Estimation of the Required Amount of Hydrological Exploration in Lignite Mining Areas on the Basis of Hypothetical Hydrogeological Models

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

ESTIMATION OF TBE REQUIRED AMOUNT OF

HYDROLOGICAL EXPLORATION IN LIGNITE MINNG AREAS ON THE BASIS OF RYPOTHETICAL KYDROGEOLOGICAI. MODELS

S. Kaden F. Reichel L. Luckner

December 1985 CP-85-47

CoLLuborative P u p e r s r e p o r t work which has not been performed solely at t h e International Institute f o r Applied Systems Analysis and which has received only limited review. Views o r opinions expressed herein do not necessarily r e p r e s e n t those of t h e Insti- t u t e , i t s National Member Organizations, o r o t h e r organizations supporting t h e work.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

PREFACE

The Regional Water P o l i c i e s p r o j e c t of IIASA focuses on intensively developed regions where above all groundwater i s t h e integrating element of t h e environment. Our r e s e a r c h i s d i r e c t e d towards t h e development of methods and models t o s u p p o r t t h e resolution of conflicts within such socio- economic environmental systems. Complex decision s u p p o r t model systems a r e under development f o r a region with intense a g r i c u l t u r e in The Nether- lands and f o r an open-pit lignite mining a r e a in t h e GDR. In both cases t h e modeling of t h e groundwater r e s o u r c e s system i s of fundamental impor- tance. Up to now i t h a s been assumed t h a t t h e groundwater r e s o u r c e s sys- t e m i s e x p l o r e d in d e t a i l and all d a t a needed f o r policy analysis are avail- able with sufficient a c c u r a c y .

Generally t h i s assumption is not fulfilled due t o t h e limits of explora- tion. Hydrogeological exploration, above all based on exploration drillings resulting in point samples only, is a v e r y costly t a s k . Consequently hydro- geological p a r a m e t e r s can b e explored with c e r t a i n precision only, depend- ing on t h e number of exploration drillings. F o r any decision being based on t h e s e p a r a m e t e r s t h i s uncertainty h a s t o b e t a k e n into t h e account. Usually i t comes to "pessimistic" decisions o r with o t h e r words t o a n "overdimen- sioning" of t h e c o t r o l units of t h e system under study. A compromize h a s to be fund between t h e amount of exploration (its precision) and t h e econmic losses d u e t o "pessimistic" decisions.

This p a p e r d e s c r i b e s a n attemp to solve such problems with special r e g a r d t o lignite mining areas. For t h e f u t u r e t h e extension of this a p p r o a c h and i t s inclusion into complex decision s u p p o r t systems becomes realistic.

S e r g e i Orlovski P r o j e c t L e a d e r

Regional Water Policies P r o j e c t

-

ⁱⁱⁱ

-

ABSTRACT

Mine drainage is a necessary but very costly precaution f o r open-pit lignite mining in sandy aquifers. Consequently, t h e minimization of t h e number of drainage wells and t h e i r optimal operation become important t a s k s in designing mine drainage systems. Comprehensive groundwater flow models have to b e used, both, f o r t h e design of drainage wells, and f o r t h e analysis of water management s t r a t e g i e s in mining a r e a s

.

The a c c u r a c y of computations with such models depends on t h e precision of t h e underlying hydrogeological informations. In o r d e r t o g e t these informations detailed and costly hydrogeological explorations have t o b e done in t h e mining regions.

The basic informations a r e obtained using exploration drilling. The cost f o r hydrogeological exploration a r e approximately a linear function of t h e number of exploration bore holes. Therefore t h e reduction of drilling gets a key r o l e in reducing costs of exploration. This might b e done by:

-

increased use of geophysical exploration methods,

-

complex analysis of exploration r e s u l t s using mathematical statistical methods,

-

precise estimation of t h e required amount of hydrogeological informa- tions.

The p a p e r describes a mathematical approach t o support t h e complex deci- sion making p r o c e d u r e of estimating t h e optimal amount of hydrogeological exploration with r e s p e c t t o a given mine drainage goal.

2. Methodological Approach

2.1

Hydrogeological Schematization and P a r a m e t e r Model 2.2 Design of t h e Drainage System

2.3 Geohydrological Calculation with Hypothetical Models 2.4 Economic Evaluation Depending on Exploration P r e c i s i o n 2.5 Optimal E x ~ ! o r a t i o n P r e c i s i o n

3. Test Example

4. Concluding Remarks R e f e r e n c e s

ESTKblATION OF

THE

REQUIRED AMOUNT OF

HYDROLOGICAL EXPLORATION IN LIGNITE MINING AREAS ON THE BASIS OF HYPOTHETICAL HYDROGEOLOGICAL MODELS

S.

ade en'.

F. 13eiche12 and L. ~ u c k n e r ~

1. Introduction

For 1985 in t h e German Democratic Republic (GDR) a n annual lignite produc- tion of 300 MLL.tohs /annum i s planned. The principal mining technology i s open-pit mining. The lignite seams are embedded in q u a r t e r n a r y / t e r t i a r y a q u i f e r systems. These a q u i f e r systems have t o b e drained to satisfy t h e geomechanical stability of t h e slopes of t h e open-pit mines. In 1984 a b o u t 1.7 _Bill. m 3 mine d r a i n a g e water h a s b e e n pumped out, operating more t h e n 7000 d r a i n a g e wells.

T h e r e f o r e , approximately 17 X of t h e t o t a l mining c o s t are r e q u i r e d . R e i s n e r and R6sch 1984.

Consequently, t h e minimization of t h e number of d r a i n a g e w e l l s and t h e i r ' optimal operation become important t a s k s in designing mine d r a i n a g e systems.

The extensive mine d r a i n a g e causes manifold impacts on t h e water r e s o u r c e s in mining a r e a s and significant conflicts between d i f f e r e n t water u s e r s , Kaden e t al. 1985. Groundwater flow models have t o b e used, both, f o r t h e design of d r a i n a g e wells, and f o r t h e analysis of water management s t r a t e g i e s in mining a r e a s , Kaden and Luckner 1984.

The a c c u r a c y of computations with comprehensive groundwater flow models depends on t h e precision of t h e underlying hydrogeological informations. In o r d e r to g e t t h e s e informations detailed and costly hydrogeological e x p l o r a t i o n s h a v e t o b e done.

