Chapter 10: Carbon Modelling

Learning Objectives

After reading this chapter, you should be able to:

  • Explain the role of carbon accounting in modern forest management

  • Calculate standing carbon stocks from forest inventory data

  • Model carbon fluxes from harvest and decomposition

  • Integrate carbon accounting into wood supply optimization

  • Understand the policy context (BC Emissions Trading System, carbon credits)

Why Carbon Matters in Forest Management

Carbon accounting has become a central concern in forest management for several reasons:

  1. Climate policy: British Columbia’s Emissions Trading System (ETS) requires forest operators to report and offset emissions

  2. Carbon credits: Forest carbon sequestration can generate revenue through carbon markets

  3. Sustainability reporting: Stakeholders demand carbon-transparent management plans

  4. Regulatory compliance: The BC Forest and Range Practices Act requires carbon considerations in planning

Carbon in the Forest System

Forest carbon exists in multiple pools:

Pool

Location

Description

Above-ground biomass

Living trees (bole, branches, foliage)

Largest pool in most stands

Below-ground biomass

Roots

Typically 15-25% of above-ground

Dead wood

Standing dead trees, coarse woody debris

Decomposes slowly (decades)

Litter

Leaf litter, fine woody debris

Decomposes moderately (years)

Soil organic matter

Humus layer

Largest but slowest-changing pool

 
        graph TD
  ATMOSPHERE["Atmosphere<br/>(CO₂)"] --> SEQUESTER["Photosynthesis<br/>(carbon uptake)"]
  SEQUESTER --> TREES["Living trees"]
  TREES --> ABOVE["Above-ground<br/>biomass"]
  TREES --> BELOW["Below-ground<br/>biomass"]
  TREES --> HARVEST["Harvest"]
  HARVEST --> PRODUCTS["Wood products<br/>(long-term storage)"]
  HARVEST --> DECOMP["Decomposition<br/>(CO₂ release)"]
  DECOMP --> ATMOSPHERE
  TREES --> DEAD["Dead wood"]
  DEAD --> DECOMP

Carbon Calculation Basics

Carbon is approximately 50% of dry biomass. The conversion from biomass to carbon uses a simple factor:

\[\begin{split}\\text{Carbon (tonnes)} = \\text{Biomass (tonnes)} \\times 0.5\end{split}\]

Biomass can be estimated from volume using species-specific factors:

\[\begin{split}\\text{Biomass (tonnes/ha)} = \\text{Volume (m³/ha)} \\times \\text{Bulk Density (tonnes/m³)} \\times (1 + \\text{Root-Shoot Ratio})\end{split}\]

Using ws3 for Carbon Calculations

ws3 doesn’t have built-in carbon functions, but you can calculate carbon stocks from the model output:

from ws3.forest import ForestModel
from ws3.core import Curve

# Define biomass curves (tonnes/ha of carbon)
df_carbon = Curve(
    label="DF-SI50_carbon",
    points=[(0, 0), (10, 1), (20, 5), (30, 12), (40, 22),
            (50, 35), (60, 50), (70, 65), (80, 78), (90, 88), (100, 95)]
)

spruce_carbon = Curve(
    label="SP-SI40_carbon",
    points=[(0, 0), (10, 0.5), (20, 3), (30, 8), (40, 15),
            (50, 25), (60, 38), (70, 50), (80, 60), (90, 68), (100, 73)]
)

# Calculate carbon stock for a development type
dt_area = 500.0  # hectares
dt_age = 60  # years

carbon_per_ha = df_carbon(dt_age)
total_carbon = carbon_per_ha * dt_area

print(f"Carbon stock: {total_carbon:.1f} tonnes C")
print(f"CO₂ equivalent: {total_carbon * 3.67:.1f} tonnes CO₂")

Carbon Fluxes from Harvest

When trees are harvested, carbon is released through:

  1. Immediate decomposition: Slash and residues decompose (5-20 year half-life)

  2. Product decay: Wood products gradually release carbon (decades to centuries)

  3. Soil disturbance: Harvesting disturbs soil organic matter

        graph TD
  HARVEST["Harvest"] --> SLASH["Slash decomposition<br/>(fast, 5-20 yr)"]
  HARVEST --> PRODUCTS["Wood products<br/>(slow, decades)"]
  HARVEST --> SOIL["Soil disturbance<br/>(very slow, centuries)"]
  SLASH --> RELEASE["CO₂ release"]
  PRODUCTS --> RELEASE
  SOIL --> RELEASE

A simple carbon accounting model:

def calculate_harvest_carbon_flux(volume_harvested_m3, species="DF"):
    """Calculate carbon flux from a harvest event."""

