Transient Models See Anderson and Woessner Chapter 7

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Transient ModelsSee Anderson and Woessner
Chapter 7
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Objectives

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

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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.

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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.

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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

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Grid Design

• Discretize
• Space (plan view and cross section)
• Time
• Difficult Task
• Redesign is a major undertaking

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DISCRETIZED HYPOTHETICAL AQUIFER
Layers may correspond to horizontal geohydrologic
intervals

• ---- Aquifer boundary
• ? Active cell
• ? Inactive cell
• ?rj Width of cell in row direction (j indicates
  column number)
• ?ci Width of cell in column direction (i
  indicates row number)
• ?vk Thickness of the cell
• ?rj?ci?vk Volume of cell with coordinates (i,j,k)

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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

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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

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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
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Information for transient simulations

• Storage Properties
• Initial Conditions
• Boundary Conditions
• Discritizing Time

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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.

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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)

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Storage Properties

• Specific storage (Ss),
• Ss ?(ßp n ßw)
• where
• ? is the specific weight of water (Nm-3 or
  ML-2T-2)
• n is the porosity of the material (dimensionless
  ratio between 0 and 1)
• ßp is the compressibility of the bulk aquifer
  material, and
• ßw 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).

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Storage Properties

• Storativity (S)
• S Ssb
• 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.

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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

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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

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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

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Compaction and head decline
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Inelastic compaction
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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

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Initial Conditions
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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.

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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.

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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.

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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

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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.
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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.
  

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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

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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, )

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Time Discretization
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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