Theory of Groundwater Flow Topics Differential Equations of Groundwater Flow Boundary conditions Initial Conditions for groundwater problems FlowNet analysis Mathematical analysis of some simple flow problems

5.1. Differential Equations Examples of useful use of flow equations in solving Hydro Problems WL drop around a well after 10 years of pumping Contaminant concentration changes after 5 years of remediation (cleanup) Change in storage of aquifer after use of 50 years

Mathematical approach Represent the GROUNDWATER process by an equation Solving the equation Result is hydraulic head (space, time)

How is it done? An illustrative example: A. A geological problem Silty Sand Sand Shale

How is it done? An illustrative example: B. Conceptualized mathematical problem Water Table: variable head boundary K = 1 Side No-flow boundary Side No-flow boundary K = 10 Top of shale = base of aquifer = NO-Flow boundary

How is it done? An illustrative example: C. Calculating the hydraulic head distribution (governing equations) h= 90 76 K = 1 86 88 84 82 78 80 K = 1 K = 10 K = 10

Deriving Groundwater flow Equations principle of mass conservation Darcy's Law GW Flow equations

Deriving Groundwater flow Equations Representative Elementary Volume (REV)  z  y  x Mass inflow rate - mass outflow rate = change in storage with time

The main equation of groundwater flow This is a linear parabolic partial differential equation It’s the main equation of groundwater flow in saturated media It is solvable only by numerical methods the solution of which yields h (x,y,z,t) in a heterogeneous, anisotropic confined aquifer. Also known as the Diffusion Equation

Simplifications of the equation (1) for homogeneous but anisotropic aquifer: (2) for homogeneous and isotropic (3) for horizontal flow (4) steady state flow

Laplace equation one of the most useful field equations employed in hydrogeology. The solution to this equation describes the value of the hydraulic head at any point in a 3-dimensional flow field Note: the mapped potentiometric surface represents "solution" to Laplace's equation for 2-dimensional flow field

5.2 Boundary conditions 3 types: Dirichlet Boundary Condition Specified head at a boundary 2. Neumann Boundary Condition Specified water flux at a boundary 3. Cauchy boundary condition Relates hydraulic head to water flux

5.3 Initial Conditions For steady state equations Only boundary conditions are needed For transient equations: Boundary and initial conditions are needed Initial condition: Provides hydraulic head everywhere within the domain of interest before simulation begins

5.4 Flownet Analysis

Flownets, general features in a 2-D flow domain 1. Streamlines are perpendicular to equipotential lines. If the hydraulic-head drops between the equip. lines are the same, the streamlines and equip. lines form curvilinear squares. 2. The same quantity of ground water flows between adjacent pairs of flow lines, provided no flow enters or leaves the region in the internal part of the net. It follows, then that the number of flow channels (known as stream tubes) must remain constant throughout the net. 3. The hydraulic-head drop between two adjacent equipotential lines is the same.

Flownets, strategies for construction Study well-constructed flownets and try to duplicate them by independently reanalyzing the problems they represent In a first attempt, use only four or five flow channels. Observe the appearance of the entire flownet; do not try to adjust details until the entire net is approximately correct. Be aware that frequently parts of a flownet consist of straight and parallel lines, result in uniformly sized squares. In a flow system that has symmetry, only a section of the net needs to be constructed because the other parts are images of that section. During the sketching of the net, keep in mind that the size of the rectangle changes gradually; all transitions are smooth, and where the paths are curved they are of elliptical or parabolic shape.

Flownets, Rules A no-flow boundary is a streamline The water table is a streamline when there is no flow across the water table, that is, no recharge or ET. When there is recharge, the water table is neither a flow line nor an equipotential line. Streamlines end at extraction wells, drains, and gaining streams, and they start from injection wells and losing streams. Lines dividing a flow system into two symmetric parts are streamlines. In natural ground-water systems, streamlines often begin and end at the water table in areas of ground-water recharge and discharge, respectively.

5.4 Flownet Analysis If we have squares: From Darcy’s Eqn in one flow channel in 2-D: If we have squares:

Example 5.3 Q = 1 x 106 ft3/day Find T?

Flow Net Exercise Draw a flow net for seepage through the earthen dam shown below. If the hydraulic conductivity of the material used in the dam is 0.22 ft/day, what is the seepage per unit width per day?

Flownets in Heterogeneous Media For same hydraulic drop, h1 = h2

Example 5.4

5.5 Flow Equations of Simple Problems Analytical solution only applied to Regular geometry Homogeneous Simple initial and boundary conditions Real-world problems can be solved with numerical methods using computers to handle: Variation in hydraulic properties Large number of wells Complicated boundary conditions Groundwater/ surface water interactions Variable recharge/ET

Groundwater flow in a confined aquifer 1-D, steady state, homogeneous:

Example 5.5

Groundwater flow in a unconfined aquifer Example 5.6

Chapter Highlights Ground-water hydrologists rely on quantitative mathematical approaches in analyzing test data and in making predictions about how systems are likely to behave in the future. The mathematical approach involves representing the flow process by an equation and solving it. The solution within some domain or region of interest defines how the hydraulic head varies as a function of space and time. The flow equations are complicated partial differential equations. Fortunately, at this introductory level, all one really needs to do is to identify the equation and extract a few details. In most applications, the solutions are available in simplified forms. To find the unknown in an equation, simply find the variable residing in the derivative term. Equations of ground-water flow can be developed, starting with an appropriate conservation statement of this form mass inflow rate, -mass outflow rate = change of mass storage with time The general approach is to apply this equation to a block of porous medium called a representative elementary volume. It is possible to replace the words in this equation by mathematical expressions, transforming it to a form that can be developed to the main equation of ground-water flow in a porous medium

Chapter Highlights The solution of differential equations requires boundary conditions. In effect, the boundary conditions stand in for the conditions outside of the simulation domain and effectively let one concentrate the modeling on the simulation domain. There are three types of common boundary conditions. The first type or Dirichlet condition involves providing known values of hydraulic head along the boundary. The second type or Neumann condition requires specification of water fluxes along a boundary. A no-flow boundary (water flux zero) is the most well-known second-type boundary condition. The third-type or Cauchy boundary condition relates hydraulic head to water flux. This boundary condition is commonly used to represent ground-water/surface-water interactions. For transient equations, in which the hydraulic head can change as a function of time, it is necessary to define the initial condition. The initial condition provides the hydraulic head everywhere within the domain of interest before the simulation begins (that is, at time zero).

Chapter Highlights A variety of mathematical and graphical approaches are available to solve ground-water- flow equations. One approach that is emphasized in this chapter is called the flownet analysis. For relatively simple, two-dimensional, steady-state flow problems, you can determine the distribution of equipotential lines graphically. Starting with an outline of the simulation domain, one adds streamlines and equipotential lines following a set of rules. For example, streamlines and equipotential lines must intersect at right angles to form a set of curvilinear squares. If you are careful, you can develop the unique pattern (and reproducible pattern) that describes flow in the domain. This chapter demonstrates how simple analytical solutions can be used to describe some simple steady-state problems of flow. We will return to analytical solutions again in Chapters 8-11 on well hydraulics and regional ground-water flow.