## Pcbased So Models And Sample Applications A Uswellsd for Systems Addressable Using Analytical Solutions

1. Model Background

US/WELLSd (Utah State extraction/injection well system for optimal groundwater management) is a deterministic version of an RM model. It uses influence coefficients based on analytical equations for potentiometric surface response to pumping and river depletion resulting from pumping. It is appropriate for systems where those analytical approaches are appropriate—presumably relatively homogeneous systems. (Of course, in the management and consulting arena, such approaches are commonly applied to heterogeneous systems, with acceptable error.)

Characteristics of US/WELLS0 are summarized in Table 2. The overview below is derived from the user's manual [15].

The objective function of the optimization module in US/WELLS is generally applicable and easily used for a variety of situations. The user can select either a linear or a quadratic form. The linear objective function is to minimize

US/WELLSd |
US/REMAX8 | |

Systems addressed |
One Layer, homogeneous |
Multilayer heterogeneous |

Stream/aquifer |
Stream/aquifer | |

Stream stage not |
Stream stage affected | |

affected by pumping |
by pumping | |

Management period |
One or two stress periods |
One or multiple stress periods |

of equal or unequal duration |
of equal duration | |

Steady state or transient |
Steady state or transient | |

Can rep. transient | ||

evolutionary era with | ||

terminal steady-state conditions. | ||

Influence coefficients |
Deterministic |
Deterministic |

Based on analytical expressions by |
Based on finite-difference | |

Theis and Glover and Balmer |
simulation (MODFLOW+STR) | |

Objective function |
Min or max pumping |
Min or max pumping |

or combination |
or combination | |

Time-varying weight for |
Diff. weight for each | |

extraction and injection |
pumping location | |

Bounds and constraints | ||

If s h < hu |
h1 | |

Cl,2 - G\,2 — C,i2 |
Ah\2 s A/i,,2 A/if 2 | |

2 (Ext) |
G\ 2 ~ G 1,2 — Gl2 | |

2 (Inj) | ||

= 1.0 > l.X |
= 1.0 | |

2(Ext)t == 2(Ext) < 2(Ext)t/ |

Notes: Superscripts L and U refer to lower and upper bounds; g = extraction or injection, [L3/T]; h = head.; A A, C| 2. V, 2 = head-difference, gradient, and velocity, respectively, between any two locations, [L], dimensionless, or [L/T]; 2 (Ext), 2 (Inj) = total extraction or injection, [L3/T]; d = stream depletion, [L3/T].

Notes: Superscripts L and U refer to lower and upper bounds; g = extraction or injection, [L3/T]; h = head.; A A, C| 2. V, 2 = head-difference, gradient, and velocity, respectively, between any two locations, [L], dimensionless, or [L/T]; 2 (Ext), 2 (Inj) = total extraction or injection, [L3/T]; d = stream depletion, [L3/T].

where WE X and W, x are the cost coefficient or weight assigned to extraction (E) or injection (/) rates in the jrth time period, [$/(L3 T)] or dimensionless; Ej x and Jkjc are extraction (E) or injection (/) rate at well j (or k) in the xth time period, [L3/T]; and J and K are number of extraction (J) or injection (K) wells.

### Potential constraints are the following.

1. Hydraulic gradient between any gradient control pair of wells at any time period must be within user-specified bounds. This can ensure that water is moving only in the desired direction. The maximum value can differ for each gradient control pair and time period. This constraint is useful, for example, when US/WELLSD is used for groundwater contaminant plume immobilization or for any situation where hydraulic gradient control is desired.

2. Extraction or injection rate at any well must be within user-specified bounds (lower and upper limits). If the user cannot decide if a certain well should be used for extraction or injection, he can locate one of each at the same location. The model will then determine either an extraction or an injection rate, or neither, for that location.

3. Hydraulic head at any injection, extraction, or observation well must be within user-specified lower and upper bounds. For example, a lower bound may be used to maintain adequate saturated thickness. An upper bound may be used to prevent surface flooding or to eliminate the need for pressurized injection. These lower and upper bounds can differ for different locations. The bounds are the same for both time periods.

4. Total import or export of water can be controlled to be within a user-specified range. The user can also completely prevent import or export of water or both. If no import or export of water is allowed, the total optimal extraction must equal the total optimal injection.

