Formula sheet

Contaminated Site Remediation and Groundwater Protection Formula Sheet

Contaminated-site remediation formulas for gradient, Darcy flow, seepage velocity, travel time, retardation, decay, mass flux, capture ratio, monitoring, and uncertainty.

Branch: Environmental Engineering
Content: Formula sheet
Updated: Jun 24, 2026
Revision: v1.0.0 · reviewed

This formula sheet collects first-pass calculations used in contaminated-site remediation and groundwater protection. Use it to make hydrogeology, plume migration, contaminant mass, hydraulic capture, treatment loading, monitoring trends and uncertainty checks traceable.

The equations are screening and review tools. They do not replace a conceptual site model, aquifer testing, regulatory criteria, toxicology, numerical groundwater modelling, laboratory quality assurance, field health and safety controls, or professional environmental review.

Before calculating, define the source, pathway, receptor, aquifer interval, time basis, concentration basis, monitoring evidence, and decision being supported. A correct formula applied to the wrong hydrostratigraphic unit can produce a confident but wrong remediation decision.

Symbols and Basis

Use consistent units. This sheet uses SI units unless a field convention is explicitly stated.

Symbol	Meaning	Common unit
$h$	hydraulic head	$\text{m}$
$\Delta h$	head difference	$\text{m}$
$L$	flow-path length or control distance	$\text{m}$
$i$	hydraulic gradient magnitude	dimensionless
$K$	hydraulic conductivity	$\text{m/s}$ or $\text{m/day}$
$q$	Darcy flux or Darcy velocity	$\text{m/s}$
$Q$	volumetric groundwater flow rate	$\text{m}^3/\text{s}$ or $\text{m}^3/\text{day}$
$A$	cross-sectional flow area	$\text{m}^2$
$b$	saturated aquifer thickness represented by the control plane	$\text{m}$
$W$	plume or control-plane width	$\text{m}$
$n_e$	effective porosity	dimensionless
$v_s$	average seepage velocity	$\text{m/s}$ or $\text{m/day}$
$C$	dissolved concentration	$\text{mg/L}$ or $\text{kg/m}^3$
$\dot{M}$	contaminant mass loading rate	$\text{kg/day}$ or $\text{g/day}$
$K_d$	soil-water distribution coefficient	$\text{L/kg}$ or $\text{m}^3/\text{kg}$
$\rho_b$	dry bulk density	$\text{kg/L}$ or $\text{kg/m}^3$
$R$	retardation factor	dimensionless
$k$	first-order decay constant	$1/\text{time}$
$t_{1/2}$	half-life	time
$Q_{ext}$	groundwater extraction rate	$\text{m}^3/\text{day}$
$R_c$	capture ratio	dimensionless
$\eta$	treatment removal efficiency	dimensionless or percent

Hydraulic Head Difference

Head difference between two monitoring points is:

\Delta h=h_{up}-h_{down}

Hydraulic gradient magnitude along the interpreted flow path is:

\displaystyle i=\frac{\Delta h}{L}

Validation Use

Use this calculation to check likely groundwater direction, capture gradients, seasonal changes, and monitoring-well consistency. It is valid only when the wells are screened in the same hydrostratigraphic unit and elevations are surveyed to adequate accuracy.

Mini-Check

If an upgradient well has:

h_{up}=103.20\ \text{m}

and a downgradient well has:

h_{down}=102.72\ \text{m}

over:

L=80\ \text{m}

then:

\Delta h=103.20-102.72=0.48\ \text{m}

and:

\displaystyle i=\frac{0.48}{80}=0.0060

The result is plausible for a low-gradient shallow aquifer. It does not prove plume direction if a utility trench, sand lens, pumping well or fracture network creates a preferential pathway.

Darcy Flux and Through-Flow

Darcy flux is:

q=Ki

Groundwater flow through a control plane is:

Q=qA=KiA

For a rectangular control plane:

A=bW

where $b$ is the represented saturated thickness and $W$ is the plume or compliance-plane width.

Mini-Check

Use:

K=2.5\times10^{-5}\ \text{m/s}

i=0.008

b=5.0\ \text{m}

W=40\ \text{m}

Darcy flux:

q=(2.5\times10^{-5})(0.008)=2.0\times10^{-7}\ \text{m/s}

Area:

A=5.0(40)=200\ \text{m}^2

Flow:

Q=(2.0\times10^{-7})(200)=4.0\times10^{-5}\ \text{m}^3/\text{s}

Convert to cubic metres per day:

Q=(4.0\times10^{-5})(86400)=3.46\ \text{m}^3/\text{day}

Engineering Comment

The calculation is order-of-magnitude. Hydraulic conductivity can vary by orders of magnitude across a site. A single $K$ value should not be used to prove capture, closure or receptor protection.

