PB32-258476
Procedures for Predictive Analysis of
Selected Hydrologic Impacts of Surface Mining
Colorado State Univ., Fort Collins
Prepared for
Industrial Environmental Research Lab.
Cincinnati, OH
Aug 82
£mBE£3Sj£52]
J..3. C-esjftr.r.'nt of Cccrt??erce
f 1 Tedrnjcal !nfom)atiofl Service
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PB02-258U76
CPA-600/7-82-055
August 1982
PROCEDURES FOR PREDICTIVE ANALYSIS OF SELECTED
HYDROLOGIC IMPACTS OF SURFACE MIMING
by
David B. McWhorter
Agricultural and Chemical Engineering Department
Colorado State University
Fort Collins, Colorado 30523
Grant No. R-80467 3
Project Officer
Roger C. wilmoth
Industrial Environmental Research Laboratory
Cincinnati, Ohio 45268
INDUSTRIAL ENVIRONMENTAL RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
CINCINNATI, OHIO 4 5268
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TECHNICAL REPORT DATA
IPlro'r i:ad Initnjcnom on the reverse Itefore complrlinfl
V RtPOHT NO. 2.
EPA 600/7-82-055
3. RECIPIENTS ACCESSION NO.
tm 25847 6
4. 7 IT Lfc AND SUBTITLE
Procedures For Predictive Analysis of Selected
Hvdrologic Impacts of Surface Mining
5. REPORT OAT €
iliKiust—1932 ....
6. performing organization toot
7 AUIKOHISI
David B. McWhorter
8. PERf-O^MlNC ORGANIZATION FIEPOHT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Agricultural and Chemical Engineering Department
Colorado State University
Fort Collins, Colorado
80523
10. PROGRAM ELEMENT NO.
EHE623
11. CONTRACT/G^. A NT NO.
R-806673
12- SPONSORING AGE NC V NAME AND ADDRESS
Industrial Environmental Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Cincinnati. Ohio 45268
ID. TYPE OF REPORT ANO PEH»00 COVtREl)
Fi na1-Sent.'76-Sept. 181
Id. SPONSORING AGtNCV CODE
EPA/600/12
IS. SUPPLEMENT ARV NOTES
16. abstract -phis report presents a methodology for the prediction of selected iiydrologic
impacts of surface coal mining. Procedures for estimation of the chemical and hydrolo-
gic parameters required by an algebraic water quality model are provided. Th? model
predicts the long terra mean dissolved solids concentration in combined direct and sub-
surface runoff froTn a watershed partially disturbed by mining. The computational pro-
cedure is demonstrated in a step-by-step calculation for a mine sire in Colorado. The
predicted results are in satisfactory agreement with snort term (2 and 3 year) obser-
va t ions.
Procedures for determining the Cransmissivicy of coal and overburden aquifers from
single-hole aquifer tests are provided. The procedures permit the analysis of reco"ery
data, affected by vell-bort storage, following a prolonged pumping period. V,'ell-bore
storage is an important effect in the recovery of low transnissivity aquifers often
encountered in coal mining related hydrology. Several approximate, closed form formulas
for estimating selected impacts of surface mining on ground vater are provided. Among
:hera are formulas for estimating ground-uater inflows to an advancing pit and to a pit
advancing parallel to an alluvial valley. Formulas for calculating the extent of the
impressed piezometric surface as a function of time ana distance from the pit are deve-
loped . These formulas can be used to assess the probable severity of corresponding
impacts and to judge the need . for additional data ana more deta-.led models in site
<--neci f ic s i r i:;: r i nn = .
17. KEY WORDS AND DOCUMENT ANALYSIS
a DESCRIPTORS
b. IDE.NTI i- ie us/ope v £ no1; o t r. rms
C. COSATl l icld/Gioup
Surface Coal Mining
Hydrology
• Mathematical Models
Boring
Runo f f
Sa1i nity
TDS
Snovme1t
13B
IB. DISTRIBUTION STATEMENT
Release to public
19. SECuRH Y CLASS (Thu Report/
I'nc lass i f ied
21 NO. Of- PAGES
96
?0. SECURITY CLASS (This pegej ,
Unc1 ass i f i ed
17. PRICE
EPA Form ?2?0 —1 (Rav. 4-77) PHE'iOUS EDmON ij ObSCLETE
88
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This Page Intentionally Blank
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DISCLAIMER
This report has been reviewed by the Industrial Environmen
tal Research Laboratory - Cincinnati, U. S. Environmental Protec
tion Agency, and approved for publication. Approval does not
signify that the contents necessarily reflect the views and
policy of the (J. S. Environmental Protection Agency, nor does
mention of trade names of commercial products constitute en-
dorsement or recommendation for use.
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FOREWORD
When energy and material resources are extracted, processed,
converted, and used, the related pollutional impacts on our
environment and even on our health often require that new and
increasingly more; efficient pollutional control methods be used.
The Industrial Environmental Research Laboratory - Cincinnati
(IERL-Ci) assists in developing and demonstrating new and im-
proved methodologies that will meet these needs both efficiently
and economically.
This report presents a methodology for predicting the post-
mining salt load in combined direct and subsurface runoff from
a watershed partially disturbed by surface mining. Aiso provided
are methods for analyzing single-well cest data collected in tests
of the low transmissivity aquifers often encountered in coal min-
ing related hydrology. Methods are developed for predicting
several hvdrologic impacts of surface mining.
David Z. Stephan
Director
Industrial Environmental Research Laboratory
Cir.ci n.iati
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ABSTRACT
The purpose of this report is to present a methodology for
the prediction of selected hyarolooic impacts of surface coal
mining. Procedures for estimation of the chemical and hydroloq-
.ic parameters required by an algebraic water quality model are
provided. The model predicts the long term mean dissolved
solids concentration in combined direct and subsurface runoff
from a watershed partially disturbed by mining. Use of the
model requires that precipitation be partitioned intc direct
runoff and infiltration. This is accomplished in this report
using the historical recorc. of daily precipitation and the estim-
ation of direct runoff by the Soil Conservation Service Curve
Number method. Long term mean values of monthly infiltration
are utilized in a soil-water balance to estimate mean annual
subsurface runoff. The resulting estimates of mean annual di-
rect. and subsurface runoff from both the mined land and the
undisturbed portion of the watershed permit computation of the
hydrologic parameters in the algebraic model. The computation-
al procedure is demonstrated in a step-by-step calculation for
a mine site in Colorado. The predicted results are in satis-
factory agreement with short term (2 and 3 year) observations.
Procedures for determining the transmissivity of coal and
overburden aquifers from r,ingle-hole aquifer tests are provided.
The procedures permit the analysis of recovery data, affected
by well-bore storage, following a prolonged pumping period.
Well-bore storage is an important effect in the recovery of
low transmissivity aquifers often encountered in coal mining
related hydrology. Several approximate, closed form formulas
for estimating selected impacts of surface mining on ground
water are provided. Among them are formulas for estimating
ground-water inflows to an advancing pit and to a pit advanc-
ing parallel to an alluvial valley. Formulas for calculating
the extent of the depressed piezometric surface as a function
of time and distance from the pit are developed. These formu-
las can be used to assess the probable severity of correspond-
ing impacts and to judge the need for additional data and more
detailed models in site specific situations.
iv
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CONTENTS
Foreword .\ . . iii
\
Abstract \ . iv
Figures \ vii
\
\
Tables viii
\
1. Introduction \
2. Conclusions 3
3. Recommendations 5
4. The Quality and Quantity of Runoff from Surface
Mined Watersheds g
A Combined Water and Salt Balance 9
Determination of Water Quality Parameters ... 11
Determination of Hydro.logic Parameters .... 13
SCS method for direct runoff 16
Direct runoff from snowmelt 23
A surface ^ater balance 24
A soil water balance 25
Summary 27
Example 28
Description of site 29
Comparison of Predicted and Measured Values . . 39
5. Single Well Aquifer Tests in Coal Hydrology .... 41
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Recovery Following an Instantaneous Drawdown
in the Well Bore 4 2
Recovery Following a Pumping Period -
Standard Theory 43
Recovery Affected by Afterflow . 44
Example #1 49
Example #2 51
Alternate Method of Analyzing Recovery Data. . 53
Example #3 . 58
Example ff4 58
6. Analysis of Selected Flow Problems Important in
Surface Mining Hydrology 60
f low to an Advancing Pit 60
Exampls if 1 65
Flow to an Advancing Pit Initiated on a Crop
Line 66
Example S2 7 0
Flow to a Pit Advancing to an Alluvial Valley. 70
F.xample #3 74
Drainage of Alluvium or Fault Zone Receiving
no Recharge 74
Example #4 77
Discharge from Spoil in Response to Vertical
Recharge 77
Example {? 5 81
Bibliography . 05
vi
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FIGURES
Number Page
-4
1 Dimensionless recovery functions for a = 10 .... 48
2 Superposition of response functions on data for
example f?l 50
3 Standard recovery analysis of data in example #1 . . 52
4 Schematic drawing of aquifer discharge as a function
of time 54
5 Definition sketch for flow to the first cut 61
6 Calculated inflow to bo:c cut for example SI 67
7 Schematic physical model for flow to a pit initiated
on a crop line 58
8 Schematic of the physical conditions for example 12 . 71
9 Schematic drawing of flow to a mine from an
alluvial aquifer 73
10 Schematic of physical model for drainage of an
aquifer or fault zone 76
11 Schematic physical model for example #4 78
12 Physical model for the buildup and discharge of
ground water in a spoil bank 79
13 Dimensionless water table height resulting from
periodic recharge 82
vii
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TABLES
Number Page
1 Runoff curve numbers for AMC II 10
2 Curve numbers for wet and dry antecedent conditions . 19
3 Calculation of average annual direct runoff from
histogram of daily precipitation - precipitation
interval = 1.27 cm 22
4 Values of degree day factor 24
5 Water use coefficionts for selected covers 27
6 Climatological characteristics for mine site .... 30
7 Summary of parameters for calculation of direct
runoff 31
8 Calculation of direct runoff from rainfall
disturbed ground (spoil) 32
9 Calculation of direct runoff from rainfall
natural ground 33
10 Summary of surface water balance 34
11 Water-use coefficients for example 35
12 Soil-water balance in spoil 37
13 Soil-water balance in natural area 37
14 Summary of predicted model parameters 4 0
15 Comparison of predicted with measured values ... 40
16 Drawdown as a function of recovery time for
a = 10~4 and various pumping times 47
17 Recovery data generated from exact solution -
10 minute intervals 56
viii
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TABLES (continued)
Number Page
18 Recovery data generated from exact solution -
5 minute intervals 57
19 Recovery and afterflov; discharge for test of
example #1 . . . 58
20 Recovery and afterflow discharge for test of
example #2 59
21 Computation of discharge to the mine of example #2 . 72
22 Discharge from spoil bank for example #5 84
ix
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SECTION 1
INTRODUCTION
. Federal and State regulations require an analysis of the
potential influence of coal mining upon the hydrologic balance
in the area affected by mining. Potential effects of coal min-
ing upon the hydrologic balance include changes in the quality of
around and surface waters and a modification of the relative
quantities of direct and ground-water runoff. Other possible
effects are the modification of recharge to regional and local
aquifers, a change in the pattern of ground-water flow, and a
shift in the magnitude and peak Ol the runoff hydrographs. The
changes that may be anticipated are different in the active min-
ing phase than in the long term, post-mil irig phase.
Implicit is the requirement that the influence of a particu-
lar mining project upon the hydrologic balance be predicted be-
fore mining is initiated. This can be accomplished only through
the use of models, even if they are conceptual. Each of the
several components of the hydrologic balance is a complex phe-
nomenon that exhibits all of the vagaries of natural processes,
and models range from simple, non-quantitative concepts through
sophisticated stochastic models to detailed, physically based
descriptions. Those that are faced with the preparation and
review of predictions relative to the hydrologic consequences of
mining must select methods or models upon which to draw conclu-
sions. The most useful set of models is that which provides re-
sults in the desired form and of suitable reliability and is, at
the same time, commensurate with the experience, technical knowl-
edge, resources, and data that can reasonably be obtained by the
user.
In keeping with this perception, this report presents a set
of methods by which the influence of surface coal mining upon the
hydrolcgic balance can be analyzed. The methods presented in
this report are not applicable in all situations, of course, nor
are they intended to be. Through the use of examples, the appli-
cation of the methods are demonstrated. It is anticipated that
interested readers will devise ways to modify the procedures for
site specific needs. It is hoped that a reasonable balance has
been struck between the degree of rigor and realism in the
methods and the knowledge, resources, and data required to apply
them. The emphasis throughout the report is on guidelines for
application rather than on theoretical justification.
].
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Section 4 develops a method by which the influence of sur-
face mining on the partitioning of mean annual watershed drain-
age into direct runoff and ground-water runoff can be estimated
in the post-mining phase. It is shown how these results can be
used to predict the mean annual dissolved solids content in
total watershed drainage. A detailed example computation is
included that utilized data from a site in Colorado. Also
available for this site are observations that permit a compari-
son of predicted results with measured values. The agreement
is found to be satisfactory for this particular area.
An important aspect of mine hydrology during the mining
phase is the prediction of mine inflows from disturbed aquifers,
the disturbance of the piezometric surface, and effects on near-
by aquifers such as those in alluvial valleys. These aspects
are of concern relative to water quality as well because of the
possible need to discharge the inflows in surface waters. Funda-
mental to the prediction of these phenomena is knowledge of the
hydraulic coefficients of the aquifers. Possibly the greatest
single cost of a hydrologic impact analysis is the construction
of wells to be used for aquifer testing and monitoring. In r^ny
cases the cost can be substantially reduced by the utilization
of single-well aquifer tests. A procedure for the analysis of
recovery data in pumped wells is developed and demonstrated in
Section b. Finally, procedures for calculating mine inflows
for several different situations are developed and demonstrated
in Section 6.
o
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SECTION 2
CONCLUSIONS
A major purpose of this report is to present a method by
which the chemical and hydrologic parameters required for use in
an algebraic water quality model can be obtained. At the outset,
the intent was to prepare a set of charts and nomographs that
could be used in conjunction with basic watershed characteristics
to determine the necessary parameters. This proved to be less
feasible than the method presented herein.
The algebraic water quality model referred to above and
elsewhere in this report requires the estimation of the mean con-
centration cf dissolved solids in combined direct and subsurface
runoff from the undisturbed area (Pn), the mean TDS concentra-
tion in direct runoff from the mined land (Psm), and the mean
concentration in subsurface runoff from the mined land (P ).
gm
The value of Pn is most, reliably determined from premine monitor-
ing on the watershed to be mined. On the assumption that the
mined land will be reclaimed, the best estimate for Pgm is the
mean concentration of dissolved solids in direct runoff from the
undisturbed area. Again this can be determined from pre-mine
monitoring. The parameter is more d'fficult to estimate.
Dissolved solids data for spoil waters in nearby areas with a
similar hydrogeochemical environment will probably provide the
most reliable estimate for P . The value of P can be estima-
gm gm
ted from the dissolved solids concentration in saturated
extracts prepared from drill cuttings or core samples of the
overburden. Large variations in individual determinations are
to be expected and many determinations may be required to pro-
duce. a meaningful average value.
The model also requires estimation of two hydrologic para-
meters, K and f , both of which require estimation of the mean
sm
partitioning of precipitation into direct runoff and infiltra-
tion. It is concluded that the Soil Conservation Service Curve
Number method can be used for this purpose. This method strikes
a reasonable balance between the tradeoff of improved simulation
of the physical'processes involved and the cost in terms of data
3
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requirements. Calculation of daily rainfall excess from the
historical preciDitation record is required. The curve number
method can be applied for each day of the historical record or a
histogram method can be used to reduce the number of computations.
The estimates of long term mean monthly rainfall excess by the
histogram method agreed closely with those calculated on a daily
basis.
The difference between precipitation and precipitation ex-
cess is used as input to the soil zone. A water balance for the
root zone is used to estimate the quantity of subsurface runoff.
It is concluded that this method yields satisfactory estimates
of long term mean subsurface runoff if the time increment for the
balance is no longer than one month. The computation is sensi-
tive to the value of available water holding capacity of the root
zone.
The algebraic water quality model was used to estimate the
mean water quality in combined direct and subsurface runoff for
an area in Colorado where data were available for comparison
with the predictions. The predicted value of P was within the
range of values observed for either two or three years on 5 sub-
watersheds in the mining areas. Prediction of combined direct
and subsurface runoff agreed satisfactorily with the observed
v.ilue, also.
