Calculations

Calculations

• Depth at a point

  The depth of water at a plan point.

  depth = WS − ground

  Flood extent on the ground surface river-extent.js

  • CivilCraft
  64-bit

• Waterline point on a triangle edge

  Where the depth is zero along an edge between a wet corner and a dry corner.

  u = d_a ÷ (d_a − d_b); point = a + u × (b − a)

  Flood extent on the ground surface river-extent.js

  • CivilCraft
  64-bit

• Flooded area and greatest depth

  Adds up the wet part of each triangle and tracks the greatest depth at a corner.

  piece area = |Σ (x_i·y_j − x_j·y_i)| ÷ 2; area = Σ piece areas

  Flood extent on the ground surface river-extent.js

  • CivilCraft
  64-bit

• Barrel flow area and top width

  The flow area, water surface width and full-barrel hydraulic radius of one culvert barrel at a given depth.

  circle: θ = 2·acos((r − y) ÷ r); A = r²(θ − sin θ) ÷ 2; T = 2√(y(D − y)); R = D ÷ 4. box: A = B·y; R = B·D ÷ (2(B + D))

  Culverts, bridges and junctions in a river model river-structures.js

  • CivilCraft
  64-bit

• Critical depth in a barrel

  The depth at which the flow in one barrel is critical, capped at the rise.

  Q²·T(y) − g·A(y)³ = 0

  Culverts, bridges and junctions in a river model river-structures.js

  • CivilCraft
  64-bit

• Headwater for one barrel's flow

  The inlet-control and outlet-control headwater depths for a flow in one barrel, and which one governs.

  HWw = (Q ÷ (Cw·B))^(2/3); HWo = Q² ÷ (Cd²·A²·2g) + D ÷ 2; φ = clamp((HWo ÷ D − 1) ÷ 0.5, 0, 1); HWin = (1 − φ)·HWw + φ·HWo; Kf = 2g·n²·L ÷ (k²·R^(4/3)); HWout = (1 + Ke + Kf)·V² ÷ 2g + ho − L·S₀

  Culverts, bridges and junctions in a river model river-structures.js

  • CivilCraft
  64-bit

• Culvert headwater with road overtopping

  The headwater at which the flow through the barrels plus the flow over the road equals the total flow.

  n·Qbarrel(HW) + Cr·Lr·(HW − crest)^1.5 = Q

  Culverts, bridges and junctions in a river model river-structures.js

  • CivilCraft
  64-bit

• Bridge opening section

  The downstream cross-section clipped to the abutments, with each pier as a strip that carries no flow.

  z(o) = z₀ + (o − o₀) ÷ (o₁ − o₀) × (z₁ − z₀)

  Culverts, bridges and junctions in a river model river-structures.js

  • CivilCraft
  64-bit

• Water surface above a bridge

  The water surface in the opening and at the section above the bridge, in low flow, pressure flow or flow over the deck.

  WS + α·Q² ÷ (2g·A²) = E₀ + L·(Q ÷ ((K₁ + K₂) ÷ 2))² + c·|Δ velocity head|; pressure: Q = Cd·An·√(2g·head); deck: Q = Cw·L·(EGL − deck)^1.5

  Culverts, bridges and junctions in a river model river-structures.js

  • CivilCraft
  64-bit

• Stepping a profile across a structure

  Takes the water surface below a structure, works out the level above it and uses that as the start of the next reach upstream.

  TW = max(0, WSd − invert down); WSu = invert up + HW

  Culverts, bridges and junctions in a river model river-structures.js

  • CivilCraft
  64-bit

• Flows and water levels at a junction

  Adds the flows of the reaches that join and carries the water surface at the head of the downstream reach up to them.

  Q = Σ Q of the joining reaches; WS of each joining reach's downstream end = WS at the head of the reach below

  Culverts, bridges and junctions in a river model river-structures.js

  • CivilCraft
  64-bit

• Area and conveyance table of a section

  Flow area and conveyance of one cross-section as functions of water level, read from a table.

  levels hᵢ = h₀ + (hi − h₀)·(i ÷ 200)²; between levels: linear interpolation; below h₀: A·f, K·f^1.6667 with f = depth ÷ (h₀ − bed)

  Unsteady flow routing down a river river-unsteady.js

  • CivilCraft
  64-bit

• Banded linear solve

  Solves the Newton step for all sections at once.

