Stage 6 / dewatering

Stage 6 dewatering.

The dewatering route couples a steady-state unconfined radial-flow model with the CPT-based effective-stress change and settlement response at the CPT location. It derives the Dupuit–Forchheimer expression for layered profiles, uses the Sichardt screening radius for the outer boundary, handles the trench case via the image method, and feeds the resulting drawdown into the same constrained-modulus integration used in the settlement route.

Scope and intent

1. Problem class and deliverable

The dewatering route is a serviceability-oriented screening model. It couples hydraulic drawdown with CPT-based effective-stress change to report pumping rate and settlement at the CPT location. It is layered by construction — the Stage 4 interpreted conductivity profile is preserved rather than collapsed to one equivalent k.

Model class. Steady-state analytical screen. Not a full transient groundwater model, not a substitute for a regional flow study when hydraulic interaction becomes critical, and not an unsaturated-flow engine.
Primary quantities
Q
Steady-state discharge from the source [m³/s].
s(r), ΔhCPT
Drawdown field and drawdown at the CPT observation point [m].
T(h)
Transmissivity of the currently saturated profile as a function of saturated thickness [m²/s].
ΔSdewatering
Settlement at the CPT induced by the drawdown [m].

Darcy's law

2. Darcy flow, radial co-ordinates, and saturated thickness

Saturated flow through a porous medium is described by Darcy's law: specific discharge is proportional to the hydraulic gradient with the conductivity as the proportionality constant. In radial co-ordinates about an axisymmetric source, continuity over a cylinder of radius r couples the specific discharge to the total discharge Q.

q = −k ∇h (Darcy's law)
Q = 2πr · h(r) · qr (continuity across radius r)
qr = −k dh/dr (radial specific discharge)
⇒ Q = −2πr k h(r) dh/dr
Notation
q, qr
Darcy specific discharge (flux per unit area), general and radial component [m/s].
h(r)
Saturated thickness above the aquifer base at radial distance r from the source [m].
k
Local horizontal hydraulic conductivity [m/s].

Dupuit-Forchheimer

3. Dupuit-Forchheimer derivation for the unconfined aquifer

For the unconfined case the upper boundary of the flow domain is the phreatic surface, which moves toward the source during pumping. Dupuit's approximation ignores the vertical flow component at the phreatic surface, so the head gradient reduces to dh/dr, independent of elevation. Integrating the continuity equation from the source (rw, hw) to the far field (R, h0) yields the classical Dupuit expression.

Q = −2πr k h dh/dr (from continuity)
∫rwR dr/r = −(2π k / Q) ∫hwh0 h dh
⇒ ln(R/rw) = (π k / Q)(h0² − hw²)
Q = πk(h0² − hw²) / ln(R/rw) (Dupuit formula, homogeneous)
Notation
rw
Source radius (well, well-point radius, or equivalent outer radius of a trench image) [m].
hw, h0
Saturated thickness at the source and at the far-field boundary [m].
R
Radius of influence — outer boundary at which h ≈ h0 [m]. See §6.
  • Dupuit assumes horizontal flow, hydrostatic pressure on verticals, and a phreatic surface shallow relative to the aquifer depth. For shallow drawdown this is tight; for deep drawdown the true phreatic surface dips steeper near the well than Dupuit predicts.
  • The expression assumes isotropic k. When vertical and horizontal k differ, the effective k in the formula is kh; vertical flow losses are a correction, not a redefinition.

Transmissivity moment

4. Transmissivity and transmissivity-moment formulation for a layered profile

The CPT column is layered, and collapsing it to a single conductivity would lose most of the information the interpretation captures. The transmissivity-moment formulation keeps the layering by generalising the Dupuit integrand: the function T(h) returns the transmissivity of the profile when the saturated thickness is h, and M(h) is its integral. Substituting M into the Dupuit derivation replaces the h·dh integral directly without any further assumption about layer geometry.

T(h) = Σi kh,i bi(h) (layered transmissivity at saturated thickness h)
M(h) = ∫0h T(ξ) dξ (transmissivity moment)
Q dr / r = 2π dM(h) ⇒ Q ln(R/rw) = 2π [M(h0) − M(hw)]
Q = 2π [M(h0) − M(hw)] / ln(R/rw)
Notation
kh,i
Horizontal conductivity of layer i [m/s].
bi(h)
Saturated-thickness contribution of layer i as a function of saturated thickness h [m].
T, M
Transmissivity [m²/s] and transmissivity moment [m³/s].

For a homogeneous aquifer (one layer, kh,1 = k), T(h) = k h, so M(h) = kh²/2 and the formulation collapses exactly to the classical Dupuit expression Q = πk(h0² − hw²) / ln(R/rw). In the general layered case the evaluation is numerical but cheap — the app integrates M(h) analytically through each piecewise-constant layer.

Drawdown profile

5. Drawdown profile between the source and the CPT

For a radial source the head profile between rw and R follows from the same transmissivity-moment integration, solved forward in r rather than at the boundaries. The drawdown at the CPT observation point is then sampled from the profile.

M(h(r)) = M(hw) + (Q / 2π) ln(r/rw)
h(r) = M−1(M(hw) + (Q/2π) ln(r/rw))
ΔhCPT = h0 − h(rCPT)
Homogeneous limit: h²(r) = hw² + (Q/(πk)) ln(r/rw)
Notation
rCPT
Horizontal distance from the source to the CPT observation point [m].
M−1
Inverse of the transmissivity-moment function. For a homogeneous aquifer, h = √(2M/k).
  • M is monotonically non-decreasing in h because T ≥ 0 everywhere, so the inverse is well-defined up to numerical precision.
  • The CPT location is treated as the observation point. For a design using multiple CPTs, the app runs one evaluation per location.

