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Effective Tools for Modelling the
Groundwater Dynamics
in Large and complex Aquifers
KAROL
KOSORIN
Institute
of Hydrology SAS
Racianska
75, 830 08 Bratislava, Slovakia
Tel.:
(+4217) 49268/255, Fax: (+4217) 259
404, E-mail: Kosorin@uh.savba.sk
ABSTRACT
The
free boundary and the model largeness and complexity problems create frequently
serious limitations in computer simulations of groundwater dynamics in real
conditions. The paper presents some basic information on the effective model
systems INKANS and SKOKY being built upon on correct solution of these
problems. The solution arises from methodological possibilities of the
hydrodynamic theory of boundaries [2]
and directly relates to solution of the problem of unknown moving boundary of
flow, see [2], [3] and [4]. Model system INKANS + SKOKY is suitable for
computer simulation of groundwater dynamics
- in large and complex aquifers
- simultaneously in regional and local scale
- accepting correct interaction with surface waters and
underground structures
- with possibility to apply it to
solution of special tasks induced by singular effects.
Both models have been successfuly applied to various simulation tasks,
e.g. INKANS to the 3-D groundwater dynamics in large aquifer with complex
interaction with channel and river system. 3-D pressure and velocity fields in
domains with flow singularities were simulated by SKOKY. Several application
samples are presented in grafical form.
Introduction
The problem of free (moving)
boundary occurs when the model domain has a free surface of water and such flow
patern, which does not allow any simplifying e.g. by shallow water theory
assumptions leading to the Boussinesq type of model. Boundary condition on free
surface is nonlinear and moreover it
contains additional unknown (free surface position) of the model Since this condition is unplesant, many
standard and commercialy successful model systems do not respect it, e.g.
MODFLOW, MIKE SHE and others, see [5] and [6]. But, of course, such avoiding
this problem usualy brings troubles with computational accuracy, (see e.g.
[6]), because this boundary condition guarantees the mass conservation in model
computations.
Problem of model largeness can
occur especially in case of models, based on implicit computation schemes,
leading to solution of linear systems of algebraic equations, if model domain is large and complex. Large
equation systems may cause computation problems inspite of some
"good" characteristics of their matrix with diagonal arrangement of
non-zero coefficients, etc. But model domains covering horizontally more than
1000 km2 and more than 100 m deep occur in practice. If
non-homogeneous hydrogeologic characteristics and interaction with surface waters is added, the number of
algebraic equations of the simultaneously solved system can exceed the value
100 thousand. However, this can induce
real problems especially with respect to computation time even in case of
high-efficiency PC.
The three-dimensional numerical
simulation models of seepage flow INKANS and SKOKY are based on the
mathematical model, consisting of equations
grad P +q/k = O (1)
div q = 0 (2)
for the pressure function
(potencial) P = y + p/g + const and three components of the velocity vector q,
where p is the hydrodynamic pressure and y is the vertical space variable (in
direction of the gravitational acceleration g). Model is completed with
correctly stated boundary condition including that for free surface. To avoid
an uncertainity in computational accuracy the requirement to have a well posed
boundary problem (in sense of Cauchy, see [1]) has been satisfied always and
over the both models.
Comprehensive formal mathematical
decription of solution of the formulated boundary problem by means of the
hydrodynamic theory of boundaries is presented in [2], [3] and [4].
Therefore only some methodical aspects and consequences of the solution are
treated here. This solution is closely and directly connected with the solution
of the problem of unknown moving boundary, which is in our case the free surface
y = h(x,z,t).
The key to solution of both
problems is the possibility to transform each three-dimensional boundary
problem for (1), (2) over the whole domain below the free surface h(x,z,t) into a two-dimensional one for this
boundary. Kosorin [4] demonstrated the
possibility and process of this transformation, including its hydrodynamic
(physical) interpretation. The first step of the trasformation yields the 3-D velocity field
q(u,w,v)=F(H,ht,hx,hz,hxt,hzt,hxx,hzz...,y)
(3)
in form of the Taylors
series for velocity components u,v,w in
the direction of y between h(x,z,t) and impermeable boundary h0(x,z).
Eq. (3) in actual explicit form has been derived in [4] using the concept
of inner and outer partial derivatives
on material boundary of flow. To complete
the simulation numeric model it is sufficient to operate with the equation of mass conservation for horizontal
streams ,
the saturated depth of the porous medium H=h-h0 and the porosity m and to
get eq.
(4)
which results from
integration of the eq. (2) according to y between h0 and h. Equation (4), after elimination of
horizontal components of the velocity vector u and w, can be transformed by means of (3)
to a form of partial
differential equation of
N-order for the free surface h(x,z,t), N>1.
The next important yield of the hydrodynamic theory
of boundaries are so-called "jump relations". Seepage velocity field in layered
porous environment is discontinual, i.e. going through layer interface it
overcoms a jump. Three jump relations between velocity components in front of
and behind interface follow from (1) and (2) using there formal relations among
inner and outer derivatives of potential P on interfaces, see [4]. The jump
relations are inportant for model concept of SKOKY as well as for numerical representation of eq. (3) in model
INKANS.