Sections 5 and 6 of this report pertain to the development
of procedures that may be used to estimate hydraulic properties
of coal or overburden aquifers and to predict hydrologic impacts
of mining for selected problems. It is concluded that single
hole recovery data taken following a prolonged pumping period
can be analyzed conveniently by either of two methods developed
in this study. Both methods account explicitly for the effects
of well bore storage that are common in tests of aquifers with
low transmissivity.
Approximate closed-form formulas for several ground water
problems commonly encountered in coal mining hydrology are
developed. These formulas can be used for predicting pit inflows
for a few different situations, the extent to which the piezomet-
ric surface will be disturbed, and some aspects of spoil bank
hydraulics. The analytic models are simple and have minimum data
requirements relative to the many numerical models that are
available. However, their use is constrained by the degree of
idealization of the flow problems, particularly with respect to
the requirement that the aquifers be homogeneous. The formulas
can be used to assess the probable severity of corresponding
hydrologic impacts and to judge the need for additional data and
more detailed models in site specific instances.
4
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SECTION 3
RECOMMENDATIONS
The following specific recommendations relate to the ur'- of
the algebraic water quality model discussed in Section 4.
1. The algebraic water quality model is recommended for r.se when
the long term mean concentration of dissolved solids in com-
bined direct and subsurface runoff is required. The model is
not appropriate for predicting temporal fluctuations (e.g.
seasonal changes).
2. Determination of the parameter should be accomplished by
pre-mine monitoring. Pn is the discharge weighted, mean con-
centration of dissolved solids in combined direct and subsur-
face runoff from the watershed to be mined. It is recommended
that the station at which Pn is to be determined be located
on a watershed drainage channel where there is reasonable
assurance that all direct and shallow subsurface runoff will
be observed. Subsurface runoff via deep regional aquifers
is not considered in the model. Continuous records of stream
flow as indicated by continuous recordings of stage on a
rated section are recommended. Samples for concentration
measurements should be collected with a frequency sufficient
to define the seasonal fluctuation of dissolved solids con-
centration. This may require daily samples during periods of
peak seasonal runoff. Streams that flow only in response to
individual precipitation events should be sampled so that the
concentration variation over the runoff period is defined.
3. The parameters P and P are the mean concentrations of
r sm gm
dissolved solids in direct runoff and subsurface runoff, re-
spectively, from the mined land. It is recommended that P
sm
be taken as being equal to the concentration of direct runoff
from the undisturbed land. This recommendation is based on
the assumption that the reclaimed lands will exhibit soil
chemical and erosive characteristics similar to those exist-
ing prior to mining. The parameter P can be estimated
from the dissolved solids concentration in saturation extracts
prepared from overburden drill cuttings and cores. A great
deal of spatial variation (both vertically and areally) is to
5
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be expected and it is recommended that sufficient determina-
tions be made to provide a statistically reliable mean. The
user of this technique should realize that the method may
provide only a rough estimate. Any existing data on spoil-
water quality from similar hydrogeochemical settings should
be examined and used as a guide to the determination of P
gm
Where such data exists, its use is preferable to the satura-
tion extract method.
4. The algebraic model requires that the unit area ratio of
.total runoff on the undisturbed portion to that on the mined
land be estimated. This is the parameter K in the model.
Also required is f , defined as the fraction of total rur.-
sm
off from the mined land that is direct (surface) runoff.
Estimation of both K and f require that the mean annual
sm
partitioning of precipitation into direct runoff and infil-
tration be estimated. This report recommends the use of the
Soil Conservation Service Curve Number methods for this pur-
pose. This method is recommended, not because of any inher-
ent superiority, but because it permits calculation of rain-
fall excess from data that are usually available. It is
recommended that rainfall excess be calculated on a daily
basis from at least 20 years of daily precipitarion data.
5. A root zone water balance calculation is recommended for
estimation of the quantity of subsurface runoff. It is
recommendsd that the time increment for the water balance
calculation be no greater than one month.
Sections 5 and 6 of this report deal with the use of analyt-
ic procedures for determination of aquifer hydraulic properties
and calculation of aquifer response to mining. The following
recommendations relate specifically to these results.
6. It is recommended that single hole aquifer tests be conducted
by pumping up to several hours and that emphasis be placed
on the collection and analysis; of recovery data. Prolonged
pumping creates a cone of influence that is substantially
larger than that created by a L^lug test and the recovery data
reflect the response in a larger volume of aquifer. It is
recommended that the Papadopulos-Cooper theory, as extended
in this report to the recovery period, be used to estimate
the hydraulic properties from recovery data. Alternatively,
an algebraic method not requiring a carve match may be used.
Both methods are useful only when well-bore storage is im-
portant.
7. Several problems of aquifer response to mining were analyzed
by approximate analytic techniques. The resulting formulas
are recommended for use when there is reasonable assurance
6
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that the idealized situations used in the analyses approxi-
mate the field problem. Probably most critical in this re-
gard is the idealization of the aquifer as homogeneous with
respect to hydraulic conductivity.
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SECTION 4
THE QUALITY AND QUANTITY OF RUNOFF FROM
. SURFACE MINED WATERSHEDS
Estimation of the effects of surface mining on the hydrolog-
ic balance of a watershed requires the prediction of both the
quality and quantity of runoff from the disturbed watershed. The
quantity of runoff from a watershed depends upon such factors as:
1) interception and evaporation of precipitation from the
surfaces of vegetal cover,
2) depression storage in surface puddles, ditches, and
other surface depressions,
3) infiltration of precipitation into the soil,
4) percolation of water through the soil under the influ-
ences of gravity and capillarity,
5) near surface perching of subsurface waters and movement
of these waters to drainage channels,
6) deep percolation and ground-water flow,
7) overland flow,
8) evapotranspiration,
9) intensity and duration of storms.
A set of even more fundamental factors affecting the runoff can
be listed under each of the above. For example, infiltration is
influenced by permeability and antecedent moisture conditions.
Evapotranspiration is affected by tempprature, solar radiation,
stage of plant growth and several other variables. Likewise, a
list of factors affecting the quality of runoff can be prepared
and would include such variables as:
1) the geo-chemical characteristics of solids contacted by
water and the distribution of these solids,
2) resident times and rates of chemical reactions,
3) weathering agents such as freeze-thaw cycles and micro-
bial activity.
Each of these parameters depends upon a host of more fundamental
variables.
Models of the post-mining water quality hydrology could
range from very detailed models that attempt to account for the
influence of all fundamental variables upon each of the compo-
nents, to simple formulations in which the effects of many varia-
bles are neglected and the influence of others is included only
implicitly in the form of lumped parameters. Usually, the
8
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generality, confidence, and reliability of the more detailed
models are obtained at the expense cf increased complexity and
increased data requirements relative to the simpler models.
Within the context of routine use by practioners, the "best"
model is that which will yield results of acceptable accuracy
and reliability at ? minimum cost. Obviously, the model selected
for use will depend upon the requirements of the application,
the available data, the available resources, time contraints,
and the knowledge and experience of the user.
Consistent with the approach used elsewhere in this report,
the developments of this section were obtained by attempting to
strike a reasonable balance between the perceived requirements
and constraints of the prospective user and the generality of
the predictive procedures that are presented.
A COMBINED WATER AND SALT BALANCE
Of principal interest is the change in the water quality
hydrology that results from disturbing a portion of a watershed
by surface mining. The change will occur as a result of a modi-
fication of one or more of the previously listed factors that
affect the water quality hydrology. Based upon a simple water
and salt balance approach, the dependence of the mean concentra-
tion of dissolved solids in total watershed runoff upon several
of the more important factors has been derived previously
(McWhorter and "ov/e, 197 6; Rowe and McWhorter, 197 8; and
N.cWhorte.", et al., 1979). The equation is
KRP_ + Pm
P. = - - (1)
t 1 + KR
where Pfc = mean concentration of dissolved solids in total
watershed runoff including both direct and sub-
surface runoff ,
Pn = mean concentration of dissolved solids in combined
direct and subsurface runoff from natural (undis-
turbed) portion of the watershed,
P = mean concentration of dissolved solids in combined
direct and subsurface runoff from mined land,
K = the ratio of total runoff per unit area (including
direct and subsurface runoff) on the undisturbed
area to that on the mined land,
R = the ratio of the area of the natural land to the
area of the mined land.
The material balances also leM to
9
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and
P = f P + (1-f )P
m sm sm sm gm
P = f P + (1-f )P
n sn sn sn gn
(2)
(3)
where f = the fraction of runoff from mined land that is
direct runoff/
sn
the fraction of runoff from natural (undisturbed)
land that is direct runoff.
sm
concentration of dissolved solids in direct run-
off from mined land,
gm
concentration of dissolved solids in subsurface
runoff from mined land,
sn
concentration of dissolved solids in direct run-
off from natural land/
P = concentration of dissolved solids in subsurface
gn runoff from natural land.
The above equations are dimensionally consistent and any
consistent set of units can be used. In this report concentra-
tions are expressed in milligrams per liter (mg/I). Volumes of
runoff are expressed in cubic meters or, when referred to a unit
area, in centimeters.
Throughout the report the word runoff will refer to the
total runoff. Runoff is comprised of two components; direct
runoff and subsurface runoff. Direct runoff is the overland
flow that makes its way into a drainage channel and contacts the
soil at the surface or within a few millimeters of the surface.
Subsurface runoff refers to waters that percolate beneath the
root zone of vegetation and eventually reappears in the drainage
channels. As used in this report, subsurface runoff may lag
direct runoff by times much greater than those usually invisioned
in storm hydrograph analysis. The term interflow is not used
for this reason. Interchange of waters in the near-surface sys-
tem with deep, regional ground waters is not considered.
The above equations were derived upon the assumption that
the net change in the volume of water stored in the watershed
is zero over the time period of interest. Relative effects of
depression storage, infiltration, and evapotranspiration upon
total runoff for each portion of the watershed are lumped into
the parameter K. In arid and semi-arid areas, evapotranspira-
tion is limited by the availability of water in the root zone.
Therefore, factors which affect the partitioning of precipita-
tion irto direct runoff and infiltration in turn affect
10
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evapotranspiration. The partitioning of total drainage into dir-
ect and subsurface runoff is accounted for in both K and the f
factors. Also, it is assumed that precipitation is uniform over
the watershed. It is emphasized that the abcve equations do not
apply to particular precipitation events, but rather, to. long-
term average conditions.
Previous use of the above equations has been to estimate the
contribution of disturbed land to the totai observed salt load
from a watershed. This was accomplished by applying the equa-
tions to individual watersheds comprising a partially mined area
and using regression techniques to determine the values of K and
f. To be useful in the predictive mode, however, the values of
all parameters must be estimated prior to mining. The required
parameters are the water quality parameters P , p , P , p
. ^ sm' gm sn' gn'
and the hydrologic parameters K, f and fgm. The parameter R
is the area of the natural land relative to the mined area as is
determined from the mining plan. The water quality and hydrolog-
ic parameters .In the simple model described above depend upon
more fundamental quantities, of course. It is the purpose of the
following subjections to outline and demonstrate a procedure by
which the required parameters may be estimated.
DETERMINATION OF WATER QUALITY PARAMETERS
Equation 3 expresses the dependence of the dissolved solids
concentration in runoff from the natural (undisturbed) portion
of the watershed upon the qualities and relative quantities of
direct and subsurface runoff. This equation is provided and
recommended for use in cases where Pn cannot be measured directly.
It is anticipated that monitoring in the pre-mining phase will
establish the value of P^ directly in most cases. In order to
establish a value for Pn by monitoring, it will be necessary to
measure the dissolved solids concentration and discharge at the
drainage point for the watershed under consideration. The appro-
priate value of P^ is the discharge weighted mean
where P^ is the dissolved solids concentration in discharge ,
and n is the number of determinations. Ideally, P^ is obtained
n
Epi°i
(4)
i=l
11
-------
by continuously monitoring the concentration and discharge, but
frequent grab samples will suffice. The frequency of sampling
and the time period over which sampling must be maintained are
site-specific considerations. Continuous water level recording
on a rated section is a necessity. On intermittently flov/ing
streams, several samples from individual runoff events will
often be required. On streams in which the major variation in
discharge is seasonal, the frequency should be greatest during
the high runoff period. It has been the author's experience
that the dissolved solids concentration in runoff from undis-
turbed watersheds decreases as the discharge increases. It is
recommended that sampling be sufficient to establish the rela-
tionship between P and Q.
A similar procedure for the determination of P is not
m
possible in the pre-mining phase. Therefore, it is necessary to
estimate Pcrri and P and use Eq. 2 to obtain P . Determination
sm gm ^ m
of f is discussed subsequently. The values of P and P aire
sm ^ J sm gm
the mean concentrations of dissolved solids in direct and subsur-
face runoff, respectively, from the mined portion of the water-
shed. A field study of post-mining water quality at a mine in
northwest Colorado (McWhorter, et al., 1979) suggests that the
concentrations of solids in saturation extracts (Richards, 1954)
prepared from drill cuttings in the overburden is a direct indi-
cator of P„. Elsewhere, however, the dissolved solids concen-
gm
trations in ground water in the cast overburden has been found to
be greater than that in saturation extracts by factors of 2 or 3
(Moran, et al., 1979; Groenewold, 1979, Pegenkopf, et al., 1977).
Furthermore, measured concentrations of dissolved solids in water
contained in spoil material is highly variable at most locations,
with the standard deviations being a large fraction of the mean.
The current state of knowledge relative to the chemistry of
waters in spoil notwithstanding, projections of the water quality
affected by mining are required in mining permit app ications.
Probably the most reasonable estimate of P can be made from a
judicious study of the quality of spoil water from nearby mines
in a similar geo-chemical environment. Sampling of springs
formed on the interface between the spoil and the undisturbed
underburder. is possible in some cases, but sampliny from wells
completed in the spoil aquifer will usually be required. In the
absence of this possibility, present experience suggests that the
dissolved solids concentration in extracts from saturated drill
cuttings will provide a reasonable lower limit for Pgm- If the
dissolved solids derive mainly from highly soluble sodium salts,
Pgm can ke exPecte<3 to be greater than the saturation extract
value, perhaps by as much as a factor of three. It is recognized
12
-------
that the above statements do not constitute a satisfactory guide
and that additional research is needed. The accumulation of data
on spoil-water quality now accompanying the expansion of surface
coal mining should permit more definitive guides in the future.
Virtually all spoil will be topsoiled on future surface
mines. In this case, the most reasonable estimate for the dis-
solved solids concentration in direct runoff Pgm from the top-
soiled mine is the value that exists on the undisturbed watershed
in which the mine will be located. Thus, P can be determined
' sm
from data collected in the pre-mining phase. Cases may be en-
countered wnere thin topscil placed on the spoil is eroded to the
extent that the spoil becomes exposed. Direct runoff in contact
with the spoil can be expected to contain dissolved solids in
concentrations that differ from that contacting only the soil.
However, the readily soluble salts in spoil contacted by direct
runoff are rapidly leached because the volume of water contacting
the spoil is large relative to the volume contacted. Dissolved
solids concentrations in direct runoff from even highly saline
materials are surprisingly small.
Before proceeding to a discussion of the hydrologic parame-
ters, a few concluding comments on the quality parameters are in
order. Sampling of waters contained in cast overburden has shown
the dissolved solids concentrations to be highly variable, even
within a single mine. The variability can be attributed to dif-
ferent leaching rates caused by variable permeability end water
availability, to differing chemical composition of materials con-
tacted, to variable clay contents and to variable degrees of
aeration, among other factors. Clearly, the single parameter P
is a lumped parameter, the value of which is a consequence of
complex interactions among several variables. Furthermore, the
dissolved solids concentration cannot be expected to be constant
in time. It has been demonstrated in many leaching studies that
the dissolved solids in the effluent decrease as the volume of
effluent increases. It seems reasonable to assume that a similar
improvement will occur in the field. The improvement is expected
to be slow, however. On the order of 1 pore volume of throughput
is required to effect significant improvement. Particularly in
arid and iiemi-arid regions where subsurface runoff is a few cen-
timeters per year at most, noticeable imorovement in P cannot
gm
be expected for several decades.
DETERMINATION OF HVDROLOGIC PARAMETERS
There are three explicit hydrologic parameters in the mode].;
K, f and fsn- The parameter fgn need be estimated only if Pn
cannot be successfully evaluated through a pre-mining monitoring
program. The following paragraphs describe a means for estimating
13
-------
K and f
sm
The same procedure applies for estimation of f
sn
should it be necessary.