Radius of influence

6. Sichardt screening radius and its limitations

The Dupuit expression requires an outer boundary condition h = h0 at r = R. In the absence of a natural boundary (a stream, a lake, a low-permeability barrier), the engineer adopts a screening radius. The most widely used rule is Sichardt (1928), which relates R to the drawdown and the effective conductivity.

R = C s √keff,h (Sichardt screen)
C ≈ 3000 (well) – 2000 (trench) with s in m and k in m/s
s = h0 − hw (drawdown at the source)
keff,h = Σ kh,i bi / Σ bi (thickness-weighted horizontal conductivity)
Transparency note. Sichardt is a practical screening rule with no mechanical derivation. The app reports it because the rule is widely used in engineering practice, but documents it as approximate and provides the option to override R with a project-specific value sourced from observed responses on similar works nearby.

Trenches and line sources

7. Trench geometry and the image method

A pumped trench is a line source rather than a point source. For steady-state axisymmetric flow, the equivalent radial form is recovered through the image method: replace the line source with an infinite sequence of point sources (or, equivalently, an integrated line sink). For short trenches compared to the radius of influence, the equivalent radius rw,eq of the trench is approximately π times the half-length divided by 2; for long trenches parallel to the aquifer boundary, a two-sided linear seepage form is used.

Short trench (L ≪ R): rw,eq ≈ π L / 2 (Dupuit form with L = trench half-length)
Long trench (L ≫ R): Q = k (h0² − hw²) · (Lt/R) (linear seepage form per unit length)
Intermediate: app linearly blends the two limits on L/R
Notation
L, Lt
Half-length [m] and full length [m] of the trench.
rw,eq
Equivalent radial source radius used in the Dupuit form [m].
  • The trench is treated as a single equivalent source; well-point arrays are modelled by this route as well, with rw,eq estimated from the array span.
  • Head losses inside the trench itself (skin losses, filter-cake effects) are not modelled explicitly — the engineer handles them by adjusting hw.

Stress and settlement

8. Effective-stress change and settlement response

Once the new phreatic level at the CPT is known, pore pressure and effective stress are recomputed and settlement is evaluated with the same constrained-modulus philosophy used in the settlement route. Drawdown loads the soil column by removing a portion of the buoyancy the water once provided.

z′w = zw,0 + ΔhCPT (new phreatic depth)
u′(z) = γw max(0, z − z′w)
σv(z) unchanged; σ′v,new(z) = σv(z) − u′(z)
Δσ′v(z) = σ′v,new(z) − σ′v,old(z) = uold(z) − u′(z)
Δεv,i = Δσ′v,i / Eoed,i (constrained-modulus constitutive relation)
ΔSdewatering = Σ Δεv,i Δzi
Notation
zw,0, z′w
Original and post-drawdown phreatic-level depth below ground [m].
uold, u′
Pore pressure before and after drawdown [kPa].
Δσ′v
Effective-stress increase from drawdown [kPa]. Positive under the phreatic fall.
Eoed,i
Oedometric stiffness in sublayer i [kPa], evaluated at the mean stress level between the old and new state.
  • The app reports two σv assumptions (saturated-weight everywhere and original layering) so the user can see how sensitive the result is to the assumed total stress in the partially submerged column.
  • Optional fine-grained time interpretation follows the same Terzaghi consolidation logic used in the settlement route.
  • The settlement output is at the CPT, not a spatial settlement field.

Limitations

9. Assumptions and boundaries of validity

  • Steady-state only. Transient drawdown (Theis, Jacob) is not in the current route.
  • Dupuit assumption: horizontal flow, hydrostatic pressure on verticals. Near the well the true phreatic surface dips steeper; Q is typically conservative relative to true 3D flow.
  • Sichardt for R is a screening rule, not a boundary condition. The engineer should override R with observed or modelled values when hydraulic sensitivity matters.
  • Aquifer bottom is assumed horizontal and impermeable.
  • Radial symmetry breaks down near streams, sheet piles, and other hydraulic boundaries. The trench image limit helps but does not reproduce partial boundary conditions.
  • Partial-penetration effects (well screen shorter than saturated thickness) are not modelled; the app assumes full penetration.
  • Unsaturated-zone transient effects in the capillary fringe are not captured.
  • The deeper audit trail remains in the full technical specification.

References

References

  • Dupuit, J. (1863). Études théoriques et pratiques sur le mouvement des eaux dans les canaux découverts et à travers les terrains perméables. Dunod, Paris.
  • Forchheimer, P. (1886). Ueber die Ergiebigkeit von Brunnen-Anlagen und Sickerschlitzen. Z. des Vereins deutscher Ingenieure.
  • Sichardt, W. (1928). Das Fassungsvermögen von Bohrbrunnen und seine Bedeutung für die Grundwasserabsenkung, insbesondere für größere Absenkungstiefen. Springer, Berlin.
  • Theis, C. V. (1935). The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground-water storage. Trans. Am. Geophys. Union 16(2), 519–524.
  • Cooper, H. H. & Jacob, C. E. (1946). A generalized graphical method for evaluating formation constants and summarizing well-field history. Trans. Am. Geophys. Union 27(4), 526–534.
  • Powers, J. P., Corwin, A. B., Schmall, P. C. & Kaeck, W. E. (2007). Construction Dewatering and Groundwater Control — New Methods and Applications. Wiley.