1. Transient Models See Anderson and Woessner Chapter 7

2. Objectives
BECOME FAMILIAR with ASPECTS OF NUMERICAL MODELING that are UNIQUE to TRANSIENT PROBLEMS
UNDERSTAND the need for and significance of INITIAL CONDITIONS

3. Transient Models Provide insight into the rate of change in a system
This is of value because we may only be interested in the temporary application of a stress to the ground-water system.
For example, the life of a mine may be 50 years and the response of the system may be slow enough that we do not even begin to approach steady state during that time frame.

4. Transient Models
The steady state solution provides the maximum impact of the stress.
The impacts during the transient period while the system is approaching steady state can only be less than those that prevail under steady state conditions.
Some problems do not have a steady state result.
For example if a basin is pumped at a rate greater than the recharge, eventually the basin will go dry and the pumping cannot continue. A balanced steady state condition cannot be reached and so a steady state solution for pumping the basin at that rate does not exist.

5. Grid Design
Numerical model needs to be divided into pieces of space and time for which the solution can be linearized and the properties and results averaged
Compromise between accuracy, cost, and effort
Smaller pieces are more accurate, but require more time and effort

6. Grid Design Discretize: Space (plan view and cross section) Time
Difficult Task
Redesign is a major undertaking

7. DISCRETIZED HYPOTHETICAL AQUIFER
Layers may correspond to horizontal geohydrologic intervals
Aquifer boundary
● Active cell
○ Inactive cell
Δrj Width of cell in row direction (j indicates column number)
Δci Width of cell in column direction (i indicates row number)
Δvk Thickness of the cell
ΔrjΔciΔvk Volume of cell with coordinates (i,j,k)

8. Spatial Dimension 2D areal 2D profile (special class)
Quasi 3D (confining layers by leakage)
Fully 3D
Aquifer viewpoint: 2D areal and quasi 3D
Flow system viewpoint: 2D profile and 3D

9. Fully 3D Models
Simulate confined and unconfined aquifers when vertical head gradients are important
Represent transient release of water from storage in confining beds by including confining bed as a layer with storage properties
Parameter arrays specified for each layer of the model

10. Layer Considerations – Purpose of the Model
Confining Unit Storage
No layers
No storage
Leakage
Multiple layers
Water in storage
Long travel times for pressure gradient
Future Transport Modeling
All of above issues
Travel time requires multiple layers
No cells for confining unit:
Multiple layers for confining unit:

11. Information for transient simulations
Storage Properties
Initial Conditions
Boundary Conditions
Discritizing Time

12. Storage Properties Don't need storage properties in steady state
Need storage properties when water released/added storage
Confined aquifers - Specific storage
Unconfined aquifers - Specific yield
Note: In 2-D or quasi 3-D, you don't consider the storage of the confining layer. Instead, you must specify the leakage rate.

13. Storage Properties
Material physical properties that characterize the capacity of an aquifer to release groundwater from storage in response to a decline in hydraulic head
Specific storage (Ss),
Storativity (S; S = Ssb),
Specific yield (Sy), and
Specific capacity (Sc)

14. Storage Properties Specific storage (Ss), Ss = γ(βp + n* βw)
where
γ is the specific weight of water (N•m-3 or [ML-2T-2])
n is the porosity of the material (dimensionless ratio between 0 and 1)
βp is the compressibility of the bulk aquifer material, and
βw is the compressibility of water (m2N-1 or [LM-1T2])
Relates a change in total or water volume per change in applied stress (effective stress) per unit volume. The compressibilities (and therefore also Ss) can be estimated from laboratory consolidation tests (in an apparatus called a consolidometer), using the consolidation theory of soil mechanics (developed by Karl Terzaghi).

15. Storage Properties Storativity (S) S = Ss*b
where
Ss is the specific storage of water
b is the thickness of the aquifer
Volume of water released from storage per unit decline in hydraulic head in the aquifer, per unit area of the aquifer
Dimensionless
Water is released from
Confined Aquifers
Aquifer compaction
Expansion of water
Unconfined Aquifers
Drainable porosity (specific yield)
Ranges between 0 and the effective porosity of the aquifer; although for confined aquifers, this number is usually much less than 0.01.

