1
SURFACE RUNOFF MODELLING IN STEEP TERRAIN AT GIS
ENVIRONMENT*
R. ARSOV
University of Architecture, Civil Engineering and Geodesy
1 Chr.Smirnenski Blvd., 1046 Sofia, Bulgaria
1. Introduction
Protection from floods and flood mutigation has been always of great concern of human
civilizations and nowadays is an important part of the hydraulic engineering. Urban
flooding in particular, is still one of the most dangerous natural events concerning the
life and leaving conditions of millions of people all over the world.
Because of its scale, complicity and multilateral dependency, urban flooding is
investigated and modelled usually taking into account the runoff, generated in the
relevant urban area only, while dependence of this event on the runoff from the
upstream part of the relevant river catchment is well known fact. Nevertheless separate
modelling of urban areas and river basins is still a current practice, motivated mainly by
the problems, associated with the different scale of the relevant objects and their spatial
discretization. As it has been proved many times at various hydrological and
morphological conditions, the urban areas flooding and silting occur in a result of runoff
generated outside the urban territory in the upsteram part of the relevant river basin
catchment. Therefore integrated approach in this sense is indispensable for adequate
modelling, prediction and control of the flood and silting events. In this respect
adequate surface runoff modelling on the steep terrain is of great importance, since this
is the most usual case for the river basins catchments, particularly in the mountainous
regions, where the most dangerous floods originated from.
Because of the complicity of the problem, all the commercially available urban
drainage and river runoff models utilise conceptual approach for surface runoff
modelling, which in consequence demands complicated calibration, availability of
relevant reliable data and long enough period for their acquisition. In this respect
physically based models offer more reliable runoff and flood prediction with fewer
amounts of data. In this case however, more detailed spatial discretization and terrain
properties definition associated of the urban and/or river catchment is needed, which
makes indispensable implementation of the contemporary information technologies and
tools.
As a complicated spatially depending problem, contemporary surface runoff
physical modelling is associated with implementation of Geographical Information
Systems (GIS), usually used in pre- and post-processing but not yet in the physical
model itself. Incorporation of GIS along with suitable spatial discretization engines in
the runoff physical models will establish new generation tools for more reliable and
adequate prediction of surface runoff and associated flood and silting events.
*Published in “Urban Water Management”, NATO Science Series, Environmental Security –
Vol.25, Kluwer Academic Publishers,2002, The Netherlands
2
The incorporation of GIS in physically based runoff models however is
associated with overcoming of specific problems. The later are based on the fact, that
the vector fields of velocity and flowrate in the relevant GIS electronic layers are by
definition presented in their horizontal projections (plane), at which the real values of
the relevant vectors are loose. This in turn reflects on the adequate description not only
of the surface runoff itself, but also on associated silting phenomena modelling, where
considering of the real surface score/deposition forces is of primary importance. It is
obvious that these problems are not important for runoff modelling from flat surfaces,
as well as in 2D shallow water bodies modelling (estuaries, lagoons, shallow channels,
etc.), where they are neglected as a rule. This is not the case however with the physical
modelling of runoff on steep terrain with GIS implementation, where the real
magnitudes of the velocity and flowrate vectors differ significantly (at extend,
depending on the slope) from their horizontal projections.
For overcoming the above problems, a specific tool is necessary to be
introduced in the model for conversion of the mentioned vector fields from their real
magnitudes (associated with their generation on the slope surface) to the relevant
horizontal projections of their in the GIS electronic layers for further processing. Such a
tool is well known in the mathematics (analytical geometry), where it is applied for
vectors presentation in two differently oriented orthogonal (Cortesian, Dekart)
coordinate systems. Application of this tool at the physically based modelling of runoff
on steep terrain however creates additional problem, associated with the mutual
influence of the magnitudes of the relevant orthogonal vector’s components in their
transfer from one coordinate system to another, as it is shown hereafter. For overcoming
this additional problem, the transformation tool has to be involved at the earliest stage
