QUANTITATIVE DESCRIPTION OF SURFACE VERTICAL KARSTIFICATION
THE MATHEMATICAL MODEL OF THE SURFACE VERTICAL KARSTIFICATION
Let us examine a joint or a fault in the karstic rock (limestone) that is approxi
mately vertical and ends at the surface. This fault gets waters flowing down upon the surface so the CO2 originates from the atmosphere and not from the soil. Let us suppose that the solvent infiltrates at v[m/s] speed and the rock is being dissolved.
Depending on the water supply, the solution process is assumed to be quasista- tionary, the flow is laminar being in thermal equilibrium with the environment.
First, the saturation process of the infiltrating solution is analysed. It is as
sumed, that the entering water does not, contain dissolved limestone or contains it in a very small amount. The infiltrating solution contacting the surface of the rock exerts its solvent ability until it becomes saturated.
Let m[kg] denote the limestone mass that can be dissolved for achieving satu
ration by the V0[m3] volume unit of the downwards moving solution at x [m] depth from the surface.
Our experience allowed to set up two hypotheses: If -dm is the decrease of m during the dx downwards moving of the solution, the C 0 2 content of which is un
changed, and it does not mix with other solutions, then
1) -dm is directly proportional to the value of m that is, the closer the solution ap
proaches saturation when infiltrating downwards, the less ability it has for further solu
tion:
2) -dm is directly proportional to the value of dx, that is the longer section the solution passes downwards, the less ability it has for solution.
Therefore
/1/ - dm = A • mdx
where Atm'1], A>0 is the constant characteristic of the solution process. Parameter A practically characterises the solvent power of the infiltrating solution. From /1/ after integration
/2/ m = m ( l e _Ax
is dissolved, when m=m0 if x=0, where m0[kg] denotes the maximum limestone mass that the unit volume of water can take up. If C, is the limestone concentration of the infil
trating solution at x depth and Ce is the equilibratory concentration of saturated solution, then function C,=C,(x) is obtainable from formula 111. According to it, in the case of unit volume of solution Ce and Ce-C, are the equivalents to quantities m0 and m, respectively.
Thus
/3/ Cc - Cj = Ce ■ e~**
that is
/4/ C, = C, ■ (I-« -* ')
is dissolved. Using the above equation, a formula is obtained which describes the shape and the temporal evolution of the formation developed during vertical karstification.
D. Rickard's, E. L. Sjöberg's (1983, 1984) and J. V. Dublyansky's (1987, 1988) differential equation is used for the mathematical description of the first stage:
dm
dt k K + Kt
■ S-(C e - C t ),
is the rate of chemical dissolution
is the rate of material transport in the boundary zone
is the surface of the dx wide zone at x depth of the vertical karstic for
mation
is the limestone mass dissolved from zone of S surface at x depth of the vertical karstic formation during dt[s] time (Figure 2) According to our experiences kKkT
« V . Let us determine the /?[m] radius of the ideal vertical karstic formation. This radius is the function of x[m] depth measured from the karstic surface and the i[s] time. Let us search for the explicit form of R=R(x,t) func
tion which is the special solution of differen
tial equation /5/ for the case when function
with the use of which equation /5/ has the 111 <77? _ k Kk T C, — C,
/ 8 / dR _ k Kk T Ce dt k K + k T p
According to J. V. Dublyansky (1987) formula /1 0/ k T = — - V Ö V
T 8 d
is valid for the rate of material transport in the boundary zone, where c/[m] is the char
acteristic measurement of the conduit, here the diameter of the conduit at x depth, that is d=2R,
0[m 2/s] diffusion constant,
v[m2/s] kinematic viscosity coefficient of the flowing solution.
Equation /9/, by using /10/, can be written in the
/11/
form, from which the explicit form of function t-t(R ) can be derived by integration:
/12/
form, where
a J _ L + i 6 _ j ) p _ e * dR { k K 85 V Ö V J C,
p_
+
aC / U*+85 VűV,
It can be seen that with data denoted in years the term linear in R is negligible in relation to the term quadratic in R, thus formula /12/ has the simpler
_A R = a ■ Vf • e 2 and
/9/
follows.
± J ± ± } _ P _ At dR \ k K k T ) Ce
/14/ flsJ “ V 5 T ^ ,
tively. The a quantity seen in formula /14/
is related to the activity of the solution ideal, symmetrical pipe narrowing downwards and is theoretically endlessly deep. Our model is valid for the description of the first stage of evolution of real shafts only when the infiltrating solution is not saturated.