')international l n s t i t u t e f o r Applied S y s t e m s Analysis, Laxenburg, Austria

Z ) ~ e s e a r c h Croup f o r Open-Pit Dewatering Problems o f t h e Crossraschen l n s t i t u t e f o r L i g n i t e Min- ing and t h e Dresden U n i v e r s i t y of Technology

Generally, hydrogeological exploration is based o n t h e following techniques:

-

e z p l o r a t i o n d r i l l i n g f o r exploration including t h e collection a n d analysis of samples of t h e m a t e r i a l in t h e b o r e hole, resulting in point informations on t h e hydrogeological s t r u c t u r e ,

-

pumping t e s t s for t h e estimation of transmissivities a n d specific yields being r e p r e s e n t a t i v e for a small region of t h e a q u i f e r (a f e w 1 0 0 m 2),

-

geophysical methods t o g e t detailed informations within b o r e holes, a n d above a l l using s u r f a c e methods a n d remote sensing techniques in o r d e r t o obtain local a n d regional informations on t h e geohydrological system.

In Figure 1 t h e outcome of t h e s e d i f f e r e n t methods is i l l u s t r a t e d .

I

Research

I

Geophysical Methods I Pumping Test

Drilling I I

I

I =

Remote sensing

I

I I I

r'

.g. Seismic refraction

Aquifer

Lignite (AquicludIAquitard)

- -

~ ~ u i f e r -

^{. .} . .

^{# ,}

^.. - - ^.

^{, ' _}

Point Samples I Regional Samples I Local Samples

(Groundwater table, I (Aquifer geometry, I

I (Transmissivit y, aquifer geometry, I rough estimates of I specific yield) aquifer parameters I permeability)

I ^I

as hydraulic conduc-

I

^I

tivity, storativity) I

I

I

Figure 1 : Methods f o r hydrogeological e x p l o r a t i o n

The basic informations are obtained using e x p l o r a t i o n d r i l l i n g . The c o s t f o r hydrogeological e x p l o r a t i o n are approximately a l i n e a r function of t h e number of exploration b o r e holes. T h e r e f o r e , t h e r e d u c t i o n of drilling g e t s a key role in reducing c o s t of e x p l o r a t i o n . This might b e done by:

-

i n c r e a s e d u s e of geophysical exploration methods,

-

complex analysis of exploration r e s u l t s using mathematical s t a t i s t i c a l methods,

-

p r e c i s e estimation of t h e r e q u i r e d amount of hydrogeological informations.

In t h e p a p e r we will c o n c e n t r a t e on t h e last mentioned a l t e r n a t i v e .

A t p r e s e n t , in t h e GDR t h e following tools a r e used f o r t h e e s t i m a t i o n of t h e r e q u i r e d a m o u n t of h y d r o g e o l o g i c d e x p l o r a t i o n :

a ) standard values in t h e field of lignite exploration according to s e v e r a l s t a g e s of exploration and according t o t h e type of t h e coal seam a n d deposit. n.n.

1976 , s e e Figure 2a.

b) catalogue of groundwater deposits, Bamberg et al. 1975; t h e p a r a m e t e r s of t h e deposits are c h a r a c t e r i z e d by statistical values (mean, dispersion, v a r i - ance). Assuming a r e q u i r e d precision of exploration, t h e r e q u i r e d amount of exploration c a n b e estimated, see Figure 2b.

Figure 2: Methods f o r t h e estimation of t h e r e q u i r e d amount of hydrogeological exploration in t h e GDR

Exploration

Both t h e tools give only rough estimates and t h e y d o not consider t h e aim of exploration, i t s r o l e in t h e complex economic system of e x p l o r a t i o n

-

^{m i n e}

d r a i n a g e

-

m i n i n g . Principally, t h e amount of exploration i s estimated by e x p e r t s taking into t h e account t h i s complexity, but more o r less on a subjective basis, see Figure 2c.

Formation of the Hydrogeologic Model a)

0 a v,

a

5 3

--_-..-

V)

b)

l~ E O W

o ~ z

0 0

< ⁵

c o o

- 5

c)

lL

:z s a

5 :

O X vl vl 0

Simulation

and Design Economic Evaluation

Distance between

Geologic basic model : Type of deposit Type of coal seam

Estimation of the benefit of exploration:

Stages of exploration

bore holes I

Numberof bore holes n

- _-_--- -- --- --- -_-- - --- ---

Hydrogeologic basic model:

Type of ground water d e w i t Required precision

Estimation of the influena of the precision of exploration on drainege design of exploration

.

Distance befween

bore h o l a Number of bore h o l a n

--- - --- --- --- --- --- ---

Estimation of:

Hydrogeologic basic model

Losses due t o insufficient , Bemfip due t o earlier

operatton of mine drainager Number of bore

h o l a n

t

4 I

Obviously, t h e o b j e c t i v e s t o minimize mine drainage c o s t (e.g. minimizing t h e number of d r a i n a g e wells) and t o minimize exploration c o s t (e.g. minimizing t h e number of exploration drillings) a r e contradictory.

-

The l e s s hydrogeological exploration i s done, t h e more uncertain a r e t h e hydrogeological p a r a m e t e r s f o r drainage well design.

In t h e case of u n c e r t a i n p a r a m e t e r s drainage wells a r e more o r less overdi- mensioned o r t h e r i s k of geomechanical a v e r a g e s i n c r e a s e s . Nevertheless, detailed hydrogeological exploration may b e omitted if t h e r i s k of a v e r a g e s (and i t s economic consequences) i s s m a l l in comparison with t h e benefits of saved exploration c a p a c i t y and e a r l i e r operation of t h e drainage system, Goldbecher e t al. 1982.

In t h e following w e p r o p o s e a mathematical a p p r o a c h t o s u p p o r t t h e complex decision making p r o c e d u r e of estimating t h e amount of hydrogeological explora- tion, s e e a l s o Reichel and Lomakin 1984.

2. Methodological Approach

Hydrogeological exploration in lignite mining areas aims above all a t t h e esti- mation of hydrogeological p a r a m e t e r s f o r calculations of t h e groundwater flow caused by mine d r a i n a g e in t h e a q u i f e r system. Such calculations include:

-

estimation of flow t o dewatering w e l l s as t h e basis of mine d r a i n a g e design,

-

estimation of t h e t o t a l d i s c h a r g e of t h e mine drainage system f o r t h e design of mine water t r e a t m e n t s plants, and f o r water management decisions,

-

estimation of groundwater t a b l e variations (both, lowering in drained a r e a s , and r i s e in abandoned mining a r e a s ) f o r water management and environmental decisions.