of the model development - derivation of basic hydraulic equations.
This paper is devoted in description and explanation of the procedures and
relationships, comprising a new generation of 2D shallow flow model, incorporating
GIS implicitly in the derived basic hydraulic equations.
2. Terrain discretization and parameters space distribution
As it is well known, the geographical information systems - GIS comprise various
databases and offer possibilities for their arrangements, analyses, processing, as well as
analytical or graphical presentation of the results of these procedures. Creation of digital
terrain model – DTM (or digital elevation model – DEM) as one of its electronic layers
is among the basic features of every real GIS. Obviously the DTM is the most important
electronic layer for the physically based runoff modelling.
Spatial distribution of any parameter, used in the model (rainfall intensity, land
use, land cover, hydraulic conductivity, slopes and their aspects, etc.) can be represented
in a separate electronic layer, closely related to the relevant mesh, discretizing the
terrain space. This give an opportunity for build-in implicitly in the runoff model the
information organized and adapted in the GIS.
Definition of the spatial distribution of above mentioned parameters, as well as
the numerical methods, applied in solution of the partial differential equations,
comprising the runoff model are based on relevant terrain discretization. For 2D objects
3
under consideration the Finite Element Method - FEM is most suitable, based on an
optimal discretization mesh. For the later, an optimal version of the well-known
discretization method of Irregular Triangular Network – TIN is convenient. An
illustration of optimal TIN application at an urban area is shown in Fig. 1.
Legend
Buildings
Grass
River
Streets
50
0
50
100
150
[m]
Figure 1. Urban area discretised through TIN
Every triangular element (pixel) of so discretized terrain represents a
horizontal projection of the relevant declined plane. The later is defined by the
coordinates of the relevant three nodes,: 1 (x1, y1, z1), 2 (x2, y2, z2) and 3 (x3, y3, z3). They
are computed and registered at the DEM in relation to an arbitrary chosen orthogonal
coordinate system xyz, accepted as a basic one, where the plane xOy is horizontal (Fig
2). Its main (directing) vector – R (A, B, C) defines space positioning of every declined
plane triangle, where
4
y1
A = y2
y3
z1 1
z2 1 ,
z1
B = z2
z3 1
z3
x1 1
x 2 1 and
x3 1
q
v'y
x1
C = x2
x3
y1 1
y2 1 .
(1)
y3 1
z,r
v'x
vy
vx
Figure 2. TIN element and water unit volume over its declined plane,
represented in local and base orthogonal coordinate systems
The DTM and the TIN along with the others electronic layers, defining the
hydraulic-related parameters space distribution, are the necessary virtual (digital) base
offered by the GIS, which is suitable for its implicit incorporation in the physically
based runoff model. The only missing element for adequate runoff modelling at this
stage is a tool, transforming the velocity vectors from their real values, associated with
the decline TIN planes of their origin, to their horizontal projections in the relevant
horizontal GIS layer.
3. Velocity vectors transform parameters
Sophisticated processes of mass and energy transfer and transformation, associated with
a surface runoff can be adequately modelled based on their balance at an elementary
volume of the water body, running over a homogenous declined terrain surface with
constant characteristics. Here this elementary water volume is oriented firstly towards
the local orthogonal coordinate system x’y’z’, associated with the relevant TIN pixel, so
that the coordinate system plane x’O’y’ coincides with the declined plane of the relevant
discrete terrain module (triangle in 3D space), as it is shown in Fig. 2. The same
elementary water volume is oriented also towards the basic orthogonal coordinate
5
system xyz, in respect of which all input and output data and results are represented in
the GIS environment. Obviously the orthogonal components v’x and v’y of the velocity
vector v’, associated with the running elementary water volume (Fig. 2) are represented
in their real values in the local coordinate system x’y’z’. Their horizontal projections vx
and vy in the basic coordinate system xyz however have less values, depending on the
slope of the relevant discrete terrain module.
As it has been already pointed out, only the horizontal components vx and vy
can be processed in the GIS environment. Therefore, the real values v’x and v’y,
obtained in a result of the physically based modelling have to be transformed into their
horizontal projections vx and vy. For this it is necessary to define the local orthogonal
coordinate systems orientations towards the basic one xyz. As it is well known, the
relevant transform operator  can be represented as follows:
where
 
