First-Principles Deconstruction of RDII in SWMM5

01 Inherited Assumptions

What almost everyone takes for granted when working with RDII — users calibrating, developers coding, educators teaching, researchers publishing. Each of these is treated as "how it works" when it's actually "how we chose to model it."

1.1 The RTK Triangular Unit Hydrograph arbitrary

1.2 Linear Time-Invariant (LTI) Superposition arbitrary

1.3 Infiltration vs. Inflow as Separate Phenomena arbitrary

1.4 Monthly Seasonal Variation as Discrete States arbitrary

1.5 Spatial Homogeneity Within a Sewershed arbitrary

1.6 Antecedent Moisture is Captured by Monthly Variation arbitrary

1.7 The System is Fully Drained Between Events arbitrary

1.8 Rainfall is the Only Forcing Function treated as sacred

1.9 Calibration as Truth arbitrary

1.10 The Model Boundary is the Pipe Network treated as sacred

02 Bedrock Truths

What is fundamentally, provably true about the physical system — independent of SWMM5, RTK, unit hydrographs, or any modeling framework. Only conservation laws, physics, and logic.

2.1 Conservation of Mass fundamental

V_rainfall = V_surface_runoff + V_evap + V_ET + V_infiltration_to_soil + V_RDII + V_other_sinks

At the pipe boundary:
  V_RDII = V_groundwater_in + V_surface_in + V_depression_overflow_in + V_other_direct_entries

This is not a modeling assumption — it's physics. Any model that violates mass balance is wrong.

2.2 Gravity and Pressure Gradients Drive Flow fundamental

Q_RDII ∝ f(H_soil - H_pipe, soil_hydraulic_conductivity, pipe_defect_geometry)

This is Darcy's Law + orifice/weir equations at defects.
Not a triangle. Not a unit hydrograph. Physics.

2.3 Soil is a Stateful, Nonlinear Reservoir fundamental

Soil state S(t) ∈ [0, S_max]
  dS/dt = P(t) - ET(t) - K(S) · ∇H - Q_deep
  where K(S) is highly nonlinear (exponential in many soil models)

The soil's "unit hydrograph" is not a triangle — it's the impulse
response of a nonlinear reservoir that changes shape with state.

2.4 Pipe Defects are Physical Orifices and Weirs fundamental

For a defect at depth d below surface:
  If H_soil(d) > H_pipe(d):
    Q_defect = C_d · A_defect · √(2g · ΔH)    [orifice flow]
  If defect is partially submerged:
    Q_defect = C_w · L_defect · (ΔH)^1.5        [weir flow]

Total RDII = Σ Q_defect(i) for all defects i
This is additive but NOT linear in rainfall — it depends on H_soil(t),
which is a state variable.

2.5 The Water Table is a Continuous Boundary Condition fundamental

2.6 Surface Hydraulics Connect to Subsurface fundamental

2.7 Time is Continuous, Not Discrete Monthly States fundamental

2.8 Spatial Variability is Real and Significant fundamental

03 Rebuild from Bedrock

Starting only from the truths in Section 02, here is the minimal viable model, its invariants, necessary trade-offs, and cleanest expression.

3.1 Minimal Viable Model

The simplest physically-grounded RDII model needs:

1. A soil moisture state variable S(t) that tracks wetness continuously
2. A groundwater head H_gw(t) at each modeled location
3. A pipe internal head H_pipe(t) from the hydraulic model
4. A transfer function from head difference to flow through defects
5. A surface pathway for direct inflow

3.2 Key Invariants

3.3 Cleanest Mathematical Expression

// STATE VARIABLES (per spatial unit)
S(t)     // soil moisture [0, S_max], volume/volume
H_gw(t)  // groundwater head [m above datum]

// FORCING
P(t)     // rainfall intensity [m/s]
ET(t)    // evapotranspiration rate [m/s]
H_pipe(t)// pipe hydraulic head [m] (from flow model)

// PHYSICS
// 1. Soil moisture balance
dS/dt = P(t) · f_inf(S) - ET(t) - K_sat · g(S) · (∂H/∂z)

where:
  f_inf(S) = infiltration fraction = (1 - S/S_max)^β   // Green-Ampt-like
  g(S)     = relative conductivity = exp(α·(S - S_0))   // exponential

// 2. Groundwater response (simplified lumped)
dH_gw/dt = (1/S_y) · [K_sat · g(S) · (∂H/∂z) - Q_drain(H_gw)]

where S_y = specific yield, Q_drain = lateral/baseflow

// 3. RDII transfer
ΔH(t) = max(0, H_gw(t) - H_pipe(t))
Q_rdii(t) = C · A_defects · √(2g · ΔH(t))   // orifice equation

// 4. Surface inflow (direct connections)
Q_direct(t) = η · [P(t) - f_inf(S)·P(t)]   // η = connection fraction

// TOTAL RDII
Q_total(t) = Q_rdii(t) + Q_direct(t)

3.4 Necessary Trade-offs

No model is the territory. Here's what we trade away honestly:

04 Contrast: Legacy vs. Rebuilt

05 Radically Different Architecture

Once the old scaffolding is removed, the software architecture changes fundamentally.

5.1 Legacy Architecture (SWMM5 RTK)

5.2 Rebuilt Architecture

5.3 Key Architectural Differences

5.4 Implementation Minimalism

The rebuilt model can be implemented with surprisingly few components:

// Pseudocode: Minimal State-Based RDII Module

class RDIISoilGWModel {
  // State
  S       // soil moisture ratio [0,1]
  H_gw    // groundwater head [m]
  
  // Parameters (physical, bounded, estimable from data)
  S_max       // max soil moisture capacity [m³/m³]
  K_sat       // saturated hydraulic conductivity [m/s]
  S_y         // specific yield [-]
  A_defects   // effective defect area [m²]
  eta         // direct connection fraction [-]
  C_d         // discharge coefficient [-]
  
  step(P, ET, H_pipe, dt) {
    // 1. Infiltration (Green-Ampt-like)
    f_inf = pow(max(0, 1 - S/S_max), 1.5)
    I = P * f_inf
    
    // 2. Soil moisture update
    K_rel = exp(10 * (S - 0.7))  // relative conductivity
    drain = K_sat * K_rel
    dS = (I - ET - drain) * dt
    S = clamp(S + dS, 0, S_max)
    
    // 3. Groundwater update
    recharge = drain
    gw_drain = 0.01 * (H_gw - H_base)  // simple linear drain
    dH_gw = (recharge / S_y - gw_drain) * dt
    H_gw += dH_gw
    
    // 4. RDII via defects
    dH = max(0, H_gw - H_pipe)
    Q_rdii = C_d * A_defects * sqrt(2 * 9.81 * dH)
    
    // 5. Direct inflow
    Q_direct = eta * P * (1 - f_inf)
    
    return Q_rdii + Q_direct
  }
}

That's ~30 lines of core logic. The RTK implementation in SWMM5 is similarly compact — but the above captures physics that RTK structurally cannot represent.

06 Interactive Comparison

See the difference. Adjust parameters and observe how the rebuilt state-based model responds to repeated rainfall events vs. the legacy RTK approach.

Key Observation

07 Summary: What's Sacred, What's Arbitrary