The parameter K represents the ratio of total runoff per
unit area on the undisturbed portion of the watershed to that on
the mined portion. In situations where evapotranspiration is
limited by water, availability, any factors affecting the parti-
tioning of precipitation into direct runoff and infiltration also
affect the value of K. Use cf Eq. 1 for predictive purposes will
be practical only if a practical method for determining K and
f„m can be devised. A review of the physical interpretation of X
is in order before proceeding to a discussion of their
In the derivation of Eq. 1 (Mc'Whorter, et al. , 1979)
it is shown that
"sm
and f
sm
estimation
h =
en
1 - f
em
(5)
where f = fraction of the precipitation per unit area that
is consumptively used on the natural area,
f = fraction of the precipitation per unit area that
is consumptively used on tne mined area.
Under the assumption that there is no net change in water stored
in the watershed, the numerator of Eq. 5 represents the fraction
of precipitation that becomes runoff from the undisturbed portion
of the watershed. The dt ..ninator is the corresponding quantity
for the mined area. Both are on a per unit area basis.
The quantity of water consumptively used will depend upon
the type and quality of the vegetal cover, the potential evapo-
transpiration, anc the timing and volume of infiltration into the
soil. In arid and semi-arid climates, the potential annual e^po-'
transpiration is larger than the mean annual precipitation.
Considering the fact that a fraction of precipitation runs off as
overland flow instead of entering the root zone, it becomes appar-
ent that the potential evapotranspiration is an even greater mul-
tiple of the volume of soil water available for plant use. At
first glance it would seem, therefore, that no subsurface runoff
would occur under such circumstances. However, the timing and
volumes of infiltration may be such that, at particular times,
the water holding capacity of the soil is exceeded and percola-
tion through the root zone occurs. This is especially true
where a large fraction of the annual precipitation is in the
form of snov; that accumulates through the winter and melts in a
relatively short time in the spring. Subsurface runoff may occur
in response to percolation below the root zone during this period,
14
-------
: even though there exists a deficit of available soil moisture on
/ the average over the year.
Because the potential annual evapotranspiration is greater
than the annual available supply in the root zone, the actual
evapotranrpiration over the year is limited by water availability.
Thus, a critical factor in the estimation of actual evapotrans-
piration is the estimation of the quantities of water that enter
the root zone. Clearly, this is directly dependent upon the
partitioning of precipitation into infiltration and direct run-
off components. Also, the fraction of total drainage from the
mined land that is overhand flow (i.e. f ) is directly depen-
sm
dent upon the partitioning of precipitation into infiltration
and overland runoff components.
The importance of the partitioning of precipitation into
infiltration and runoff components dictated that the focus of
this portion of the study be on methods by which this partition-
ing can be estimated as -1 function of precipitation, infiltra-
tion capacity of the soil, antecedent moisture conditions, and
depression and interception storage. Substantial effort went
into an attempt to prepare ncmographs from which, not only the
partitioning could be estimated, but also the values of K and
fsm- For this approach to be practical the independent varia-
bles used to enter the nomographs must be determinable by in-
spection of the watershed, coupled with pertinent measurements
that are relatively easy to make. Several available hydrologic
models were investigated with the intent of generating ground-
water and overland runoff data for a wide variety of conditions
in which selected key variables would be systematically varied.
For these data correlations of the ground-water and overland
runoff with the pertinent independent variables would be estab-
lished. The USDA HL-74 hydrologic model was selected and this
approach pursued. A difficulty with this procedure is that the
USDA model (and others investigated)are basically oriented
toward hydrograph prediction, while our main interest, was on the
mean annual volumes of ground-water and overland runoff. Never-
theless, some limited success was achieved but it became appar-
ent that results of equal suitability could be achieved by the
use of simple and widely used methods for estimating runoff
volume.
An overview of the water-balance procedures used to estimate
the hydrologic parameters is presented in the following para-
graph. This is followed by several subsections that discuss
methods by which the variables and parameters in the water-
balance equations can be estimated. Finally, the use of the
procedures are demonstrated and an indication of the accuracy
and reliability of the method is provided by application to an
existing mine site.
15
-------
Use of Eq. 1 requires that K, f and f be known. These
^ ^ sn sn
parameters can be calculated directly from the long-term average
quantities of annual direct and subsurface runoff. Subsurface
runoff, is estimated herein from a monthly subsurface water
balance which, in turn, requires that long term average values
of monthly direct runoff be estimated. Therefore, the first
step is to estimate long term average values of monthly direct
runoff. The second step is to compute infiltration from
li = Pi - Qi (6)
where 1^ = average infiltrated volume per unit area for month
i ,
P. - average precipitation available for direct runoff
or infiltration in month i,
= average
direct runoff in month i.
The infiltration values determined from Eq. 6 are input to the
monthly soil-water balance:
"i " !i - Etai " 4Si ">
where W. = volume of water per unit area passing below the
root zone in month i,
Et i = volume per unit area consumptively used from the
root zone in month i (i.e., actual evapotrans-
piration) ,
AS- = change in volume of water stored in the root
zone per unit area in month i •
The summation of is the average annual direct runoff. Under
the assumption that net changes in storage are zero for average
annual conditions, the summation of is the average annual sub-
surface runoff. A large time lag may exist between the time a
particular W. occurs and the time at which the water appears as
r'inoff. However, the timing of subsurface runoff is not neces-
sary for the application cf this model. The above computations,
performed for both the disturbed and undisturbed lands, permit
calculation of K, fsm, and fgn directly from their definitions.
The SCS Method for Direct Runoff
A widely used method for computing the volume of direct run-
off is the SCS curve number method described in detail in "Hy-
drology", Supplement A to Section A of the SCS National
16
-------
Engineering Handbook and summarized elsewhere (e.g., Viessman,
et al., 1977; Hahn and Barfield, 1978). The PCS method has been
used to estimate direct runoff from daily precipitation data and
from individual storms. Daily precipitation values are used in
this report. The volume of direct runoff per unit area from a
24 hr precipitation value, P, is given by
P - 1P-"-2S|2 (6)
P + 0. 8S
where Q = direct runoff, volume per unit area (e.g., cm) ,
P = precipitation, volume per unit area (e.g., cm),
S = potential maximum retention, volume per unit area
(e.g. , cm) .
The parameter S is related to a curve number CN by
s = ~ 10" (9)
The reader i;; cautioned that Eq. 9 gives S in inches, anc must
be converted to the appropriate unit if other than inches are
being used. The curve number CN is in turn related to such
factors as the hydrologic classification of the soil, the hydro-
logic condition of the watershed, the type and quantity of vege-
tal cover and the antecedent moisture conditions. In the
interest of completeness, the proceduias by which values of CN
are determined are included in this report, even thcugh they are
readily available el-.ewhere.
The curve number for an average antscedent moisture condi-
tion (i.e., AMC II) is obtained from Table 1. The hydrologic
soil group designation refers to the following conditions:
A) Soils exhibiting high .infiltration rates, even when
thoroughly wetted; deep, well drained, sands and gravels.
Low runoff potential.
B) Soils exhibiting moderate infiltration rates when
thoroughly wetted; moderately well to well-drained
soils; medium to moderately coarse textured soils. Mod-
erate runoff potential.
C) Soils exhibiting low infiltration rates when thoroughly
wetted; restricted drainage by impeding layer or low
permeability; medium to fine textured soils. Moderate
to high runoff potential.
17
-------
D) Soils exhibiting very low infiltration rates; soils with
swelling clays, clay pan near the' surface, water table
near the surface, shallow soils over nearly impervious
surface. High runoff potential.
A large number of soils throughout the United States have been
given hydrologic classification. Also, detailed soils investiga-
tions are required as part of mining permit applications in most
states,'and data from the soils study should be used to assist,
in determining the hydrologic classification.
The curve number applicable for a particular runoff event
depends upon the antecedent moisture condition as well as the
factors listed in Table 1. The curve numbers corresponding to
antecedent moisture conditions other than the average value of
II can be determined from Table 2. The classification of ante-
cedent moisture conditions is given below.
AMC I - Soils are dry but not to the wilting point
AMC II - Average soil moisture condition
AMC III- Soil is wet, nearly saturated from previous precipi-
tation
Table 1
RUNOFF CURVF NUMBERS FOR AMC II
Land Use Treatment Hydrologic Hydrologic Soil Group
Or Cover
Or Practice
Condition
A
B
C
D
Fallow
Straight Row
77
86
91
94
Row Crops
Straight Row
poor
72
81
88
91
good
67
78
85
89
Contoured
poor
70
79
84
88
good
65
75
82
86
Contoured
s> Terraced
poor
66
74
80
82
good
62
71
78
81
Small Grain
Straight Row
poor
6 5
76
84
88
good
63
75
83
87
Contoured
poor
63
74
82
85
good
61
73
81
84
(continued)
18
-------
TABLE 1 (continued)
Land Use Treatment Hydrologic Hydrologic.Soil Group
Or Cover Or Practice
Condition
A
B
C
D
Legumes,
Rotation
Meadow
Contoured
& Terraced
Straight Row
poor
good
poor
good
61
59
66
58
72
70
77
72
79
78
85
. 81
82
81
89
85
Contoured
poor
64
75
83
85
Contoured
& Terraced
good
poor
good
55
63
51
69
73
67
78
80
76
83
83
80
Pasture,range
poor
68
79
86
89
fair
49
69
79
84
good
39
61
74
80
Contoured
poor
47
67
81
88
fair
25
59
75
83
good
6
35
70
19
Meadow
good
30
58
71
78
Woods
poor
45
66
77
83
fair
36
60
73
79
Roads
(including
right-of-way)
Dirt
Hard Surf ice
good
25
72
74
55
82
84
70
87
90
77
89
92
Table 2
CURVE NUMBERS FOR WET
AND DRY ANTECEDENT
CONDITIONS
CN for
CN for
CN for
AMC II
A!'-iC I
AMC III
100
100
100
(continued)
19
-------
TABLE 2 (continued)
CN for
CN for
CN for
AMC II
AMC I
AMC III
95
87
98
90
78
96
85
70
94
80
63
91
75
57
88
70
51
85
65
45
82
60
40
78
55
35
74
50
31
70
45
26
65
40
22
60
35
18
55
30
15
50
25
12
43
20
9
37
15
6
30
10
4
22
5
2
13
As noted previously, the long term average values of month:y and
annual direct runoff are required. A straight forward way to
estimate these quantities is to compute direct runoff from Eq. 8
for each day for which precipitation exceeded 0.2S as determined
from the historical precipitation record. The individual values
of direct runoff are summed by month and divided by the number
of years of record used to determine the average direct runoff
by month. This is a laborious computation to be made by hand,
especially for situations in which daily precipitation frequently
exceeds 0.2S. However, the computation can be made readily by
almost any desk top computer. The salient features of an algorithm
by which the average quantities of annual and monthly direct run-
off can be computed are given below.
Before beginning the computation, the values of S corres-
ponding to antecedent moisture conditions I, II, and 111 are
determined. It is also necessary to quantify the conditions in
which AMC I, II, and III are expected. For example:
A) AMC I applies if there has been zero precipitation for
x days prior to the day in question,
B) AMC II applies if total precipitation in x days prior to
the day in question is greater than zero but less than y.
20
-------
C) AMC III applies if total precipitation, in x days prior
I to the day in question is equal to or greater than y.
Specific values for x and y must be determined on a site-by-site
basis using the best judgement of the user. With known values of
S corresponding to each antecedent moisture condition and an
unambiguous method by which the antecedent moisture condition is
determined, the direct runoff can be calculated.
1) Determine if precipitation occurred on day j. If not,
proceed to day j +1. Otherwise,
2) determine applicable antecedent moisture condition and
determine if precipitation on day j is greater than the
corresponding value of 0.25. If not, proceed to day
j +1. Otherwise,
3) compute direct runoff from Eq. 8.
4) Go to day j + 1.
A less satisfactory method for estimating the average values
of annual and direct runoff that does not require a computation
of direct runoff for each day can be used when the above proce-
dure is deemed impractical fcr some reason. The first step is
to determine the value of S corresponding to AMC II. Intervals
of daily precipitation are selected and a histogram of daily
precipitation for each monLh is prepared from the historical
record. Only precipitation values greater than 0.2S need be
considered. The average direct runoff for month j is computed
from
(P^ .-0.2S)2
°j =£Nij p/^+0.85 <101
where Qj = average direct runoff for month j
P.. = midrange precipitation for histogram interval i
in month j
N. .= mean number of daily precipitation values in
i'3 interval i for month j
m = number of histogram intervals
Eq. 10 also applies for.calculation of average annual direct run-
off by simply dropping the index j. The use of Eq. 10 on a
monthly basis is demonstrated subsequently in the application to
an existing mine.
21
-------
Questions concerning the degree to which Eq. 10 agrees with
calculation of direct runoff from each daily precipitation value
in the historical record were investigated. This was accomplished
by calculating average annual direct runoff by Eq. 10 and by
individual days using 10 years of precipitation record from a
humid region. Table 3 contains the histogram data and the cal-
culation using Eq. 10. The value of S used in the calculation
was 10.16 cm. Daily precipitation values greater than 0.2S =
2.03 cm result in direct runoff, therefore. In the ten years of
record, there were a total of 185 days on which precipitation
was greater than 2.03. During the ten years of record, there
were 104 days on which the precipitation was between 2.03 and
3.30 cm, 28 days on which the precipitation was between 3.30 and
4.57 cm and so on. It will be noted that the precipitation
interval used in Table 3 is 1.27 cm (0.5 inches). For each
value of P^ in the second column, a value of direct runoff was
computed and multiplied by the corresponding value of N to obtain
the values in right-hand column. The sum of values in the right-
hand column is average annual direct runoff and is 9.1 cm for
this example.
TABLE 3
CALCULATION OF AVERAGE ANNUAL DIRFCT RUNOFF FROM
HISTOGRAM OF DAILY PRECIPITATION - PRECIPITATION INTERVAL= 1.27 cm
Histogram
P.
Number of Days
Average Number
Qi
Interval
1
In 10 Yr
Of Days Per
cm
cm
Record
Year, N-^
cm
2. 03 -
3. 30
2.67
10 4
10.4
0.4
3. 30 -
4 . 57
3.94
28
2.8
0.9
4 . 57 -
5.84
5.21
23
2 . 3
1.8
5.84 -
7.11
6.48
18
1.8
2.5
7.11-
8. 38
7.75
8
0.8
1.7
8. 38 -
9.65
°.02
1
0.1
0.3
9.65 -
10.92
10 .2?
0
0
0
10.92 -
12.19
1.1.56
2
0.2
0.9
12.19 -
13.40
1 2 . t! 3
0
0
0
13.46 -
14.73
14 .10
l
0.1
0.0
IS5
IB. b
5TT
The direct runoff for each of the 185 days for which
P > 2.03 cm was calculated and summed. The average annual direct
runoff by this method is 8.7 cm. To obtain this estimate
required 185 individual calculations using Eq. 8. The estimate
of 9.1 cm was obtained by application of Eq. C only 10 times.
To investigate the effect of the size of the- precipitation inter-
val, the histogram method was again used, but with an interval
of 0.51 cm. The histogram method using an interval of 0.51 cm
22
-------
resulted in averaye annual direct runoff of 8.8 cm which is near-
ly identical to that obtained using an interval of 1.27 cm.
It is concluded that calculation of direct runoff by sum-
ming the daily direct runoff values is preferred over the histo-
gram method. The histogram method will greatly reduce the number
of calculations required and will yield estimates that are within
the confidence interval associated with the SCS direct runoff
formula. However, the histogram method is less flexible with
respect to adjustment of S in accordance with the antecedent
moisture conditions that prevail.
Direct Runoff From Snowmelt
The procedure discussed above is applicable to direct run-
off from precipitation in the form of rainfall. Snowfall con-
tributes significantly to the total annual precipitation in many
surface coal mining regions and must be accounted for in the
water balance computations. Water that becomes available for
direct runoff or infiltration as the result of snowmelt usually
does so much more slowly than from precipitation: Seasonal run-
off hydrographs resulting from the meltinq of snowpacks do not
exhibit the sharp peaks and short durations that are character-
istic of rainfall-runoff hydrographs. This is because the di-
rect runoff contribution to the seasonal snowmelt hydrograph is
usually small relative to the subsurface contribution.
