UNIT - II Well Hydraulics Darcy’s law – Groundwater Flow Equation .pptx

Governing Equations for Well Hydraulics:
1. Theis Equation (Unsteady-State Flow in Confined Aquifers)
Definition: A solution for transient flow to a well in a confined aquifer, often referred to as the "well function."
Equation:
𝑠
=
𝑄
4
𝜋
𝑇
⋅
𝑊
(
𝑢
)
s=
4πT
Q
​
⋅W(u)
where:
𝑠
s = drawdown (m)
𝑄
Q = pumping rate (m³/s)
𝑇
T = transmissivity (m²/s)
𝑊
(
𝑢
)
W(u) = Theis well function
𝑢
=
𝑟
2
𝑆
4
𝑇
𝑡
u=
4Tt
r
2
S
​

𝑟
r = radial distance from the well (m)
𝑆
S = storage coefficient (dimensionless)
𝑡
t = time since pumping started (s)
Significance: Used to predict how drawdown changes over time and distance from the well in confined aquifers.
2. Thiem Equation (Steady-State Flow in Confined Aquifers)
Definition: Used to calculate steady-state drawdown in a confined aquifer.

Equation:

𝑠
=
𝑄
2
𝜋
𝑇
ln
⁡
(
𝑟
2
𝑟
1
)
s=
2πT
Q
​
ln(
r
1
​

r
2
​

​
)
where:

𝑠
s = drawdown (m)
𝑄
Q = pumping rate (m³/s)
𝑇
T = transmissivity (m²/s)
𝑟
1
r
1
​
= distance from the well where the drawdown is observed (m)
𝑟
2
r
2
​
= distance from the well where no drawdown is observed (m)
Significance: Helps in determining the transmissivity and well efficiency in confined aquifers.

3. Dupuit’s Equation (Steady-State Flow in Unconfined Aquifers)
Definition: Describes radial flow to a well in an unconfined aquifer under steady-state conditions.

Equation:

𝑄
=
𝜋
𝐾
(
ℎ
1
2
−
ℎ
2
2
)
ln
⁡
(
𝑟
2
/
𝑟
1
)
Q=
ln(r
2
​
/r
1
​
)
πK(h
1
2
​
−h
2
2
​
)
​

where:

𝑄
Q = pumping rate (m³/s)
𝐾
K = hydraulic conductivity (m/s)
ℎ
1
h
1
​
,
ℎ
2
h
2
​
= water table elevations at distances
𝑟
1
r
1
​
and
𝑟
2
r
2
​
(m)
Significance: Used to estimate drawdown in unconfined aquifers, accounting for changing saturated thickness.

Well Performance and Efficiency:
1. Specific Capacity (C)
Definition: The rate of water pumped per unit of drawdown.
Formula:
𝐶
=
𝑄
𝑠
C=
s
Q
​

Significance: Often used as a measure of well performance. Higher specific capacity means a more efficient well.
2. Well Losses
Definition: Drawdown caused by factors other than aquifer properties, such as well construction or turbulent flow near the well.
Types:
Aquifer Loss: Due to hydraulic resistance of the aquifer.
Well Loss: Due to inefficiencies near the well, like skin effect or improper well design.
3. Well Efficiency
Definition: The ratio of theoretical drawdown (based on aquifer properties) to the actual observed drawdown.
Formula:
Efficiency
=
Theoretical Drawdown
Observed Drawdown
×
100
Efficiency=
Observed Drawdown
Theoretical Drawdown
​
×100
Significance: High well efficiency means minimal energy loss and more water for the effort expended in pumping.

1. KONGUNADU COLLEGE OF ENGINEERING AND TECHNOLOGY (Autonomous) Department of Agriculture Engineering Course: Groundwater and Well Engineering Topic : WELL HYDRAULICS By Mr. M.Sundharan , Assistant Professor, Department of Agricultural Engineering, Kongunadu College of Engineering and Technology

2. DARCY’S LAW In fluid dyanamics and hydrogeology, Darcy’s law is a phenominal derived constitutive equation that describes the flow of a fluid through the porous medium. The law was formulated by Henry Darcy in 1856 on the flow of water through beds of sand. (Filtration)

3. Principle: Darcy’s law states the principle which governs the movement of fluid in the given substance. Darcy’s law equation describes the capability of the liquid to flow via any porous media like a rock. The law is based on the fact according to, the flow between two points is directly proportional to the pressure differences between the points, the distance, and the connectivity of flow within rocks between the points. Measuring the inter-connectivity is known as permeability.

4. Objectives Able to identify the direction of flow Able to measure the amount of water discharge through permeability

5. Darcy’s law equation

6. Flow of Fluids

7. Aquifers

8. Groundwater Flow Equation Important Symbols Used in Groundwater Flow b - Aquifer thickness h - Piozometric head K - Hydraulic Conductivity S - Storage co-efficient S s - Specific stoarage T - Transmissibility t - Time X, Y, Z - Co-ordinate directions

9. Flow Concepts A. Homogeneous Rocks B. Heterogenous Rocks i) Anisotropic Aquifer ii) Isotropic Aquifer 1) Confined aquifer 2) Un-confined aquifer

10. Steady Groundwater Flow Steady Flow in “Confined and Unconfined Aquifer” Under the conditions of a steady flow in both the “Confined and Unconfined Aquifer”, the level of groundwater remain steady or constant over the time. It needs to be highlighted that when the rate of discharge of groundwater is equal to the rate of recharge of groundwater, then there can be observed a steady flow in both confined and unconfined aquifers.

11. Steady Flow Equation

12. Dupuit Forcheimer

13. Unsteady Groundwater Flow In both the confined and unconfined aquifer, an unsteady flow refers to the condition where the quantity of groundwater flowing per second does not remain always constant. Furthermore, from various researches, it can be observed that an unsteady flow is considered as the “transient phenomenon”. As time passes the unsteady flow in both the confined and unconfined aquifer can become zero or steady flow. On the other hand, in this type of flow, the rate of recharge of groundwater is not equal to the rate of groundwater discharge.

14. Theis Method

15. Jacob method

16. Image Well Theory "Image well theory" in groundwater refers to a conceptual approach used in hydrogeology to understand and model the behavior of wells and groundwater flow systems, particularly in the vicinity of boundaries and interfaces. The image well theory is based on the concept of mirroring or "image" wells that are imagined to exist across boundaries or interfaces. This theory is often used in situations where the actual hydrogeological conditions may not be straightforward due to the presence of geological complexities.

17. Partial Penetration of wells Partial penetration of wells refers to a situation where the screened or perforated section of a wellbore does not extend through the entire thickness of the aquifer. Instead, the well penetrates only a portion of the aquifer's thickness.