 
 cosxˆ' z ; n
l1
l2
l3
  m1
n1
m2
n2
m3 ,
n3
 
 
 cosyˆ' z ; n
(2)
 
 
 coszˆ' z .
l1  cos xˆ' x ; l 2  cos yˆ' x ; l3  cos zˆ' x ;
m  cos xˆ' y ; m  cos yˆ' y ; m  cos zˆ' y ;
1
n1
2
2
3
3
Because the local coordinate system’s plane x’oy’ coincides with the declined
terrain plane, associated with the relevant TIN pixel, the local main (directing) vector –
R (A, B, C) of the later is parallel to the axis oz’. Therefore, A = l 3 , B = m3 and C =
n3 . Taking this into account and assuming a definite mutual orientation of local and
base coordinate systems (with a symmetry, for instance such as to insure
relationship (2) is transformed into the following one:
l1
  m1
l2
l1
A
B.
n1
n2
C
l1  m2 ), the
(3)
Based on the matrix mathematics, the following system of five equations can
be obtained for definition of the unknown parameters l1 , l 2 , m1 , n1 and n 2 in
relationship (3):
6
l12  l 22  A 2  1
m12  l12  B 2  1
l1 .m1  l 2 .l1  A.B  0
l1 .n1  l 2 .n2  A.C  0
m1 .n1  l1 .n2  B.C  0
(4)
After solvating the above equations system, the following expressions are
obtained for the parameters of matrix (2):
l1  A.B /( A12  B12 ) ;
(5)
l 2  1  A 2 .B 2 /( A 2  B 2 ) 2 ;
(6)


m1  A2 .B 2  ( A2  B 2 )(1  B 2 / A.B( A2  B 2 ) 2 ;


(7)

n1 
C. A  C A 2 .B 2 (1  B)  (1  B 2 ) . 1  A 2 (1  B 2 ) ;
A.B 2 2 A.B 2  (1  A 2 ).( A 2  B 2 )
n2 
C.( A 2  B 2 ) A .B 2 .(1  B)  (1  B 2 ) .
B. 2 A 2 .B 2  (1  A 2 ).( A 2  B 2 )






(8)
(9)
For convenience in the explanations hereafter the following substitutions are
assumed:
i xx  cos( xˆ' x)  l1 ;
i yy  cos( yˆ' y)  m2  l1 ;
i  cos( zˆ' z)  n  C ;
zz
3
i yx  cos( yˆ' x)  l 2 ;
i  cos( xˆ' y )  m ;
xy
1
i zx  cos( zˆ' x)  l3  A ;
i  cos( z ˆ' y)  m  B .
zy
3
(10)
(11)
(12)
(13)
(14)
(15)
(16)
The following relationships are valid for the unit vector orthogonal
components i g , x and i g , y , defining respectively the slopes of the declined terrain plane
(associated with the relevant TIN pixel) along the axes Ox and Oy of the base
coordinate system xyz:
7
i g , x  sin( xˆ' z )  (1  n1 ) 0,5 ;
i  sin( yˆ' z )  (1  n ) 0,5 .
g,y
(17)
(18)
2
The parameters defined by equations (10)  (18) are unit vectors, needed for
transformation of the runoff velocity vectors from the local to the basic orthogonal
coordinate systems for further processing in the GIS environment. For obtaining the
adequate values the runoff velocity vectors, the relevant hydraulic equations have to be
derived firstly, based on the mass and momentum conservation relationships.
The water currents mass and momentum conservation relationships, derived for
different conditions are usually based on the relevant balances regarding an elementary
water volume. For the case of runoff modelling on a steep terrain, such balances here
are made regarding to the acting forces, shown in Fig. 3. Explanations are given in the
text.
'x
v 'x
h dy
y '
'
h
y '
'
h d x
'z,y
x '
h
x '
h
q
'y
'y,z
'x,y
'y,x
'o,x
'y
'
y,z y'dy'
'
'
v y d y '
'y+ y'
'
'z y
dy '
G
q
v 'y
z 'y dx '
'

'z,y x
o,y
 dx
y' 
'y ',y x '
,z
z
d

y ' y'
dx
 dy '
'x y'
x'
v
z,r
'
x' 
'x
x ' dx '
x ' dx '
vx'+ v
'x
z,f
Fig. 3. Stresses applied at running water unit volume, represented
in the basic orthogonal coordinate system
4.
Continuity equation
The elementary water volume mass m is defined by equation (19):
m   .dx'.dy '.z ' ,
(19)
where  is water density.
The rate of change dm/dt of the mass m can be represented by equation (20),
depending on the intensities of the rainfall – qz,r and infiltration – qz,f, which are vertical
vectors:
8
dm / dt   .[( q z ,r  q z , f ).i zz .dx'.dy '(v x' / x' (.( z '.dx'.dy '
 (v 'y / y ' ).( z '.dx'.dy ' )]
(20)
Taking into account equations (19) and (20), after some rearrangements the
following equation is obtained:
z '.(v x' / x'v 'y / y ' )  v x' .z ' / x'v 'y .z ' / y 'z ' / t 
(21)
 (q z ,r  q z , f ).i zz
Taking into account that
v x  v x' .i xx  v 'y .i yx , v y  v 'y .i yy  v x' .i xy ,
dx  dx'.i xx , dy  dy '.i yy , dz  dz '.i zz and derived relationship z’ = f(h) =
= h.i zz /(1  i g , x .h / x'i g , y .h / y ' ) , after neglecting the derivatives of second
order and some rearrangements, equation (21) can be finally transformed into the
following one:
h(
i yy  i xy .i yx
v x i yy  i xy .i yx v y i yx v y i xx  i yx .i xy v x i xy
.