Generally, t h e installation a n d operation of mine d r a i n a g e systems becomes t h e most costly t a s k f o r water r e l a t e d decisions in mining a r e a s . For t h e design of mine drainage systems t h e most detailed hydrogeological informations, t h e i r highest a c c u r a c y is needed. Consequently i t i s reasonable t o assume, t h a t t h e a m o u n t of hydrogeological e z p l o r a t i o n i s estimated depending o n the r e q u i r e d explora- .tion for the d e s i g n of t h e m i n e d r a i n a g e s y s t e m . For a n extention see Section 4.

Our a p p r o a c h is based on t h e following principal presumptions:

-

For t h e area u n d e r consideration tentative geological and hydrogeological investigations h a v e been done including preliminary (rough) exploration.

-

The objectives of d r a i n a g e are defined (location and depth of t h e lignite mine) and t h e d r a i n a g e system is d r a f t e d (type of t h e d r a i n a g e system, i t s d e p t h and location).

The working s t e p s of t h e proposed a p p r o a c h a r e summerized in Figure 3.

2.1. H ydrogeological Schematization and Parameter Model

The complicated a n d p a r t l y random s t r u c t u r e of hydrogeolocial systems necessitates t h e i r schematization because:

-

hydrogeological e x p l o r a t i o n r e s u l t s e i t h e r in rough estimates of t h e s t r u c - t u r e (geophysical methods) o r in random samples (drilling, pumping t e s t s ) ,

-

calculations of t h e groundwater flow system r e q u i r e a schematized and more o r less simplified hydrogeolocial model of t h e real system.

Figure 3 : Working s t e p s

Let us assume t h a t t h e principle v e r t i c a l s t r u c t u r e (stratification) of t h e hydro- geological system under study i s known. Usually t h i s knowledge i s w e l l based on g e n e r a l geological informations and preliminary explorations. From t h a t w e obtain a v e r t i c a l schematization. Define P

=

( p ( ' ) , p ( 2 ) ,

- .

)' t h e set of hydrogeological p a r a m e t e r s needed t o quantify t h i s schematization. The horizontal p a r a m e t e r dis- tribution i s called p a r a m e t e r f i n c t i o n fp (Z , y ).

Simulation and Design Economic Evaluation Exploration

In t h e c o u r s e of hydrogeological exploration (exploration drilling) point samples of t h e p a r a m e t e r function are obtained.

Formatjonof_(he Hydrogeologic Model

with nb

-

number of b o r e holes

H y d ~ l o g i c b ~ i c model

Based on t h e s e point samples a n approximative p a r a m e t e r function h a s t o b e estimated.

Numbn of

=eG& n

This i s a widely studied a n d discussed problem in hydrogeological r e s e a r c h and p r a c t i c e . From o u r point of view only two concepts a r e practically important:

-

t h e k r i g i n g i n t e r p o t a t i o n technique of b e s t l i n e a r unbiased estimation of a regionalized v a r i a b l e , f o r a review s e e Virdee and Kottegoda 1984,

Cost of.explorat:lo!l

a, a function of n

-

t h e concept of g e o h y d r a u t i c a l t y r e p r e s e n t a t i v e p a r a m e t e r s .

The kriging interpolation technique and o t h e r s a r e used to estimate Local parame- ters, whereas t h e second c o n c e p t r e s u l t s in regional r e p r e s e n t a t i v e p a r a m e t e r s . Our a p p r o a c h i s based on t h e second concept.