  L·U·x = b with row pivoting inside the band

  Unsteady flow routing down a river river-unsteady.js

  • CivilCraft
  64-bit

• Continuity and momentum residuals of one interval

  The two equations that must be zero between a pair of neighboring sections for one time step.

  continuity: Δ(Aᵢ + Aⱼ) ÷ (2Δt) + (θ·ΔQ_new + (1 − θ)·ΔQ_old) ÷ Δx; momentum: Δ(Qᵢ + Qⱼ) ÷ (2Δt) + Δ(Q|Q| ÷ A) ÷ Δx + g·Ā·Δh ÷ Δx + g·Ā·S̄f, Sf = Q|Q| ÷ K²

  Unsteady flow routing down a river river-unsteady.js

  • CivilCraft
  64-bit

• Newton iteration for one time step

  Finds the water levels and flows at the end of a time step.

  J·Δu = −F; J by forward differences with steps 1e-6·Href and 1e-6·Qref; stop when max(|Δh| ÷ Href, |ΔQ| ÷ Qref) < 1e-8

  Unsteady flow routing down a river river-unsteady.js

  • CivilCraft
  64-bit

• Time stepping, peaks and Froude check

  Advances the run step by step, records the peak level and flow at each section and flags supercritical flow.

  V_in += Δt·(θ·Q_new + (1 − θ)·Q_old); T ≈ (A(h + 0.001·Href) − A(h)) ÷ (0.001·Href); flag when |Q| ÷ A > √(g·A ÷ T)

  Unsteady flow routing down a river river-unsteady.js

  • CivilCraft
  64-bit

• Volume balance error

  The share of water not accounted for between inflow, outflow and the change of storage in the reach.

  storage = Σ ½(Aᵢ + Aᵢ₊₁)·Δxᵢ; error = (V_in − V_out − (S₁ − S₀)) ÷ max(V_in, V_out, |S₁ − S₀|, 1e-12)

  Unsteady flow routing down a river river-unsteady.js

  • CivilCraft
  64-bit

• Gravity and Manning constant per unit system

  The gravity and the Manning equation constant every river calculation uses.

  m: k = 1, g = 9.80665; ft: k = 1.486, g = 32.174

  River model: stations, cut sections and reach lengths river.js

  • CivilCraft
  64-bit

• River station of a point

  The station of a point, measured along the centerline from its downstream end, and its offset to the side.

  rs = L − d

  River model: stations, cut sections and reach lengths river.js

  • CivilCraft
  64-bit

• Cross-section from a cut line

  Turns a line drawn across the river into a cross-section: station, ground points by offset, and first bank positions.

  rs = round₆(L − d_crossing); o = round₆(offset); banks = lowest point ± bank distance, clipped to the section

  River model: stations, cut sections and reach lengths river.js

  • CivilCraft
  64-bit

• Sections at a spacing along the centerline

  Cuts square cross-sections every set distance along the centerline and at both ends.

  dₖ = k × spacing while k × spacing < L − 1e-9; rs = round₆(L − dₖ)

  River model: stations, cut sections and reach lengths river.js

  • CivilCraft
  64-bit

• Reach lengths between sections

  The channel and overbank distances from each section to the next one downstream.

  channel = rsᵢ − rsᵢ₋₁; overbank = √(Δx² + Δy²) between the middles of the overbank parts of the two cut lines

  River model: stations, cut sections and reach lengths river.js

  • CivilCraft
  64-bit

• Flow at a station

  The flow that applies at a station, from the profile's flow and its flow changes.

  River model: stations, cut sections and reach lengths river.js

  • CivilCraft
  64-bit

• First guess of the boundary slope

  The slope offered for a normal-depth boundary when one is first chosen, from the lowest points of the end sections.