16. Storage Properties Specific yield (Sy) Sy = Vwd / VT
where
Vwd is the volume of water drained, and
VT is the total rock or material volume
drainable porosity,
ratio, less than or equal to the effective porosity
primarily used for unconfined aquifers,
since the elastic storage component, Ss, is relatively small and usually has an insignificant contribution

17. Storage Properties Specific capacity (Sc) Sc = Q/ (h0 − h )
where
Sc is the specific capacity ([L2T−1]; m²/day or USgal/day/ft)
Q is the pumping rate ([L3T−1]; m³/day or USgal/day), and
h0 − h is the drawdown ([L]; m or ft)
quantity that which a water well can produce per unit of drawdown

18. Storage Properties in Simulations
Specific yield (Sy)
Specific storage (Ss)
Elastic
Inelastic
First two components Sy and elastic specific storage are reversible
Inelastic specific storage
compaction of the fine-grained deposits or permanent reduction of pore space – land subsidence
inelastic specific storage are much larger than those of elastic specific storage

19. Compaction and head decline

20. Inelastic compaction

21. Initial Conditions
Some boundary conditions may be time dependent, h(x,y,z,t)
(a) Static steady state
Head is constant in space and time
(b) Dynamic average steady state
Head is constant in time
Head is not constant in space
(c) Dynamic cyclic
Head varies in space and time
Must calibrate to a hydrograph

22. Initial Conditions

23. Initial Conditions Transient analytical solutions
use relatively simple hydrostatic conditions,
often yield solutions in terms of drawdown, and
use superposition to apply the results to alternative initial conditions if the problem was linear
If the solution is expressed in terms of head rather than drawdown, then the initial heads must be defined.
Numerical modeling is conducted in terms of head and allows us to define complex initial conditions.

24. Initial Conditions Points to consider:
(1) material properties and boundary conditions must be consistent with the initial heads
If you start with initial heads that are contoured from field measurements and you do not apply a stress to the system, the heads will adjust to the properties and boundaries, so you are inadvertently introducing a stress by defining inconsistent values for starting conditions. The most common way to deal with this problem is to calculate a pre-stress steady state solution for use as initial heads.

25. Initial Conditions Points to consider:
(2) If the field system being simulated is not in equilibrium, an earlier equilibrium condition can be identified and defined as a starting point.
All subsequent stresses must be simulated from the time when equilibrium prevailed until time when the initial conditions are needed is reached, then the early stresses must continue along with the new stress of interest if the early stresses continue in the field.

26. Initial Conditions Points to consider:
(3) If we cannot use a steady state initial condition because our problem is dependent on a short term response during a particular time of year, you may be able to start with a rather arbitrary initial condition but simulate the cycle long enough such that you simulate the same values at the same times in subsequent cycles

27. Initial Conditions Points to consider:
(4) If you do not have sufficient data to establish an acceptable steady initial condition to commence our cyclic equilibrium, then you may be able to start with a rather arbitrary initial condition but simulate the cycle long enough such that we simulate the same values at the same times in subsequent cycles.

28. Initial Conditions Points to consider:
(5) It may be that there is enough information in your transient data to estimate initial conditions. (6) It is useful to note that the further, in time, the simulation is from the initial conditions the less influence those initial conditions have on the simulated values.

29. Boundary Conditions
A specific boundary condition determines a dynamic average steady-state calibration which forms the initial condition for the transient case
Make sure transient stresses aren't influenced by boundaries
Checking for change in flow rates across specified heads
Checking heads along boundary for specified flux
You can switch between spec. head and spec. flux to evaluate effects

30. Discritizing Time TIME STEPS: temporal equivalent of grid cells
Small when stresses change and increase in length to a constant, convenient size until the stresses change
STRESS PERIODS: groups of time steps during which stresses do not change
Temporal data compiled at these increments (ie pumping, recharge, …)

31. Time Discretization

32. Time Discretization Considerations
Difficult to decide on initial time step size
MODFLOW requires the time period, number of steps and a multiplier to gradually increase steps
Multiplier is typically 1.1 to 1.5