4

v

E&o~tip

~~~~ ^:

Hypothetic- hydrogeologic Gmhydraulic celculatiw with hypohtic

models models

Design of ~ezinagc system for h y p o m k a l modds

H ~ ~ ~ t h e t i c a I lossas

-

^{a, b} function of n

O p t i m c n b a of booholes

"opt

b

Minimizingrum of hy-ical l--plu

con of exploration

4 +

The a r e a under consideration

C

( a r e a being influenced by t h e mining) should b e divided into N s u b a r e a s A&.

C = C l u C 2 u u C N

(3)

Each s u b a r e a is c h a r a c t e r i z e d by a uniform stratification and an accompanying s e t of hydrogeological p a r a m e t e r s .

For t h e p u r p o s e of simplification w e consider in t h e following only one param- e t e r pi f o r e a c h s u b a r e a i , being t h e decisive p a r a m e t e r f o r drainage design. The a p p r o a c h c a n b e extended f o r s e v e r a l p a r a m e t e r s without principle difficulties.

The p a r a m e t e r distribution within t h e s u b a r e a i s assumed t o b e random, and t h e s u b a r e a should a p p e r t a i n statistically to a basic totallity with r e g a r d to t h e pi f e a t u r e .

The r e s u l t s of hydrogeological explorations a r e used t o estimate empirical distribution functions of t h e p a r a m e t e r p i . E.g. s t a t i s t i c a l investigations of t h e empirical distribution functions of t h e hydraulic conductivity f o r a l a r g e number of a q u i f e r s and s u b a r e a s have r e v e a l e d different distributions ranging from t h e Normal distribution t o a logarithmic bell-shaped distribution, Beims 1974.

F o r geohydraulic calculations, t h e random p a r a m e t e r pi h a s t o b e r e p l a c e d by a constant geohydraulically representative parameter p ~ , , .

According to Beims and Luckner 1975, t h e p a r a m e t e r PR,, i s geohydrauli- c a l l y representative, if computations of t h e flow system (aimed at t h e design of t h e drainage system) with t h e r e p r e s e n t a t i v e value a r e adequate t o computations with t h e r e a l hydrogeological p a r a m e t e r p, ( z

.

^{y ) as a} function in s p a c e . This means, t h e r e a l hydrogeological system in t h e s u b a r e a i s r e p l a c e d by a schematized horizon- t a l s t r a t i f i e d s u b a r e a being homogeneously within each stratum, s e e Figure 4.

Frequently, as a n estimate of t h e geohydraulically r e p r e s e n t a t i v e value t h e a r i t h - metic mean

5,

from a l l samples in t h e s u b a r e a i s used, assuming normal distributed p a r a m e t e r s . F o r t h e p a r a m e t e r holds:

W e need a n estimate

5,

of t h e mean. Because

5,

i s normal distributed, t h e estimate of t h e mean will b e normal distributed:

with o,

-

v a r i a n c e

n , ^,(

-

number of samples ( b o r e holes).

Under t h e given assumptions a confidence interval may b e estimated as defined below.

with a

-

e r r o r probability

t

.,,,-

p a r a m e t e r of t h e t-distribution f o r unilateral questioning mi

-

d e g r e e s of freedom

=

^{nb ,t}

-

¹

Earth Surface

k : hydraulic conductivity s: storativity

Figure 4 : Schematized hydrogeological s t r u c t u r e of s u b a r e a s st

-

estimate of t h e variance.

A generalized method for t h e estimation of geohydraulically r e p r e s e n t a t i v e param- e t e r s (hydraulic conductivity, transmissivity) h a s been developed by Beims, 1974.

The method although r e n d e r s t h e estimation of t h e confidence of t h e calculated p a r a m e t e r s . F o r t h e geohydraulically r e p r e s e n t a t i v e p a r a m e t e r s holds:

and for t h e confidence limits w e obtain:

with c

-

^reduction factor v

-

v a r i a t i o n coefficient

K

-

transformation coefficient

Both t h e simple a r i t h m e t i c mean and more s o p h i s t i c a t e d p a r a m e t e r models for geohydraulically r e p r e s e n t a t i v e p a r a m e t e r s open t h e possibility of confidence estimates.

With Eq. (8) we have a r e l a t i o n s h i p between:

-

t h e e z p l o r a t i o n p r e c i s i o n in terms of t h e confidence of t h e hydrogeological p a r a m e t e r t o b e e x p l o r e d and

-

t h e a m o u n t of e z p l o r a t i o n in terms of t h e number of exploration drillings.

Generally, t h e p a r a m e t e r model h a s to be chosen depending on t h e goal of explora- tion and on t h e e x p e c t e d hydrogeological situation. Other comprehensive parame- t e r models (with a u t o c o r r e l a t i o n ) a r e proposed by Bamberg e t al. 1975, and Stoyan 1973, 1974.

****

Es i s t zu p r u e f e n , o b d i e s e Modelle sich in d a s vorgeschlagene Konzept einord- nen lassen, Darstellung ggf. a u s f u e h r l i c h e r oder ganz weglassen.

2.2. Design of the Drainage System

The common technology for mine drainage in sandy a q u i f e r s a r e v e r t i c a l dewa- t e r i n g wells ( b o r d e r and field well galleries). In g e n e r a l t h e i r design i s based on uncertain d a t a due t o t h e random s t r u c t u r e of t h e hydrogeological system and i t s r e s t r i c t e d explorability

.

Furthermore, e r r o r s caused by schematizations and numerical calculations h a v e to b e taken into t h e account.

In open-pit mining t h e economic losses due to insufficient mine d r a i n a g e (caus- ing water inrushes) may b e tremendous and may e x c e e d t h e losses resulting from

"over-design". The lower t h e r i s k of a breakdown of mine drainage t h e h i g h e r i s t h e r i s k of uneconomic design. Nevertheless, t h e r i s k of damages due t o "under- design" should b e sufficiently small.

Principally, t h e system of dewatering wells h a s t o b e designed in such a way t h a t t h e technologically r e q u i r e d groundwater depression within t h e mine will b e satis- fied with a c e r t a i n reliability. Hence, t h e design depend on t h e reliability of t h e input data.

According t o o u r principal presumptions we assume a fixed construction of drainage wells (drilling diameter, w e l l s c r e e n and f i l t e r , capacity of pumps), as well as a given location of t h e dewatering galleries. Based on t h a t , t h e number of wells, t h e i r distribution along t h e galleries become t h e decisive design values.

