Capture zone analysis, Theis equation, and cost-benefit comparison of remediation approaches
2026-03-05
The largest pump-and-treat system in North America operates at the Rocky Mountain Arsenal near Denver, Colorado — a former chemical weapons manufacturing site contaminated with nerve agent precursors, pesticides, and aviation fuel. The system has been pumping and treating since 1982, handling up to 6,000 gallons per minute through a treatment train that includes air stripping, activated carbon adsorption, and biological treatment. After 40 years and hundreds of millions of dollars, it continues to operate — not because it has failed, but because the contamination was so severe and so deeply embedded in the aquifer that complete cleanup will take generations.
Pump-and-treat is not a silver bullet. It is a hydraulic containment technology that, in the best case, prevents plume migration to receptors while natural attenuation and time slowly reduce contaminant mass. Understanding its mathematics — capture zones, pumping rates, and the economics of operation over decades — is essential for realistic site management.
What pumping rate is required to capture a contaminant plume and prevent it from migrating to a receptor? How does a single pumping well affect the surrounding water table? What in-situ chemical treatment can destroy contaminant mass? And how do we compare remediation alternatives economically over a 30-year project life?
A pumping well in an aquifer with regional groundwater flow creates a cone of depression — a radially symmetric drawdown zone that deflects the regional flow lines toward the well. The capture zone is the upgradient area whose flow lines converge on the well. Any contaminant within the capture zone will be pumped to the surface; any contaminant outside will continue to migrate downgradient.
The width of the capture zone at the well is determined by the pumping rate relative to the regional flow rate. Insufficient pumping allows the plume to pass between wells; excessive pumping wastes energy and treatment costs.
For a single well pumping at rate Q [m³ s⁻¹] in a confined aquifer at steady state, the drawdown s = h_0 - h (where h_0 is undisturbed head) at distance r [m] from the well is described by the Thiem equation:
s(r) = s(r_1) - \frac{Q}{2\pi T}\ln\!\left(\frac{r}{r_1}\right)
where T = Kb [m² s⁻¹] is transmissivity (hydraulic conductivity × saturated thickness), and r_1 is a reference distance where head is known.
For an unconfined aquifer (water table), the equivalent is:
h(r)^2 - h(r_1)^2 = \frac{Q}{\pi K}\ln\!\left(\frac{r}{r_1}\right)
Before steady state is reached, drawdown evolves with time. The Theis equation (1935):
s(r,t) = \frac{Q}{4\pi T} W(u)
where W(u) = -\text{Ei}(-u) = \int_u^\infty \frac{e^{-t}}{t}dt is the well function and:
u = \frac{r^2 S}{4Tt}
S is the storativity (confined: 10⁻⁵–10⁻³; unconfined = specific yield S_y \approx 0.1–0.3), and t is time.
Cooper-Jacob approximation (valid for u < 0.05, i.e., large t or small r):
s(r,t) \approx \frac{Q}{4\pi T}\left[\ln\!\left(\frac{2.25Tt}{r^2 S}\right)\right] = \frac{2.303Q}{4\pi T}\log_{10}\!\left(\frac{2.25Tt}{r^2 S}\right)
For Q = 0.01 m³/s, T = 0.002 m²/s, r = 50 m, S = 0.001, t = 24 hours = 86,400 s:
u = 50^2 \times 0.001 / (4 \times 0.002 \times 86400) = 2500 \times 0.001/691.2 = 0.00362 (< 0.05 ✓)
s = 0.01/(4\pi \times 0.002) \times \ln(2.25 \times 0.002 \times 86400/(2500 \times 0.001)) = 0.0398 \times \ln(155.52) = 0.0398 \times 5.047 = 0.201 m drawdown at 50 m after 24 hours.
