FLOW ANALYSIS AROUND UNDERGROUND STORAGE CAVERNS INCORPORATING SPATIAL NONUNIFORMITY OF HYDRAULIC CONDUCTIVITY

FLOW ANALYSIS AROUND UNDERGROUND STORAGE CAVERNS INCORPORATING SPATIAL NONUNIFORMITYOF HYDRAULIC CONDUCTIVITY

IL MOON, CHUNG

Water Resources Division, Korea Institute of Construction Technology,

Daewha Dong 2311, Ilsan, Korea, 411-410.

Phone : 82-344-9100-271

Fax : 82-344-9100-251

E-mail Address : IMJUNG@KICT.RE.KR

WONCHEOL, CHO and JUN-HAENG, HEO

Department of Civil Engineering, Yonsei University, Seoul, Korea

ABSTRACT

In this study, the finite element methodology combined with random field theory was developed to analyze the influences due to the spatial nonuniformity of hydraulic conductivity. Monte Carlo technique was applied to obtain an approximate solution for two-dimensional steady flow in a stochastically defined nonuniform medium. The uncertainty in model prediction depends on both the spatial nonuniformity in hydraulic conductivity and the natures of the flow system such as water curtains and boundary conditions. Especially, we relate these uncertainties with well known gas tightness condition.

Keywords: Storage Caverns, Spatial Variability, Gas Tightness, Uncertainity

INTRODUCTION

Reservoirs for liquified gas must be located so deep below the natural groundwater level in which the hydrostatic pressure of the groundwater exceeds the vapour pressure of the gas at its maximum temperature in the reservoir. This gas thightness condition is suggested by Aberg (1977).

In numerical analysis, groundwater flow around caverns is mainly dependent on the mathematical solution of deterministic model, in which porous media property is greatly simplified. This simplification has great differences between modeling and real situation. Therefore, we suggested the uncertainty of deterministic solution from spatial variability of hydraulic conductivity. Also, we can relate the gas tightness condition with this uncertainty.

DETERMINISTIC MODELING AND GAS TIGHTNESS

The governing differential equation for is the Laplace equation in two dimensional space. The main assumption in this mathematical treatment is that both in hard rock with small fissures and in sedimentary porous formations, the law of motion is properly described by Darcy's law. Theoretical calculations showed that the groundwater above the cavern must flow in the downwards direction with a hydraulic gradient I > 1.0 if bubbles of gas cannot be able to leak out from the cavern upwards through the fractures of the rock and also that I > 1.0 cannot be satisfied by means of the natural flow of groundwater to the cavern unless it is located unreasonably deep. The Sealing water shall instead be recharged from an array of boreholes -so called water curtains- in the rock above the cavern.

The deterministic modeling problem is treated as a two dimensional flow in a vertical plane. The FEM model used in this study is already developed by Chung et al. (1997). The modeling area is 206 m wide and 84.5 m deep. The number of finite elements is 660 and representative hydraulic conductivity is 1.6X10^-7m/sec. The water curtains are located at El. -90m below sea level, and the shape of cavern is horseshue of which the height is 22.5 m and the width is 18m. The corresponding boundary conditions are natural groundwater heads (=El.1.5m), boundaries of caverns (=gas pressure+elevation head) and water curtains (=El. 5 m). In Fig. 1, are the deterministic contours of hydraulic head based on a solution using a uniform hydraulic conductivity value everywhere in the flow domain. According to Aberg's criteria, present gas tightness is secured by water curtains, because the hydraulic gradients above caverns are greater than 1 as shown in Fig. 2.

Fig. 1. Deterministic head distribution

Fig. 2 Result of gradient distribution

In this study, random field theory for the generation of soil permeability properties with a fixed mean and standard deviation, have been combined with finite element method to perform Monte Carlo simulations as presented in Smith and Freeze (1979 a,b).