. 
.

. )  (v x .
x
i yy
x i yy
y
i xx
y i xx
i xx .i yy
i xx  i yx .i xy
i xy
h
h h
 vy .
)i xx .  (v y .
 vx .
)i yy . 
 q z ,r  q z , f  0
i xx .i yy
dx
i xx .i yy
i xx .i yy
dy t
,
i yx
(22)
where h is piesometric head.
The relationship (22) is the investigated mass conservation (continuity) equation for
the surface runoff on steep terrain, represented in horizontal plane (2D space) of a GIS
layer.
5.
Dynamic equations
As it is usual for similar cases, momentum conservation balance for the elementary
water volume, shown in Fig. 3 is applied hereafter for the case of runoff on a steep
terrain. The following acting forces, applied along the axes of the local coordinate
system x’y’z’ are considered:

'
Mass forces - Pm (represented only by the axes’ components of gravity -
Gi )
9

Surface forces shear stress -
Pf' (depending on the axes components of water body
 i', j , water body normal stress -  i'
on the contact surface water/solid bed 
Inertial forces -
'
in,i (depending
P

and tangent shear stress
'
0 ,i )
'
on the axes components of velocity - v i
and acceleration - vi / t )
'
According to the well known definitions in hydraulics, the following
relationships are valid for the components on axis O’x’ of the above stresses:
 x'  p   ' , x  .g (h  z'.i zz / 2)  2..v x' / x' ;
 x' , y   y' , x  .(
v 'y
x'

v x'
);
y '
 0' , x  .g.z '.I x'  .g.z '.
where
 ' ,x
. x (v x' ) 2
8.g.z '
(23)
(24)
;
(25)
is normal stress component on O’x, reflecting viscosity impact on velocity
changes;
p - hydraulic pressure;
 - viscosity;
g - gravity acceleration;
I x' - hydraulic gradient;
. x
- hydraulic resistance coefficient.
Taking into account some geometrical relations, following relationship can be
written about hydraulic head h :
h
z'
h
h
.(1  i g , x .
 ig , y . ) .
i zz
x'
y '
(26)
For convenience the formula of Fedorov (similar to one of Kolebrook-White)
is used here for definition of  , which is given explicitly there:
1
. x
 2 lg(
a1
a
a
a .
 2' )  2 lg( 1  ' 2 ) ,
'
z ' v x .z '.
R x Re x
(27)
10
where a1 and a 2 are constants, depending on the terrain roughness and its shape,
respectively.
Taking into account equations (23)  (27), the forces, applied at the water unit
volume (Fig. 3) can be defined by the following relationships:
Pm'  Gx  .g.z'.dx'.dy'.i g , x ,
(28)
 z' 
h
h  z '.i
2 v x' 
Px'   x' .z '.dy'  .g.z '. .1  i g , x .
 i g , y .   zz 
.
 dy' ; (29)
x'
y' 
2
.g x' 
 i zz 
 v 'y v x' 
'
'
.z '.dx' ;
(30)
T y , x   y , x .z '.dx'  .

 x' y ' 


To', x   0' , x .dx'.dy' 
Pin' , x  m.
a
a 
1
..(v x' ) 2 . lg 2  1  ' 2 . .dx'.dy' ;
32
 z ' v x .z '  
 v '
dv x'
v ' 
  , z '. x  v x' . x .dx'.dy' ,
dt
x' 
 t
(31)
(32)
'
where Px is the hydraulic force component on axis O’x’;
T y' ,x - water body shear force component on axis O’x’;
T0', x - component on axis O’x’ of shear force on the contact surface
water/solid bed;
'
in,x
T
- inertial force component on axis O’x’ (D’Alambert’s force equivalent of
momentum).
'
The derivative component of the surface force Px , applied at the water unit
volume on the axis O’x’ (Fig.3) is defined by the following equation, based on
relationship (29):
2 
 z '
Px'
h
h 
.dx'  .g.z '. .1  i g , x .
 i g , y .   i zz . .dx'.dy' ; (33)
x'
x'
y' 
 i zz 
 x'
The investigated momentum conservation equation in respect to the axis O’x’
is based on the following balance of the forces, applied at the elementary water volume
(Fig.3):
Pm' , x  Pf' , x  Pin' , x  0 ,
or
(34)
11