In a uniform regional flow field with Darcy flux q_0 = K \cdot i [m/day], a pumping well of rate Q [m³/day] creates a capture zone whose maximum width y_{\max} at the well (the total width far upgradient) is:
y_{\max} = \frac{Q}{q_0 b}
where b is aquifer saturated thickness. This is the width of the entire capture zone — the region whose groundwater is captured by the well.
The stagnation point — the upgradient point where the regional flow exactly balances the wellward flow — is at distance:
x_s = -\frac{Q}{2\pi T i \cdot b} = -\frac{Q}{2\pi q_0 b}
(negative = upgradient). Beyond this point, all groundwater flows toward the well.
The capture zone boundary is the streamline that divides captured from escaped flow. Its shape is:
y = \pm\frac{Q}{2q_0 b}\left(1 - \frac{x}{|x_s|}\right) \quad \text{(approximate, for large }x\text{)}
For complete capture of a plume of width W_{\text{plume}}: need y_{\max} \geq W_{\text{plume}} → required pumping rate:
Q_{\min} = W_{\text{plume}} \cdot q_0 \cdot b
ISCO destroys contaminants in the subsurface by injecting a strong oxidant — permanganate (KMnO₄), persulfate (Na₂S₂O₈), or Fenton’s reagent (H₂O₂ + Fe²⁺). The stoichiometric requirement for each contaminant is derived from the balanced chemical equation.
PCE destruction by permanganate:
$3\text{C}_2\text{Cl}_4 + 4\text{KMnO}_4 \rightarrow 6\text{CO}_2 + 4\text{KCl} + 4\text{MnO}_2 + 2\text{Cl}_2$
Stoichiometric ratio: 4 mol KMnO₄ per 3 mol PCE.
Molar masses: PCE (C_2Cl_4) = 165.8 g/mol; KMnO₄ = 158.0 g/mol.
Mass ratio: \frac{4 \times 158.0}{3 \times 165.8} = \frac{632}{497.4} = 1.27 kg KMnO₄ per kg PCE.
Practical demand: The subsurface natural oxidant demand (NOD) — from naturally occurring reduced compounds (organic matter, Fe²⁺, Mn²⁺, sulfide) — typically requires 1–10× the theoretical ISCO demand. Total oxidant mass:
M_{\text{oxidant}} = (S_R \cdot M_{\text{contaminant}} + \text{NOD} \cdot V_{\text{treatment}}) \cdot \text{safety factor}
where S_R is stoichiometric ratio, M_{\text{contaminant}} is contaminant mass to destroy, \text{NOD} is oxidant demand per unit aquifer volume, and the safety factor is typically 2–5 to ensure reaction completion.
Monitored natural attenuation (MNA) is accepted as a remedial strategy when:
Lines of evidence demonstrate active biodegradation:
Plume stability demonstrated: L_{\text{plume}} < L_{\text{receptor}} (plume length does not reach nearest receptor)
Risk-based cleanup targets reachable within a reasonable timeframe (typically 30–50 years for residential use)
The attenuation rate constant \lambda is estimated from the spatial decrease in concentration along the plume centreline:
\ln C(x) = \ln C_0 - \frac{\lambda}{v_c} x
If \ln C vs. x is linear, the slope gives -\lambda/v_c, and \lambda = -\text{slope} \times v_c.
Remediation costs extend over decades. Net present value (NPV) converts all future costs to present-day equivalents:
\text{NPV} = \sum_{t=0}^{T} \frac{C_t}{(1+r)^t}
where C_t is cost in year t, r is discount rate (typically 3–7% for public projects), and T is project horizon.
For a pump-and-treat system with capital cost C_0 and annual O&M cost C_{\text{OM}} for T years:
\text{NPV}_{\text{P\&T}} = C_0 + C_{\text{OM}} \cdot \frac{1-(1+r)^{-T}}{r}
The factor [1-(1+r)^{-T}]/r is the present value annuity factor (PVAF).
