Flow to Wells – 1 Steady flow to a well in a confined aquifer Groundwater Hydraulics Daene C. McKinney

Summary Some Information on Wells Steady flow to a well in a confined aquifer

Some Information on Wells

Domestic Hand Pumped Well Domestic dug well with rock curb, concrete seal, and hand pump ~20 m depth > 1 m diameter < 500 m3/day Hand dug well in Beirut, Lebanon Hand dug well in Trets, France

Augers Hand-driven augers ~15 m depth > 20 cm diameter Power-driven augers ~30 m depth > 1 m diameter

Power Auger Auger drilling is done with a helical screw driven into the ground with rotation; cuttings are lifted up the borehole by the screw ~ 30 m depth < 15-90 cm diameter < 500 m3/day

Drilled Well - Cable Tool Traditional way of drilling large diameter water supply wells. The Rig raises and drops the drill string with a heavy carbide tipped drill bit that chisels through the rock and pulverizes the materials. 8 – 60 cm 600 m

Mud/Air Rotary Rotary drilling relies on continuous circular motion of the bit to break rock at the bottom of the hole. Cuttings are removed as drilling fluids circulate through the bit and up the wellbore to the surface.

Drilling Mud Circulation Lift cuttings from the borehole and carry to pit; Cuttings drop out in the pit; Length of drill pipe is added; Film on the borehole wall prevents caving; Seals borehole wall to reduce fluid loss; Cools and cleans bit; and Lubricates bit, bearings, mud pump and drill pipe .

Well Completion After drilling, must “complete” the well Placement of casing Placement of well screen Placement of gravel packing Open hole

Well Construction Well casing Lining to maintain open hole Seals out other water (surface, formations) Structural support against cave-in

Well in Limestone Surface casing From ground surface through unconsolidated upper material

Placing the Pack

Steady Flow to a Well in a Confined Aquifer

Flow to a Well in a Confined Aquifer

Drawdown in a Well Drawdown in a pumped well consists of two components: Aquifer losses Head losses that occur in the aquifer where the flow is laminar Tme-dependent Vary linearly with the well discharge Well losses Aquifer damage during drilling and completion Turbulent friction losses adjacent to well, in the well and pipe

Drawdown to a pumping well in a confined aquifer Uncased hole Screened well Gravel-packed well Bear, 1979, Fig. 8.3

Groundwater Notation Confined aquifer 𝛻∙𝑇𝛻ℎ= 𝜕 𝜕𝑥 𝑇 𝑥 𝜕ℎ 𝜕𝑥 + 𝜕 𝜕𝑦 𝑇 𝑦 𝜕ℎ 𝜕𝑦 =S 𝜕ℎ 𝜕𝑡 General equation, rectangular coordinates 𝛻 2 ℎ= 𝜕 2 ℎ 𝜕 𝑥 2 + 𝜕 2 ℎ 𝜕 𝑦 2 =0 Steady state flow, homogeneous and Isotropic aquifer 𝛻∙𝑇𝛻ℎ= 1 𝑟 𝜕 𝜕𝑟 𝑇 𝑟 𝑟 𝜕ℎ 𝜕𝑟 + 1 𝑟 2 𝜕 𝜕𝜃 𝑇 𝜃 𝜕ℎ 𝜕𝜃 =𝑆 𝜕ℎ 𝜕𝑡 General equation, radial coordinates 𝛻 2 ℎ= 1 𝑟 𝜕 𝜕𝑟 𝑟 𝜕ℎ 𝜕𝑟 + 1 𝑟 2 𝜕 2 ℎ 𝜕 𝜃 2 =0 Steady state flow, homogeneous and Isotropic aquifer

Steady Flow to a Well in a Confined Aquifer 2rw Ground surface Bedrock Confined aquifer Q h0 Pre-pumping head Confining Layer b r1 r2 h2 h1 hw Observation wells Drawdown curve Pumping well Governing equation in radial coordinates 𝛻 2 ℎ= 1 𝑟 𝜕 𝜕𝑟 𝑟 𝜕ℎ 𝜕𝑟 + 1 𝑟 2 𝜕 2 ℎ 𝜕 𝜃 2 = 𝑆 𝑇 𝜕ℎ 𝜕𝑡 Homogeneous and Isotropic aquifer 𝑑 𝑑𝑟 𝑟 𝑑ℎ 𝑑𝑟 =0 Steady state flow