For the purposes of the application intended in this report,
the time and space distribution of snowmelt is of interest only
insofar as it affects direct runoff. It is assumed that no di-
rect runoff from snowmelt occurs if the daily rate of snowmelt
is less than the base intake rate of the soil (i.e., less than
the rate of infiltration after prolonged wetting). In those
cases in which the rate of snowmelt exceeds the base intake
rate, the volume of direct runoff is calculated by multiplying
the difference between the rate of snowmelt and the intake rate
by the time interval over which the rate of snowmelt exceeds the
intake rate. In many cases the most important influence of snow-
melt will be to cause the antecedent moisture condition to be
AMC III, thus causing increased direct runoff from rainfall that
may occur.
Basin-wide snowmelt rates are difficult to estimate. A
rather crude method, but one for which the required data is al-
most always available, is the degree day index method. The
equation is
M = k D (11)
where M = cm of melt over the snow covered area
D = degree days with 0°C as base
23
-------
k = degree day factor
Table 4 contains values for k adapted from the U.S. Corps of
Engineers (1956). The degree day factors in Table 4 are average
values obtained from several years of record. The values are
appropriate for use with the mean monthly temperature. Other in-
vestigators have found the degree day factor to range from 0.23
to 0.69 cm per degree day based on a mean temperature above 0°C
(llortori, 1945, Gartska, et al.f 1958). The melt rates are great-
est in open area and least in densely forested areas.
TABLE 4
VALUES OF DEGREE DAY FACTOR*
Basin**
Forested Area
%
Degree
April
Day Factor
May
CSSL
40
0.407
0.457
UCSI.
90
0.169
0.329
WBSL
93
0.178
0.192
* cm of melt over the snow covered area per degree day
above 0°C
** designates Central Sierra, Upper Columbia, and Willamette
Basin Snow Laboratories, respectively
Computations of snowmelt in this application are made by
determining the water equivalent in the snow pack at the begin-
ning of the snowmelt season. This water equivalent is reduced
in time by subtracting the volume of snowmelt from the remaining
water equivalent. An average loss of 1.2 cm/mo (or more) water
equivalent due to evaporation/sublimation may occur during the
winter months (Gartska, 1964). Losses greater than 1.2 cm/mo
are common u -er windy conditions. Evaporation/sublimation
losses may significantly reduce the water equivalent at the
beginning of snowmelt from the total accumulated snow precipita-
tion .
A Surface Water Balance
A surface water balance on a monthly basis is used to com-
pute the volume of water per unit area infiltrated into the soil.
Evapotranspiration and subsurface runoff are components of the
soil-water balance and arc addressed in the next subsection.
The surface water balance is expressed by
h ' pi " Qi 1121
24
-------
where 1^ = infiltrated volume per unit area in month i,
P. = precipitation available for runoff or infiltration
in month i and includes snowmelt,
= direct run .ff in month i.
Equation 12 is intended to apply in the warm season months when
water becomes available for infiltration or runoff by rainfall
and snowme3.t.
A Soil Water Balance
The volume of infiltrated water computed from the surface
water balance is used as input to the root zone. Again, a
monthly balance is used and is expressed by
Wi = :i " Etai ~ ASi
where W. = volume of water per unit area passing below
1 the root zone in month i,
E£?j_ = volume per unit area consumptively used from
the root zone in month i (i.e. actual evapo-
transpiration),
AS = change in volume of stored water in the root
zone per unit area during month i.
There is a maximum value of the volume per unit area that
can be stored in the root zone for prolonged periods. This
volume, called the available water capacity (AWC), is equal to
the difference in volumetric water content at field capacity and
that at permanent wilting, multiplied by the effective depth of
the root zone {McWhorter and Sunada, 1977). It is assumed that
water in excess of that required to make the available water
(AW) equal to AWC and to satisfy the demands of evapotranspira-
tion in any particular month will pass through the root zone.
Therefore, application of Eu. .13 requires an accounting of the
available water because jS can never be larger than AWC-AW for
a given month. For example, suppose that at the beginning of a
particular month, the AW = 5 cm and that during the month I=10 cm
and E^a - 3 cm. Therefore, during the month there has been a net
input of 7 cm to the root zone. Part of this net input of 7 cm
will go toward satisfying any deficit in AW and, if there re-
mains any left over, that part will pass below the root zone and
eventually become ground-water runoff. Suppose that AWC=10 cm,
then 5 of the 7 cm of net input will go into storage to make
AW = AWC and 2 cm will become ground-water rur.off. On the other
hand, if AWC= 14 cm, the entire net input would go into storage
brinqincj the AW to 12 cm and there would occur no ground-water
runoff.
25
-------
It should be pointed out that use of the term ground-water
runoff for W is not meant tc imply that this quantity, of water
will appear as watershed drainage in the month for which it is
computed. Appearance of W as drainage may lag the time it occurs
substantially. However, under the assumption that the system is
in dynamic equilibrium (i.e. there is no net change in water
storage during the year) all of the water represented by W will
eventually drain from the watershed.
Use of Eq. 11 requires that E be known. Wymore (1974)
to
has made adjustments in the original Jensen-Haise (Jensen and
Haise, 1963) equation that may be conveniently used for computing
the potential evapotranspiration The evapotranspiration
under conditions of plentiful water Efc is obtained by multiplying
Etp ^ a water-use coefficient kc> The Jensen-Haise equation as
modified by Wymore (197 4) is
E = [0.25T - 0.123 + 0.13E£]Rg (14)
tp L <-J s
where E^. = potential evapotranspiration (cm)
T = temperature in degrees centigrade
= elevation in thousands of meters
Rg = evaporation equivalent of solar radiation (cm).
Application of Eq. 14 on a monthly basis requires the mean
monthly temperature T and the total cumulative evaporation equiv-
alent of solar radiation. The solar radiation is normally meas-
ured in calories per cm" per unit of time. The unit of time of
interest in this application is the month. The evaporation
equivalent of the solar radiation is obtained by dividing the
solar radiation by the latent heat of vaporization of water (535
calories/cm"^) .
Water is not always consumed at the potential rate, even
when available water is not limiting. The evapotranspiration in
the case when water is not limiting is given
Et = KcEtp (15)
where Kc is a water-use coefficient that depends upon the vege-
tal cover and time of the year. Wymore (1974) provides the
values of Kc shown in Table 5. Wymore assumed that phreatophytes
would use water at the potential rate.
26
-------
TABLE 5
WATER USE COEFFICIENTS FOR SELECTED COVERS
Type of Cover Kc
Apr May Jun Jul Aug Sep Oct
Sagebrush-grass 0.60 0.80 0.80 0.80 0.71 0.53 0.50
Pinyon-Juniper 0.70 0.80 0.80 0.80 0.80 0.69 0.65
Mixed Mt. Shrubs 0.67 0.81 0.85 0.52 0.74 0.65 0.60
Conifer forest 0.71 0.80 0.80 0.80 0.79 0.75 0.71
Aspen forest 0.67 0.85 0.90 0.86 0.75 0.65 0.60
Rockland & misc. 0.60 0.65 0.65 0.65 0.60 0.50 0.50
Phreatophytes 1.00 1.00 1.00 1.00 1.00 1.00 1.00
(After Wymore, 1974)
Under most arid and semi-arid conditions, the cumulative
evapotranspiration for the season as computed by summing the
monthly values from Eq. 15 would exceed the total precipitation.
This is because Eq. 15 is based 3n the assumption that water
available is not limiting. Procedures have been presented
(Jensen, et al., 1970; Hall, et al., 1979) to modify the value
of Kc to include the effects of depletion of stored water. For
the purposes of this report in which monthly values are used, it
is simply assumed that, if Efc as calculated from Eq. 15 is
larger than the sum of the AW and I for a particular month, then
the actual evapotranspiration E is equal to AW + I. When
AW+I is greater than E,, then E. =E..
t ta t
Readers may find methods for the computation of evapotrans-
piration other than that above to be more convenient because of
experience or data availability. The publication entitled
Consumptive Use cf Water and Irrigation Water Requirements
(Jensen, 1973) presents a thorough and comparative discussion of
many different methods.
SUMMARY
The above paragraphs describe a method by which the mean
annual precipitation is partitioned into direct runoff and
ground-water runcff by months. The annual direct and ground-
water runoff quantities are computed by summing the monthly
values. It is a simple matter to compute K, fR » and fgn
directly from their definitions and the quantities of runoff in
each component. The calculation is best summarized by the de-
tailed example computation that follows.
-------
Before proceeding to the example, it is convenient to sum-
marize the data requirements in the following list.
1) Historical record of daily precipitation
2) Mean monthly temperatures
3) Mean monthly solar radiation
4) Elevation
5) Soil data to permit hydrologic classification
6) Vegetal type and cover
7) Land use
8) Base intake rate of soils
9) Available water capacity and rooting depth
10) Estimates of P and P
gm sm
11) Estimates of from pre-mining monitoring
12) Fraction of watershed area to be minevl
EXAMPLE
The example calculation presented in this subsection is
intended to provide both a demonstration of the computational
procedure and an indication of the accuracy that can be expected
from the method. It is important to understand that the purpose
of the model is to predict the dissolved-solids concentration
in the combined direct and subsurface runoff from a watershed
that has been partially disturbed by mining. The predicted con-
centration is that which is expected on a long-term average
basis. Comparison of the predicted value of P with that meas-
ured over a short period of record provides only an indication
of the reliability. A long-term record (perhaps 20 yrs or more)
is required for a thoroughly valid test. The method recommended
in the previous subsections of this report for calculating the
hydrologic parameters involves estimating such individual quan-
tities as direct runoff and combined direct and subsurface run-
off. If should be recognized that considerable error in the
estimation of these individual quantities can be tolerated,
provided that the required ratios are satisfactorily accurate.
Certainly if. both the required ratios and the individual quan-
tities defining the ratios are accurately predicted, increased
confidence in the procedures is warranted.
28
-------
The method recommended in the foregoing portions of this
report are applied to a partially mir.cd watershed comprising
part of the Edna Mine in northwest Colorado. This'site was
selected for the example application primarily because measure-
ments of have been made previously and are available for com-
parison with the value of Pfc predicted herein. At the time the
field studies were conducted at the Edna Mine, the .-\ethod pro-
vided in this report for computation of the hydrolouic parameters
was not contemplated. Therefore, all of the data required by the
method was not obtained, at least in the detail that is desirable.
For this reason the example application provides a demonstration
•of some of the judgements that may be required in other applica-
tions. Data form the Edna Mine field study include both meas-
ured values for P and measured total runoff. Therefore, it is
possible to compare predicted values of both and total runoff
u
to measured values. The latter comparison will provide an indi-
cation of the degree of reliability of the water balance compu-
tations .
Pescription of Site
Descriptions of the Edna Mine site are detailed elsewhere
(McWhorter, et al., 1979; Rowe and McWhorter, 1978; McWhorter..
et al., 1975) and only the features salient to the present appli-
cation are repeated here. The portion of the watershed in which
the Edna Mine is located ranges in elevation from 2195 m to
2500 m above mean sea level. The mined portion of area has not
been regraded or top-soiled at the time of the above referenced
field studies. Vegetation on the spoil was sparse relative to
that on the undisturbed ground and consisted of short grasses
and weeds. Vegetation on the undisturbed portion was a mixture
of native grasses, oak brush, sage brush, and some aspen and
conifers on the highest portions of the area.
Other characteristics of the site are given in the follow-
ing paragraphs where they relate to a particular computation.
1. The first step in the computation is to assemble the
climatological data that is required. It is recommended
that no less than 20 years of record be used. In this
example compuation the precipitation record for Pyramid,
Colorado is used for precipitation on the mine site.
The Pyramid station is at elevation 2465 m and located
only a few miles from the study area and in the same
watershed. Unfortunately, there exists no temperature
record; either on the mine site or at Pyramid. Tempera-
ture records at Steamboat Springs, Colorado (elevation
2034 m) were utilized and adjusted to ..he mean eleva-
tion of the mine site (2350 m) Dy decreasing the tempera-
tures at the rate of O.S5°C per 100 m of elevation. No
29
-------
solar radiation data were available for the mine site
and data from Fort Collins, Colorado were used. Fort
Collins is nearly the same latitude as the mine site.
The long-term mean climato.logical data that are estima-
ted to apply at the mine site are given in Table 6.
Also shown in Table 6 sre values of potential evapo-
transpiration, Etp- These were calculated from Eq. 14
using the mean temperatures and R values as shown. In
s
addition to the data in Table 6, the daily precipita-
tion values are required for calculation of direct
runoff. These data are presented in the next step.
TABLE 6
CLIMATOLOGICAL CHARACTERISTICS FOK MINE SITE
Month
Total
Snow
Rain
Aver
R
Efn
P
Prec
Prec
Temp
s
tp
cm
cm
cm
°C
cm
cm
Jan
4.6
4.6
0
-11.8
12.1
Feb
4.5
4.5
0
- 9.3
14 .7
Mar
5.6
5.6
0
- 4.9
23.3
Apr
5.0
3.7
1.3
1.5
25.0
5.4
May
3.6
0.8
2.8
6.8
28.3
10.0
Jun
3.8
0
3.8
10. 7
33.0
14.8
Jul
3.3
0
3.3
14.4
30.3
16.4
Aug
4.3
0
4.3
13. 5
27.1
14 .1
Sep
4.4
0.6
3.8
9.1
23.4
9.6
Oct
4.2
1.3
2.9
3.9
19.1
5.3
Nov
4.0
3.0
1.0
- 4.0
11.8
Dec
5.1
5.1
0
- 9.9
10.5
52.4
29. 2
23.2
258 .6
75.6
II. The second step is the estimation of direct runoff
from precipitation in the form of rainfall. Determina-
tion of the applicable curve numbers is required and is
accomplished from Tables 1 and 2, together with knowl-
edge of the cover and hydrologic classification of the
soil. The mined area in the present example had not
been topsoiled or revegetated at the time this computa-
tion applies. Weathering of the fragmented shale and
sandstone spoil on the surface had tended to produce a
layer of fairly low permeability and a moderately high
runoff potential. The hydrologic soil group classifi-
cation believed to best represent the spoil is C.
Also, the spoil is range land judged to be in good to
fair hydrologic condition. From Table 1, the curve
number for the spoil wc.s estimated to be 8 0 for AMC II.
30
-------
The undisturbed ground has a high infiltration rate
(McWhorter, et al., 1975) but is only moderately well
drained. The hydrologic soil group classification was
judged to be between A and B. For range land in fair
hydrologic condition it was estimated that CN = 62 for
the undisturbed ground at AMC II.
Snowmelt occurs mainly in April on the mine site
and it was estimated that AMC III is the prevailing
antecedent moisture condition in April. It was also
judged that AMC II was appropriate for May and that
AMC I applied during the remainder of the summer.
Table 2 was used to determine the curve numbers for
AMC I and AMC III for both the mined and unmined por-
tions, and the corresponding values for S were calcula-
ted from Eq. 9. All of these data are summarized in
Table 7.
TABLE 7
SUMMARY OF PARAMETERS FOR CALCULATION
OF DIRECT RUNOFF
Month
Sooil
Natural
AMC
CN
S (cm)
AMC
CN
S (cm)
April
III
90
2.54
III
80
6.35
May
II
80
6.35
II
62
15.21
June - Oct
I
62
15.20
I
43
33.78
It was noted from Eq. 8 that direct runoff occurs
for i5 greater than 0.2S. In April direct runoff will
occur from the spoil for P greater than 0.5.1 cm, and
for P greater than 1.27 cm in May. Similarily, the
threshold value of rainfall can be computed for each
period for both the spoil and the natural ground. In
this example the histogram method was used to compute
the direct runoff. Table 3 shows the computation for
the spoil. The historical record of rainfall was
examined to determine the number of days that rainfall
amounts were in the intervals 0.51-1.27 cm, 1.27-
2.03 cm and etc. for each month. The 20 years of record
showed that there were 54 days in April for which rain-
fall was between 0.51 cm and 1.27 cm and 9 days for
which rainfall was between 1.27 and 2.03 cm, for example.
There were only 2 days in the 20 year record for which
rainfall was greater than 2.79 cm, and direct runoff
for these two days was computed directly from Eq. 8.
There were no days in June, July, or August for which
rainfall exceeded the threshold of 3.05. In September
31
-------
and October there were a total of three days and direct
runoff was computed for each individual day as shown in
the table. The same calculations for the natural
ground are shown in Table 9.