Comparison example: Three options for a PCE plume at Q = 0.002 m³/s pump rate:
| Option | Capital (M) | Annual O&M (M/yr) | Duration (yr) | NPV at 5% ($M) | |
|---|---|---|---|---|
| Pump-and-treat | 2.0 | 0.4 | 30 | 2.0 + 0.4×15.37 = 8.15 |
| ISCO | 1.5 | 0.05 | 5 | 1.5 + 0.05×4.33 = 1.72 |
| MNA with monitoring | 0.1 | 0.08 | 30 | 0.1 + 0.08×15.37 = 1.33 |
PVAF (5%, 30yr) = [1-(1.05)^{-30}]/0.05 = 15.37; PVAF (5%, 5yr) = 4.33.
MNA has the lowest NPV if the site qualifies — but requires demonstrating regulatory criteria. ISCO has higher NPV than MNA but lower than P&T and achieves faster cleanup of source mass. P&T is by far the most expensive option over a 30-year horizon — yet it remains the most common technology at major sites because it provides immediate hydraulic containment and regulatory certainty.
Setting: A TCE plume at a Calgary industrial site. Aquifer: K = 5\times10^{-4} m/s, b = 12 m, i = 0.003, T = Kb = 6\times10^{-3} m²/s. Plume width at leading edge: 35 m. Design a capture well.
Step 1: Regional Darcy flux
q_0 = K \cdot i = 5\times10^{-4} \times 0.003 = 1.5\times10^{-6} \text{ m/s} = 0.130 \text{ m/day}
Step 2: Minimum pumping rate for full capture
Q_{\min} = W_{\text{plume}} \cdot q_0 \cdot b = 35 \times 1.5\times10^{-6} \times 12 = 6.3\times10^{-4} \text{ m}^3/\text{s} = 54.4 \text{ m}^3/\text{day}
Step 3: Stagnation point location
x_s = -\frac{Q_{\min}}{2\pi q_0 b} = -\frac{6.3\times10^{-4}}{2\pi \times 1.5\times10^{-6} \times 12} = -\frac{6.3\times10^{-4}}{1.131\times10^{-4}} = -5.57 \text{ m}
The stagnation point is 5.6 m upgradient of the well — a relatively small capture zone for this plume width, meaning the well is positioned close to the leading edge.
Step 4: ISCO requirement (TCE source mass = 80 kg)
TCE + permanganate: $2_4 + _2_3 _2 + 2_2 + 2 + $
Stoichiometric ratio: (2 \times 158.0)/(131.4) = 316/131.4 = 2.41 kg KMnO₄/kg TCE
M_{\text{KMnO}_4,\text{stoic}} = 2.41 \times 80 = 193 kg
With NOD of 3 g/kg aquifer solids and treatment volume 1000 m³ at bulk density 1800 kg/m³:
\text{NOD mass} = 3 \times 10^{-3} \times 1800 \times 1000 = 5400 kg
M_{\text{total}} = (193 + 5400) \times 2 \text{ (safety factor)} = 11{,}186 kg KMnO₄
At \$1.50/kg, reagent cost ≈ $$16{,}800.
