Groundwater Systems D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 Water Resources System Modeling

Objectives D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 2  To briefly learn about groundwater hydrology  To discuss about the governing equations of groundwater systems  To discuss simulation and optimization of groundwater systems

Introduction D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 3 Groundwater management: Deals with planning, implementation, development and operation of water resources containing groundwater Numerical-simulation models have been used extensively for understanding the flow characteristics of aquifers and evaluate the groundwater resource Simulation models attempt various scenarios to find the best objective Optimization models directly consider the management objective taking care of all the constraints

Groundwater Hydrology D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 4 Subsurface water is stored underneath in subsurface formations called aquifers Unconfined aquifer: Upper surface is water table itself Confined aquifer is confined under pressure greater than the atmospheric Confined aquifer may be confined between two impermeable layers An aquifer serves two functions: Storage and Transmission. Storage function: Exhibited through porosity Φ, specific yield S y and storage coefficient S. Transmission function: Exhibited through the permeability property (coefficient of permeability K).

Groundwater Hydrology… D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 5 Porosity: Measure of the amount of water an aquifer can hold Specific yield: Water drained from a saturated sample of unit volume Specific retention: Water retained in the unit volume Porosity is the sum of specific yield and specific retention Storage coefficient: Volume of water an aquifer releases or stores per unit surface area per unit decline of head Permeability: Measure of the ease of movement of water through aquifers Coefficient of permeability or hydraulic conductivity: Rate of flow of water through a unit cross-sectional area under a unit hydraulic gradient

Groundwater Hydrology… D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 6 Transmissivity T: Rate of flow of water through a vertical strip of unit width extending the saturated thickness of the aquifer under a unit hydraulic gradient T = K b for a confined aquifer where b is the saturated thickness of aquifer T = K h for a confined aquifer where h is the head (saturated thickness) Darcy’s law Flow thorough an aquifer is expressed by Darcy’s law Can be expressed as (1) where v is the velocity or specific discharge, l is the length of flow along the average direction and is the rate of headloss per unit length. Then, the total discharge, q is (2) Darcy’s law states that flow rate through a porous media is proportional to the head loss and inversely proportional to the length of flow path

Simulation of Groundwater Systems D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 7 Governing equations: Darcy’s law in terms of transmissivity is for confined aquifers (3) for unconfined aquifers (4) Considering a two-dimensional horizontal flow as shown by a rectangular control volume element. q2q2 q4q4 q1q1 q3q3 ΔyΔy ΔxΔx i i+1 i - 1 j - 1 j j +1

Simulation of Groundwater Systems… D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 8 Governing equations: General equations of flow can be expressed as: The flow discharge q = Av for four sides (5) (6) (7) (8) where A = Δx. h for unconfined case or A = Δx. b for confined case and assuming constant transmissivities along the x and y directions. are the hydraulic gradients at sides 1,2,… respectively. q2q2 q4q4 q1q1 q3q3 ΔyΔy ΔxΔx i i+1 i - 1 j - 1 j j +1

Simulation of Groundwater Systems… D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 9 Governing equations: The rate at which water is stored or released in the element is (9) where S is the storage coefficient of the element. The flow rate for constant net withdrawal or recharge for time Δt (10) Applying continuity law, (11) Substituting eqns. 5 – 10 in above eqn, and dividing by Δx Δy and simplifying, the final form of eqn. 11 will be (12) where W = q / Δx Δy.

Simulation of Groundwater Systems… D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 10 Governing equations: The above equations can be written in finite difference form and solved for each rectangular element. Partial derivatives in eqns. 5-9 can be expressed in finite difference form as, (13) These can be substituted in eqn. 12 and solved using finite element methods.

Optimization Model D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 11 Optimization models for hydraulic management for groundwater have been developed based on three approaches: Embedding approach: Equations from a simulation model are incorporated into an optimization model directly Optimal control approach: Simulation model solves the governing equations, for each iteration of the optimization. It works on the concept of optimal control theory Unit response matrix approach: A unit response matrix is generated by running the simulation model several times with unit pumpage at a single node. Total drawdowns are then determined by superpositions. In this lecture, we will discuss only about the embedding approach for steady-state one-dimensional confined and unconfined aquifers.

Embedding approach: Steady state one-dimensional confined aquifer D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 12 Consider a confined aquifer with penetrating wells and flow in one-dimension From eqn. 12, the governing equation is where (14) q1q1 q2q2 q3q3 ▼ ▼ ΔxΔx ΔxΔx ΔxΔx ΔxΔx h0h0 h4h4

Embedding approach: Steady state one-dimensional confined aquifer… D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 13 Eqn. (14) can be expressed in finite difference form as (using central difference scheme) (15) Optimization problem can be stated as Maximize (16) where i is the number of wells. Subject to (17) where W min is the minimum production rate for each well. The pumpage can be finally determined from the relation q i = W.Δ x i 2.

Example: Steady state one-dimensional confined aquifer D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 14 Formulate an LP model for the above confined aquifer for maximum heads. The boundaries have a constant head h 0 and h 5. The distance between the wells is Δ x. Solution: Objective function: Maximize Subject to: Acc. to eqn. 15 And acc. to eqns. 17 The unknowns are h 1, h 2, h 3, h 4 and W 1, W 2, W 3, W 4.

Embedding approach: Steady state one-dimensional unconfined aquifer D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 15 Governing equation can be written as (18) where T = Kh. Substituting w=h 2 and assuming K is constant, the finite difference form can be written as (19)

Embedding approach: Steady state one-dimensional unconfined aquifer… D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 16 Optimization problem is to Maximize (20) where i is the number of wells. Subject to (21) After solving the above problem, the heads

Example: Steady state one-dimensional unconfined aquifer D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 17 Formulate a LP model to determine the steady state pumpages of an unconfined aquifer shown below Solution Optimization problem is Objective function: Maximize q1q1 q2q2 q3q3 ▼ ▼ h0h0 h4h4 Δx

Example: Steady state one-dimensional unconfined aquifer… D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 18 Subject to: Acc. to eqn. 19 And acc. to eqns. 21 The unknowns are w 1, w 2, w 3, w 4 and W 1, W 2, W 3, W 4.

D Nagesh Kumar, IISc Water Resources Planning and Management: M8L3 Thank You