Form ula/13/ postulates continuous water supply. Practically, it is intervallical, thus a h proportion coefficient can be determined for the given pipe. If during a fixed T0 time - e.g. for t0 time during a year - water infiltrates into the examined sinkhole, the 77
The mathematical description of the second stage of the evolution is discussed in the followings. At the end of the first evolutional stage the ideal pipe obtains the hori
zontal dimensions when the solvent is not able to fill it up during the solutional periods, but runs down on its walls.
Simplifying the argument the shape of the pipe developed by the end of the first evolutional stage is regarded as a cone narrowing downwards, the sides of which enclose a angle with the horizontal. If the radius of the orifice of the cone is R0 and the intensity of the incoming water is 7, then according to L. D. Landau and E. M. Lifsic (1980):
/18/ / = 2R0nhaghg sin a
3 v
where Ao=2R07ih0 is the flow section at the orifice, ghl sin a
v = — - ---3 v
is the rate of flow at the same location, h0 is the thickness of the
solution, g is the gravitational acceleration and v is the kinematic viscosity coefficient (Figure 4) If / is known (measurable or calculable), the h0 can be determined by /18/:
/19/ K = -3
3 v 7
2Ran g ú n a
If (S*[m] denotes the thickness of the Kármán-boundary zone in the solution flowing down on the wall of the pipe, and jc'[m] denotes the distance from the orifice covered by the solution, then according to F. M. White (1979):
/20/ 5 ' _ 1,721 Fig. 4. Diagram o f the water flow on the wall o f the ideal pipe
where
/21/ R e, vx'
V
is the local Reynolds number. Then, 8* by using /20/ and /18/
/22/ o 1.72 lx ’ 2.981V
<5* =
ghl sin a , h„Jg sin a 3v
where
/23/ x = x'- sin a
is the depth from the orifice. As the rate of material transport is
/ 2 4 /
Dublyansky's (1987) formula /5/ equation /9/ was obtained, from which, after substitu
tion by /25/ the
differential equation is obtained. After integration and substitution of g=9.81 m/s2 with fixed x formula
In further descriptions the relation of the angle of the walls and the depth could be regarded, though the method above gives satisfactory results as we shall see in the followings.
Similarly to formula /13/, /28/ also postulates continuous water supply. With the help of t] proportion coefficient explained in /16/ formula /28/ obtains the
( 1 0.952 v A p h „
/2 9 /
form.
The second stage of the evolution of the pipe described above needs much more time, therefore the length of the first "embryonic" stage is negligible for the esti
mation of the age.
The possible improvement of the mathematical model of vertical karstification is discussed regarding the horizontal growth of the pipe as well. The value of t] propor
tion coefficient introduced in /16/ was supposed to be constant and unrelated to the momentary R0 radius of the orifice characterising the horizontal dimensions of the pipe.
Actually, solution of increasing intensity is needed for the quickening of the solution as the geometrical measurements of the shaft increase, that is rj decreases with the growing dimensions. Under given climatic circumstances the developing shaft achieves the inac
tive stage, when due to its size, even the heaviest rain fails to start a global solution process. A global solution which includes the total surface of the pipe postulates the inequality between the 8* thickness of the boundary zone given in /22/ and the h thick
ness of the water flow on the wall of the pipe:
Both members of the inequality depend on x depth measured from the surface.
/33/ K < — Rn
Here K represents the parameters character
ising the given pipe K is a constant independent o f * depth (0<x<m). That is, the minimal water intensity
I mi„ necessary for the global solution of the pipe is directly proportional to R0 radius of the orifice, that is
/34/ I K •/?„
The ages of pipes can be estimated by for
mula /29/. First the parameters of formula /29/ are determined.
For the determination of Ce equilibratory concen
tration of the saturated solution we start from the concentration of the C c0i absorbed in the water from the atmosphere.
On the basis of Henry-Dalton's law
/35/ Cco_ = 1.9634Lp
Fig. 6. Precipitation time o f a given karstic area as the func
tion o f the intensity o f precipi
tation t=t(I)
where p is the partial pressure of the C 0 2 in the atmosphere, L is the solution coefficient of C 0 2, which is the decreasing function of temperature.
Function L=L(t) is seen in Figure 7, constructed on the basis of L. Jakucs's data (1971).
The C co concentration consists of two parts; concentration of absorved in calcium carbonate Ck [kg/m3] the equilibratory concentration C r [kg/m3] necessary to keep the calcium carbonate in solution. According to J. Tillmans (1932) the relation of the two concentrations is
/36/ c = C 3
" K,
where K, [kg2/m6] is a factor conditional on an absolute temperature. Following D.