  S = (z_min upstream − z_min downstream) ÷ (rs upstream − rs downstream), to 3 significant digits; 0.001 when S ≤ 1e-5

  River model: stations, cut sections and reach lengths river-ui.js

  • CivilCraft
  64-bit

• Least-cost search

  Searches the grid for the cheapest allowed path and returns its points and total cost.

  grade = |z₂ − z₁| ÷ length ≤ limit; cost = length × unit cost × factor × (1 + w·(grade ÷ limit)²) + area × length × earth cost

  Best route across the terrain route.js

  • CivilCraft
  32-bit in part

• Path points and their elevations

  Turns the chain of grid cells into the route's points, with the two ends at the picked points and collinear points removed.

  x = minX + (i + 0.5) × cell; collinear when |cross product| < 1e-9 × cell²

  Best route across the terrain route.js

  • CivilCraft
  32-bit in part

• Route length, steepest grade and earthwork

  The length, steepest grade, cut and fill along a finished route.

  length = Σ √(Δx² + Δy²); grade = |Δz| ÷ l; s = |z_left − z_right| ÷ width; area = width²·s ÷ 4 ÷ max(0.05, 1 − sideSlope·s); volume += area × l ÷ n

  Best route across the terrain route.js

  • CivilCraft
  64-bit

• Curve radius at each bend

  The turn angle and the radius implied at each bend of the route.

  turn = atan2 of the leg after − atan2 of the leg before, wrapped to (−π, π]; radius = (l₁ + l₂) ÷ (2|turn|)

  Best route across the terrain route.js

  • CivilCraft
  64-bit

• Route profile

  Distance along the route against elevation, for the profile drawing and the point table.

  dᵢ = dᵢ₋₁ + √(Δx² + Δy²)

  Best route across the terrain route.js

  • CivilCraft
  64-bit

• Sample stations

  The stations where sections are cut: every multiple of the step, both ends, the curve ends and any extra stations.

  sta = round₆(k × step) for whole k; neighbors closer than 1e-6 merged

  Cross sections and end-area volumes along an alignment sections.js

  • CivilCraft
  64-bit

• Sample line at a station

  The two ends of the line square to the alignment at a station.

  right = (sin dir, −cos dir); a = c − right × left; b = c + right × right offset

  Cross sections and end-area volumes along an alignment sections.js

  • CivilCraft
  64-bit

• Surface along a sample line

  The ground or design line along a sample line, with a point at every triangle edge it crosses.

  z at a + (b − a)·t for every edge-crossing parameter t; o = offA + (offB − offA)·t

  Cross sections and end-area volumes along an alignment sections.js

  • CivilCraft
  64-bit

• Segment crossing parameter

  Where one segment crosses another, as a fraction along the first.

  den = r × s; t = ((q − p) × s) ÷ den; u = ((q − p) × r) ÷ den

  Cross sections and end-area volumes along an alignment sections.js

  • CivilCraft
  64-bit

• Elevation of a section line at an offset

  The elevation of a section at any offset, between its points.

  z = z₀ + (z₁ − z₀)·(o − o₀) ÷ (o₁ − o₀)

  Cross sections and end-area volumes along an alignment sections.js

  • CivilCraft
  64-bit

• Cut and fill area of a section

  The area of cut and of fill between the ground line and the design line of one section.

  per interval: area = (e₀ + e₁) ÷ 2 × w; where the lines cross: o× = o₀ + w·e₀ ÷ (e₀ − e₁), two triangles

  Cross sections and end-area volumes along an alignment sections.js

  • CivilCraft
  64-bit

• Feature line crossings

  The offset and elevation where each corridor feature line crosses the sample line.

  o = −left + (left + right)·t; z = z_p + (z_q − z_p)·u

  Cross sections and end-area volumes along an alignment sections.js

  • CivilCraft
  64-bit

• Spans inside the graded regions

  The parts of a sample line that lie inside the graded regions, which is where the design counts.

  Cross sections and end-area volumes along an alignment sections.js

  • CivilCraft
  64-bit

• Volumes between sections

  Cut and fill volume between neighboring sections and the running totals.

