplified solvers for showing preliminary results
3. Collect data from the Flow Country bog-pools area
4. Test models against the data
5. Develop tool (look-up table style, intuitive software with GUI) for developers to use
6. Policy influence...