Water balance, infiltration, and plant-available water in the root zone
2026-02-26
You should know
That soil can gain water from precipitation and lose water through drainage and evaporation, and that storage changes when inputs and outputs differ.
You will learn
How simple bucket models represent soil moisture, why wet soils and dry soils respond differently, and how water balance shapes ecosystems.
Why this matters
Soil moisture sits at the centre of drought, agriculture, flooding, and vegetation health. It is one of the most practical state variables in environmental modelling.
If this gets hard, focus on…
Track three things only: water coming in, water leaving, and water stored in the soil. That balance carries the chapter.
The summer of 2021 brought a heat dome to the Pacific Northwest that killed an estimated 619 people in British Columbia alone. But the heat amplified an existing condition: soils across the region were already at or near their permanent wilting point — the moisture level below which plant roots cannot extract water regardless of how much suction they apply. Soil moisture was so depleted that even irrigated agricultural land showed stress. The heat dome broke records partly because dry soils meant there was no evaporative cooling to moderate surface temperatures. Soil moisture and air temperature formed a positive feedback that turned a hot spell into a lethal event.
Soil moisture is the amount of water held in the pore spaces between soil particles, and it controls more processes than almost any other single variable in the land surface system. It determines whether precipitation runs off or infiltrates, whether plants can transpire or are forced to close their stomata, whether soil microbes decompose organic matter quickly or slowly, and whether the land surface warms quickly (dry soil) or moderately (wet soil with evaporative cooling). The soil moisture bucket model — tracking the balance between rainfall inputs and evapotranspiration and drainage outputs — is one of the most useful tools in hydrology. This model builds it from first principles, introduces the key soil parameters (field capacity, wilting point, hydraulic conductivity), and shows how moisture controls the transition from rain to runoff.
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?
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}}
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
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.
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 occurs when: 1. Soil is at saturation (S = S_{\text{sat}}) 2. 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}})
At each time step:
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.
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.
Below is an interactive soil moisture simulator.
<label>
Soil type:
<select id="soil-type-sm">
<option value="sand">Sandy Soil</option>
<option value="loam" selected>Loam</option>
<option value="clay">Clay</option>
</select>
</label>
<label>
Root zone depth (m):
<input type="range" id="depth-slider" min="0.5" max="2" step="0.1" value="1">
<span id="depth-value">1.0</span> m
</label>
<label>
Growing season:
<select id="season-select">
<option value="wet">Wet Season (frequent rain)</option>
<option value="normal" selected>Normal Season</option>
<option value="dry">Dry Season (drought)</option>
</select>
</label>
<div class="button-group">
<button id="run-simulation">Run 90-Day Simulation</button>
<button id="add-rain">Add 30mm Rain Event</button>
</div><p><strong>Current moisture:</strong> <span id="current-s"></span> mm</p>
<p><strong>Field capacity:</strong> <span id="fc-display"></span> mm</p>
<p><strong>Wilting point:</strong> <span id="wp-display"></span> mm</p>
<p><strong>Water stress (Ks):</strong> <span id="ks-display"></span></p>
<p><strong>Plant available water:</strong> <span id="paw-display"></span> mm</p>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.
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
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.
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.
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.
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).
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.
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).
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.
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
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.