These values depend strongly on t h e specific groundwater flow t o t h e well gallery, t h e specific pumpage r e s p e c t i v e l y , f o r a given groundwater depression t a r g e t .

For t h e p u r p o s e of simplification t h e drainage well galleries should b e divided in a finite number of s e c t i o n s Agj , j =I.

...,

J with a constant specific pumpage qj

.

Assuming a homogeneous horizontal hydrogeological s t r u c t u r e along t h e g a l l e r y , t h e well distance Awj f o r t h e section j will b e constant and for t h e number of wells holds

with k j

-

length of section Agj

For a given well construction and drainage t a r g e t t h e well distance Awj i s a mul- tidimensional function, depending on t h e specific pumpage qj and on t h e hydrogeo- logical p a r a m e t e r p j ( t h e transmissivity) f o r t h e g a l l e r y section j. W e assume s t e a d y s t a t e conditions.

Frequently t h e well distance f o r single a q u i f e r s is estimated from t h e following implicit mathematical model, Luckner et a l . 1969:

with q 5

-

specific pumpage f o r s e c t o r j ,

Tj -

a c t u a l transmissivity f o r sector j ,

H,

, j

-

piecometric head a t t h e d r a i n a g e c o n t o u r of s e c t o r j, HWL5

-

minimum piecometric head in t h e d r a i n a g e wells,

rom5

-

hydraulic e f f e c t i v e r a d i u s of t h e wells.

The specific pumpage i s a function of t h e hydrogeological p a r a m e t e r of t h e s u b a r e a A&.

Q j =.fr(P1nP2*

- . .

~ P M ) From Eq. (12), (101, (9) w e would g e t

nw .5

=

P , ( P 1 n ~ e l

. -

^{,PM) (13)}

Such a relationship would give us t h e interdependency between t h e reliability of t h e hydrogeological p a r a m e t e r s t o b e e x p l o r e d and t h e dasign value of t h e d r a i n a g e system. Unfortunately such a function can not b e found f o r generalized problems. This i s a consequence of t h e nonlinear, implicite w e l l functions (e.g. Eq.

( l l ) ) , and of t h e numerical solution of t h e groundwater flow problem. This will b e discussed in t h e n e x t section.

2.3. Geohydraulic Calculation with Hypothetic Hydrogeological Models The specific pumpage q j , j =I,.

. .

^, J h a s to b e estimated by t h e help of geohy- d r a u l i c calculations. In g e n e r a l , f o r such calculations comprehensive system- d e s c r i p t i v e groundwater flow models are used. Analytical solutions are r a r e l y applicable. Consequently, t h e i r is no explicit relationship between t h e hydrogeo- logical p a r a m e t e r s of t h e system and t h e s p e c i f i c pumpage. F u r t h e r m o r e , w e will not g e t a n e x p l i c i t e relationship between random hydrogeological p a r a m e t e r s and random s p e c i f i c pumpage (both in t e r m s of a probability distribution function or as expectation values with confidence interval).

Empirical a t t e m p t s to obtain empirical distributions (and confidence i n t e r - vals) have been made using Monte-Carlo simulations, e.g. Reichel 1979, or applying numerical a p p r o a c h e s , Reichel e t a l . 1982. The p r a c t i c a l applicability is r e s - t r i c t e d d u e t o t h e l a r g e amount of computations needed.

In t h e following, we p r o p o s e a r e d u c e d deterministic a p p r o a c h , based on t h e t h e o r y of s t a t i s t i c a l experiments, S c h e f f l e r 1974.

Instead of a s t o c h a s t i c p a r a m e t e r model ( Section 2.1 ) a d i s c r e t e lumped p a r a m e t e r model is used. Each p a r a m e t e r i s d e s c r i b e d as a finite number of possi- b l e realizations, chosen by t h e help of s t a t i s t i c a l methods.

PC : (PJ1),pJ2),

. . .

.PC ( I ) ) (14) Through simulation experiments with a n a p p r o p r i a t e groundwater flow model t h e effect,of d i f f e r e n t realizations of t h e p a r a m e t e r s

-

called h y p o t h e t i c a l hydrogeo- Logical model

-

on t h e specific pumpage q 5 a n d finally t h e design value Awj c a n b e estimated.

If a l i n e a r relationship between t h e hydrogeological p a r a m e t e r s p( and t h e design value Awj holds t r u e , two s t e p experiments a r e needed. Two s t e p experi- ments a r e a l s o reasonable as a f i r s t approximation f o r nonlinear relationships, s e e below.

Each p a r a m e t e r is d e s c r i b e d by i t s expected lower and u p p e r bounds:

f 1

These values may b e estimated by previous hydrogeological e x p l o r a t i o n s and p r a c - tical experiences.

Define m t h e number of independent p a r a m e t e r s t o b e explored. Conse- quently, Zm experiments with hypothetical models a r e needed t o c o v e r a l l possible combinations of p a r a m e t e r s . This might b e with one p a r a m e t e r in m s u b a r e a s o r N p a r a m e t e r s in m s u b a r e a s with m =N.m *. H e r e t h e problem of numerical e f f o r t in c a s e of more t h a n 4 p a r a m e t e r s becomes obvious (e.g. m =8 r e s u l t s in 256 e x p e r - iments).

To overcome t h i s problem two a l t e r n a t i v e s should b e investigated:

-

application of simulation models with low computational e f f o r t ,

-

application of t h e t h e o r y of s t a t i s t i c a l experiments t o r e d u c e t h e number of experiments.

The outcome of t h e f i r s t a l t e r n a t i v e i s r e s t r i c t e d . In p r i n c i p l e , t h e system- d e s c r i p t i v e groundwater flow models with distributed p a r a m e t e r s h a v e t o b e used.

That means, numerical models based on Finite-Differences-Methods o r Finite- Elements-Methods a r e needed. A c e r t a i n number of nodes (elements) i s r e q u i r e d t o satisfy a a c c e p t a b l e a c c u r a c y . F u r t h e r m o r e , this number might i n c r e a s e with t h e number of p a r a m e t e r s t o b e investigated.

In case of nonlinear relationships t h e deviation from t h e l i n e a r behavior c a n b e

-

estimated

-

with a n additional experiment f o r t h e c e n t r a l point (minp{ - .

+

maxpi)/2. For s t r o n g nonlinearities more detailed computations - . are n e c e s s a r y , e.g. 3m experiments in t h e case of a quadratic dependency.

From t h e experiments with t h e hypothetical models we obtain a n evaluation t o t h e e f f e c t of t h e hydrogeological p a r a m e t e r

&

in t h e s u b a r e a i on t h e d r a i n a g e design, ~ ,, j j=l,.