Seepage Velocity and Travel Time

Darcy flux is not the average pore-water velocity. A common seepage velocity screen is:

\displaystyle v_s=\frac{q}{n_e}=\frac{Ki}{n_e}

Travel time over a path length $L$ is:

\displaystyle t=\frac{L}{v_s}

Mini-Check

Using:

q=2.0\times10^{-7}\ \text{m/s}

and:

n_e=0.25

gives:

\displaystyle v_s=\frac{2.0\times10^{-7}}{0.25}=8.0\times10^{-7}\ \text{m/s}

Convert to metres per day:

v_s=(8.0\times10^{-7})(86400)=0.069\ \text{m/day}

For a receptor:

L=120\ \text{m}

the travel time is:

\displaystyle t=\frac{120}{0.069}=1739\ \text{days}=4.8\ \text{years}

Engineering Comment

This is a water-particle travel time. A contaminant may move faster or slower depending on sorption, density, degradation, diffusion into low-permeability zones, non-aqueous phase liquid, preferential pathways and source persistence.

Retardation by Linear Sorption

For a linear equilibrium sorption screen:

\displaystyle R=1+\frac{\rho_bK_d}{n_e}

Contaminant velocity is:

\displaystyle v_c=\frac{v_s}{R}

Retarded travel time is:

\displaystyle t_c=\frac{L}{v_c}=\frac{LR}{v_s}

Unit Warning

If $K_d$ is in $\text{L/kg}$ , use $\rho_b$ in $\text{kg/L}$ . If $K_d$ is in $\text{m}^3/\text{kg}$ , use $\rho_b$ in $\text{kg/m}^3$ .

Mini-Check

Use:

\rho_b=1.65\ \text{kg/L}

K_d=0.12\ \text{L/kg}

n_e=0.30

Then:

\displaystyle R=1+\frac{1.65(0.12)}{0.30}=1.66

If:

v_s=0.069\ \text{m/day}

then:

\displaystyle v_c=\frac{0.069}{1.66}=0.0416\ \text{m/day}

For:

L=120\ \text{m}

the retarded travel time is:

\displaystyle t_c=\frac{120}{0.0416}=2885\ \text{days}=7.9\ \text{years}

Engineering Comment

The result is only as defensible as the sorption model. Linear $K_d$ may fail for changing pH, redox, organic carbon, concentration, salinity, competing ions, colloids or non-equilibrium transport.

First-Order Decay or Transformation

For a first-order decay or transformation screen:

C(t)=C_0e^{-kt}

Half-life is:

\displaystyle t_{1/2}=\frac{\ln 2}{k}

or:

\displaystyle k=\frac{\ln 2}{t_{1/2}}

Mini-Check

If the reviewed contaminant has an estimated half-life:

t_{1/2}=2.0\ \text{years}

then:

\displaystyle k=\frac{\ln 2}{2.0}=0.347\ \text{year}^{-1}

For a retarded travel time:

t_c=7.9\ \text{years}

the concentration ratio is:

\displaystyle \frac{C}{C_0}=e^{-0.347(7.9)}=0.064

Engineering Comment

Do not use decay credit unless field conditions support it. Some contaminants degrade only under specific redox, electron-donor, temperature, pH and microbial conditions. Transformation products may be more mobile or more toxic than the parent compound.

Longitudinal Dispersion Screen

A common longitudinal dispersion coefficient screen is:

D_L=\alpha_Lv_s+D_m

where $\alpha_L$ is longitudinal dispersivity and $D_m$ is effective molecular diffusion.

A Peclet number check is:

\displaystyle Pe=\frac{v_sL}{D_L}

High $Pe$ means advection dominates. Low $Pe$ means dispersion and diffusion strongly smear the plume front.

Validation Use

Use dispersion formulas to understand plume spreading, not to hide uncertainty. Dispersivity depends on scale, geology, measurement method and model structure.

Contaminant Mass Loading

Mass loading rate from a water stream is:

\dot{M}=QC

When:

$Q$ is in $\text{m}^3/\text{day}$ ;
$C$ is in $\text{mg/L}$ ;

then:

\dot{M}\ [\text{g/day}]=Q\ [\text{m}^3/\text{day}] C\ [\text{mg/L}]

because $1\ \text{m}^3=1000\ \text{L}$ and $1000\ \text{mg}=1\ \text{g}$ .

Total mass over a time interval is:

M=\int_0^T Q(t)C(t)\,dt

For discrete monitoring intervals:

M\approx \sum_j Q_jC_j\Delta t_j

Mini-Check

If extraction flow is:

Q=18\ \text{m}^3/\text{day}

and influent concentration is:

C=1.4\ \text{mg/L}

then:

\dot{M}=18(1.4)=25.2\ \text{g/day}

Engineering Comment

Mass loading is more useful than concentration alone. Falling concentration with falling extraction flow may not mean the remedy is removing more mass.

Mass Flux Through a Control Plane

Dissolved contaminant mass flux per unit area can be screened as:

J=qC

Total mass discharge across a control plane is:

\dot{M}_{plane}=KiAC

with unit conversions handled consistently.

Validation Use

Mass discharge is useful for comparing source strength, plume stability, natural attenuation, pump-and-treat removal, and downgradient receptor loading. It requires representative concentration distribution across the control plane, not only one well.