MODEL INKANS
Model INKANS is based on the numeric integration of
system (3), (4) for 3-D flow with free surface. Numeric integration applies the method of mean values, by means
of which the system (3), (4) has been transformed in a system of ordinary
differential equations of the 1 st order each of them related to one element of
domain.This system was integrated by the method of the predictor-corrector type
of the 2nd order with variable time step.
When numericaly applying the velocity field (3) in
the resulting eq. (4) it is important that the Taylor series members with
respect vertical variable in one layer can be limited to two (linear) members.
Obviously, for transition through nonparallel and not plane interfaces among
discrete layers of model domain the obtained velocity jump relations are
sufficient to use.
Model INKANS was used for
simulation of the 3-D seepage velocity field in estimating the impact of the
hydropower project Gabčíkovo on groundwaters of the Danube Island Zitny
ostrov, influenced by the interaction with the river Danube and with the whole
canal network of the concerned area, see Figs. 1, 2 and 3. Horizontal area of
the Zitny ostrov covering about 1450 km2 was divided into
quadrangles with 84x48=4032 nodal points. Vertical division considers 10
layers, giving 11x4032=44352 computational nodes. No problems with
computational time occur particulary due to explicit nature of INKANS.
Fig. 1
Zitny ostrov area; Influences of
the Gabcikovo reservoir filling on
groundwater dynamics; Vector field
of the horizontal streams 3
months after start of filling. Model INKANS
Fig. 2
Vertical profile of the velocity vector field in projection B-B through the
Zitny ostrov aquifer (see Fig. 1); time 8 months after start of reservoir
filling. Model INKANS
Fig. 3
Groundwater elevation in profile B-B` of the Zitny ostrov aquifer 12 and 40 days after start of the reservoir
filling; Numeric simulation of free surface h(x,z,t) by INKANS compared
with monitored results (sings x).
MODEL SKOKY
Concept of discontinual velocity
field and above mentioned jump relations are base of the model SKOKY
algoritm. In one element of model domain the permeability
supposed to be constant. Velocity distribution is assumed to has linear and
pressure function P quadratic distribution in the same element. Using this by
means of chosen approximation function in system (1) and (1) one obtains the
first part of linear equation of the model. The second part is given by
velocity jump relation valid on interfaces among elements. Final part of
equations arises from boundary conditions. Therefore result scheme is implicit.
Horizontal discretization of model
domain is rectangular with nonhomogeneous size of rectangles. Interfaces of
layers in vertical discretization can be nonparallel. The discontinual velocity
concept and elastic discretization of model domain allow to apply this model
for groundwater dynamics simulation in complex domains with singularities of
various kind. Some of them are seen in Figs. 4, 5, 6.
Fig. 4
Asymetric groundwater-surface water interaction along partly silted channel
hed. Simulation by model SKOKY.
Fig. 5
By-passing a sealing wal below leve`e. The prefferential ways and the boil
effects. Velocity field. Model SKOKY.
Fig. 6
Groundwater flow around
navigation lock chamber with
left wall gapped; leakage effect; velocity field. Model SKOKY.
Conclusions
Building of model system INKANS +
SKOKY utilizes a possibility to transform
the boundary problem from
originally N-dimensional domain into task,
defined upon its N-1 dimensional boundaries. The crucial
fact allowing this transformation is,
that knowledge of the shape and motion
of free surface of seepage flow is sufficient to describe the velocity field
below this surface. The next
important possibility is to consider the discontinual velocity field and to use
obtained velocity jump relations.
Model INKANS has been applied to
simulate the influence of the Gabcikovo reservoir filling on the groundwater
dynamics of the 1450 km2
Zitny ostrov aquifer. Simulated movemnet of the free water surface shows very
good accordance with field observations, see Fig. 3.
Model SKOKY can be used as
solitary as well as submodel of INKANS for local scale simulations. Its recent
applications on the groundwater dynamics in domains with flow singularities
including leakage, preferential ways, boils and other effects show real
possibility to utilize it for special tasks, e.g. for localization of gaps in
underground structures or prefferential ways in leve`e body.
REFERENCES
[1] COURANT,R.-HILBERT,D.:Methods of
mathematical physics. Vol.
II. Intercience,
New York-London 1962
[2]. Kosorin, K .:
Hydrodynamic Theory of Boundaries; DrSc-thesis; ÚH SAV, Bratislava 1983.
[3] Kosorin , K .: Movement
on free surface flow boundaries;
Vodohosp. časopis, Vol.26,1978,No.2, pp.154-163.
[4] Kosorin, K.:Modelling of groundwater surface water
interactions by means of hydrodynamic theory of boundaries; J. Hydrolog.
Hydromech., 41, 1993, 2-3,.
[5] Mc Donald,M.G.-HARBAUGH,A.W.: A modular 3-D finite-diference
groundwater flow model, US Geological Survey report, book 6, Washington 1988
[6] REFSGAARD,J.C. et all:
Danubian Lowland-Ground Water model,
PHARE/EC/ WAT/1, Draft Final
Pasport, Vol. 1,2,3, sept 1995