. . ^,J.

Obviously. t h e e f f e c t depends above all on t h e d i s t a n c e of t h e s u b a r e a from t h e location of t h e d r a i n a g e system.

Define a hypothetical model M S ( ) as a model c h a r a c t e r i z e d by a s p e c i a l p a r a m e t e r

F(

f o r s u b a r e a i and any values according t o Eq. (15) f o r t h e parame- ters of t h e o t h e r s u b a r e a s 1 , 1 =1,

...,

N ; 1 t i .

I 1

Analogously w e define M(&

,Fk

) as

In using t h i s notation we c a n define m a i n

sects

and i n t e r a c t i o n efpects.

The m a i n efpect ^A&,, of t h e p a r a m e t e r

F(

is defined as t h e d i f f e r e n c e between t h e mean f o r a l l p a r a m e t e r combinations f o r maxp( minus t h e mean f o r a l l p a r a m e t e r combinations f o r minp(.

with K 1

-

number of models (

=zN"-')

This relationship implies t h a t t h e p a r a m e t e r s of t h e s u b a r e a s L ,L

=I,. .

.,N , L ti are considered as a mean. Consequently for a l i n e a r dependency between

and

5{

A n w a j is t h e number of w e l l s

-

f o r section

-

j being additionally n e c e s s a r y if t h e p a r a m e t e r

F{

changes from minp{ t o maxp{.

The i n t e r a c t i o n e m t A?&: i s defined as

with K2

-

number of models (

= z ~ " - ~ ) .

In t h e case of 3m experiments or more a v a r i a n c e analysis h a s to b e applied. The number of experiments c a n b e r e d u c e d neglecting some of t h e interaction e f f e c t s , see S c h e f f l e r 1984.

With Eq. (18) and (19) w e h a v e got a measure f o r t h e e f f e c t of p a r a m e t e r changes o n t h e design values. A s l a r g e r t h e s e main e f f e c t s for t h e p a r a m e t e r

5{

^of

a s u b a r e a AX{ a r e , as more important i s t h e exploration in t h a t s u b a r e a . L a r g e i n t e r a c t i o n e f f e c t s A&:, indicates, t h a t t h e p a r a m e t e r

F{

^{h a s} a significant influ- e n c e on t h e design with a s p e c i a l constellation of p a r a m e t e r

Fk.

According t o t h e s e numbers a step-wise exploration might b e implemented. W e will i l l u s t r a t e t h i s in Section 3 f o r a n example.

2.4. Economic Evaluation Depending on Exploration Precision

F o r t h e time being w e will not consider t h e economic evaluation of t h e r i s k due t o insufficient design of mine d r a i n a g e , as explained above. W e will b a s e on t h e economic evaluation of t h e d r a i n a g e design and of t h e exploration. In Figure 5 t h e principle economical relationships are depicted.

A s lower t h e exploration precision is, as h i g h e r are t h e c o s t f o r mine d r a i n a g e t o b e e x p e c t e d in o r d e r t o minimize t h e r i s k of "under-design". On t h e o t h e r hand, t h e c o s t of e x p l o r a t i o n i n c r e a s e s with t h e exploration precision.

Define e x

Cd

t h e c o s t of mine d r a i n a g e being n e c e s s a r y f o r t h e given system i t means t h e system would h a v e been optimal designed with e x a c t p a r a m e t e r s . The additional c o s t f o r d r a i n a g e design due to insufficient exploration precision w e define as h y p o t h e t i c a l Losses L

.

To develop a relationship between t h e exploration precision ( t h e number of e x p l o r a t i o n drillings) and t h e hypothetical losses is r a t h e r difficult. The cost for mine d r a i n a g e as well as t h e hypothetical losses depend l i n e a r on t h e number of e x p l o r a t i o n drillings. But t h e number of drillings is a nonlinear function of t h e p a r a m e t e r s , see Eq. (11).

A s a f i r s t approximation w e assume t h a t t h e number of well drillings and con- sequently t h e hypothetical losses depend l i n e a r o n t h e p a r a m e t e r s p"{ , i = I , .

..

, N . This assumption c a n b e c h e c k e d estimating t h e number of exploration drillings f o r t h e lower and u p p e r p a r a m e t e r bounds of Eq. (15) and f o r t h e mean p a r a m e t e r .

Consider o n e p a r a m e t e r

&

with e x p e c t e d lower and u p p e r bounds rninp"{,max&. Define

pt

i t s "pessimistic" value used for t h e design of t h e d r a i n a g e system. Hypothetical losses d u e to p a r a m e t e r ${ will equal z e r o , if t h e p a r a m e t e r

pt

i s equal t o t h e r e a l value p"{, and will r e a c h t h e maximum, if

&

i s equal to t h e

"optimistic" value pp.

I I

Exploration precision Optimal exploration precision

Figure 5 : Economical evaluation The losses a r e defined as

with

max Cd

-

^minCd

L G O

= I 5

- P ~ I

maxp',

-

^minp',

In Figure 6 t h e function i s illustrated.

For t h e p a r a m e t e r p f we apply t h e generalized p a r a m e t e r model from Eq. (8).

P f

= 5, * ^A5,

^{( n b}

,, ¹

⁽²²⁾

That means, f o r t h e design e i t h e r t h e lower o r u p p e r confidence limit (for a given probability) is used.

Inserting Eq. (22) into Eq. (21) we obtain

Distribution function of L

Figure 6 : Cost of design and hypothetical losses in relation t o t h e p a r a m e t e r

p", f o r pp=minp",

.

For t h e normal distributed p a r a m e t e r model we get with ^{Eq. (7)}

In Figure 7 this functional relationship is illustrated.

Because @, is normal distributed, L (p",) is normal distributed with t h e mean

I 1

E [L*,) ] = L 6 0 and t h e dispersion

An open problem is t h e evaluation of t h e cost of drainage. Let c, b e t h e specific

.

cost of one drainage well. Now w e will r e p l a c e the cost by using t h e eflects, being defined in t h e previous section (Eq. (18), (19)).

(

L

Hvpothetical losses

Parameter to be explored

Number of bore holes Figure 7 : Hypothetical losses f o r normal d i s t r i b u t e d p a r a m e t e r s

A s explained a b o v e , bnASj is t h e number of w e l l s being additional n e c e s s a r y , if t h e p a r a m e t e r

5c

c h a n g e s from minp", to maxp"(.

-

And t h i s is t h e meaning of maxCd

-

minCd !

Inserting in Eq. (22) w e obtain

In t h e case of normal d i s t r i b u t e d p a r a m e t e r s w e c a n r e p l a c e maxp",

-

^minfi,

maxp",

-

minp"(

2

=

t 'St

with st

-

v a r i a n c e

t

-

measure of t h e quality of p r e v i o u s informations Hence w e g e t for Eq. (23) with Eq. (27)

