modelling Level 3

Soil Moisture Dynamics

Soil moisture controls plant water stress, runoff generation, and evaporation. This model builds a simple water balance model: precipitation inputs, evapotranspiration and drainage outputs, with storage limited by field capacity and wilting point. We'll track soil moisture over time and calculate water stress effects on plants.

Prerequisites: water balance, soil water potential, field capacity, wilting point

26 Feb 2026 Updated 12 Mar 2026 13 min read

1. The Question

How much water is available to plants, and when do they experience drought stress?

Rainfall adds water to soil. Plants extract it through roots. Gravity drains excess water downward. The balance determines:

Soil moisture content — how wet the soil is
Plant water stress — whether plants can meet transpiration demand
Runoff generation — when soil is saturated, rain becomes runoff
Groundwater recharge — deep drainage replenishes aquifers

The mathematical question: How do we model soil moisture dynamics from precipitation, evapotranspiration, and drainage?

2. The Conceptual Model

Soil Water Balance

The root zone is the soil layer where most roots extract water (typically 0–1 m depth).

Water balance equation:

\[\frac{dS}{dt} = P - ET - D - R\]

Where (all in mm/day or mm/time):

$S$ = soil moisture storage (mm)
$P$ = precipitation (mm/day)
$ET$ = evapotranspiration (mm/day)
$D$ = drainage (deep percolation, mm/day)
$R$ = runoff (surface flow, mm/day)

Storage limits:

Maximum storage (field capacity): $S_{\text{FC}}$
Minimum usable storage (wilting point): $S_{\text{WP}}$

Key Soil Moisture States

Saturation ($S_{\text{sat}}$):
All pore space filled with water. Typical: 40–60% of soil volume.

Field capacity ($S_{\text{FC}}$):
Water held against gravity after ~2 days of drainage. Typical: 25–45% volume.
This is the practical maximum for agricultural modelling.

Wilting point ($S_{\text{WP}}$):
Plants cannot extract more water (high tension required). Typical: 10–25% volume.
Below this, plants wilt permanently.

Plant-available water (PAW):

\[\text{PAW} = S_{\text{FC}} - S_{\text{WP}}\]

Typical values (for 1 m root zone):

Sandy soil: PAW ≈ 50–100 mm
Loam: PAW ≈ 150–200 mm
Clay: PAW ≈ 150–250 mm

Water Stress Factor

As soil dries, plants experience stress. Define a water stress coefficient (0 ≤ K_s ≤ 1):

\[K_s = \begin{cases} 1 & \text{if } S > S_{\text{crit}} \\ \frac{S - S_{\text{WP}}}{S_{\text{crit}} - S_{\text{WP}}} & \text{if } S_{\text{WP}} < S < S_{\text{crit}} \\ 0 & \text{if } S \leq S_{\text{WP}} \end{cases}\]

Where $S_{\text{crit}}$ is the threshold below which stress begins (typically 50–70% of PAW).

Actual evapotranspiration:

\[ET_{\text{actual}} = K_s \times ET_{\text{potential}}\]

When stressed ($K_s < 1$), plants close stomata → transpiration decreases → photosynthesis decreases.

3. Building the Mathematical Model

Drainage

Drainage occurs when soil moisture exceeds field capacity.

Simple linear drainage:

\[D = \begin{cases} k_d (S - S_{\text{FC}}) & \text{if } S > S_{\text{FC}} \\ 0 & \text{if } S \leq S_{\text{FC}} \end{cases}\]

Where $k_d$ is the drainage coefficient (day⁻¹, typically 0.1–1.0).

Fast drainage (sandy soil): $k_d \approx 1$ (excess drains in 1 day)
Slow drainage (clay): $k_d \approx 0.1$ (excess drains in 10 days)

Runoff

Runoff occurs when:

Soil is at saturation ($S = S_{\text{sat}}$)
Precipitation exceeds infiltration capacity

Simple bucket model:

\[R = \begin{cases} P - I_{\text{max}} & \text{if } P > I_{\text{max}} \text{ and } S = S_{\text{sat}} \\ 0 & \text{otherwise} \end{cases}\]

Where $I_{\text{max}}$ is infiltration capacity (mm/day, typically 10–100 mm/day).