function theis_drawdown(r, t, Q, T, S):
u = r**2 * S / (4 * T * t)
W_u = well_function(u) # -Ei(-u)
return Q / (4 * pi * T) * W_u
function capture_width(Q, q0, b):
return Q / (q0 * b)
function stagnation_x(Q, q0, b):
return -Q / (2 * pi * q0 * b)
function npv(capital, annual_om, years, rate):
pvaf = (1 - (1 + rate)**(-years)) / rate
return capital + annual_om * pvaf
function isco_mass(contaminant_mass_kg, stoic_ratio, NOD_g_kg,
V_treatment_m3, rho_b_kg_m3, safety=2):
theoretical = stoic_ratio * contaminant_mass_kg
nod_demand = NOD_g_kg/1000 * rho_b_kg_m3 * V_treatment_m3
return (theoretical + nod_demand) * safetyimport numpy as np
from scipy.special import exp1
def well_function(u):
return exp1(u) # -Ei(-u) = E1(u)
def theis(r, t, Q, T, S):
u = r**2 * S / (4*T*t)
return Q / (4*np.pi*T) * well_function(u)
def cooper_jacob(r, t, Q, T, S):
u = r**2 * S / (4*T*t)
if u > 0.05: return np.nan # approximation invalid
return Q/(4*np.pi*T) * np.log(2.25*T*t/(r**2*S))
def capture_analysis(Q, K, b, i):
q0 = K * i
width = Q / (q0 * b)
xs = -Q / (2*np.pi*q0*b)
return q0, width, xs
def npv_cost(capital, annual_om, years, r=0.05):
pvaf = (1-(1+r)**(-years))/r
return capital + annual_om * pvaf
# Calgary TCE site
K, b, i = 5e-4, 12.0, 0.003
T = K * b
W_plume = 35.0
Q_min = W_plume * K * i * b
print(f"Minimum pumping rate: {Q_min*86400:.1f} m³/day = {Q_min*1000:.2f} L/s")
q0, cap_w, xs = capture_analysis(Q_min, K, b, i)
print(f"Capture zone width: {cap_w:.1f} m")
print(f"Stagnation point: {xs:.1f} m upgradient")
# Theis drawdown at various distances after 7 days
S = 0.0002
t7 = 7*86400
print(f"\nTheis drawdown after 7 days (Q={Q_min:.2e} m³/s, T={T:.3e} m²/s):")
for r in [1, 5, 10, 25, 50, 100]:
s = theis(r, t7, Q_min, T, S)
print(f" r={r:4d}m: s={s:.3f} m")
# NPV comparison
print("\nNPV comparison (5% discount rate, r in million $):")
for name, cap, om, yrs in [
("Pump-and-treat", 2.0, 0.40, 30),
("ISCO", 1.5, 0.05, 5),
("MNA + monitoring",0.1, 0.08, 30)]:
nv = npv_cost(cap, om, yrs)
print(f" {name:25s}: NPV = ${nv:.2f}M")Capture zone width scales linearly with pumping rate and inversely with the product q_0 b (natural groundwater flux × saturated thickness). A fast aquifer (high q_0 b) requires much higher pumping to capture the same plume width. This explains why pump-and-treat is economically challenging in high-transmissivity aquifers: the energy cost of pumping against a large regional flux is prohibitive, and the required pumping rate may exceed practical limits.
The capture zone analysis reveals a fundamental constraint on pump-and-treat: the pumping rate must exceed the natural groundwater flux through the entire plume cross-section. In a transmissive aquifer with a 50 m wide plume and q_0 b = 0.05 m²/day: Q_{\min} = 50 \times 0.05 = 2.5 m³/day. This is achievable at modest cost. But in a highly transmissive alluvial aquifer with q_0 b = 0.5 m²/day: Q_{\min} = 25 m³/day — still manageable. The challenge becomes acute when the plume is large (hundreds of metres wide) in a fast aquifer.
The NPV comparison shows that pump-and-treat is economically justified primarily when containment is legally required (imminent receptor risk) or when the site generates ongoing income that offsets remediation cost. For sites where receptor risk is manageable and MNA criteria are met, monitored natural attenuation is typically 5–10× less expensive in present value terms. The regulatory decision between these alternatives is therefore not purely technical — it involves risk tolerance, timeline commitments, and institutional trust in models that predict future concentrations.
Pump-and-treat asymptotically approaches but never achieves the cleanup goal. Aquifer heterogeneity creates high-concentration residuals in low-permeability zones that slowly diffuse back into the pumped water. The characteristic “tail” behaviour — rapid initial concentration decline, followed by extremely slow decline toward the cleanup standard — has been observed at virtually every long-running P&T site. Cost projections based on initial concentration decline rates consistently underestimate the total project cost.