  V = (A₁ + A₂) ÷ 2 × (sta₂ − sta₁); cumulative = Σ V

  Cross sections and end-area volumes along an alignment sections.js

  • CivilCraft
  64-bit

• Section table as CSV

  Writes station, areas, volumes and running totals to a CSV file.

  areas to 2 decimals; volumes to 1 decimal; feet projects: volume ÷ 27 (cubic yards)

  Cross sections and end-area volumes along an alignment sections.js

  • CivilCraft
  64-bit

• US gallon in cubic feet

  The factor that turns gallons into cubic feet in feet projects.

  1 gal = 0.133680556 ft³

  Sanitary sewer design flows and pipe sizes sewer.js

  • CivilCraft
  64-bit

• Peaking factor

  The ratio of peak to average sanitary flow for a population.

  Harmon: 1 + 14 ÷ (4 + √P); Babbitt: 5 ÷ P^0.2; P in thousands; held between pfMin and pfMax

  Sanitary sewer design flows and pipe sizes sewer.js

  • CivilCraft
  64-bit

• Average flow from people

  People times per-person daily flow, in project flow units.

  m: P × L/day ÷ 1000 ÷ 86400; ft: P × gal/day × 0.133680556 ÷ 86400

  Sanitary sewer design flows and pipe sizes sewer.js

  • CivilCraft
  64-bit

• Infiltration flow

  Tributary area times the infiltration allowance, in project flow units.

  m: ha × L/s per ha ÷ 1000; ft: ac × gal/day per ac × 0.133680556 ÷ 86400

  Sanitary sewer design flows and pipe sizes sewer.js

  • CivilCraft
  64-bit

• Design flow, slope and velocity per pipe

  For each pipe in flow order: the people and area above it, the peak flow, the slope, the full capacity, the normal depth and the velocity.

  S = (inv up − inv down) ÷ length; Q = q_dom × pf + q_inf; ratio = Q ÷ Q_full; V = Q ÷ A(yn)

  Sanitary sewer design flows and pipe sizes sewer.js

  • CivilCraft
  64-bit

• Pipe size for the design flow

  The smallest listed pipe size whose allowed share of full capacity carries the design flow.

  first size with Q_full(size) × maxFull ≥ Q

  Sanitary sewer design flows and pipe sizes sewer.js

  • CivilCraft
  64-bit

• Catchment runoff hydrograph

  The runoff from one catchment at each time step for a constant storm.

  q(t) = k·C·A·(depth(t) − depth(t − w)) ÷ w; depth(τ) = i·clamp(τ, 0, duration); k = 1 ÷ 360 (m), 1 (ft)

  Storm hydrographs routed through the pipe network storm-route.js

  • CivilCraft
  64-bit

• Muskingum coefficients

  The three weights of the routing equation for a reach.

  d = K(1 − X) + Δt ÷ 2; c₀ = (−KX + Δt ÷ 2) ÷ d; c₁ = (KX + Δt ÷ 2) ÷ d; c₂ = (K(1 − X) − Δt ÷ 2) ÷ d

  Storm hydrographs routed through the pipe network storm-route.js

  • CivilCraft
  64-bit

• Routing one reach

  The outflow hydrograph of a pipe from its inflow hydrograph.

  O₂ = max(0, c₀·I₂ + c₁·I₁ + c₂·O₁) on a step of max(Δt, 2KX); values between steps by linear interpolation

  Storm hydrographs routed through the pipe network storm-route.js

  • CivilCraft
  64-bit

• Travel time and weighting of each pipe

  The routing constants K and X of each pipe from its length, slope and rational peak flow.

  c = 1.5 × max(V, 0.1); K = L ÷ c ÷ 60; X = clamp(½(1 − Q ÷ (T·S·c·L)), 0, ½)

  Storm hydrographs routed through the pipe network storm-route.js

  • CivilCraft
  64-bit

• Summing flows and reading peaks

  Adds the hydrographs arriving at each structure and reads each pipe's peak in, peak out, attenuation and share of capacity.

  att = max(0, 1 − peak out ÷ peak in); ratio = peak out ÷ capacity; over when peak out > capacity × (1 + 1e-9)

  Storm hydrographs routed through the pipe network storm-route.js

  • CivilCraft
  64-bit