Hydraulic Capture Ratio

A first-pass capture ratio is:

\displaystyle R_c=\frac{Q_{ext}}{Q_{through}}

where:

Q_{through}=KiA

Mini-Check

If:

Q_{ext}=35\ \text{m}^3/\text{day}

and:

Q_{through}=3.46\ \text{m}^3/\text{day}

then:

\displaystyle R_c=\frac{35}{3.46}=10.1

Engineering Comment

A high capture ratio does not prove hydraulic capture. It only says extraction is large relative to a bulk through-flow estimate. Capture must be validated with groundwater elevations, inward gradients, plume trends, sentinel wells, pumping stability and boundary conditions.

Treatment Removal Efficiency

Treatment removal efficiency is:

\displaystyle \eta=\frac{C_{in}-C_{out}}{C_{in}}

Percent removal is:

\eta_{\%}=100\eta

Mini-Check

For:

C_{in}=1.4\ \text{mg/L}

and:

C_{out}=0.006\ \text{mg/L}

then:

\displaystyle \eta=\frac{1.4-0.006}{1.4}=0.9957

or:

\eta_{\%}=99.57\%

Engineering Comment

Percent removal is not the only compliance metric. The treated effluent must meet the actual discharge limit, sampling uncertainty must be considered near the limit, and media breakthrough or fouling must be tracked.

Mixing and Dilution

For two water streams mixing without reaction:

\displaystyle C_{mix}=\frac{Q_1C_1+Q_2C_2}{Q_1+Q_2}

Validation Use

Use this only for hydraulic mixing checks. It should not be used to justify dilution as a remedy unless the regulatory and receptor context explicitly permits it.

Monitoring Trend Slope

A simple concentration trend slope is:

\displaystyle m=\frac{C_2-C_1}{t_2-t_1}

Percent change relative to baseline is:

\displaystyle \Delta C_{\%}=100\frac{C_2-C_1}{C_1}

Mini-Check

If a compliance well changes from:

C_1=0.080\ \text{mg/L}

to:

C_2=0.104\ \text{mg/L}

over:

t_2-t_1=90\ \text{days}

then:

\displaystyle m=\frac{0.104-0.080}{90}=2.67\times10^{-4}\ \text{mg/(L day)}

and:

\displaystyle \Delta C_{\%}=100\frac{0.104-0.080}{0.080}=30\%

If the action level is a $25\%$ increase, this trend triggers review.

Engineering Comment

Trend interpretation should consider sampling variability, laboratory uncertainty, seasonal water levels, detection limits, non-detect handling, well condition and plume movement. A two-point trend is a trigger, not a statistical proof.

Signal-to-Noise Check for Monitoring Change

A simple signal-to-noise ratio for a measured change is:

\displaystyle SNR=\frac{|\Delta C|}{u_{\Delta C}}

For independent concentration uncertainties:

u_{\Delta C}=\sqrt{u_{C1}^2+u_{C2}^2}

Validation Use

If $SNR$ is near 1, the claimed change is similar to measurement noise. Use more data, better sampling control, or a wider decision band before making a closure claim.

Rebound Ratio

After shutdown or pulsed operation, a simple rebound ratio is:

\displaystyle R_b=\frac{C_{rebound}}{C_{shutdown}}

where $C_{shutdown}$ is concentration near the end of active pumping or treatment and $C_{rebound}$ is concentration after the specified rebound period.

Engineering Comment

Rebound indicates residual source, diffusion from low-permeability zones, desorption, hydraulic redistribution or incomplete treatment. It should be interpreted with water levels and source-zone evidence.

Uncertainty Guard Band

For a measured result $x$ with expanded uncertainty $U$ , a conservative upper decision value is:

x_{upper}=x+U

For a concentration limit $C_{limit}$ , a simple pass condition is:

C+U\le C_{limit}

Validation Use

Guard bands are useful near discharge limits, closure criteria and action levels. The uncertainty basis should state whether it includes sampling, laboratory, calibration, field variability or only instrument precision.

Common Misuses

Misuse	Why it is risky
using a single monitoring well as a plume boundary	the plume may bypass the well screen or interval
treating Darcy through-flow as capture proof	capture needs head and gradient evidence under pumping
using $K_d$ without unit conversion	retardation can be wrong by orders of magnitude
applying decay credit without redox evidence	degradation may not occur under site conditions
using concentration decline without flow	mass removal may be falling while concentration appears lower
ignoring low-permeability zones	rebound can occur after shutdown
comparing samples with different methods or limits	trend evidence may be artificial
averaging across hydrostratigraphic units	the calculation may mix unrelated flow systems

Minimum Validation Package

A defensible remediation calculation package should include:

conceptual site model boundary and receptor statement;
well construction, screened intervals and surveyed elevations;
hydraulic gradient basis and seasonal range;
hydraulic conductivity source and uncertainty;
porosity, bulk density and sorption assumptions;
concentration data quality and detection limits;
mass loading or mass discharge basis;
treatment loading and discharge comparison;
monitoring trend and uncertainty check;
field evidence that could disprove the calculation.

The formulas are useful only when the evidence loop is closed: source understood, pathway represented, receptor defined, calculation traceable, uncertainty visible, and validation data tied to the decision.

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