TABLE 8
CALCULATION OF DIRECT RUNOFF FROM RAINFALL
DISTURBED GROUND (SPOIL)
Month
Histogram
Interval
cm
Pi
cm
Number of
Days in 20 vr
Record
Avg No of
Days per
Year,
Qi1
cm
April
May
June
July
Aug
Sept
Oct
0.51-1.27
1.27-2.03
2.03-2.79
2.95
3.18
1.27-2.03
3.35
>3.05
>3.05
>3.05
7.37
3.30
3.81
0.89
1.65
2.41
2.95
3.18
1.65
3.35
7.37
3.30
3.81
54
9
4
1
1
0
0
0
1
1
1
2.7
0.45
0.20
0.05
0.05
0.45
0.05
0
0
0
0.05
0.05
0.05
0.13
0.16
0.16
0.06
0.07
0.58
0.01
0.03
0.04
0
0
0
0.05
~0
~0
0.67
(- 0.51 ) (Pi-1.27\/
April, Qi p +2.03 ' May' Qi = P- + 5. 08
June through Oc
t 0 -Ci-3'05)2
c' ui -P- + TT793
i
32
-------
TABLE 9
CALCULATION OF DIRECT RUNOFF FROM RAINFALL
NATURAL GROUND
Month
1
Histogram
Number of
Avg No of
Qi*
Interval
Days in 20 yr
Days per
cm
cm
Record
Year,
cm
April
1.27 - 2.03
1.65
9
0.45
2.03 - 2.79
2.41
4
0.20
0.16
2.95
2.95
1
0.05
0.06
3.18
3.18
1
0.05
0.07
0.29
May
3.35
3.35
1
0.05
0.03
June
>6.86
0
0
0
July
>6. 86
0
0
0
Aug
>6. S6
0
0
0
Sept
7.37
7.37
1
0.05
~0
Oct
>6. 86
0
0
0
(Pi"1-27)'
* April, + 2.03
_(pi"3-05)2
May ' Qi P. +11.93
(p. -6.76y
June through Oct, Qi = p1 + 27.03
0.32
III. The third step in the computation is estimation of dir-
ect runoff from snowmelt. In this example, it is estim-
ated that sublimation/evaporation losses are 1.5 cm per
month for each month that continuous snow cover exists
(i.e., November through April). Total precipitation in
the form of snowfall for these 6 months is 26.5 cm
(Table 6). The loss is 9.0 cm leaving an estimate water
equivalent of 17.5 cm to become available for direct
runoff or infiltration during the spring sncwmelt season.
The degree day factor applicable to this site is
not known. Using the largest value of k for April from
Table 4 and a mean April temperature of 1.5°C from
33
-------
Table 6 yields an average daily melt rate of 0.6 cm/d.
At this average rate, the 17.5 cm of water equivalent
would require 29 days to melt. Experience at the mine
site suggests that snowmelt is completed within the
first 10 days or uo of May. This indicates that a melt
rate of 0.6 cm/d is probably somewhat too large. Even
so, the calculated melt rate is believed to be substan-
tially less than the base intake rate of the spoil and
is much less than the base intake rate of the natural
ground, Therefore, it is assumed that the entire
17.5 cm of water equivalent melts in April and that no
direct runoff is produced from the melt, either on the
spoil or natural ground.
IV. The fourth step in the computation is the surface water
balance calculation from which monthly values of infil-
tration are estimated. This is accomplished using
Eq. 12. The computation for the Edna Mine example is
shown in Table 10. The calculation shows that a total
of 42.7 cm infiltrate the spoil and 43.1 cm infiltrate
the natural soil.
TABLE 10
SUMMARY OF SURFACE WATER BALANCE
Month
Total
Loss From
Available Prec
Direct
Runoff
Infiltration
Prec
Snow Pack
Snow
Rain
Spoil Natural
Spoil Natural
cm
cm
cm
cm
en
cm
cm
cm
Jan
4.6
1.5
0
0
0
0
0
0
Feb
4.5
1.5
0
0
0
0
0
0
Ma.r
5.6
1.5
0
0
0
0
0
0
Apr
5.0
1.5
17.5
1.3
o. sa
0 .
29
18.2
18.5
May
3.6
0
0.8
2.8
0.C4
0.
03
3.6
3.6
Jun
3.8
0
0
3.3
0
0
3.8
3.8
Jul
3.3
0
0
3.3
0
0
3.3
3.3
Aug
4.3
0
0
4.3
0
0
4.3
4 .3
Sep
4.4
0
0.6
3.8
0.05
0
4 . 3
4.4
Oct
l. .2
0
1.3
2.9
0
0
4.2
4.2
Nov
4.0
] . 5
0
1.0
0
0
1.0
1.0
Dec
5.1
1.5
_0_
0
0
0
0
0
52.4
9 . C
20 . 2
23 . 2
0.7
0 .
3
42.7
4 3.1
V. The fifth step in the computation is the soil-water
balance. In order to mike the soil-water balance cal-
culations it is necessary to estimate the available
water capacity. In the case of the spoil, laboratory
measurements indicate that the available water for the
34
-------
spoil is about 0.11 cm/cm. It is estimated that the
effective rooting depth is 60 cm yielding a value of
)\WC equal to 6.6 cm. The available water per centimeter
for the soil is probably about the same as for the
spoil, but the effective rooting depth is judged to be
somewhat greater. The available water capacity for the
soil is estimated at AWC=8.0 cm.
In addition to AWC it is also necessary to deter-
mine the water-use coefficients that apply. Table 11
gives the values for Kc that are deemed most reasonable
for the spoil and for the natural ground at the Edna
Mine.
TAELE 11
WATER USE COEFFICIENTS FOR EXAMPLE
Month Kc
Spoil
Natural
April
0. 60
0.67
May
0.72
0.83
June
0.72
0.88
July
0.72
0.84
Aug
0.65
0.75
Sept
0. 51
0.65
Oct
0. 50
0.60
The values in Table 11 for spoil are average values of
the Wymores water-use coefficients for rockland and
sagebrush-grass, (Table 5). The values for the natural
ground are average values from Table 5 for aspen forest
and mixed mountain shrubs.
The calculations of the soil-water balance for the
spoil and the natural ground are shown in Tables 12 and
13, respectively. It is necessary to guess at the quan-
tity of available water in storage at the beginning of
the month for which the calculation is initiated. The
calculation for the entire season will not balance un-
less the initial guess is correct. If balance is not
achieved, then a new guess is made and the computations
repeated until a balance is achieved. In arid and semi-
arid regions where the potential for water use is great-
er than the quantity available, it is usually possible
to eliminate the trial and error. This is accomplished
by initiating the calculation for a month that is pre-
ceeded by months of low precipitation and high potential
35
-------
water use. In this case the best first estimate for the
quantity of stored water is zero.
The computations shown in Tables 12 and 13 were
initiated for the month of October because it appeared
reasonable that the available water would be completely
depleted by the beginning of October. Transfer of
water to or from the root zone is assumed to be zero
from December through March. Therefore, the first
active month.following November is April, followed by
May and so on. The next step is to fill in the columns
headed I (infiltration), (potential evapotranspira-
tion) and (evapotranspiration with water availability
not limiting). In this example, values of I were given
previously in Table 10, and values of E were listed
in Table 6. The values for Efc shown in Tables 1.2 and
13 were obtained by multiplying Et by the appropriate
water-use coefficient (Table 11).
The column head AW is the available water in stor-
age at the beginning of the month. The deficit column
is simply the quantity of water required to bring the
root zone to maximum storage capacity (i.e. AW+deficit
= AWC). As explained above, the calculation is initia-
ted by assuming a value for AW. A value of AW equal to
zero was used in this example for October 1. Thus, the
deficit on Oct. 1 is 6.6 cm for the spoil (Table 12).
During October, 4.2 cm of infiltration occurs. In the
same time period the evapotranspiration demand (Et) was
2.6 cm. Because infiltration exceeded demand, the
entire demand of 2.6 cm was satisfied and the actual
evapotranspiration (Eta) equalled Efc. The difference
between infiltration and evapotranspiration went toward
making up the deficit in soil-water storage. Thus, the
change in storage (AS) was positive and equal to 1.6 cm.
No percolation (W) occurred because the storage deficit
was sufficient to accommodate the entire excess of in-
filtration over demand.
Th^ 1.6 cm of water that went into storage in
October causes AW to equ.Hl 1.6 cm at the beginning of
November. The corresponding deficit is 5.0 cm. There
is no evapotranspiration demand in November so the 1 cm
of infiltration goes into storage, making the AW equal
to 2.G cm at the beginning of April. Infiltration dr.
April is 18.2 cm and is more than sufficient to meet
36
-------
the demand of 3.2 cm. Therefore, E. = E =3.2 cm,
td . t
leaving an excess of 15.0 cm. Four centimeter's of this
excess go to make up the remaining deficit. So AS = +4.0.
The remainder is 11.0 cm and becomes percolation below
the root zone.
TABLE 12
SOIL-WATER BALANCE IN SPOIL*
Month.
AW**
Deficit**
I
EtP
El-
Efa
AS
W
cm
cm
cm
cm
cm
cm
cm
cm
Oct
0
6.6
4.2
5.3
2.6
2.6
+ 1.6
0
Nov
1.6
5.0
1.0
0
0
0
+ 1.0
0
Apr
2.6
4.0
18.2
5.4
3.2
3.2
+ 4.0
11.0
May
6.6
0
3.6
10.0
7.2
7.2
-3.6
0
Jun
3.0
3.6
3.8
14 .8
10.7
6.8 ,
-3.0
0
Jul
0
6.6
3.3
16.4
11.8
3.3
0
0
Aug
0
6.6
4.3
14.1
9.2
4.3
0
0
Sep
0
6.6
4.3
9.6
4.9
4.3
0
0
42.7
75.6
49.6
31.7 .
0
11.0
* AWC = 6.6 cm
** Evaluated at beginning of month
TABLE
]3
SOIL-V.'ATER
BALANCE
IN NA
T'JRAL
AREA
Month
AW * *
Dificit**
I
EtP
Et
Eta
AS
W
cm
cm
cm
cm
cm
cm
cm
cm
Oct
0
8.0
4.2
5.3
3.2
3.2
+ 1.0
0
Nov
1.0
7.0
1.0
0
0
0
+ 1.0
0
Apr
2.0
6.0
18. 2
5.4
3.6
3.6
+ 6.0
8.6
May
8.0
0
3.6
10.0
8 . 3
8.3
-3.3
0
Jun
4.7
3.3
3.8
14 .8
13.0
8.5
-4.7
r\
Jul
0
8 . 0
3.3
16.4
13.8
3.3
0
0
Aug
0
8 . 0
4 . 3
14 .1
10.6
4 . 3
0
0
Sep
0
8.0
4 . 3
9.6
6.2
4.3
0
0
42.7
75.6
58.7
35.5
0
8.6
37
-------
May is begun with a 6.6 cm of water in storage.
The evapotranspiration demand for May is 7.2 cm. This
demand is greater than the infiltration of 3.6 cm but
the sum of infiltration and available water is suffi-
cient to meet the demand. Thus, Eta = Et = 7.2 cm. To
meet this demand, 3.6 cm of stored water were required,
so AS=- 3.6 cm and no percolation occurrs.
The available water at the beginning of June is
3.0 cm since 3.6 cm of stored water were used in May.
The water-use demand in June is 10.7 cm, a quantity that
exceeds the sum of AW and I. Therefore, the actual
evapotranspiration is limited to AW + I = 6 . 8 cm and all
remain stored water is used (i.e. AS = -3.0 cin). For the
remainder of the season, the evapotranspiration demand
exceeds the infiltration so Efc =I and AW remains equal
to zero through September. The guess that AW=0 on
October 1 was correct and no new trial is required.
The water balance computations are now complete.
The procedure predicts that average annual direct run-
off from the spoil is 0.7 cm and subsurface runoff is
11.0 cm for a total runoff value of 11.7 cm. The cor-
responding values on the natural ground are 0.3 cm,
8.6 cm, and 8.9 cm.
VI. Once the water balance calculations are complete, it is
a simple matter to calculate fsn, f and K, directly
from their definitions:
* Q _ 0.3 cm ^ ,
fsn = Q"T¥ " 8.9 cm = °-04' natural ground
fsm = 0~T~TC = TT77 = °-06' spoil
v _ total runoff from natural ground _ 8. 9 _ n nc-
total runoff from spoil 11.7
VII. The final step is to determine the values of the chemi-
cal parameters P , P , P and P . It is anticipated
c sm' gm sn gn c
that pre-mine monitoring will usually permit Pn to be
estimated directly. In those cases, it is not necessary
to estimate f , P or P Mining at the Edna Mine
sn sn gn ^
preceeded the field studies, however, and it is neces-
sary to estimate.the parameters for the natural ground
as well as for the mined portion.
Field plots were established in the spoil at the
Edna Mine from which both direct runoff and subsurface
38
-------
runoff was collected (McWhorter, et al., 1979). Data from these
plots suggested that P? 15C mg/I and Pgm = 3030 mg/I. In the
same study samples of surface and subsurface waters indicate that
P _ = 150 mg/t and P = 460
sn J/ gn
Equations 2 and 3 are used to calculate P and P ,' respec-
m n ^
tively, using f and f as determined in Step VI. Finally,
Pfc is computed using the value of K from Step VI together with
the appropriate value for R.
COMPARISON OF PREDICTED AND MEASURED VALUES
Table 14 summarizes the values of the model parameters as
determined above for the Edna Mine. The field studies conducted
at the Edna Mine resulted in measured values for P on 4 individ-
ual watersheds comprising a portion of the Edna Mine. Annual
data for ei.ther 2 or 3 years were utilized to compute P . The
fraction of each watershed mined was different and, therefore,
resulted in a different value for R. The model parameters in
Table 14 were used to calculate predicted values of P for each
watershed. The predicted and measured values are presented in
Table 15. As explained by McWhorter, et al., (1979) some inter-
change of water between watersheds C9 and CIO was suspected.
For this reason, data from the combined watershed C9+CIO is also
presented. It is believed that the degree of agreement between,
the predicted and measured values is satisfactory. The fact
that the predicted values are above the measured values in some
cases and below in others is noteworthy.
Further insight into the reliability of the computational
procedure is provided by comparing the predicted values of total
runoff with the measured valuer. Adequate total runoff measure-
ments are available for watersheds C3, C9 and CIO. Together,
C 0
these three watersheds total 8.06x10 m . In calendar year 1975,
the measure runoff per unit area was 8.7 cm. The fraction of
this total area that consisted of spoil was 0.467, the remainder
being undisturbed ground. The water balance calculations pre-
dict runoff of 8.9 cm on natural ground and 11.7 cm on spoil.
Thus, the predicted runoff is
(0.467) (11.7) + (0. 533) (8.9) = 10.2 cm.
The predicted total runoff is 1.5 cm greater than the meas-
ured value for 1975. The 1975 precipitation was 2.8 cm less
than the mean vlaue used in the prediction. This difference may
provide a partial explanation for the difference between meas-
ured and predicted runoff. However, the confidence interval
about the predicted value, while not known, is undoubtedly at
39
-------
least as large as the difference between the measured and pre-
dicted values.
TABLE 14
SUMMARY OF PREDICTED MODEL PARAMETERS
Hydrologic
Chemical
Calculated
f =0.04 P = 150mg/£ P =2850 mg/I; Eq.2
sn sm m . ^
f =0.06 P = 3030 mg/I P = 450 mg/-£; Eq. 3
sm gm J n r 1
K = 0.76 P 150 mq/l
sn ^
P = 460 mg/I
gn ^
TABLE 15
COMPARISON OF PREDICTED Pfc WITH MEASURED VALUES
Watershed
R
Predicted
Avg. Meas.
Range of
No.