Steady Flow to a Well in a Confined Aquifer 𝑑 𝑑𝑟 𝑟 𝑑ℎ 𝑑𝑟 =0 2rw Ground surface Bedrock Confined aquifer Q h0 Pre-pumping head Confining Layer b r1 r2 h2 h1 hw Observation wells Drawdown curve Pumping well 𝑟 𝑑ℎ 𝑑𝑟 = 𝐶 1 Darcy’s law 𝑄=𝐴𝑞= 2𝜋𝑟𝑏 𝐾 𝑑ℎ 𝑑𝑟 𝑟 𝑑ℎ 𝑑𝑟 = 𝑄 2𝜋𝑏𝐾 =𝐶 1 𝑑ℎ= 𝑄 2𝜋𝑇 𝑑𝑟 𝑟 ℎ= 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 + 𝐶 2

Steady Flow to a Well in a Confined Aquifer ℎ= 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 + 𝐶 2 2rw Ground surface Bedrock Confined aquifer Q h0 Pre-pumping head Confining Layer b r1 r2 h2 h1 hw Observation wells Drawdown curve Pumping well Boundary condition: ℎ= ℎ 1 @ 𝑟= 𝑟 1 ℎ 1 = 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 1 + 𝐶 2 𝐶 2 =ℎ 1 − 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 1 ℎ= 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 + ℎ 1 − 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 1 ℎ= ℎ 1 + 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 𝑟 1

Example - Steady Flow to a Well in a Confined Aquifer Q = 400 m3/hr b = 40 m Two observation wells, h1 = 85.3 m (@ r1 = 25 m) h2 = 89.6 m (@ r2 = 75 m) Find: Transmissivity (T) 2rw Ground surface Bedrock Confined aquifer Q h0 Confining Layer b r1 r2 h2 h1 hw Pumping well ℎ= ℎ 1 + 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 𝑟 1 ℎ 2 = ℎ 1 + 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 2 𝑟 1

Drawdown, s ℎ= ℎ 0 @ 𝑟= 𝑟 0 ℎ= ℎ 0 + 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 𝑟 0 𝑠= ℎ 0 −ℎ ℎ= ℎ 0 @ 𝑟= 𝑟 0 ℎ= ℎ 0 + 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 𝑟 0 𝑠= ℎ 0 −ℎ 𝑠= ℎ 0 − ℎ 0 + 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 𝑟 0 𝑠(𝑟)= 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 0 𝑟

Example - Steady Flow to a Well in a Confined Aquifer 1-m diameter well Q = 113 m3/hr b = 30 m h0= 40 m Two observation wells, h1 = 38.2 m (@ r1 = 15 m) h2 = 39.5 m (@ r2 = 50 m) Find: Head and drawdown at the well 2rw Ground surface Bedrock Confined aquifer Q h0 Confining Layer b r1 r2 h2 h1 hw Pumping well Drawdown ℎ= ℎ 1 + 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 𝑟 1 𝑠 1 = ℎ 0 − ℎ 1 𝑠 2 = ℎ 0 − ℎ 2 𝑠 1 − 𝑠 2 = ℎ 2 − ℎ 1 = 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 2 𝑟 1 𝑇= 𝑄 2𝜋 𝑠 1 − 𝑠 2 𝑙𝑛 𝑟 2 𝑟 1 = 113 𝑚 3 /ℎ𝑟 2𝜋 1.8−0.5 𝑚 𝑙𝑛 50 15 =16.66 𝑚 2 /ℎ𝑟

Example - Steady Flow to a Well in a Confined Aquifer 2rw Ground surface Bedrock Confined aquifer Q h0 Confining Layer b r1 r2 h2 h1 hw Drawdown @ well ℎ= ℎ 1 + 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 𝑟 1 ℎ 𝑤 = ℎ 1 + 𝑄 2𝜋𝑇 𝑙𝑛 𝑟 𝑤 𝑟 1 ℎ 𝑤 =38.2 𝑚+ 113 𝑚 3 /ℎ𝑟 2𝜋 16.66 𝑚 2 /ℎ𝑟 𝑙𝑛 0.5 15 ℎ 𝑤 =34.5 𝑚 Head at the well 𝑠 𝑤 = ℎ 0 −ℎ=40−34.5 𝑚=5.5 𝑚 Drawdown at the well

Flow to a Partially Penetrating Well in a Confined Aquifer

Summary Some Information on Wells Steady flow to a well in a confined aquifer