T y' , x


P '
G x'  Px'   Px'  x .dx'   T y' , x   T y' , x 
.dy '   T0', x  Pin' , x  0 . (35)


x'
y '




Considering equations (31)  (33) and neglecting the derivatives of second
order, equation (35) can be transformed into the following one:
v x'
v ' z ' 
2
h
h 
 v x' . x 
.g.i zz  1  i g , x .
 i g , y .   g.i g , x 
t
x' x' 
i zz 
x'
y ' 
(v x' ) 2  2  a1
a 

. lg   ' 2 .   0
32.z '
 z ' v x .z '  
.
(36)
Relationship (36) is actually the investigated dynamic equation, represented on
the axis O’x’ of the local coordinate system x’y’z’. For its transforming to the base
coordinate system xyz the following relationships have to be considered:
dx' dx / i xx ;
(37)
v x  v x' .i xx  v 'y .i yx ;
(38)
v y  v 'y .i yy  v x' .i xy .
(39)
Taking into account relationships (37)  (39) as well as equation (26) with
neglecting derivatives of second order, the following expressions are obtained,
respectively:
v x'
v i xx  i yx .i xy v y i yx
 x.

. ;
x'
x
i yy
x i yy
v 'y
y'

(40)
v y i xx  i yx .i xy v x i xy
.

. ;
y
.i xx
y i xx
(41)
1

z ' h
h
h 
 .i xx .i zz .1  i g , x .i xx .  i g , y .i yy .  .
x' x
x
y 

(42)
For simplification of the expression, obtained in result of transformation of
equation (31) in respect to the basic coordinate system, it is assumed that
h / x'  0, h / y '  0
and
i yx  i xy  0 .
Then
z '  h.i zz , v x'  v x / i xx ,
v 'y  v y / i yy and equation (31) can be transformed into the following one:
12
T0, x 
.v x2
 a
a .i . 
. lg 2  1  2 xx  .
32.i xx
 h.i zz v x .h.i zz . 
(43)
The above simplifications have insignificant impact on the value of T0, x , which is
comparable with the accuracy of selected values of the roughness coefficients
a1 and
a 2 . Taking into account relationships (37)  (43), equation (36) can be finally
transferred as follows:
i  i .i
i
vx iyy  ixy .iyx v y iyx
v i  i .i v i
.
 .
 (vx . yy xy yx  v y . yx )( x . yy xy yx  y . yx ) 
t ixx .iyy
t ixx .iyy
ixx .iyy
ixx .iyy x
iyy
t iyy .
 g.ixx (2  izz2 )
(44)
h
v
a
a .i .
 2
lg  2 ( 1  2 xx )  g.ig , x  0;
x 32.ixx .izz .h
izz .h izz .h.vx .
2
x
Following the same approach, the following final version is obtained,
concerning the relevant forces balance on the axis Oy of the basic coordinate system:
v y i xx  i yx .i xy v x i xy
i yy  i yx .i xy
i xy v y i xx  i yx .i xy v x i xy
.
 .
 (v y .
 vx .
)( .
 . )
t
i xx .i yy
t i xx .i yy
i xx .i yy
i xx .i yy y i xx .i yy
y i xx
v y2
a 2 .i yy .
a
h
 g.i yy .(2  i zz2 )  2
lg 2 ( 1 
)  g.i g , y  0
y 32.i yy .i zz .h
i zz .h i zz .h.v y
. (45)
Relationships (44) and (45) are the investigated dynamic (momentum
conservation) equations, which along with the continuity (mass conservation) equation
(22) fully describe the surface runoff on steep terrain (2D space), represented in a plane
layer of GIS environment.
6.
Conclusion
The GIS implicit involvement in hydraulic modelling is still not a usual practice. The
attempts for introducing such approach may face definite problems, overcoming of
which needs new specific developments.
Such specific approach has been discussed in this paper for the case of surface
runoff modelling on steep terrain in GIS environment. Crucial problem in this case is
the velocity vectors transfer from their real values, associated with the decline planes of
their origin, to horizontal GIS layer. Equations (22), (44) and (45) are contribution in
this respect and are alternative to usual approach of employing of 1D equations of
Saint-Venant for hydraulic modeling in 2D space, which is incorrect in principal.