For simplicity, assume:

Runoff occurs when $S$ would exceed $S_{\text{sat}}$
Any excess precipitation becomes runoff

\[R = \max(0, S + P - S_{\text{sat}})\]

Complete Water Balance

At each time step:

Add precipitation: $S’ = S + P \Delta t$
Check for saturation excess runoff:
If $S’ > S_{\text{sat}}$: $R = S’ - S_{\text{sat}}$, set $S’ = S_{\text{sat}}$
Calculate stress factor $K_s$ from current $S’$
Remove ET: $S’’ = S’ - K_s ET_{\text{pot}} \Delta t$
Calculate drainage:
If $S’’ > S_{\text{FC}}$: $D = k_d (S’’ - S_{\text{FC}})$
Update storage: $S(t + \Delta t) = S’’ - D \Delta t$
Enforce bounds: $S \geq 0$

4. Worked Example by Hand

Problem: A 1 m deep root zone has:

Field capacity: $S_{\text{FC}} = 200$ mm
Wilting point: $S_{\text{WP}} = 80$ mm
Critical moisture: $S_{\text{crit}} = 140$ mm
Initial storage: $S(0) = 150$ mm
Drainage coefficient: $k_d = 0.5$ day⁻¹

Over 5 days:

Day 1: $P = 0$ mm, $ET_{\text{pot}} = 5$ mm/day
Day 2: $P = 0$ mm, $ET_{\text{pot}} = 5$ mm/day
Day 3: $P = 40$ mm, $ET_{\text{pot}} = 4$ mm/day
Day 4: $P = 0$ mm, $ET_{\text{pot}} = 6$ mm/day
Day 5: $P = 0$ mm, $ET_{\text{pot}} = 6$ mm/day

Calculate $S$ at each day and determine stress.

Solution

Day 0 → 1:

$S_0 = 150$ mm
No stress: $K_s = 1$ (since $S_0 > S_{\text{crit}} = 140$)
$S_1 = 150 - 1 \times 5 = 145$ mm
Below FC, no drainage.

Day 1 → 2:

$S_1 = 145$ mm
No stress: $K_s = 1$
$S_2 = 145 - 1 \times 5 = 140$ mm

Day 2 → 3:

$S_2 = 140$ mm (exactly at critical point)
No stress yet: $K_s = 1$
After ET: $S’ = 140 - 4 = 136$ mm
Add rain: $S_3 = 136 + 40 = 176$ mm
Below FC (200 mm), no drainage.

Day 3 → 4:

$S_3 = 176$ mm
No stress: $K_s = 1$
$S_4 = 176 - 1 \times 6 = 170$ mm

Day 4 → 5:

$S_4 = 170$ mm
No stress: $K_s = 1$
$S_5 = 170 - 1 \times 6 = 164$ mm

Summary:

Day	S (mm)	K_s	ET_actual (mm)	Notes
0	150	1.0	—	Initial
1	145	1.0	5	Drying
2	140	1.0	5	At critical
3	176	1.0	4	Rain recharged
4	170	1.0	6	Drying
5	164	1.0	6	Drying

No water stress throughout this period (all $K_s = 1$).

If we continued without rain, stress would begin when $S < 140$ mm.

5. Computational Implementation

Below is an interactive soil moisture simulator.

Soil type: Root zone depth (m): 1.0 m Growing season:

Current moisture: mm

Field capacity: mm

Wilting point: mm

Water stress (Ks):

Plant available water: mm

Try this:

Dry season: Watch moisture decline steadily, stress increases
Wet season: Frequent rain keeps moisture near field capacity
Sandy soil: Low water holding capacity, dries quickly
Clay soil: High capacity, but also higher wilting point
Add rain event: See immediate moisture spike
Notice: When moisture drops below critical point (orange line), stress factor (Ks) decreases

Key insight: Plant-available water is the difference between field capacity and wilting point—not total moisture.

6. Interpretation

Drought Tolerance

Plants differ in critical moisture threshold:

Drought-tolerant (desert shrubs):

Low $S_{\text{crit}}$ (can extract water at high tension)
Deep roots access moisture below 1 m
Stress begins near wilting point

Drought-sensitive (shallow-rooted crops):

High $S_{\text{crit}}$ (stress begins early)
Shallow roots (0–0.5 m)
Require frequent irrigation

Irrigation Scheduling

Goal: Maintain $S > S_{\text{crit}}$ to avoid stress.

