Theis Non-Equilibrium Well Drawdown Calculator - Free Tool: Free Theis Non-Equilibrium Well Drawdown Calculator tool: calculate groundwater well drawdown s(r,t), Theis Well Function W(u), transmissivity.
1. The Physics of Transient Groundwater Flow to a Well
Groundwater represents over 95% of the planet's unfrozen liquid freshwater reserves. When a water supply or environmental remediation well begins pumping, water is initially released from storage in the immediate vicinity of the borehole. As pumping continues, groundwater heads decline, propagating a cone of depression outward in all radial directions until a steady state or recharge boundary is encountered.
In 1935, Charles V. Theis published the fundamental transient solution for radial non-equilibrium flow to a fully penetrating well in a confined aquifer by drawing an analogy between Darcy's law in porous media and Fourier's law of heat conduction in solid conductors.
2. Mathematical Derivation of the Theis Equation
The governing partial differential equation for two-dimensional horizontal radial flow in homogeneous isotropic porous media is:
Subject to boundary conditions: initial drawdown $s(r, 0) = 0$, $s(infty, t) = 0$, and constant discharge boundary at the wellbore $lim_{r to 0} left(r frac{partial s}{partial r}right) = -frac{Q}{2pi T}$.
The exact analytical solution yields:
Where $s$ is the hydraulic head drawdown ($m$), $Q$ is constant volumetric extraction rate ($m^3/s$), $T$ is aquifer transmissivity ($m^2/s$), and $W(u)$ is the Theis Well Function evaluated at the dimensionless Boltzmann variable $u$:
Where $r$ is radial distance from the pumping well ($m$), $S$ is dimensionless aquifer storativity, and $t$ is elapsed pumping duration ($s$).
3. The Well Function $W(u)$ Series Expansion
Mathematically, $W(u)$ is the exponential integral $-Ei(-u)$:
Where $gamma approx 0.5772156649$ is the Euler-Mascheroni constant:
4. The Cooper-Jacob Semi-Logarithmic Approximation
When the dimensionless parameter $u$ is small ($u le 0.05$ or $0.03$), all higher-order terms in the infinite series ($u, u^2, dots$) become negligible compared to the logarithmic term:
Substituting into the Theis equation gives the Cooper-Jacob equation:
Plotting drawdown $s$ against $log_{10}(t)$ produces a straight line whose slope over one log cycle of time ($Delta s$) directly reveals aquifer transmissivity: $$T = frac{2.3026 Q}{4pi Delta s}$$
5. Confined Storativity vs Unconfined Specific Yield
- Confined Aquifers ($S approx 10^{-5} - 10^{-3}$): The aquifer is sandwiched between low-permeability aquitards under pressure. Pumping does not drain pore space; instead, water is produced via slight elastic expansion of the water ($beta_w$) and compaction of the aquifer matrix ($alpha_s$): $$S = rho_w g b (alpha_s + n beta_w)$$ Because $S$ is tiny, the cone of depression propagates rapidly over thousands of meters within hours.
- Unconfined Aquifers ($S = S_y approx 0.05 - 0.30$): The water table forms the upper boundary. Pumping physically dewaters pore space by gravity drainage. Because specific yield $S_y$ is hundreds of times larger than elastic storativity, the cone of depression expands much slower, and drawdowns are shallower for equivalent discharge.
6. Step-by-Step Worked Numerical Example: Municipal Well Pumping Test
Consider a municipal water well pumping from a confined gravel aquifer:
- Pumping rate $Q = 60.0text{ m}^3/text{h}$ ($0.01667text{ m}^3/text{s}$ or $1{,}440text{ m}^3/text{day}$)
- Aquifer transmissivity $T = 300.0text{ m}^2/text{day}$ ($0.003472text{ m}^2/text{s}$)
- Storativity $S = 4.0 times 10^{-4}$ ($0.00040$)
- Observation well distance $r = 75.0text{ m}$
- Elapsed pumping duration $t = 24.0text{ hours}$ ($1.0text{ day} = 86{,}400text{ s}$)
- Aquifer thickness $b = 20.0text{ m}$
- Calculate Dimensionless Parameter $u$: $$u = frac{r^2 S}{4 T t} = frac{(75.0)^2 times 0.00040}{4 times 300.0 times 1.0} = frac{5{,}625 times 0.00040}{1{,}200.0} = frac{2.25}{1{,}200.0} = 0.001875$$ Since $u = 0.001875 le 0.05$, the Cooper-Jacob logarithmic approximation is valid.