    # Conversion: m³ to tonnes biomass (approximate)
    bulk_density = {"DF": 0.45, "SP": 0.40, "CE": 0.35}
    biomass_factor = bulk_density.get(species, 0.40)

    biomass_tonnes = volume_harvested_m3 * biomass_factor
    carbon_tonnes = biomass_tonnes * 0.5

    # Split into pools
    slash_fraction = 0.3  # 30% goes to slash (fast decomposition)
    product_fraction = 0.6  # 60% goes to products (slow decomposition)
    soil_fraction = 0.1  # 10% from soil disturbance

    slash_carbon = carbon_tonnes * slash_fraction
    product_carbon = carbon_tonnes * product_fraction
    soil_carbon = carbon_tonnes * soil_fraction

    return {
        "slash": slash_carbon,
        "products": product_carbon,
        "soil": soil_carbon,
        "total": carbon_tonnes
    }

flux = calculate_harvest_carbon_flux(1000, species="DF")
print(f"Slash carbon: {flux['slash']:.1f} tonnes C")
print(f"Product carbon: {flux['products']:.1f} tonnes C")
print(f"Soil carbon: {flux['soil']:.1f} tonnes C")
print(f"Total flux: {flux['total']:.1f} tonnes C")

Carbon in Optimization

Carbon can be incorporated into the optimization objective:

from ws3.opt import Problem

prob = Problem("carbon_optimization")

# Decision variables
harvest_var_names = {}
for dt_code in ["DF-SI50", "SP-SI40"]:
    for period in range(20):
        var_name = f"harv_{dt_code}_p{period}"
        prob.add_var(var_name, vtype="continuous", lb=0)
        harvest_var_names[(dt_code, period)] = var_name

# Objective: maximize NPV + carbon revenue
# z() takes a dict keyed on variable names
timber_price = 50  # $/m³
carbon_price = 50  # $/tonne CO₂ (ETS price)
discount_rate = 0.05

npv_coeffs = {}
for (dt_code, period), var_name in harvest_var_names.items():
    volume_per_ha = 200  # m³/ha
    timber_revenue = volume_per_ha * timber_price
    carbon_revenue = volume_per_ha * 0.45 * 0.5 * 3.67 * carbon_price
    coeff = (timber_revenue + carbon_revenue) * (1 + discount_rate) ** (-period * 5)
    npv_coeffs[var_name] = coeff
prob.z(coeffs=npv_coeffs)

# Constraint: carbon budget (max allowable emissions)
max_carbon_budget = 10000  # tonnes CO₂ over 100 years
carbon_per_m3 = 0.45 * 0.5 * 3.67  # tonnes CO₂ per m³
carbon_coeffs = {}
for var_name in harvest_var_names.values():
    carbon_coeffs[var_name] = 200 * carbon_per_m3
prob.add_constraint("carbon_budget", coeffs=carbon_coeffs, sense="leq", rhs=max_carbon_budget)

# Set solver and solve
prob.solver("highs")
prob.solve()

Carbon Reporting

For regulatory compliance, you need to report carbon stocks and fluxes:

import pandas as pd

# Calculate carbon stocks by period
carbon_by_period = []
for period in range(20):
    age = 20 + period * 5  # starting age + periods elapsed
    carbon_stock = df_carbon(age) * 500  # tonnes C
    carbon_by_period.append({
        "period": period,
        "age": age,
        "carbon_stock_tonnes_C": carbon_stock,
        "carbon_stock_tonnes_CO2": carbon_stock * 3.67
    })

df_carbon_report = pd.DataFrame(carbon_by_period)
print(df_carbon_report)

# Export for reporting
df_carbon_report.to_csv("carbon_report.csv", index=False)

Policy Context

British Columbia’s carbon pricing framework:

  • Carbon tax: Applied to fossil fuel combustion (not directly to forestry)

  • Emissions Trading System (ETS): Cap-and-trade for large emitters

  • Forest Management Carbon Budget: Each FMU has an allowable carbon budget based on projected stock changes

Key references:

  • BC Ministry of Forests: Carbon Accounting Guidelines for Forest Management

  • BC Emissions Trading Scheme: Forest Sector Participation

  • IPCC: Good Practice Guidance for Land Use, Land-Use Change and Forestry

Limitations

Carbon modelling in ws3 has limitations:

  1. Simplified pools: Only above-ground biomass is typically modelled

  2. No soil dynamics: Soil carbon changes are approximated

  3. No product substitution: Benefits of wood substitution for concrete/steel are not modelled

  4. Static prices: Carbon prices are assumed constant

For more sophisticated carbon accounting, consider coupling ws3 with specialized tools like CO2STATS or the BC Forest Carbon Calculator.

Exercises

Exercise 1 (Easy): Calculate the carbon stock for a 500-hectare Douglas-fir stand at ages 40, 60, 80, and 100.

Exercise 2 (Medium): Extend the carbon calculation to include below-ground biomass (assume root-shoot ratio of 0.2) and calculate total ecosystem carbon.

Exercise 3 (Hard): Formulate an optimization problem that maximizes NPV subject to a carbon budget constraint. Compare the optimal harvest schedule with and without the carbon constraint.

Further Reading