Pt
Pt
Meas. Pfc
mg/£
mg/£
mg/£
C 3
0.47
2220
1840
1610 - 2030
C 5
0
2860
2910
2830 - 3080
C 9
1.86
1450
1240
1120 - 1290
C10
1.27
1670
1850
1850 - 1860
C 9 + C10
1.44
1600
1550
1520 - 1580
40
-------
SECTION 5
SINGLE-WELL AQUIFER TESTS IN COAL HYDROLOGY
Aquifer tests are the primary means of determining the hy-
draulic parameters of water-bearing strata that are required for
projecting the effect of mining on the ground-water regime and
for estimating the quantities of ground-water inflow that can be
anticipated in the mine workings. Ideally, aquifer tests are
performed by pumping at one location and observing the response
in one or more observation wells. Experience in the Rocky
Mountain and Northern Great Plains coal fields suggest that the
water-bearing strata (including the coal seams) can be expected
to exhibit low permeabilities. Aquifer tests in these materials
require very low pumping rates over a long time period to insure
that meaningful responses will be obtained in the observation
wells. It is not unusual to find that pumping rates on the
order of 1 to 5 liters/min are required to keep the pumped well
from dewatering before meaningful responses are observed in even
nearby observation wells. Control of such small pumping rates
at a constant value is difficult and, even when successfully
accomplished, analysis of the data is complicated by the effects
of well-bore storage that are almost always important in such
cases.
The above described difficulties suggest that many of the
advantages that are attributed to multi-well aquifer tests re-
lative to single-well tests under normal circumstances, often
are not obtained in practice in.the low-permeability materials
encountered in coal hydrology studies. Furthermore, the addi-
tional drilling and completion v/ork required for multi-well
tests makes them more costly than single-well tests.
Single-well tests include a number of specific tests.
Among them are: a) the slug test in which water is "instanta-
neously" added to or displaced from the well-bore and the return
of the water level towards its equilibrium position is monitor-
ed, b) the pumping/recovery test in which the well is pumped for
a period and the water level is monitored in the well during and
after pumping, c) the pressure-pump-in method in which water ia.
pumped into the well and discharge and pressure are monitored,
and a) the auger-hole method which is very similar to the slug
method. Of particular interest in this report are the slug test
and the pumping/recovery test.
41
-------
The standard theories and procedures for analyzing the slug
test and the pumping/recovery test are discussed in the follow-
ing sub-sections for i_he purposes of setting the stage for the
development and understanding of a new procedure for analyzing
the recovery under conditions where the standard methods are in-
appropriate.
RECOVERY FOLLOWING AN INSTANTANEOUS DRAWDOWN IN THE WELL-BORE
Methods of performing and analyzing slug tests are well
developed and well-known, and no new insights to this test will
be pre ided here. The main interest in the slug test for the
purposes of this report derives from the fact that it represents
a limiting case for the pumping/recovery test which is discussed
in detail subsequently.
Slug tests are nonnally performed by adding to or displac-
ing from the well-bore a quantity of water. The return of the
water level toward its initial equilibrium position is monitored
and this recovery is analyzed by fitting a theoretical response
function to the data. The theoretical response functions are
solutions to the differential equation of ground-water flow and
are based upon the condition that the initial drawdown or build-
up of water level in the well is created instantaneously. Solu-
tions for the slug test have been derived by Hvorslov (19 51) , by
Cooper et al. (1967), and by Bouwer and Rice (1976). These
solutions differ somewhat because of different assumptions made
concerning the importance of aquifer storage on the flow induced
in the aquifer.
Of the three solutions referenced above, the most rigorous-
ly correct response function is that of Cooper et al. and is
given by
00
w
(1)
o
where
drawdown (or buildup) in the well
6
o
initial drawdown (or buildup) in the well
a
S
storage coefficient
r
radius of well
w
r
c
radius of casing
TtAc2
42
-------
transmissivity
t = time
u = integration variable
Au = a combination of Bessel functions
Cooper et al. provide a table of values of the integial which
permit the solution to be displayed graphically in the form cf a
2
family of curves of *v;A0 vs Tt/r with a as the curve para-
meter. This solution rigorously includes the effects of well-
bore storage as well as the effects cf aquifer storage, and from
this solution it is apparent that the particular flow under con-
sideration here is insensitive to the aquifer storage coeffi-
cient in the range of values for S normally encountered in con-
fined aquifers. This conclusion tends to justify the assumption
implicit in the Hvorslov (1951) method that the distribution of
piezometric head in the aquifer is that given by the appropriate
solution to the Laplace equation which does not account for
aquifer storage. It is not surprising, therefore, that the
Cooper et al. and the Hvorslov methods usually yield results in
close agreement with one another.
The characteristic of this test that is particularly impor-
tant in subsequent developments in this report is that flow to
the well is initiated by cr?ating a drawdown at the wall-bore at
t = 0 by displacing water from the well-bore only. It is assum-
ed that no water has been withdrawn from the aquifer at the be-
ginning of recovery in this test.
RECOVERY FOLLOWING h PUMPING PERIOD - STANDARD THEORY
While the response functions for the slug test are based on
the assumption that the drawdown in the well at the beginning of
recovery is created without withdrawing any water *rom the aqui-
fer, the standard theory for analyzing the recovery of water
level following a pumping period assumes that all of the pumped
water is derived from aquifer storage. Furthermore, the stan-
dard theory of recovery following a pumping period assumes that
aquifer discharge is zero during recovery (i.e., well-bore stor-
age is neglected) whil^ the slug theories account explicitly for
the discharge to the well-bore required for the recovery of
water levels. Thus, the two theories represent opposing ex-
tremes .
The Theis (1935) solution for the drawdown in a well pumped
at constant rate Q is
A = J2_ [ exp(-x) d (2)
w 4ttT / X
B
43
-------
where Q = pumping rate
B = Srw2/4Tt
X = dummy variable of integration
and other parameters are as defined previously. For values of
6 less than about 0.05, the integral can be closely approximated
by the first two terms in a series expansion and
V" " 4?f{lnf^ " °-5772i (3>
' w "
The theory of superposition in the time domain (McWhorter
and Sunada, 1977) is used to write
V-> " wf j1" ^72 " °-5,72| - gf I1" - °-5,"j
W W (4)
for the drawdown in the well after pumping ceases. The para-
meter tQ is the pumping period. A simplification results in
V'1 - 4§f ln trr- ' k <5'
o
which is widely used to determine the transmissivity T using the
recovery data. Particularly important, in the context of this
report, is the implicit assumption that aquifer discharge ceases
immediately upon the cessation of pumping.
RECOVERY AFFECTED BY 7\FTERFLOW
The aquifer discharge to a well is not zero immediately
following the cessation of pumping, because some discharge is
required to refill the well-bore as the water level rises.
Throughout the remainder of this report, the aquifer discharge
to the well-bore after pumping ceases will be referred to as
afterflow. There are, indeed, many situations in which the
afterflow discharge is very small relative to the aquifer dis-
charge during the pumping period and Eq. 5 will yield a valid
estimate of transmissivity T. However, in cases where the aqui-
fer transmissivity is small, the afterflow discharge can be of
the same order as the aquifer discharge during pumping. The
limiting case is that in which the aquifer discharge during
pumping is zero and the only aquifer discharge is afterflow.
This limiting case is represented by the slug test. The re-
sponse function developed below accounts for the effects of
afterflow of any relative importance and incorporates the
44
-------
Cpoper et al. (1967) slug test response and Eq. 5 as limiting
conditions at the two extremes.
The drawdown in a homogeneous, isotropic aquifer caused by
pumping a well at a constant rate is described by the solution
to the following boundary value problem.
2
3 i , 1 3i ¦ S 3i v
T~2 Fair = Tit' r-rw (6)
0 L
6 (<*>,t) = 0 (7)
& (r,o) = 0 , r > r (8)
— w
34 (r ,t) _ d
-------
*w(t)
4ttT
jFp(aft) - Fr(aft)j
t > t
(12)
where
32a f
*2 J.
1 - exp
r4Tt->
lar
x3&x
<*X
(13)
and
32j_ J
If J-
1 - exp
( 4T(t-t )i
I
3 .
X Ax
dX
(14)
The solution repvesented in Eq. 12 gives the drawdown dur-
ing the recovery period and gives full consideration to the re-
lative effects of afterflow. Furthermore it is ar. exact solu-
tion entailing no approximations other than those inherent in
the differential equation (Eq. 6). It is, of course, necessary
to evaluate the integral functions by numerical integration and
to provide the solution in graphical and/or tabular form. This
was accomplished and the results are presented in Table 16 in a
form consistent with the format used by Cooper et al. (1967) to
present their solution for the slug test case. Explicitly, the
ratio of the drawdown at a particular time to the drawdown <6o at
the beginning of recovery is expressed as a function o. the
2
dimensionless recovery time Tt /r for different dimensionless
2 re
pumping times, Tt /r . The response functions are shown graph-
ically in Figure 1. The curve in Figure 1 corresponding to a
pumping period of zero duration is the slug test response. It
is clear that the slug test response is a natural limiting mem-
ber of the family of curves representing the recovery following
different pumping periods. It will be noted that the response
functions in Table 16 and shown graphically in Figure 1 corre-
-4
spond to a value of a equal to 10 . A separate table and fam-
ily of curves must be prepared for each value of a, strictly
speaking. However the response functions are much less sensi-
tive to values of a than to transnissivity and, in practice, it
is probably not necessary to consider other values of a. The
-4
value of a = 10 is typical for confined aquifers unless the
radius of the casing r is much smaller than the radius of the
c
well, r .
W
The procedure for determining the transmissivity by this
method is one of curve matching. Values of are prepared
46
-------
TABLE 16
DRAWDOWN AS A FUNCTION OF RECOVERY TIME FOR
a = 10"4 AND VARIOUS PUMPING TIMES
i / 6
w o
Ttr
2
r
c
Value of
Tto
r2
0
2.5
25
250
1.00
0.0
0.0
0.0
0.0
0.95
0.083
0.130
0.175
0.21
0.90
0.195
0.287
0.375
0.438
0.80
0.470
0.601
0.795
0.941
0.70.
C. 601
0.990
1. 305
1.61
0.60
1.21
1.48
1.93
2.49
0.50
1.72
2.09
2. 72
3.72
0.40
2. ?-9
2.83
3.77
5. 50
0.30
3.33
3.94
5. 40
8.75
0.25
4.00
4.80
6.65
11.9
0.20
4.82
5.95
8.40
18.0
0.15
5.90
7.70
11.0
0.10
7.60
10.6
15.3
as a function of recovery time from the measured values of water
level during recovery. The value of <5q is the drawdown at the
beginning of recovery. These results are plotted on semi-log
paper with 4 /& on the coordinate scale and recovery time on
w o
the log scale. This plot is superimposed upon the response
function curves shown in Figure 1, and adjusted laterally until
a "best fit" is achieved. The best fit may be achieved with a
curve that is visually interpolated between the curves shown.
Once the best.fit is achieved, the dimensionless recovery time
that corresponds to a particular recovery time is noted and the
transmissivity is computed.
47
-------
10
S rw2/rc2 = 10
0-8
0-6
o
in
0-4
0-2
250
I00C
100
•01
0-1
- 4
Figure 1. Dimensionless Recovery Functions for a = 10
-------
Example #1
The1recovery data measured during a single-well aquifer
test in a coal hydrology study are tabulated below, along with
other pertinent information concerning the test.
r = 4.87 cm, r = 4.61 cm, t = 320 minutes, Q = 10.2 t/min
W v O
Totjal vol. pumped = 3220 I, vol. pumped from Csg - 104 I,
Vol. pumped from aquifer = 3116 I, = 15.542 m
RECOVERY
TIME, MIN
0.57 1.05 1.55 2.10 3.42 4.50 5.60 6.48
6 /t,
w o
0.950 0.909 0.861 0.G18 0.721 0.653 0.592 0.549
7.63 8.45 10.27 12.20 .13.48 15.68 20.42 26.02 33.75
0.500 0.470 0.412 0.363 0.338 0.300 0.248 0.209 0.176
38.62 45.10
0.162 0.148
The response functions of Fig. 1 are shown superimposed upon
these data in Fig. 2. A dimensionless recovery time of 0.40
corresponds to an actual recovery time of 1 min for what is
judged to be the "best" fit. As shov/n on the figure, the cal~
2
culated transmissivity is 8.5 cm /min. The best fit is judged
to be on a visually interpolated curve between the curves for
which the dimensionless pumping times are 25 and 250. Once the
data have been matched to a particular curve and the transmis-
sivity calculated, it is possible to calculate the dimensionless
pumping time and check if it corresponds with the curve used for
the match. For the example at hand, the dimensionless pumping
time is TtQ/rc^ = (8.5) (320)/(4.61)^ = 128 which corresponds to
a curve between the two shown.
It is instructive to analyze these data as if the recovery
resulted from an instantaneous drawdown of 15.542 m (i.e., as a
slug test). Using response function curves prepared from the
table presented by Cooper et al. (1967) and superimposing them
on the data of this example yields an estimated transmissivity
2
equal to 4.6 cm /min; a value which is substantially too low.
Furthermore, it is not possible to find a satisfactory match;
the data plot indicating a less rapid rate of rise than predict-
ed by any of the response curves. The less rapid rate of rise,
of course, is because the gradients causing inflow to the well
49
-------
OS
0-6
o
0-4
0-2
Tt,
. (0-40)(4-6!)
= 8-5 cm /min
10
= 25
0-1
1-0
100-0
Recovery Time, minutes
Figure 2. Superposition of Response Functions on Data Plot for
Example 1.
-------
are much less in this test than those which would be produced by
ve'ry rapidly drawing the water level down by 15.542 m. The less
rapid rise of the water level, which is actually due to smaller
gradients, is approximately accounted for by a reduced trans-
missivity when the slug test response functions are used.
It is also constructive to analyze the data of this example
using the standard recovery analysis in which the drawdown is
plotted against the time variable t/t on semi-log paper and the
transmissivity calculated from the slope of the resulting line.
The data and analyses are shown in Fig. 3. Clearly the data do
not form a single straight line and two interpretations are pos-
sible. The steeper of the two lines yields an estimated T much
too low and even the line with a small slope yields a value that
is somewhat low. The failure of the standard analysis to pro-
vide reliable estimates in this case is attributable to afterflow
discharge which is of the same order as the aquifer discharge
during the pumping interval. The effects of afterflow become
smaller with increasing recovery time and the standard analysis
yields increasingly better results as shown by the line with the
smaller slope. In many cases, the slope of the data plot is ill-
defined at large recovery time, however, and use of thia method
is not recommended.
Example £2
The recovery daJ.:a in this example is also from a coal hydro-
logy study, but the transmissivity is approximately an order of
magnitude smaller than for Example #1.
r = 4.87 cm, rc = 4.61 cm, t = 1508 minutes, Q = 2.24 -£/min
Total vol. pumped = 3381 I, vol. pumped from Csg = 172 I,
Vol. pumped from aquifer = 3209 t, = 25.774 m
RECOVERY
TIME, MIN
4.00
6.50
9.40
14.00
19.^0
32.00
40.40
¦6 /i
w o
0.960
0.937
0.913
0. 878
0.842
0.763
0.724
52.00 67.10 100.40 127.80 191.80 237.00 1368.0
0.665 0.606 0.503 0.438 0.332 0.281 0.038
The data and response curves are superimposed as in Example
2
#1 and the transmissivity is determined to be 0.81 cm /min.
51
-------
to
0 8
'As = 1038 cm/cycle
T_ 2-3030 . 2-303(l0-2xIP3)
4ttAs " 4tt(I038)
= i-8 cm2/min
0-6
o
0-4
As = 256-4 cm/cycle
2 303 (10 2 x IP3) .
02
= 7-3cmz/min
4tt (256-4)
1000
100
1-0
t
Figure 3. Standard Recovery Analysis of Data in Example 1.
-------
Analysis of this same data as if it were a slug test yields the
2
too low value of T = 0.36 cm /min.
ALTERNATE METHOD OF ANALYZING RECOVERY DATA
In those cases where the volume of water pumped from the
aquifer is 70 percent or more of the total volume pumped, the
standard recovery theory can be modified to account, approxi-
mately, for the effects of. afterflow. Figure 4 shows a schematic
of the aquifer discharge to the well-bore. The declining dis-
charge indicated after pumping ceases is the afterflow.
The basis for the analysis procedure to be developed here
is the principle of superposition that permits the drawdown in
response to a time varying discharge to be calculated by con-
volution. An appropriate form for a discretized variation in
aquifer discharge is given by McVJhorter and Sunada (1977) and by
/ ax, t > tn (15)
Matthews and Russell (1967).
4 ~ ^ i=l AQi
where
Sr 2
w
4T(t-ti)
AQ. - Q. - Q • ,
i i l-l
For the purposes of this analysis, the integral in Eq. 15 will
be approximated by In (1/6^) - 0.5772.
