First-Principles Deconstruction of RDII in SWMM5

01 Inherited Assumptions

What almost everyone takes for granted when working with RDII. 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 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 Captured by Monthly Variation arbitrary

1.7 System Fully Drains 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 — independent of SWMM5, RTK, or any framework.

2.1 Conservation of Mass fundamental

V_rain = V_runoff + V_evap + V_ET + V_soil_infil + V_RDII + V_other
At the pipe boundary:
  V_RDII = V_gw_in + V_surface_in + V_depression_in + V_other_entries

2.2 Gravity and Pressure Gradients Drive Flow fundamental

Q_RDII ∝ f(H_soil - H_pipe, K_soil, defect_geometry)
This is Darcy's Law + orifice/weir 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
  K(S) is highly nonlinear (exponential in many models)
The soil's "unit hydrograph" changes shape with state.

2.4 Pipe Defects are Physical Orifices and Weirs fundamental

For defect at depth d:
  Q_defect = C_d · A_defect · √(2g · ΔH)   [orifice]
  Q_defect = C_w · L_defect · (ΔH)^1.5      [weir]
Total RDII = Σ Q_defect(i) — additive but NOT linear in rainfall.

2.5 Water Table is a Continuous Boundary Condition fundamental

2.6 Surface Hydraulics Connect to Subsurface fundamental

2.7 Time is Continuous fundamental

2.8 Spatial Variability is Real fundamental

03 Rebuild from Bedrock

3.1 Minimal Viable Model

The simplest physically-grounded RDII model needs:

1. A soil moisture state S(t) tracking wetness continuously
2. A groundwater head H_gw(t) at each location
3. A pipe internal head H_pipe(t) from the hydraulic model
4. A transfer function from head difference to defect flow
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]
H_gw(t)    // groundwater head [m]

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

// 1. Soil moisture balance
dS/dt = P(t)·f_inf(S) − ET(t) − K_sat·g(S)·(∂H/∂z)
  f_inf(S) = (1 − S/S_max)^β        // Green-Ampt-like
  g(S)     = exp(α·(S − S₀))         // relative conductivity

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

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

// 4. Surface direct inflow
Q_direct(t) = η·[P(t) − f_inf(S)·P(t)]

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

3.4 Necessary Trade-offs

04 Contrast: Legacy vs. Rebuilt

05 Radically Different Architecture

5.1 Legacy Architecture

5.2 Rebuilt Architecture

5.3 Key Architectural Differences

5.4 Implementation Minimalism

class RDIISoilGWModel {
  S, H_gw                              // state
  S_max, K_sat, S_y, A_defects, C_d   // physical params

  step(P, ET, H_pipe, dt) {
    f_inf  = (1 - S/S_max)^1.5
    I      = P * f_inf
    K_rel  = exp(10 * (S - 0.7))
    drain  = K_sat * K_rel
    S     += (I - ET - drain) * dt
    H_gw  += (drain/S_y - 0.01*(H_gw - H_base)) * dt
    dH     = max(0, H_gw - H_pipe)
    return C_d * A_defects * sqrt(2*9.81*dH) + η*P*(1-f_inf)
  }
}

~15 lines of core logic. RTK is similarly compact — but this captures physics RTK structurally cannot represent.

06 Interactive Comparison

See the difference. The rebuilt model shows less RDII for the first storm (dry soil) and more for subsequent storms (wet soil). RTK produces the same response every time.

07 What's Sacred, What's Arbitrary

08 How to Share This

This is a single self-contained HTML file — no dependencies, no build step, no server required. Here's every way to share it: