III. Ground-Water Management Problem Used for the Exercises

Ground-Water Flow System The flow system is comprised of two confined aquifers separated

by a confining unit.

Inflow occurs primarily as areal recharge; there is also a very small

amount of inflow across the boundary with the hillside.

At steady-state, outflow occurs only as discharge to the river.

Management Issues

Ground-water management issues related to this flow system:

Pumping wells for water supply are being completed in aquifers 1 and

2. The effect of pumping on the river is of concern, because there is a

minimum required discharge from the ground-water system to the river.

A landfill is proposed in one corner of the study area.

The landfill developers claim

(a) the landfill is outside the capture zone of the wells and

(b) any leaking effluent will reach the river sufficiently diluted.

We are developing a flow model to help evaluate these claims.

Will effluent go to well or river?

?

?

Predicted transport from landfill

Ground-Water Flow Model

First, steady-state model without pumping will be developed and

calibrated using available measurements of hydraulic-heads and discharge

to the river, and will be used for preliminary evaluation of the issues.

Then, pumping wells will be installed, a long-term aquifer test will be

conducted using the wells, and a transient model of the system will be

recalibrated using the steady-state measurements as well as the additional

drawdown and river discharge data collected during the test.

See flow model setup in Figure 2-1b (page 32) of Hill and Tiedeman.

See model parameter definition and starting values in Table 3-1 (page 38).

See true simulated conditions in Figures 2-1c and 2-1d (page 22-23).

MODFLOW Packages Used

Basic (BAS) and the Discretization input file Define model grid, including rows, columns, and layers Define types of model layer (confined; convertible)

Layer Property Flow (LPF) Define hydraulic properties – K, Ss, Sy, Anisotropy

Recharge (RCH) Define distribution of areal recharge

River (RIV) General-Head Boundary (GHB) Well (WEL)

Define pumpage Used here for prediction runs only

Spatial Discretization -- Ground-Water Flow Model Grid

Figure 2-1b of Hill and Tiedeman (page 22)

Layer-Property Flow Package

Figure 2-1a of Hill and Tiedeman (page 34)

Define hydraulic properties – K, Ss, Sy, Anisotropy

Model layer 1 is homogeneousModel layer 2 has K that increases linearly from beneath the river to the hillside

Recharge Package

Can apply recharge to model layer 1 or top active cell at each row, column location

Input a flow rate (L/T). The program multiplies by area to get volume per time.

Layer data. Parameters are used to define the recharge rate. For this problem, use 2 zones; no multiplication arrays.

GHB

DRN RIV

Positive qn

indicates flow into

the subsurface

Negative qn

indicates flow out of

the subsurface Hn

qn = 0

Slope = -Cn = -(KnAn)/Dn

Positive qn

indicates flow into

the subsurface

Negative qn

indicates flow out of

the subsurface

En Hn

qn = 0

Slope = -Cn = -(KnAn)/Dn

Slope = -Cn = -(KnAn)/Dn

Hn

Negative qn

indicates flow out of

the subsurface

qn

qn

qn

hn

hn

hn

(C)

(A)

(B)

EXPLANATION

qn the simulated flow rate at one cell (L3/T)

(negative for flow out of the ground-water system)

Kn the hydraulic conductivity (L/T) of, for

example, the riverbed or lakebed

Dn the thickness (L) of, for example, the riverbed

or lakebed

An the area of the water body within the finite-

difference cell (L2)

Cn the conductance calculated using Kn, Dn, and An.

hn is the simulated hydraulic head in the ground-

water system adjacent to the head-dependent boundary (L); and

Hn is the water level in the water body or the

elevation of the drain (L)

En is the bottom of the streambed

(C)

qn = 0

Positive qn

indicates flow into

the subsurface

Negative qn

indicates flow out of

the subsurface Hn

qn = 0

Slope = -Cn = -(KnAn)/Dn

Positive qn

indicates flow into

the subsurface

Negative qn

indicates flow out of

the subsurface

En Hn

qn = 0

Slope = -Cn = -(KnAn)/Dn

Slope = -Cn = -(KnAn)/Dn

H

Positive qn

indicates flow into

the subsurface

Negative qn

indicates flow out of

the subsurface Hn

qn = 0

Slope = -Cn = -(KnAn)/Dn

Positive qn

indicates flow into

the subsurface

Negative qn

indicates flow out of

the subsurface

En Hn

qn = 0

Slope = -Cn = -(KnAn)/Dn

Slope = -Cn = -(KnAn)/Dn

Hn

Negative qn

indicates flow out of

the subsurface

qn

qn

qn

hn

hn

hn

(C)

(A)

(B)

EXPLANATION

qn the simulated flow rate at one cell (L3/T)

(negative for flow out of the ground-water system)

Kn the hydraulic conductivity (L/T) of, for

example, the riverbed or lakebed

Dn the thickness (L) of, for example, the riverbed

or lakebed

An the area of the water body within the finite-

difference cell (L2)

Cn the conductance calculated using Kn, Dn, and An.

hn is the simulated hydraulic head in the ground-

water system adjacent to the head-dependent boundary (L); and

Hn is the water level in the water body or the

elevation of the drain (L)

En is the bottom of the streambed

(C)

qn = 0

The River (RIV) and General-Head Boundary (GHB)Packages are used

Both define head-dependent boundaries: q=C(H-h)

GHB

DRN RIV

Well Package

List data – layer, row, column, pumpage rate Input volume of discharge per time. Outflow is NEGATIVE. For this problem, no pumpage parameters need to be

defined, but you can use the parameter capability to define them if you like. [Why might you want to do that?]

Parameters and Starting Values

Table 3-1 of Hill and Tiedeman (page 38)

Flow-system property Parameter

name Starting

value

Horizontal hydraulic conductivity of layer 1, in m/s HK_1 3.0 x 10-4

Vertical hydraulic conductivity of confining bed, in m/s VK_CB 1.0 x 10-7

Horizontal hydraulic conductivity of layer 2 in columns 1 and 2, in m/s HK_2 4.0 x 10-5

Hydraulic conductivity of the riverbed, in m/s K_RB 1.2 x 10-3

Recharge in recharge zone 1, in cm/yr RCH_1 63.072

Recharge in recharge zone 2, in cm/yr RCH_2 31.536

True Simulated Conditions

Figure 2-1 c and 2d of Hill and Tiedeman (page 22-23)

Observations

10 head observations, 5 in each model layer.

Observed values shown in Table 3-2 (page 38) of Hill and Tiedeman.

1 steady-state flow observation: ground-water discharge to the river of –4.4 m3/sec.

Head Observations

Well identifier Observation name

Layer Row Col Observed head (m)

Variance of well elevation measurement

error (m2)

Variance of water-level

measurement error (m2)

Variance of the observation

(m2)

1 1.ss 1 3 1 101.804 1.00 0.0025 1.0025

2 2.ss 1 4 4 128.117 1.00 0.0025 1.0025

3 3.ss 1 10 9 156.678 1.00 0.0025 1.0025

4 4.ss 1 13 4 124.893 1.00 0.0025 1.0025

5 5.ss 1 14 6 140.961 1.00 0.0025 1.0025

6 6.ss 2 4 4 126.537 1.00 0.0025 1.0025

7 7.ss 2 10 1 101.112 1.00 0.0025 1.0025

8 8.ss 2 10 9 158.135 1.00 0.0025 1.0025

9 9.ss 2 10 18 176.374 1.00 0.0025 1.0025

10 10.ss 2 18 6 142.020 1.00 0.0025 1.0025

Table 3-2 of Hill and Tiedeman (page 38)

Weighting is accomplished using UCODE_2005

Here we break for a lecture introducing UCODE_2005

Weights on Observations

Weighted residuals are squared and summed in the objective function:

2

1))('()( bhhbS ii

nh

ii

2

1))('( bqq ii

nq

ii

2

1))('( bPP ii

npr

ii

PriorHeads Flows

Need to weight the observations for two primary reasons:

So that residuals for observations with different units (e.g. heads and

flows) can be summed in the objective function.

To account for different observations having different degrees of

measurement error. Weighting is used to reduce the influence of

observations that are less accurate, and to increase the influence of

observations that are more accurate.

Weights on Observations

Defining the weights as being proportional to the inverse of the variance of measurement error meets both these needs:

21

i

The variance of measurement error is an estimate of the uncertainty of the observation. How do we quantify this?

Need to quantify the uncertainty using:1. A range that is symmetric about the measurement used for the

observation.2. A probability with which the true value is expected to occur within the

range.

Weights on Observations: Example 1

A head observation is thought to be “good to within 3 meters”.