5. Depletion from the river must be within user-specified bounds (lower and upper limits). This is applicable only if a river exists in the considered system.

6. Constraint 3 is modified such that the probability that the actual change in head at any point in the groundwater system is not less than the change calculated by the model or is not greater than the change calculated by the model and is at least equal to the reliability level specified by the user. (This ability is found only in an alpha-test chance-constrained version of the model, US/WELLSs, which considers the stochastic nature of hydraulic conductivity. The utilized chance constraint is more accurate than previously reported formulations.)

Optionally, US/WELLSD can use a quadratic objective function to minimize

2 [WWE.* 2 Ei'* H>* + W*>* 2 Ei-* + W<-* 2 (2)

where Hjx is the dynamic lift, the difference between ground surface elevation and optimal potentiometric head resulting at extraction well j at the end of the *th time period, [L]; and WWF x is the weight assigned to the power used for extraction in the jcth time period, [$/L-T)].

The weighting factors can be used to emphasize different criteria and different time periods. For example, assume a problem of minimizing the total extraction using the linear objective function. If the second time period is chosen to be much longer than the first time period and the weights assigned to extraction and injection in the second time period are larger than those used for the first time period, then the solution will tend to minimize steady-state extraction/injection rates, and less attention will be given to the short-term transient rates. Through the weighting factors, US/WELLSD can also be used for maximizing pumping rates for water supply problems.

### 2. Application and Results

Here we illustrate the use of US/WELLSD to determine the optimal time-varying sequence of extraction and injection of water in prespecified locations needed for first immobilizing and then extracting a groundwater contaminant plume. In this example, the user specifies potential locations of extraction and injection wells around the contaminant plume (Figure 1). US/ WELLS0 then determines optimal extraction and injection rates for different time periods.

To illustrate model flexibility, four potential extraction wells and five potential injection wells are considered for placement outside the contaminant plume during the first period. In the second time period, three extraction wells are considered for placement inside the plume (to extract contaminated water) and five potential downgradient injection wells are considered. During both periods, the resulting hydraulic gradients (between 10 pairs of head observation locations) must be toward the center of the plume. Alternatively, the user could choose to minimize the pumping needed to capture the plume using only internal extraction wells in one or both periods.

• Potential Extraction Well 0 Potential Injection Well « Observation Well

Contaminated 6 Plume *

Initial

Groundwater Gradient

Contaminated 6 Plume *

16 H

16 H

Confined Aquifer

Aquifer Saturated Thickness = 20 m Storativlty = 0.00003 K = 75 m/day (isotropic)

Figure 1 Hypothetical study area for Example A, addressable with US/WELLSD.

Here, the quadratic objective function is used and employs greater weights for the second time period than the first period. This supports the fact that the second period is much longer than the first. In addition, neither export nor import of water is allowed—total injection must equal total extraction in each period. All the above considerations are incorporated within the model via the input data [15]. The user also specifies lower and upper bounds on head and pumping rates.

Figure 2 shows US/WELLSD output, in meters and m3/day. This contains, in addition to the input bounds (L. Bound and U. Bound), the optimal values of the decision variables (pumping), state variables (head and gradient), and marginal values.

The marginal is defined as the value by which the objective function will change if a tightly bounded variable changes one unit. If a variable's optimal value is not equal to either its lower or upper bound, its marginal will be zero. That is, the marginal will be nonzero only if the optimal value of the variable equals one of its bounds. In this case, the marginal shows the improvement of the value of the objective function resulting from relaxing this bound by one unit. Marginals are valid only as long as no other variable also changes in value. Thus they might be valid for only a small range of change in the bound.

To illustrate, the output file (Figure 2) shows that the marginal of the optimal injection rate in the first time period at injection well 3 is —45.3. The objective function value was 334,668.1. If the upper bound on injection in the first time period is relaxed by one unit at the mentioned well (that is, the new upper bound is 901 instead of 900), one would expect the value of the objective function to change by about -45.3 to 334,622.8. If this change is actually made and the model is rerun, the resulting change in objective function value is -45.4.

Marginals are useful in determining how to refine an optimal strategy. They help one to decide which bounds or constraints should be looked at more closely and perhaps relaxed. They also indicate the trade-off between that bound and objective achievement. They show how much one is giving up in terms of objective attainment to satisfy that restriction.