- Evaluate Well Function $W(u)$: $$W(u) approx -ln(0.001875) - 0.57721566 = 6.2797 - 0.5772 approx 5.7025$$
- Calculate Drawdown at Observation Well ($s$): $$s = frac{Q}{4pi T} W(u) = frac{1{,}440.0text{ m}^3/text{day}}{4pi times 300.0text{ m}^2/text{day}} times 5.7025 = frac{1{,}440.0}{3{,}769.91} times 5.7025 approx 0.38197 times 5.7025 approx 2.18text{ meters} quad (7.14text{ ft})$$
- Calculate Hydraulic Conductivity ($K$): $$K = frac{T}{b} = frac{300.0text{ m}^2/text{day}}{20.0text{ m}} = 15.0text{ m/day} quad (1.736 times 10^{-4}text{ m/s})$$
7. Hydrogeological Best Practices & Environmental Safeguards
- Wellhead Protection Areas (WHPA): Environmental protection zones are established around pumping wells based on calculated time-of-travel (TOT) captures (e.g. 50-day and 5-year capture zones using Theis velocity vectors $v = K i / n_e$) to prevent biological pathogen and solvent contamination.
- Saltwater Intrusion: In coastal aquifers, excessive well drawdown disrupts Ghyben-Herzberg hydrostatic equilibrium ($z = 40 times h_w$). Every 1 meter of freshwater table decline causes the underlying saline water table to rise 40 meters, risking saline upconing.
- Well Efficiency & Skin Effect: Turbulent entry losses and clogged well screens produce additional wellbore head loss ($Delta s_{text{well}} = C Q^2$). The actual pumping well drawdown is always greater than the theoretical formation drawdown ($s_{text{actual}} = s_{text{formation}} + s_{text{skin}}$).
8. Boundary Effects & The Method of Image Wells
The basic Theis solution assumes an infinite lateral aquifer expanse. In reality, aquifers encounter physical hydrogeological boundaries:
- Recharge Boundary (Constant Head, e.g., Perennial River or Lake): When the cone of depression intersects a surface water body, water infiltrates into the aquifer, halting further drawdown expansion. Under the Method of Images, this boundary is mathematically simulated by placing an imaginary recharge well of equal flow rate ($-Q$) equidistant on the opposite side of the river boundary. The net drawdown is $s_{text{net}} = s_{text{pump}} - s_{text{recharge}}$, causing the time-drawdown slope to flatten.
- Impermeable Barrier Boundary (Zero Flux, e.g., Bedrock Valley Wall or Fault): When flow cannot cross an impermeable boundary, drawdown steepens. This is modeled by placing an imaginary pumping well ($+Q$) equidistant across the fault line. The net drawdown doubles: $s_{text{net}} = s_{text{pump}} + s_{text{image}}$, causing the Cooper-Jacob semi-log slope to double ($2Delta s$) as the cone reflects off the barrier.
9. Theis Recovery Test Method ($s'$ vs $t / t'$)
When a pumping test shuts down at time $t_p$, residual drawdown $s'$ is monitored during aquifer recharge:
Where $t$ is total time since pumping began, and $t'$ is time since pump shutoff ($t = t_p + t'$). Plotting residual drawdown $s'$ against the dimensionless ratio $t / t'$ yields a clean straight line that is completely immune to turbulent wellbore skin losses ($CQ^2$), providing the most accurate and reliable field determination of formation transmissivity.