If it is supposed that aquifer discharge truly becomes zero
at t = tQ, then the drawdown at t = t^ is given by
Q ( 4Tt, ) Q ( 4T(t -t ) )
4i * 4ifr ln —i" °-5772 - dH1" 2 - °-5772
1 t «. f w»,
^o ^1
*1 = ln t-=t" ' (17)
1 C1 ro
from which one can compute the transmissivity T, using the mea-
sured value of drawdown at time t^. However, since aquifer dis-
charge did not cease at t , the transmissivity so calculated
will not be the true value and, therefore, is given the subscript
53
-------
pump dischorge = Q(
c
QJ
o*
a
•j
«/>
i5
pumping
ceases af f.
3
cr
<
0
<4
2
3
O
Time
Figure 4. Schematic Drawing of Aquifer Discharge as a
Function jf Time
-------
corresponding to the time for which it is computed. It is not
expected that is a good approximation to the actual trans-
missivity because a very significant afterflow has been neglect-
ed.
It is now supposed that aquifer discharge ceases r.t t = t^
and the drawdown at t = is given by
C> t_ Q. t,-t
*2 = 4iT7 ln t--t + 4tFt7 ln t.-t. (18)
2 2. O 2. IX
obtained by application of Eg. 15 and the logarithmic approxi-
mation to the integral. Again, can be calculated from Eq. 18
by using the measured drawdown at t = t2 and a value of cal-
culated from the average rate of rise of the water level in^the
well during the period from t to t^. Explicitly Q.^ = it rc
(/(t^-tQ) which is generalized to become
2 (4i-l"'4i) /in,
Qi = "c t.-t. . * (19)
• x i-l
As before, it is not expected that T2 is a good approximation to
the true value but it does represent an improved estimate as
compared to T^. This is true because a somewhat less signifi-
cant afterflow was neglected.
If the above process is continued up to t = t and the re-
sult solved for T :
n
Tn " 4v&~ jQo ln r=t" + °i ln F2^1! ' l20)
n | no i=l n l J
As explained previously, the Tn calculated from Eq. 20 for each
n are only approximate because the afterflow has been truncated
in each case. The larger the value of n, the better the approx-
imation and the nearer the calculated T will be to the actual
value. This is because the magnitude of the afterflow that is
neglected becomes less with increasing n.
Two questions immediately arise: 1) what accuracy can be
expected from this analysis and 2) how is the accuracy affected
by the size of time steps and the number of steps included in
the computation. These questions were investigated by generat-
ing recovery data from the exact analytical solution presented
55
-------
previously in this report, calculating T from Eq. 20 for various
sizes of the time increment and for various recovery times, and
comparing the calculated T with the actual value used to gener-
ate the data. It is concluded that:
a) t should be the time at which recovery is at
least 90 percent.complete.
b) the recovery tine considered (i.e. t -t ) should
no
be divided into a minimum of 5 equal time intervals.
c) in tests which 80 percent of the total volume
pumped is derived from the aquifer, and when
guidelines a and b are followed, the approximate
calculation will introduce an error of less than
10 percent.
The above statements are demonstrated for a particular case
using the exact solution to generate recovery data under the
following conditions:
2
T = 15.7 cm /min, tQ = 100 min, Qq = 14.16 £/min, rw = rc =
7.S25 cm and S = 10
For this situation, the volume pumped is 1416 I, the volume
pumped from casing storage is 183 I, and the volume pumped from
the aquifer is 1233 I or 87 percent of the total volume pumped.
The drawdown data as a function of time using 10-minute time
intervals are shown in Table 17. The discharge values also
shown in the table were calculated from Eq. 19.
TABLE 17
RECOVERY DATA GENERATED FROM EXACT SOLUTION - 10-MIN INTERVALS
Time, min
¦6 , cm
3
P, cm /min
100
926.6
14160
110
4 87.0
8674
120
297.8
3733
130
200.1
1769
140
152.9
1090
150
120.6
637
1G0
94.7
510
From Eq. 20, the transmissivity is calculated:
56
-------
1 = 4W94.7) {14160 ln M1 + 6674 ln § + 3733 ln Iff
+ 637 ln = 14.9 cra^/'rain.
This value of T is just 5 percent less the exact value of lb.7
2. .
cm /mm.
Table 18 shows the corresponding data using 5 nunute time
increments.
TABLE 18
RECOVERY DATA GENERATED FROM EXACT SOLUTION - 5-MIN INTERVALS
Time, min
6 , cm
Q, cm"Vmi
100
926.6
14160
105
656.1
10672
110
487.0
6675
115
372. 5
4517
120
297.8
2948
125
246.2
2036
130
208.1
1501
135
178. 7
1161
140
152.9
\0 20
145
134.9
708
150
120.6
567
155
107. 7
509
160
94.7
508
The calculated value of . is:
T = 4,(94.7) f14160 ln 10672 ln S+ 6675 ln If + •••
+ 509 ln ^j = 15.1 cm^/min,
which is very close to the value calculated using 10-minute tin
intervals.
57
-------
Example #3
The procedures of this section will now be used to deter-
mine T for the data rriven in Example #1. The drawdown data is
first plotted to facilitate determining the drawdown at selected
values of time. The results for 5 minute intervals are shown in
Table 19.
TABLE 19
RECOVERY AND AFTERFLOW DISCHARGE FOR TEST OF EXAMPLE #1
Time, min
-------
TABLE 20
RECOVERY AND AFTERFLOW DISCHARGE FOR TEST OF EXAMPLE #2
Time, min
6 , cm
Q, cm"Vmin
1508
2577.4
2240.0
1608
1306.7
848.0
1708
829.9
318,0
1808
605. 7
149.7
1908
463. 9
94.7
2008
366.0
65.4
2108
304.1
41.4
2208
250.6
35.7
The value of transmissivity calculated from Eq. 20 is
T ¦ 4itt56.(S) {2240 ln TOT + 648 ln 5TO + ••• + 41-4 ln irol
2
= 0.9 cm /min
2
This value compares with 0.81 cm /min determined by curve match-
ing in Example #2.
The above developments and example applications demonstrate
how recovery data from single-hole aquifer tests can be analyzed
to provide estimates of transmissivity- The developments of
this section do not overcome some of the disadvantages inherent
in single-hole tests. For example, single-hole test data rarely
provide a reliable estimate for storage coefficient and may be
significantly affected by well losses. Well losses in properly
constructed wells penetrating low transmissivity aquifers can be
expected to be less important than in aquifers with high trans-
missivity, however. Single-hole test data do not provide i.nform-
mation on anisotropy or leakage through aquitards. When these
are important, multi-well tests must be used.
59
-------
SECTION 6
ANALYSIS OF SELECTED FLOW PROBLEMS IMPORTANT IN
SURFACE MINING HYDROLOGY
Aspects of ground-water hydrology that may be important dur-
ing the mining phase are several and include: 1) the quality and
quantity of inflows to pits, shafts or other excavations, 2) the
resultant lowering of the piezometric surface in affected aqui-
fers, 3) inflow to the mine from fault zones, 4) the lowering of
water levels in infrequently recharged alluvial aquifers adja-
cent to the mine, and 5) sustained inflows from frequently re-
charged alluvial aquifers adjacent to the mine. In this section,
the ground-water hydraulics of several such problems are analyz-
ed. Each mine encounters problems that are specific to that
particular project, and it is not possible to provide general
solutions that will be applicable everywhere. The intent of
this section is to provide approximate analytic analyses and
solutions that are specifically oriented toward problems known
to have been encountered in surface mining projects.
FLOW TO AN' ADVANCING PIT
The case analyzed in this sub-section is that in which a
surface mine is to be initiated by excavating the box cut through
aquifer materials that extend laterally to great distances from
the area to be mined. The major questions to be addressed are
the inflows to the first cut as the pit advances and the extent
to which the piezometric surface is lowered in affected aqui-
fers. The analysis for a particular aquifer intersected by the
pit is given here. This aquifer may be the coal seam itself or
any other water-bearing stratum between the coal seam and the
land surface. If more than one independent aquifer is incised
by the pit, the analysis can be applied to each aquifer, indi-
vidually .
Figure 5 shows an idealized cross-section drawn in a plane
normal to the axis of the advancing pit. Only one side is shown
in the figure; it being assumed that the conditions are identi-
cal on both sides of the pit. The initial piezometric surface
is height H above the top of the aquifer. When the pit incises
the aquifer, a portion becomes unconfined adjacent to the pit
and the height of the water table above the bottom of the aqui-
fer is h in this unconfined portion. The distance to which the
60
-------
Ground Surface
Piezomefric
Surface,
Highwall
Water Table
5 Aquifer
Figure 5. Definition Sketch for Flow to the First Cut.
-------
un,confined portion extends into the aquifer, du, becomes larger
with increasing time. The fact that the pit is advancing with
time is ignored for the moment and flow to the pit in the plane
shown in Fig. 5 is analyzed.
The procedure used to obtain a solution is that of a suc-
cession of steady states. In this method, the discharge is
assumed to be independent of the space variable at a particular
instant in time, thus permitting one to calculate the distribu-
tion of piezometric head as if the flow were steady. As the
discharge changes with time, the distribution of head also
changes, but in accordance with that predicted from the appropri-
ate steady state equation. This method is approximate, of
course, but has the important property that mass continuity is
accounted for exactly. The method has been used by Bear (1972) ,
Polubarinova-Kochina (1952) and Arawin and Numerov (1953, 1965),
among others.
At a particular instant in time, the volume of water that
has been produced into the pit in a unit length of pit is given
by the exact expression
d d +d
u c u
Vd = Sya f (k~li)dx + S J (HQ-H)dx (1)
d
u
where = volume drained to pit per unit length of pit
S = apoarent specific yield
ya
S = storage coefficient
b - aquifer thickness
h = height of the water table above floor of aquifer
Hg = piezometric height above top of aquifer
d = extent of unccnfined zone
u
d = distance between the boundary of the confined
° and unconfined zone and the point at which
H = Hq
In the unconfined zone, the flow is given by
q = Kh 3— 0 < x < d (2)
dx — — u
62
-------
where q = discharge to the pit per unit length of pit
K = hydraulic conductivity (permeability)
and other symbols are as defined previously. The succession of
steady states permits Eg. 2 to be integrated
X
q J" dx = K £ h dh (3)
0
or h = 0 £ X i du
For x = Eq. 3 gives
Kb2 ,c,
q = 2d" (5)
u
In a similar way, the distribution of head in the confined zone
is computed to give
H = (x-d ) , d < Y < d + d (6)
T u u — K — u c
T «o
and q = —^— (7)
c
Equations 4 and 6 are substituted into Eq. 1 and the in-
tegrations performed to yield
bd d H«
V, = S + S CJ) (8)
d ya 3 2
The variables du and dc in Eq. 8 are replaced in terms of q by
substituting with Ecs. 5 and 7 resulting in
(S Tb2 STH 2) .
Vd/- +^-\k (9)
It is now noted that dV^/dt = q so that Eq. 9 can be written as
the ordinary differential equation
IS Tb2 STH-2) , ,
+
-------
Integration subject to q = » at t = 0 yields
¦1
S Tb2 ST!!,.2 1 15 %
~^I2 + I ' <">
Should the first cut pit be established in its entirety at
the instant t = 0,. then Eq. 11 could be multiplied by the length
of the pit to yield the inflow discharge from one side. The
entire pit is not established instantaneously, of course, and
the discharge so calculated would be much too large. To account
for the fact that the length of pit increases with time, it is
noted that Eq. 11 gives the discharge per unit length of pit
from a slice of aquifer that became exposed in the pit at time
zero. Thus, in Eq. 11, t is the time interval over which the
aquifer has been exposed in the pit. This time interval is more
conveniently expressed by t-T wherein t is time measured from
the initiation of pit excavation and t is the time at which a
particular slice of.aquifer becomes exposed in the pit. Kence,
(12)
where A =
The differential discharge to the pit from a differential slice
of aquifer is, therefore:
dQ = q dy = A (t-x) ^ dy (13)
v;here y is a coordinate measure along the pit axis from the
point of beginning, and dQ j.s the differential discharge.
It is supposed that the length of pit increases linearly
with time at an average rate of advancement equal to R. In this
case
y = Rt (14)
from which dy = R di (15)
and Eq. 13 becomes dQ = A (t-x) ^ R dx . (16)
The discharge to the pit is obtained by integrating over x from
0 to t to yield
which holds for all times during the advancement of the pit
(i.e. for t <_ li/R, where L is the maximum length of the pit) .
64
-------
Once the pit has reached its maximum length and elongation cea-
ses, the discharge is calculated by integrating Eq. 16 over t
from 0 to L/R and results in
o'. ^ jt>: (tw.). * > * (l8,
The discharge to the pit decreases sharply once the maximum
length has been attained as will be shown in a subsequent ex-
ample. The discharge in Eqs. 17 and 18 must be multiplied by
2 to obtain the discharge fi^m both sides of the cut.
. All of the equations required to calculate the distance to
which the piezometric surface remains undisturbed are now avail-
able. Specifically the distances d and d are given by
TbJS Tb2 STH^C^ H y , t (19)
- T 12 + I } R -
!
du
and
dc TK0 } 12
SyaTb2 t STH„2 t (20)
The total distance to the edge of the disturbed zone is the sum
of d and d .
u c
Probably the most limiting aspect in the foregoing analysis
is the implicit assumption that flow occurs only in plcines nor-
mal to the long axis of the advancing cut. This is not strictly
true, of course, but permits the estimation of discharge in a
relatively simple manner that does account for all the major,
first-order factors influencing the discharge. The discharge to
the pit from aquifers exposed in the highwa.ll as calculated from,
the above equations will be conservatively 1=,.rge.
Example #1
The box cut of a surface coal mine will incise an unconfin-
ed aquifer perched above the target coal seam. The saturated
thickness of the aquifer is 18 m2and the transmissivity and ap-
parent specific yield are 0.92 m /d and 0.05, respectively. The
5 3
mine is expected to produce coal at the rate of 3.62 x 10 m /yr
from a pit that is 30 m in width penetrating a coal seam that,
averages 2.09 m thick. Under these circumstances the average
rate of advance of the box cut is
„ _ 3.62xl05 in3/yr _ , c Q ,,
(365 d/vr) (30~~mj (2.09 m) " ^ '
The maximum length of the pit is 915 m. The inflow to the box
cut and the distance to which the water table will be effected as
functions of time are desired.
65
-------
Because the aquifer is unconfined, the parameter Hn is zero
in all of ;the equations developed above. Other parameters are
T = 0.52 m^/d, S = 0.05, b = 18 n, L = 915 m, and R = 15.8 m/d.
ya
The full pit length is achieved in 915/15.8 = 58 d. The dis-
charge computed from Eq. 17 for t < 58 d and from Eq. 18 for t >_
58. The results are shown in Figure 6, and include inflow from
both sides of the pit.
FLOW TO AN ADVANCING PIT INITIATED ON A CROP LINE
A schematic cross-section of the physical conditions assumed
to exist for this case is shown in Figure 7. The first cut is
assumed to be made near the crop line and extended to full length
along the strike of the coal bed. The second cut is made in the
down-dip direction from the first cut and spoil is cast in the
up-aip direction. In this case it is necessary to account for
both the elongation of the cuts and the fact that the highwali
is advancing down-dip with time. An important effect of the
sloping aquifer is that the drawdown <&0 (i.e. the difference be-
tween the initial piezometric surface elevation and the head at
the face of the highwali) increases incrementally as the high-
wall advances down-dip.
If the wp.ter-bearing stratum intersected by the cut is an
unconfined aquifer then in an aquifer slice normal to the cut,
the volume drained per unit length of pit is
d
vd = sya f i ¦ (21)
0
where i> is the drawdown as shown in Fig. 7. Under consideration
now is the box cut only, or actually, the first for which the
bottom of the pit will be below the initial water table eleva-
tion. The drawdown produced at the highwali by this cut is
If the effect of the sloping aquifer floor on the saturated
thickness h is neglected, the distribution of '4 with x can be
calculated as in the previous subsection, substituted into Eq.
21, and the integration carried out to yield
iMd
V, = S ¦ * U . (22)
d ya 3
K^oi)2
Also du = ¦ 2^X (23)
The drawdown is a constant for the first cut so the '
66
-------
600r
500
4O0-
E 300
200
100
60
70
80
90
100
20
30
40
50 (
Time, days
Figure 6. Calculated Inflows to Box Cut for Example 1.
-------
Line
Ground Surface
Highwall
Spoil
Initial Water Toble^
1
Figure 7. Schematic Physical Model for Flow to a Pit Initiated
on a Crop Line.