For t h e cost of e z p l o r a t z o n w e assume a l i n e a r dependency on t h e number of b o r e holes. For instance, we might use t h e following simple e x p r e s s i o n

N

c, = c,,,

^{+ Cb}

.

n b ,{

{ = 1

(31) with _C, ,,

-

c o n s t a n t c o s t of e x p l o r a t i o n

cb

-

s p e c i f i c c o s t of exploration (cost of o n e b o r e hole).

2.5. Optimal Exploration Precision

We o b t a i n t h e optimal e x p l o r a t i o n precision in minimizing t h e sum of t o t a l hypothetical l o s s e s and c o s t of exploration. For a p a r a l l e l e x p l o r a t i o n of a l l s u b a r e a s holds:

N N

V

=

EL(&)

+ c,,, +

c b . E n b , { + Mzn.!

The hypothetical l o s s e s L (&) depend on t h e number of exploration b o r e holes in t h e s u b a r e a i only (compare Eq. (28), (29)). Consequently, t h e minimum problem Eq. (32) c a n b e s e p a r a t e d in a set of minimum problems, without consideration of C, ,,

.

For e a c h s u b a r e a a p a r t of t h e constant c o s t C, ,, may b e added, as i t i s done in t h e example (Section 3.). But t h i s d o e s not c h a n g e t h e optimal number of e x p l o r a t i o n drillings.

With Eq. (28) w e g e t

An e x p l o r a t i o n is only t h e n r e a s o n a b l e , if i t l e a d s t o more p r e c i s e informations in comparison with t h e previous informations being fixed in maxp", , minp",. For t h e minimum number of b o r e holes minnb ,{ w e obtain by using t h e confidence limit

From Eq. (35) minnb ,{ may b e obtained.

Hence, Eq. (34) h a s t o b e investigated f o r

n b ,{

>

minnb ,{

Eq. (34) does not d e s c r i b e a simple minimization problem due t o t h e s t o c h a s t i c p a r a m e t e r @(. According t o Schneeweiss, 1967 t h e decision should b e based on t h e minimization of e x p e c t a t i o n values (Bayes-decision). The e x p e c t a t i o n value f o r is

pi.

^Hence w e obtain f r o m Eq. (34) with Eq. (25) a deterministic problem being solvable.

If t h e function of losses is nonlinear, t h e expectation value of losses might not b e obtained explicitly. Then t h e Monte-Carlo method should b e used t o estimate t h e expectation value of losses depending on t h e p a r a m e t e r model.

Finally l e t s c o n s i d e r t h e case of normal d i s t r i b u t e d p a r a m e t e r s . F o r t h e minimum number of b o r e holes holds

With Eq. (29) w e g e t

t

,,,,

h a s t o b e t a k e n from t a b l e s with

mi

=

n b ,i -1 For n b ,(

>

minnb ,( w e obtain from Eq. (37) and (30)

Eq. (41) cannot b e solved explicitely because ta,- depends o n n b , ( (Eq. (40)). The simplest way for t h e solution i s t o calculate t h e values s t a r t i n g f r o m n b ,t

=

minnb ,i

until t h e minimum is r e a c h e d .

T ~ e d d ' i r c e t h e number of calculations, Eq. (41) might b e solved f o r an estimate of tan-.

The optimal n b

,,

h a s t o b e determined precisely in t h e vicinity of n b

-

,i with t h e help of Eq. (41).

3. T e s t Example

W e will demonstrate t h e developed a p p r o a c h for a simplified example of t h e design of a mine d r a i n a g e g a l l e r y . The schematized system is d e p i c t e d in Figure 8.

W e consider a confined a q u i f e r , schematized into 4 homogeneous s u b a r e a s (with t h e transmissivities

T I , T2, T3, TI).

The v a r i a n c e of t h e r e a l transmissivities within t h e s e s u b a r e a s is e x p e c t e d to b e small, t h e mean value

?

is approximately equal to t h e r e p r e s e n t a t i v e value

TR .

The well g a l l e r y h a s to b e designed t o satisfy a given groundwater d e p r e s s i o n next t o t h e mine. Two a l t e r n a t i v e s of t h e location of t h e g a l l e r y are u n d e r con- sideration, as Figure 8 indicates (Example 1, Example 2). The piecometrtc h e a d inside t h e d r a i n a g e c o n t o u r h a s to b e equal 20 m .

Well Gallery

/

^Example1

1

1 5 0 0 F L - - h

Contour I

.I

Well Gallery

F

= r

^{Example 2}

\ ^L

\ T4

s I

^T3

\

\

= _I

8

0

:

C .

^{.-.-.-. X}

Figure B : Scheme of t h e test example, a ) horizontal plan b) cross section The transmissivities are e x p e c t e d to b e in a given r a n g e .

2 . 3 ~ 1 0 ~ S T1,TB,TS,T4 6 2.3.10-' m 2 / s e c .

The d r a i n a g e g a l l e r y i s divided into t w o sections (contour I, c o n t o u r 11). F o r t h e s e s e c t i o n s a uniform distribution of t h e wells is assumed. F o r more detailed investi- gations t h e d r a i n a g e c o n t o u r might be divided into more s e c t i o n s depending on t h e s p a t i a l variability of t h e groundwater flow (compare Figure 9, below).

For t h e geohydraulic calculations hypothetical hydrogeological models have b e e n composed f r o m t h e minimal and maximal e x p e c t e d transmissivities.

minT1 ,...,

, ⁼

2 . 3 - l o 4 m 2 / sec. (44)

In Table 1 t h e hypothetical models are listed.

F o r e a c h hypothetical model t h e specific pumpage ql,qIl (steady state values) had to b e estimated. W e used a finite difference groundwater flow model, Reichel and Lomakin 1984. The computational r e s u l t s a r e independent on t h e p a r a m e t e r T,.

, T h a t means, t h e r e s u l t s for t h e models 9-16 are t h e same as t h o s e f o r t h e models 1 - 8 (in t h e same o r d e r ) . F u r t h e r m o r e , t h e r e s u l t s f o r Example 1 and 2 are equal, assuming t h a t t h e d i f f e r e n c e in t h e location of t h e g a l l e r y i s negligible.

Table 1 : Hypothetical hydrogeological models f o r simulation experiments

I '

^{7 - I I- - 1}

;

^T~

ⁱ

^rnin

^I

^rnin

^I

^rnin

^I

^rnin

^j

^rnax

¹

^rnax

^I

rnax rnax

I

^rnin

1

rnin i rnin

/

^rnin

I

^rnax

i

m a m a x I rnax

I

The computational r e s u l t f o r t h e 8 experiments are illustrated in Figure 9 . The specific pumpage w a s normalized by t h e maximum piecometric d i f f e r e n c e

A H = 6 0 m .

For d r a i n a g e c o n t o u r I t h e maximum pumpage a p p e r t a i n s t o model 4 ⁽ maximum values for Tl,T2). A s e x p e c t e d f o r c o n t o u r I1 t h e maximum pumpage is r e a c h e d f o r t h e models 7 and 8 (maximum values of T2,T,). The l a r g e r t h e pumpage is, t h e l a r g e r becomes i t s dependency on s p a c e .