Balázs (1966), this function is
/37/ K t = 1.835-102 . e -fl-029T
Whereas the CK concentration of the fixed C 0 2 is directly proportional to the C,.
equilibratory concentration according to formula
/38/
C C02 ~ )
Ce =2.278 CK
thus after D. Balázs (1966) by using formulas /35/ - /38/ the function of Cco and Ce can be constructed seen in Figure 8. In the case of the studied atmos
pheric CCOi values (0< Cah <10'3 kg/m3). Cr is neg
ligible, thus formula
/39/ C' = 2.278 ■ CCOj
is approximately valid since Cco =Cg. Parameters kK D and v can be determined by the Arrhenius equations.
EK /40/ k K = A K e R T ,
Fig. 8. Function Ck=Ck(Cc) and
C =C (Cc)
'-co , '-c o2 v e'
whereA K -5 ,3 6 \ 0 5m / s , E K = 5 ,4 1 1 0 47/m o l, R'= 8 ,3 1 4 7 /molK\
Ep
/41/ D = A D e R T ,
where A D =2.37 ■ 10‘3m2 I s , E D = 3,72 • 104 J / mol;
E v
/42/ v = A v e
w here-A v =2.59 • 10_9m2 I s, , E v = 1.46-10 * J I mol,
and p=2930 kg/m3 is the density of the limestone. R' in formulas /40/ - /42/ is the univer
sal gas constant while EK ED and E v are the virtual empirical activation energies charac
terising chemical solution, diffusion and viscosity on the basis of Sjöberg and Rickard's (1983) results.
For the estimation of parameter A let y[m] denote the depth from the surface, where the infdtrating water is 99 % saturated. Then, according to /4/
/43/ 0.99Cf = (7,(1 — e Xy) from which
/44/ 2 In 10 _ 4.605
y y
follows. Having the parameters above, calculations are done for the increase of R0 radius of the pipe orifice. Using /29/
/45/
P
is obtained. The results are shown in Figure 9 and Tables 1 and 2. (Calculations were based on the 0,0003 pC 02 normal atmospheric partial C 0 2 pressure and we supposed 500 hour/year activity time in 0-30 °C range.
Fig. 9. Pipe diameter d as the function o f t age at 0°-30°C temperature with 100-400 hour water inflow time
Table 1. Diameters o f avens evolved during 103 - 1 0 s years in 0°C - 30°C temperature interval with
Table 2. The necessary time fo r development of avens with 1-10 m diameters in 0°C - 30°C tempera
ture interval with 500 hours rainfall time (in 1000 years) Tempe
0 14.8 29.7 44.5 59.4 74.2 89.1 103.9 118.8 133.6 148
5 11.6 23.3 34.9 46.5 58.2 69.8 81.4 93.1 104.7 116 planes widen without debris production) provides information about the growth rate and the development of the shape of the vertical karst formations. Let us emphasise that we started from a carbonate solution where the CO2 content of the solvent is determined by the partial C 0 2 pressure of the atmosphere. Therefore our model does not calculate with the biogenic C 0 2. Its complicated alteration could only be followed by adequate meas
uring series. In vertical alpine karst formations waters drained from rock surface absorb only biogenic and atmospheric C 0 2 - thus the control calculations can be done. In time scale the calculated age of development and the expectable age converge (Tables 1 and 2)
(The expectable age of development is maximum 10,000 years, since the de
velopment of formations in glacier valleys could only start after the ice had retreated.)
The rate of development of the vertical karst formation depends on the duration of the water inflow and the water temperature. Its depth and shape depends on the rate of saturation. The developed shape is inherited during further evolution if the values of the listed factors are unchanged. of the vertical karst formations does not decrease considerably even if achieving larger size they get the previous quantities of precipitation according to the smaller size.
Contrary to the previous theory, the increase of the temperature of the solvent does not decrease the rate of development, but enhances it. (The quantity of the equilibratory C 0 2 increases at higher temperatures, that is the quantity for the solution is decreased. This effect is largely compensated by the rate of the chemical solution growing with temperature.)
When the vertical karst formation is covered by soil (temperate karsts, tropical karsts) biogenic C 0 2 also influences the development in addition to the abiogenic C 0 2.
In this case the rate of development is changing (according to the daily or sea
sonal fluctuation of the biogenic C 0 2) or is different in each formation.
The C 0 2 production depends on the properties of the local soil and on the amount of solvent arriving indirectly through the soil and directly from the surface at a given moment. There exists a correlation between the C 0 2 content and the rate of devel
opment in this type of karstification as well (equations 39, 45).
Therefore, each time the solvent contains more C 0 2 the less time is needed for the development of a certain size (provided the other parameters are unchanged.)
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N o w a v a ila b le !