-------
results of the previous sub-section apply and the discharge to
the first cut is
^ -a^Ol)15 t's t < 4 (24)
and
°1 = 2R j 12 ] * ' ' " ^
Q1 = 2R (Sy'l2 "I ( ^ " "-V'l'J '
(25)
t > Lj/R^
where equals length of the first cut and R^ is the rate at
which it is elongated.
The second cut will be made down-dip from the first one and
will produce an additional increment of drawdown tx + t2 (27)
In a similar way the incremental discharge for each successive
cut is calculated. The total discharge at any time is the sum
of the incremental discharges for each cut. The incremental
drawdown caused by advancing the pit down-dip is computed from
= W tan G (28)
where W is the width of the cut and 8 is the dip angle.
The calculations outlined above are simplified substantial-
ly for the special case in which t^ = t2 ... tn, and 4^^ = =
... 40n. In this case the total discharge to the pit becomes
Q = 2R (-^Y2—• (29>
G9
-------
Example #2
Mining will be initiated near the crop line of a coal-seam
that dips 6 degrees. The mine will be located in a valley that
narrows in the down-dip direction and the length of successive
cuts, oriented parallel to the strike, will become less as the
valley narrows. A schematic plan and section view of the area
to be mined is shown in Fig. 8. The width of each cut will be
30 m and th*1 rate of increase of length will be 10 m/d. The
length of the first cut will be 600 m, and the length of each
successive cut will be 30 m less than the previous one. Other
2
required parameters are S = 0.05, K = 0.1 m/d, T = 2.6 m /d,
j u
4q1 = 4q2 = ••• = 30 tan 6^ = 3.153 m.
Table 21 shows the computation for times up to the comple-
tion of the 4th cut. For this case t^ = 60 d, t2 = 57, t^ = 54
t4 = 51 and so on. The discharges indicated under the columns
headed were computed from
Q3 = 6.563 (t-117) 117 < t < 171
Q3 = 6. 563 (T-117)*5 - (t-171)55 t >_ 171
and was computed from
.-¦s
Q4 = 6. 563 (t-171) 171 <_ t < 222
FLOW TO A PIT ADVANCING PARALLEL TO AN ALL'JVIAL VALLEY
Ground water may be withdrawn from an alluvial aquifer into
the mine through permeable strata that connect the mine and the
alluvium. An idealzed crosr-section is shown in Fig. 9. It is •
assumed that the transmissivity of L.he alluvial aquifer is large
relative to that of the strata providing hydraulic communication
between the alluvium and the mine, so that a constant head is
maintained as x = a. Unlike the situations analyzed previously,
a steady state flow from the alluvium to the mine can be estab-
lished following a transient period during which the effects of
inflow to the mine propagate outward from the hicjhwall. The
analysis is carried out by neglecting the transient period.
At steady state the discharge per unit of pit length through
the confined portion is equal to that through the unconfined por-
tion so that
TH,
a-d
70
(30)
-------
Limits of
Mining
Topogrophic
Contours (meters)
10
600m
Highwoll for
3rd 2nd
1st cut
Water Table
Aquifer
Cool Seom
Figure .8. Schematic of the Physical Conditions for Example 2.
-------
TABLE 21
COMPUTATION OF DISCHARGE TO THE MINE OP EXAMPLE #2
CUT
NO.
TIME
DAYS
Ql.
EQ. 24
Q1
EQ. 25
°2
EQ. 26
°2
EQ. 27
Q3
°3
°4
^total
m3/d
1
0
16
36
60
26
39
51
0
0
0
26
39
51
2
70
90
34
26
21
36
55
62
117
21
50
0
0
71
3
132
152
. 20
18
30
24
25
39
. 75
81
171
17
21
48
0
0
86
4
190
16
19
27
29
91
210
15
17
22
41
95
222
14
16
20
47
97
from which du = 2^b+b (31)
so that the expression f.:.r the steady discharge per unit of pit
length is
Tb (2H<>+b)
cis = 2- HnH (32)
where qg is the steady discharge per unit of pit length. As done
previously
Tb P"
dc = qs
-------
Aluvial
Aquifer
Figure 9. Schematic Drawing of Flow to a Mine from an
Alluvial Aquifer.
-------
the pit becomes constant after the maximum length is attained and
is given by
Q . • <35)
Discharge to the pit will be somewhat greater during the
transient phase than predicted by the above developments for the
steady phase. The approximate time period required for the
steady state to develop can be estimated by calculating the time
at which the transient discharge per unit of pit length becomes
equal to the steady value. From Eq. 11 with q given by Eq. 32
2 2
I 1^1
s
i 2 fS Tb STHn )
Tb(2H0+b) | | yi2 + ~A J (36)
where t is the time required to reach steady state after a
particular point has been exposed in the pit.
Example #3
2
A stratum 11 m thick with a transmissivity of 2.5 m'/d sub-
crops in the alluvium associated with a perennial stream. The
stratum will be cut through by the pit of a surface mine. The
pit will be advanced to a maximum length of 600 m parallel to
the stream at a distance of 50 m from the edge of the alluvial
aquifer. The elevation of the witer table in the alluvium is
3 rr. greater than the elevation of the top of the subcropping
stratum. Estimate the maximum steady discharge form the alluvium
to the pit.
The maximum steady discharge will occur when the pit has
reached its maximum length and is given by
Q =
Tb /2Vb
T~ (—EE—
(2.5) (11) /2(3)+ll
(50)(11)'600
255 m3/d
DRAINAGE OF ALLUVIUM OR FAULT ZONE RECEIVING NO RECHARGE
The flow problem under consideration in this subsection is
similar to that analyzed in the previous one. If the alluvium
74
-------
receives no recharge, however, it will drain and does not con-
stitute a constant head boundary. Figure 10 shows the physical
model for the problem of interest in this subsection. The in-
frequently or non-recharged alluvial aquifer shown in Fig. 10
could also be a fault zone or even a lake or reservoir.
If the short period of transient flow, in which water is
derived from storage in the connecting stratum, is neglected
then from the previous subsection
- £(!*«)
H > 0
(37)
where all symbols are as previously defined. In Eq. 37, the
head H is a function of time. The flow through the connecting
stratum as given by Eq. 37 must also be given by
= - W S ^
ya dt
(38)
where W is the width and S is the apparent specific yield of
ya
the alluvial aquifer or fault zone. If the water is supplied by
a lake or reservoir, then S = 1. Equation 38 expresses rela-
ya
tionship between the decline of the water level and the rate of
drainage. Equating the discharges in Eqs. 37 and 3C and integra-
ting the resulting ordinary differential equation yields
(|+«) = (b/2 ~ H0) (39)
where Hq is the initial value of H. The result in Eq. 39 is sub-
stituted into Eq. 37 to yield
* = I (l+ Ho) fcxp (s^r;) !40)
v/hich gives the discharge per unit length of pit.
The effect of the advancement of the pit is accounted for
by replacing t with t-T and noting that dQ = Rq dT, where R is
the.mean rate cf elongation of the pit. Thp results are
and
75
-------
Highwoll
Figure 10. Schematic of Physical Model for Drainage of an Aquifer
or Fault Zone.
-------
Q = M J ~ H0) w Sya jexp ^
Example #4
- exp
(42)
The equations developed above can be used to compute the
rate of filling of a mined area just as well as the drainage of
an aquifer or fault zone. Consider the physical situation shown
in Fig. 11, in which flow is occurring from the alluvial aquifer
into the backfilled mine irea. This flow will cease when the
water taDle in the spoil has risen to the elevation of that in
the alluvial aquifer.
For the following conditions, estimate the time required
for the water table in the spoils to rise to within 1 m of the
elevation of the water table in the alluvial aquifer: = 20 m,
2
T = 2.5 m /d, S (spoils) = 0.1, a = 50 m, and the slope of the
y a
impervious layer is 8 degrees from the horizontal.
In this problem, no consideration of an advancing pit is
required and Eq. 39 is the appropriate starting point. Because
the aquifer is unconfined, b = 0 and T is an average value being
equal to KHQ/2. Also, the value of W to be used in the computa-
tions must be an average value. The appropriate average value
of W is given by
W = i —*«¦ = 71 m.
tan8
V.'hen the water table in the spoils is within 1 m cf that in the
alluvium, H = 1 m. Thus, from Eq. 39
aWS H
ya
t = (50) (71) (0.1) in ,('£) , „25 days
DISCHARGE FROM SPOIL IK RESPONSE TO VERTICAL RECHARGE
Under consideration in this subsection is the accumulation
and discharge of ground water from a spoil bank in response to
vertical infiltration. The physical model is shown schematically
in Fig. 12. Solutions to the linearized Boussinesq equation
77
-------
Buried Hiahwoll
Spoil
Backfiii
H(t)
Alluvial
Aquifer
Figure 11. Schematic Physical Model for Example 4.
-------
KO
.Recharge Rale = i
I I I I I I I I I I I I
/ // // // /
Figure 12. Physical Model for the Buildup and Discharge of
Ground Water in a Spoil Bank.
-------
provide approximate results of engineering utility for this
case. The linearized Boussinesq equation is
32h
Jya 3x'
3h
Jt
(43)
where T is the product of permeability K and an averaged satu-
rated thickness that will be calculated subsequently.
A fundamental solution to Eq. 4 3 for boundary conditions
of h(0,t) = 0 and 3h/3x = 0 at x = a is
00
h =
41 V"*
irS La
ya n=l,3,5.
(1/n) sin 5^ exp
6 3
IT Tt
(44)
where I is an increment of recharge (volume per unit area) that
reached the water table at t = 0. Thus, Eq. 44 represents the
fundamental response function to a finite increment of recharge
that reaches the water table instantaneously at t = 0.
Maasland (1959) showed how the water table response to in-
termittent recharge can be calculated from Eq. 44. McWhorter
(1977) used this approach to calculate the v/ater-table response
to an arbitrary sequence of recharge events that occur in a time
interval t and are then repeated indefinitely. Of particular
interest in mine spoil hydrology is the long term pattern of dis-
charge and water table fluctuations that can be anticipated in
the spoil bank.
An arbitrary sequence of instantaneous recharge events, 1^,
l2,...Im> are considered. This arbitrary pattern of recharge
occurs in one year and is assumed to occur in an indentical
fashion in all subsequent years. It can be shGwn (McVJhorter,
1977) that the water table elevation hc at the point x = a is
given by
m
E
i=l
I.
x
S
ya
M.
l
(45)
where
Hi
7 £
n=l,3,5..
(_!) (n-l)/2 {exp
r t . i
exp
- <»„V <4
-exp
n 0
(46)
80
-------
2 •
n it T ....
.= 727- ^46)
4a S
ya
¦tQ = period of repeating recharge pattern - 1 year
and t^ is the time interval between the time of interest and the
time at which event i last occurred. Thus, the time variable t.
can take on all values between 0 and tg. 1
The discharge per unit length of spoil bank can be obtained
using the above theory and is (McWhorter, 1977)
m °° r 1
_ 2T -C-* ^ fexp L~(lJntOTi/to)
2
For values of (Tt.)/4a greater than about 0.06, the first
1 y a
terms in the infinite series in Eqs. 46 and 47 dominate and the
approximate discharge formula
irTh
q = (48)
is derived by combining Eqs. 46 and 47 using the first terms
only. The mean value of transmissivity, calculated from a time
and space weighted averaging procedure (McWhorter, 1977) is
given by
T = jp + | {(Kb)2 + 4P° K | ^ (49)
where P = average recharge rate =
Figure 13 is a plot of the response function M. for various
2 1
values of the parameter Ttft/4a S . In most applications it
u y a
will be found sufficiently accurate to compute the discharge
from Eq. 48 after h^ has been computed from Eq. 45 using Fig. 13.
The procedure is demonstrated in the following example.
Example #5
Analysis of precipitation and mateorological records, coupl-
ed with a water balance calculation on the root zone of a spoil
bank, yield the following estimates for long term mean recharge
81
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20
Tt,
2 10
0-8
<0-11
£>-125
0-6
,015
0-4
017!
OZ
0-2
0-25
0-4
02
0-6
08
0-4
Figure 13. Dimensionless Water Table Height
Resulting from Periodic Recharge
of Magnitude 1^.
82
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in the reclaimed mine: 1^. = 10 cm in April and = 4 cm in
May. The relatively large recharge in April is due to snow melt
and recharge in May is due to rain during the spring period when
the moisture content in the root zone is still high from the
snow melt. In this particular example, precipitation events
during other periods of the year are insufficient to brir.? the
depleted soil moisture content up to a level where deep pet eola-
tion occurs. All infiltration from such events is consumed by
evapotranspiration. All of the estimated recharge in April is
assimulated into a single value, assumed to occur instantaneous-
ly on April 15. Likewise, the entire 4 cm of May recharge is
assumed to occur instantaneously cn May 25. As explained pre-
viously, these are regarded as long mean values that are assumed
to repeat year after year.
The spoil banv for this problem is in the form of a rec-
tangle 360 m long (L = 360 m) and 100 m wide (a = 100 m). The
permeability of the spoil is 1.3 x 10 ^ m/d, S ¦ = 0.05, and b =
y "
2 m. The first step in the computation is to calculate the mean
transmissivity from Eq. 49 with P = (10 + 4)/365 = 0.0411 cn/d =
4.11 x 10"4 m/d.
T = U.3X1Q-1) + 1 ,2.6^0-1)2
^4(4.11»10-')(100)2(1.3x10-1)|1' . „ „ ^
Therefore, the curve parameter is
Tt0 = (0.57) (365) = 0>1
4a2S 4(1002(0.05)
The computation of discharge is demonstrated in Table 22.
The calculation was begun or. April 15, immediately follow-
ing recharge event number one. Thus, zero time had elapsed
since event one ofipjjfored., but 325 days had elapsed since the
May 25 recharge eveltfc&oceurred in the previous year. On April
25, ten days had e£ap§|ed since the April 15 event and 335 days
since the May 25 evetft. Once and t2 were determined for each
date, the ratios r^/tg and "^/t^ were computed and the corre-
sponding values of and read from Fig 13. The values of
hc^ were obtained by multiplying by and h^ =
S3
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TABLE 22
DISCHARGE FROM SPOIL BANK FOR EXAMPLE #5
Date
T1
M1
cl
T2
M2
hc2
h
c
Q
O
. i
Days
m
Days
m
m
m
m /d
Apr.
15
0
1.755
3.51
325
0.842
0.67
4.18
13.5
,
25
10
1.735
3.47
335
0.820
0.66
4.13
13.3
May
5
20
1.715
3.43
345
0.800
0.64
4.07
13.1
15
30
1.695
3.39
355
0.78C
0.62
4 .01
12.9
25
40
1.675
3.35
0
1.755
1.40
4 .75
15. 3
June
4
50
1.650
3.30
10
1.735
1.39
4.69
15.1
14
60
1.630
3.26
20
1.715
1.37
4 .63
14.9
24
70
1.602
3.20
30
1.695
1.36
.4.56
14.7
July
4
80
1.575
3.15
40
1.675
1.34
4.49
14.5
Aug.
3
110
1.485
2.97
70
1.602
1. 28
4.25
13.7
Sept.
2
140
1.375.
2.75
100
1.512
1.21
3.96
12.7
Oct.
2
170
1.272
2.54
130
1.415
1.13
3.67
11.8
Nov.
1
200
1.170
2.34
160
1.310
1.05
3.39
10.9
Dec.
1
230
1.085
to
-J
190
1.207
0.97
3.14
10.1
31
260
1.000
2.00
220
1.110
0.39
2.89
9.3
Jan.
30
290
0.925
1.85
250
1.028
0.82
2.67
8.6
Mar.
1
320
0.855
1.71
280
0.950
0.76
2.47
8.0
31
350
0.790
1.58
310
0.878
0.70
2.28
7.3
(l2/Sya)M2. Corresponding values of hc^ and were added to
obtain hc from which the discharge values were computed using
Eq. 48 multiplied by the length of the spoil bank (i.e. 360 m).
Use of Eq. 48 to calculate the discharge results in predicted
values of q that are less than the values calculated from the
more rigorously correct Eq. 47.
One might well question the use of instantaneous recharge
events when it is known that the recharge accumulates gradually
over a finite period of time. However, the above computation
illustrates the tremendous damping effect of the ground-water
reservoir. . Little substantive change in the discharge hydro-
graph would result if the same quantities of recharge had been
distributed over a period of a few days, but the computation
procedure v/ould heve been complicated significantly.
-------
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86
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