ISCO can mobilise contaminants. Injecting oxidant into a NAPL source zone can increase contaminant concentrations downgradient during and after treatment by: (1) temporarily increasing NAPL dissolution rate (the oxidant creates a concentration gradient that drives faster dissolution), and (2) oxidant propagation beyond the target zone destroying monitored natural attenuation capacity (e.g., eliminating electron donor supply needed by reductive dechlorinators). ISCO should not be applied without careful assessment of downgradient receptor risk during the treatment period.
MNA requires long-term institutional memory. A 30-year MNA programme requires maintaining monitoring wells, records, and institutional knowledge across multiple ownership and regulatory transitions. Sites that were accepted for MNA in the 1990s have in some cases lost their monitoring programmes, been redeveloped without proper assessment, and created new exposure pathways. MNA is not a “walk away” strategy — it requires sustained commitment that many organisations and regulatory agencies are poorly equipped to maintain.
Discount rates systematically underweight future costs. A 5% discount rate reduces a $1M cost in year 30 to $231K present value — an 77% reduction. This means long-duration remediation (pump-and-treat for 50 years) appears less expensive in NPV terms than it actually is in undiscounted terms. The choice of discount rate is not politically neutral: lower rates favour long-duration approaches; higher rates favour front-loaded capital investment (like ISCO). For projects with environmental and public health consequences, zero or near-zero discount rates for benefit and harm are defensible in principle, even if unconventional in practice.
Pump-and-treat hydraulic containment requires a minimum pumping rate Q_{\min} = W_{\text{plume}} \cdot q_0 \cdot b to capture a plume of width W_{\text{plume}} in an aquifer with Darcy flux q_0 and saturated thickness b. The Theis equation s = QW(u)/(4\pi T) describes transient drawdown around a pumping well; the Cooper-Jacob approximation simplifies this for u < 0.05.
ISCO stoichiometry determines oxidant reagent mass from balanced chemical equations, adjusted upward for natural oxidant demand and safety factors. MNA is accepted when spatial concentration decline demonstrates active biodegradation and plume stability. NPV cost comparison discounts all future cash flows: \text{NPV} = C_0 + C_{\text{OM}} \cdot [(1-(1+r)^{-T})/r]. Over 30-year horizons, pump-and-treat typically costs 5–10× more than MNA in NPV terms.
Key equations:
Q_{\min} = W_{\text{plume}} \cdot q_0 \cdot b
y_{\max} = Q/(q_0 b), \quad x_s = -Q/(2\pi q_0 b)
s(r,t) = \frac{Q}{4\pi T}W(u), \quad u = \frac{r^2 S}{4Tt}
\text{NPV} = C_0 + C_{\text{OM}}\cdot\frac{1-(1+r)^{-T}}{r}
The well function W(u). The Theis well function is W(u) = \int_u^\infty e^{-t}/t\,dt — the exponential integral E_1(u). For small u (u < 0.05), the approximation W(u) \approx -\gamma - \ln u \approx -0.5772 - \ln u works well (the Cooper-Jacob simplification). Plotted against \log u, W(u) decreases from large values at small u to near zero at large u — large u corresponds to short times or large distances where drawdown has not yet propagated. The Theis type curve is used in aquifer pump tests to estimate T and S by matching observed drawdown data to the theoretical W(u) curve.
Present value and the annuity factor. The present value of a stream of equal annual payments C for T years at discount rate r is C \times [(1-(1+r)^{-T})/r]. This annuity factor equals the number of years T when r = 0 (no discounting); it is less than T for r > 0 because future payments are worth less. For r = 5\% and T = 30, the factor is 15.37 — equivalent to 15.4 years of payments in present value terms, even though 30 actual payments will be made. This compression of future costs into a smaller present value is the fundamental reason why long-duration liabilities appear more manageable in economic analyses than they do to communities that must actually live with them for 30 years.