When to irrigate: When $S$ drops to $S_{\text{crit}}$ (typically 50–70% depletion of PAW).

How much: Refill to field capacity:

\[I = S_{\text{FC}} - S_{\text{current}}\]

Example: FC = 200 mm, current = 120 mm:

\[I = 200 - 120 = 80 \text{ mm}\]

Apply 80 mm of irrigation.

Groundwater Recharge

Recharge rate = drainage from root zone.

Annual recharge (arid region):
If $P = 200$ mm/year, $ET = 180$ mm/year:

\[D = 200 - 180 = 20 \text{ mm/year}\]

Only 10% of precipitation recharges groundwater.

Humid region:
$P = 1000$ mm/year, $ET = 500$ mm/year:

\[D = 1000 - 500 = 500 \text{ mm/year}\]

50% recharges.

7. What Could Go Wrong?

Assuming Homogeneous Root Zone

Real soil has layers with different properties. Roots concentrate in topsoil (0–30 cm).

Layered model: Divide root zone into multiple layers, each with own water balance.

Ignoring Capillary Rise

In shallow water table conditions, water moves upward from saturated zone.

Effect: Reduces drainage, can maintain moisture even without rain.

Our model assumes deep water table (no capillary rise).

Neglecting Preferential Flow

Water can move rapidly through macropores (cracks, worm holes, root channels).

Effect: Fast drainage bypasses matrix → less storage → more runoff.

Clay soils with cracks: observed drainage faster than simple model predicts.

Constant ET Potential

Real $ET_{\text{pot}}$ varies with weather (solar radiation, humidity, wind from Model 18).

Dynamic model: Calculate $ET_{\text{pot}}$ daily from meteorological data (Penman-Monteith).

8. Extension: Soil Water Potential

Water potential ($\psi$, kPa or MPa) describes how tightly water is held.

Components:

Matric potential ($\psi_m$): Tension from soil particles (negative)
Gravitational potential ($\psi_g$): Elevation effect
Osmotic potential ($\psi_o$): Dissolved salts

Total potential:

\[\psi = \psi_m + \psi_g + \psi_o\]

Water moves from high to low potential (like heat from hot to cold).

Retention curve: Relates moisture content $\theta$ to matric potential $\psi_m$:

\[\psi_m = \psi_e \left(\frac{\theta}{\theta_s}\right)^{-b}\]

Where:

$\psi_e$ = air-entry potential (kPa)
$\theta_s$ = saturated moisture content
$b$ = pore-size distribution index

At field capacity: $\psi_m \approx -33$ kPa (1/3 bar)
At wilting point: $\psi_m \approx -1500$ kPa (15 bar)

Next model will integrate photosynthesis (Model 20) and water balance to model carbon cycling and Net Primary Productivity.

9. Math Refresher: Water Balance as Stock-Flow

Stock-Flow Diagram

Stock: $S$ (soil moisture, mm)

Inflows: $P$ (precipitation)

Outflows: $ET$ (evapotranspiration), $D$ (drainage), $R$ (runoff)

Differential equation:

\[\frac{dS}{dt} = P - ET - D - R\]

Analogy:

Bank account: Stock = balance, inflows = deposits, outflows = withdrawals
Bathtub: Stock = water level, inflows = faucet, outflows = drain

Steady State

At equilibrium (long-term average):

\[\frac{dS}{dt} = 0 \implies P = ET + D + R\]

All incoming water eventually leaves as ET, drainage, or runoff.

Seasonal variation: $S$ fluctuates but average is constant.

Summary

Soil water balance: $dS/dt = P - ET - D - R$
Field capacity ($S_{\text{FC}}$): Maximum practical storage after drainage
Wilting point ($S_{\text{WP}}$): Minimum extractable water
Plant-available water: PAW = $S_{\text{FC}} - S_{\text{WP}}$
Water stress factor ($K_s$): Reduces ET when $S < S_{\text{crit}}$
Drainage occurs when $S > S_{\text{FC}}$
Runoff occurs when $S > S_{\text{sat}}$
Sandy soils: Low PAW, fast drainage
Clay soils: High PAW, slow drainage
Irrigation timing: Apply water when $S$ reaches $S_{\text{crit}}$