Based on Eq. (11) a n d on mean values of t h e specific pumpage f o r t h e d r a i n a g e con- t o u r s I and I1 from t h e simulation experiments (Figure 9) t h e optimal well distance h a s been estimated using t h e following d a t a :

Example 1 : TWsI

=

T1 , T,,*

=

T, Example 2 : TWsI

=

T,,*

=

T, The optimal number of wells is

BI Bn

n,

=

^{~ W , I}

+

n,,a

=-

+-

Awl Aun

with B1.a

-

length of d r a i n a g e contour ( 1500 m ) Awl,*

-

optimal well distance

In Table 2 t h e r e s u l t s are summerized.

.

3 a s e a on Eq. (18) and (19) t h e effects of t h e p a r a m e t e r s TI,..,, on t h e design param- eter n, have been estimated. In Table 3 s e l e c t e d r e s u l t s a r e listed.

F o r o u r example w e assume 1

=

3 and a

=

0.001 as well as a

=

0.01. The specific c o s t are c, =1.0 a n d cb

=

0.2.

From Eq. (39) a n d (40) w e obtain t h e minimum number of b o r e holes f o r T,.

The calculations are given in Table 4. The f u r t h e r calculations will b e demon- s t r a t e d f o r Example 2 , s u b a r e a 4 (T,) with a

=

0.01.

The optimal value of nb ,, i s estimated based on Eq. (41) with ( An,'

I =

13.375.

with a

=

0.01. In Table 5 t h e r e s u l t s are depicted.

Table 6 shows t h e r e s u l t s f o r a l l s u b a r e a s in t h e case of exploration in one s t e p .

Figure 9: Specific pumpage f o r hypothetical models a) contour I b) contour I1

Table 2 : Drainage d e s i g n f o r t h e h y p o t h e t i c a l models

Table 3 : E f f e c t s of t h e p a r a m e t e r s TI,..,, on t h e design p a r a m e t e r n, a ) Main e f f e c t s

b ) I n t e r a c t i o n e f f e c t s

I ! I I

1

Ex. I

^nw,l

¹

^{0.375 -0.375}

¹

^-4.875

¹

^-0.625

ⁱ

^{-1.625 0.125}

1

2

1

^{n,,~ i 0.000}

¹

^0.250

ⁱ

^{0.250 2.750}

¹

^-2.750

ⁱ

^-2.500

ⁱ

i n, 1 0.375 1 -0.125 -4.625 1 2.125

!

^-4.375

1

^-2.375

Finally some t h o u g h t s on t h e estimation of ezploration profitability b y t h e h e l p of t h e c a l c u l a t e d e f f e c t s (Table 3).

If w e a r r a n g e t h e e f f e c t s a c c o r d i n g t o t h e i r values f o r n, (Table 7), w e g e t a gen- e r a l impression of t h e i m p o r t a n c e of d i f f e r e n t s u b a r e a s f o r e x p l o r a t i o n .

F o r example 1 t h e e x p l o r a t i o n p r o f i t a b i l i t y d e c r e a s e s in t h e s e q u e n c e T,, T,, T

,.

The values are in t h e same r a n g e a n d t h e i n t e r a c t i o n e f f e c t s are small. Conse- quently f o r Example 1 t h e s u b a r e a s could b e e x p l o r e d t o g e t h e r .

-

²¹

-

Table 4: Calculation of t h e minimum number of b o r e holes

The optimal value of nb4 i s estimated based on Eq. (41) with

I ~4 1 =

^13.375.

Table 5: Calculation of t h e optimal number of b o r e holes f o r

T4

Table 6: Optimal number of b o r e holes f o r a l l s u b a r e a s

i

^{'I I! Example 1}

11

^{Exemple 2}

I

I !I m i n Tu.( opt nb i l m i n

V,

opt nb 1

! a u ~ a r e a :I I

I

! ^I'

i

^I

1 (Parameter) a-0.01 I a -0.001 a -0.01 i a -0.001 1 1 -0.01

1

^{(1 -0.001 j -0.01 1 -0.001}

i

1 1 4 ~4

ii -

^{; _ i _ ! I}

-

^4.45

1

^5.48

1

⁹

I

¹¹

1

In Example 2 t h e main e f f e c t d e c r e a s e from T 4 , T l , T , up t o T , . Both, t h e high main effect f o r T 4 , and t h e high interaction e f f e c t s f o r T , , T , indicates t h a t s u b a r e a 4 should b e e x p l o r e d f i r s t . This a l s o becomes obvious looking at Table 6.

A f t e r t h a t should b e decided on t h e exploration of t h e o t h e r s u b a r e a s .

In t h e case of step-wise exploration, t h e computed r e s u l t s f o r t h e hypothetical models (e.g. Figure 9) could b e interpolated f o r t h e estimated p a r a m e t e r f o r t h e f i r s t e x p l o r e d s u b a r e a . The next working s t e p s remain t h e same as b e f o r e

-

^with

reduced number of hypothetical models.

Table 7 : Selection of main effects and interaction e f f e c t s

1

^Xain effects

!I

I n t e r a c t i o n e f f e c t s ^I

/ E x . T 2 ~ n : _I ^{I 4.17 i I I}

T ³ A n 2 1 -3.25 1

11

~ n , 2 * ~ -2.00

I

i j

^{A n f -2.25}

¹¹

^-2.25

i I

I

I

^{h 4 s 3}

I

^-1.25 _I

1

EX.

/

^{T ,}

^{/ ~ n d I}

^-13.375

^{ii I}

2 1 T i i A n f

'

^{6.875 1 -4.625} I

I

I

1 T P An: i ^I 5.625

1

^{An:" I} -4.375

i

I

1 ⁱ

^I

i I

i

^{An:.' 1 0.375}

I I T 3

1

An; ^I 3.125

1

^{An:" i -2.375}

^{j I}

I ^I ! i ti 1 i

i ! 1 I / j -0.125

I

I i _I ^i, An$2

I

^2.125

4. Concluding Remarks

The proposed a p p r o a c h i s a n attempt t o objectify t h e decision p r o c e s s in designing exploration programs. Based on calculations with hypothetical hydrogeo- logical modei t h e precision of planned exploration is economically evaluated.

The optimal exploration precision (number of r e s e a r c h drillings) i s estimated by minimizing t h e sum of cost of exploration and hypothetical losses, depending on t h e exploration precision. Although t h e method h a s been d e s c r i b e d with s p e c i a l r e g a r d t o mine d r a i n a g e systems t h e a p p r o a c h is applicable to o t h e r problems too, e.g. f o r t h e exploration f o r groundwater e x t r a c t i o n (water work), f o r environmen- t a l p r o t e c t i o n measures needed d u e to mine drainage, etc. In g e n e r a l t h e hypothet- ical losses of a l l a c t i v i t i e s depending on exploration precision h a v e to b e summar- ized.

The profitability of t h e exploration may b e estimated, as a b a s i s f o r step-wise exploration.

The demonstrated a p p r o a c h c a n b e used f o r d i f f e r e n t p a r a m e t e r models, if t h e relationship between amount and precision of exploration c a n b e quantified.

The method is a p p l i c a b l e f o r d i f f e r e n t goals of exploration, if t h e economic e f f e c t of t h e exploration c a n b e quantified.

F u r t h e r r e s e a r c h should b e c o n c e n t r a t e d on t h e analysis of more complicated p a r a m e t e r models and nonlinear economic functions. Another important t a s k i s t h e integration of t h e p r e s e n t e d a p p r o a c h in complex model systems, as t h e system f o r analysis of water policies in open-pit lignite mining areas, u n d e r development at t h e Institute f o r Applied Systems Analysis, Kaden et al. 1985.

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