Quantify this as “there is a 95-percent chance the true value falls within 3

meters of the measurement.”

Use a normal probability table to determine that a 95-percent confidence

interval is a value plus and minus 1.96 times the standard deviation, .

Thus, 1.96 x = 3 m, so = 1.53 m, and the variance 2 = 2.34 m2.

Weights on Observations: Example 2

A measurement of stream loss to a ground-water system is derived by subtracting two streamflow measurements, an upstream value of 3.0 m3/s and a downstream value of 2.5 m3/s .

Quantify uncertainty: The first measurement is considered to be slightly worse than the second. For the upstream measurement, the hydrologist believes that there is a 90% chance that the true value falls within ± 5% of the measured value; for the downstream measurement, there is a 95% chance that the true value falls within ± 5% of the measured value.

Using values from a normal probability table, a 90% confidence interval is a value ±1.65 times the standard deviation, .

Upstream measurement: 1.65 x = 0.05 x 3.0 m3/, so = 0.091 m3/s.

Downstream measurement: 1.96 x = 0.05 x 2.5 m3/, so = 0.064 m3/s.

The loss of streamflow in the reach between these two measurements is 0.5 m3/s. How accurately is this loss known? Add variances!!

The variance of the loss is 0.0912 + 0.0642 = 0.0124 (m3/)2.

Weights for the Steady-State Problem

Head Observations:

Elevation of each observation well has a variance of measurement error of 1.00 m2.

Each water-level measurement has a variance of measurement error of 0.0025 m2.

Thus, total variance measurement error = 1.0025 m2

Flow Observation:

Coefficient of variation of 10%; thus = 0.1 x 4.4 m3/s = 0.44 m3/s

DO EXERCISE 3.2d: Calculate weights on hydraulic-head and flow observations.

Model Fit Using Starting Parameter Values

DATA AT HEAD AND FLOW LOCATIONS

OBSERVATION MEASURED SIMULATED WEIGHTED NAME VALUE VALUE RESIDUAL WEIGHT**.5 RESIDUAL

hd01.ss 101.804 100.225 1.579 0.999 1.577 hd02.ss 128.117 139.331 -11.21 0.999 -11.20 hd03.ss 156.678 174.363 -17.68 0.999 -17.66 hd04.ss 124.893 139.331 -14.44 0.999 -14.42 hd05.ss 140.961 157.132 -16.17 0.999 -16.15 hd06.ss 126.537 139.632 -13.10 0.999 -13.08 hd07.ss 101.112 102.868 -1.756 0.999 -1.754 hd08.ss 158.135 173.956 -15.82 0.999 -15.80 hd09.ss 176.374 190.300 -13.93 0.999 -13.91 hd10.ss 142.020 157.041 -15.02 0.999 -15.00 flow01.ss -4.40000 -4.86060 0.4606 2.27 1.047

STATISTICS FOR THESE RESIDUALS: MAXIMUM WEIGHTED RESIDUAL: 0.158E+01 Observation: hd01.ss MINIMUM WEIGHTED RESIDUAL: -0.177E+02 Observation: hd03.ss AVERAGE WEIGHTED RESIDUAL: -0.106E+02 # RESIDUALS >= 0. : 2 # RESIDUALS < 0. : 9 NUMBER OF RUNS: 3 IN 11 OBSERVATIONS

SUM OF SQUARED WEIGHTED RESIDUALS : 1752.4

SUM OF SQUARED WEIGHTED RESIDUALS WITH PRIOR: 1752.4

SUM OF SQUARED, WEIGHTED RESIDUALS: DEPENDENT VARIABLES: 1752.4

From UCODE-2005 output file.

DO EXERCISE 3.3: Evaluate model fit using the starting parameter values.

Least squares objective function

Example Least-Squares Objective Function

HEADS  

FLOWS 

PRIOR

2

1))('()( bhhbS ii

nh

ii

2

1))('( bqq ii

nq

ii

2

1))('( bPP ii

npr

ii

Basic Steps

for Model

CalibrationItalicized steps

are affected by

using regression

Consider predictions

Alternative models

Parameter estimation

Model(s), Parameters

Adjust parameter values and model

construction

Predictions

Compare simulated & measured values using

objective function

Evaluate predictions and prediction uncertainty

Observations Related to model output. Use to calibrate model

Societal decisions

System information related to model inputs, use in model development

Evaluate model fit and estimated parameter values