MODEL STATUS : OPTIMAL SOLUTION FOUND VALUE OF OBJECTIVE FUNCTION 334668.1

OPTIMAL EXTRACTION RATES

FIRST TIME |
PERIOD | |||

Well No |
L.Bound |
Optimal |
U.Bound |
Marginal |

1 |
0.00 |
745.42 |
900.00 |
0.000 |

2 |
0.00 |
447.60 |
900.00 |
0.000 |

3 |
0.00 |
448.71 |
900.00 |
0.000 |

4 |
0.00 |
747.86 |
900.00 |
0.000 |

5 |
0.00 |
0.00 |
0.00 |
0.000 |

6 |
0.00 |
0.00 |
0.00 |
0.000 |

7 |
0.00 |
0.00 |
0.00 |
0.000 |

SECOND TIME |
PERIOO | |||

Well No |
L.Bound |
Optimal |
U.Bound |
Marginal |

1 |
0.00 |
0.00 |
0.00 |
81.955 |

2 |
0.00 |
0.00 |
0.00 |
81.627 |

3 |
0.00 |
0.00 |
0.00 |
81.605 |

4 |
0.00 |
0.00 |
0.00 |
81.913 |

5 |
0.00 |
426.53 |
900.00 |
0.000 |

6 |
0.00 |
883.77 |
900.00 |
0.000 |

7 |
0.00 |
428.90 |
900.00 |
0.000 |

OPTIMAL INJECTION RATES

OPTIMAL INJECTION RATES

FIRST TIME PERIOD

Well No L.Bound Optimal

U.Bound Marginal 900.00 0.000 900.00 0.000

900.00 -45.342 < = = = explained In text 900.00 0.000 900.00 0.000

SECOND TIME PERIOO

Uell No L.Bound Optimal U.Bound Marginal

OPTIMAL HEADS AT OBSERVATION WELLS

Well No |
L.Bound |
Optimal |
U. Bound |
Marginal | ||||

1 |
30. |
.00 |
35. |
.69 |
40, |
.00 |
0. |
.000 |

2 |
30. |
.00 |
35. |
.54 |
40. |
.00 |
0. |
.000 |

3 |
30, |
.00 |
35. |
.60 |
40, |
.00 |
0. |
.000 |

4 |
30, |
.00 |
35. |
.55 |
40. |
.00 |
0. |
.000 |

5 |
30. |
.00 |
35. |
.70 |
40. |
.00 |
0. |
.000 |

6 |
30, |
.00 |
35, |
.79 |
40 |
.00 |
0, |
.000 |

7 |
30, |
.00 |
35, |
.92 |
40 |
.00 |
0. |
.000 |

8 |
30. |
.00 |
35, |
.88 |
40. |
.00 |
0, |
.000 |

9 |
30, |
.00 |
35, |
.84 |
40. |
.00 |
0, |
.000 |

10 |
30. |
.00 |
35, |
.77 |
40. |
.00 |
0, |
.000 |

11 |
30. |
.00 |
35, |
.65 |
40, |
.00 |
0, |
.000 |

12 |
30. |
.00 |
35. |
.60 |
40, |
.00 |
0. |
.000 |

13 |
30. |
.00 |
35. |
.65 |
40. |
.00 |
0, |
.000 |

14 |
30. |
.00 |
35. |
.79 |
40. |
.00 |
0. |
.000 |

15 |
30. |
.00 |
35. |
.88 |
40. |
.00 |
0, |
.000 |

16 |
30. |
.00 |
35. |
.77 |
40. |
.00 |
0, |
.000 |

Figure 2 US/WELLSD output file for Example A.

Well No |
L. Bound |
Opt ima I |
U. Bound |
Marginal | |||

1 |
30. |
00 |
35. |
.62 |
40. |
.00 |
0.000 |

2 |
30. |
00 |
35. |
,62 |
40. |
.00 |
0.000 |

3 |
30. |
.00 |
35. |
.68 |
40. |
.00 |
0.000 |

4 |
30. |
.00 |
35. |
.62 |
40. |
.00 |
0.000 |

5 |
30. |
.00 |
35. |
.63 |
40. |
.00 |
0.000 |

6 |
30. |
.00 |
35. |
.66 |
40. |
.00 |
0.000 |

7 |
30. |
.00 |
35. |
.75 |
40. |
.00 |
0.000 |

8 |
30. |
.00 |
35. |
.74 |
40, |
.00 |
0.000 |

9 |
30. |
.00 |
35, |
.71 |
40, |
.00 |
0.000 |

10 |
30. |
.00 |
35. |
.64 |
40, |
.00 |
0.000 |

11 |
30. |
.00 |
35. |
.56 |
40, |
.00 |
0.000 |

12 |
30. |
.00 |
35, |
.54 |
40, |
.00 |
0.000 |

13 |
30. |
.00 |
35, |
.56 |
40, |
.00 |
0.000 |

14 |
30. |
.00 |
35, |
.66 |
40, |
.00 |
0.000 |

15 |
30. |
.00 |
35, |
.74 |
40, |
.00 |
0.000 |

16 |
30. |
.00 |
35, |
.64 |
40, |
.00 |
0.000 |

OPTIMAL HEADS AT EXTRACTION WELLS

OPTIMAL HEADS AT EXTRACTION WELLS

Well |
No |
L.Bound |
Optimal |
U.Bound |
Marginal |

1 |
30.00 |
35.09 |
40.00 |
0.000 | |

2 |
30.00 |
35.29 |
40.00 |
0.000 | |

3 |
30.00 |
35.29 |
40.00 |
0.000 | |

4 |
30.00 |
35.09 |
40.00 |
0.000 | |

5 |
30.00 |
35.61 |
40.00 |
0.000 | |

6 |
30.00 |
35.62 |
40.00 |
0.000 | |

7 |
30.00 |
35.61 |
40.00 |
0.000 | |

SECOND TIME |
PERIOD | ||||

Well |
No |
L.Bound |
Optimal |
U. Bound |
Marginal |

1 |
30.00 |
35.69 |
40.00 |
0.000 | |

2 |
30.00 |
35.72 |
40.00 |
0.000 | |

3 |
30.00 |
35.73 |
40.00 |
0.000 | |

4 |
30.00 |
35.70 |
40.00 |
0.000 | |

5 |
30.00 |
35.24 |
40.00 |
0.000 | |

6 |
30.00 |
34.90 |
40.00 |
0.000 | |

7 |
30.00 |
35.25 |
40.00 |
0.000 |

OPTIMAL HEADS AT INJECTION WELLS

OPTIMAL HEADS AT INJECTION WELLS

FIRST TIME PERIOD

Well No 1

Well No 1 2

L.Bound

30.00

30.00

30.00

30.00

30.00

L .Bound

30.00

30.00

30.00

30.00

30.00

Optimal

35.90

36.03

36.46

36.47 35.83

SECOND TIME PERIOD

Optimal

35.65

35.88

36.33

36.08

35.67

U. Bound

40.00

40.00

40.00

40.00

40.00

U.Bound

40.00

40.00

40.00

40.00

40.00

Marginal 0.000 0.000 0.000 0.000 0.000

Marginal 0.000 0.000 0.000 0.000 0.000

OPTIMAL HYDRAULIC GRADIENTS

FIRST TIME PERIOD

From To

L. Bound Optimal (J. Bound Marginal

0.00000 0.00055 0.01000 0.000

0.00000 0.00003 0.01000 0.000

0.00000 0.00019 0.01000 0.000

0.00000 0.00000 0.01000 1.17E+7

0.00000 0.00157 0.01000 0.000

0.00000 0.00000 0.01000 3.26E+7

0.00000 0.00087 0.01000 0.000

0.00000 0.00000 0.01000 1.17E+7

SECOND TIME PERIOD

From To

L.Bound Optimal U.Bound Marginal

0.00000 0.00082 0.01000 0.000

0.00000 0.00241 0.01000 0.000

0.00000 0.00027 0.01000 0.000

0.00000 0.00014 0.01000 0.000

0.00000 0.00115 0.01000 0.000

0.00000 0.00000 0.01000 2.88E+8

0.00000 0.00081 0.01000 0.000

0.00000 0.00000 0.01000 0.000

Figure 2 Continued.

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