THE STUDY ON THE SPATIAL VARIABILITY OF HYDRAULIC CONDUCTIVITY

To determine the probabilistic distribution type of real field hydraulic conductivity data various types of probability distribution such as gamma-2, gamma-3, GEV, Gumbel, lognormal-2, lognormal-3, log-Pearson type III were adopted. Method of probability weighted moments(PWM) was applied to estimate the parameters of probabilistic distributions(Hosking, 1986). To obtain the probabilistic distribution type of hydraulic conductivity data in studied domain, 45 hydraulic conductivity data (Korea Petroleum Development Cooperation, 1985) were used. Goodness of fit test for the seven probability models were performed using three types of test (C-square test, Kolmogorov-Smirnov test, Cramer von Mises test). Gamma-2, GEV, Gumbel, lognormal-2 distributions showed good agreement with the data collected. According to past experimental study, the two parameter lognormal distribution was selected as an appropriate for the spatial nonuniformity of studied data.

The method used to generate two dimensional finite elemental realizations with a known spatial mean (M) and standard deviation (S) can be summarized as follows. First, the uncorrelated random number (RN) is generated from a normal distribution with mean zero and standard deviation one N[0,1], and the log transformed mean M then added to each member of Y in Eq. (1).

Y(i)=M+RN S(i) (1)

Finally, the lognormally distributed conductivity values, K are obtained by applying the exponential transform like Eq. (2).

K(i)=exp[2.3026Y(i)] (2)

The permeability field is obtained through this transformation. In this study, the log transformed mean and standard deviation of hydraulic conductivity from 45 field data are -4.79(cm/sec) and 0.19, respectively. Three standard deviations of 0.5, 1.0, 1.5 for the degrees of nonuniformity were assumed. Monte Carlo method was performed to compute the variability of results from a large number(1,000) of simulations. By repeating the analysis over a series of runs, the frequency distributions of hydraulic head at the nodal points in the finite element grid can be analyzed to obtain estimates of the mean and standard deviation of head and gradient.

UNCERTAINTY IN HYDRAULIC HEAD

Now, the stochastic problem is considered to analyze the uncertainty in hydraulic head. S(h)-the standard deviations of hydraulic head- are contoured in Fig. 3(a)-3(c). In these cases, the standard deviations are 0.5, 1.0, 1.5, respectively. As the degree of nonuniformity in hydraulic conductivity increases, so too does the degree of uncertainty associated with the predicted hydraulic head values S(h) is the greatest in the upper part of the caverns.

(a)

(b)

(c)

Figure 3. The distribution of STD(standard deviation) of hydraulic head

(a) STD of K is 0.5 (b) STD of K is 1.0 (c) STD of K is 1.5

(a)

(b)

(c)

Figure 4. The distribution of standard deviation of hydraulic gradient

(a) STD of K is 0.5 (b) STD of K is 1.0 (c) STD of K is 1.5

S(g)-the standard deviations of hydraulic gradient- are greater in the upper region of caverns where the mean hydraulic gradients are relatively large. As uncertainty in hydraulic gradient increases, the gas tightness condition may not be secured, because the standard deviation of hydraulic gradient may greater than 1 by increase of spatial variability of hydraulic gradient.

CONCLUSION

The two parameter lognormal distribution is selected a proper distribution type of real field hydraulic conductivity data. The standard deviations in hydraulic head are strongly influenced by the spatial variation of hydraulic conductivity and nature of flow system such as water curtains and boundaries. Also, the spatial variability of hydraulic conductivity directly affects on the gas tightness condition.

REFERENCES

Aberg, B. (1977). " Prevention of gas leakage from unlined reservoir in rock." Rockstore 77. pp.399-413.

Chung, I.M. , Cho, W.C. and Bae, D.H. (1997). "Establishment of numerical model for the groundwater flow (water curtain) analysis around underground caverns." Journal of Korea Water Resources Association. Vol 30. No. 1, pp.63-73.

Hosking, J.R.M. (1986). "The theory of probability weighted moments." Research Report RC12210, IBM T.J. Watson Research Center, Yorktown Heights, New York.

Korea Petroleum Development Cooperation. (1985). The report of basic invesigation of "A"storage cavern.

Smith, L. and Freeze, R.A. (1979a). "Stochastic analysis of steady state groundwater flow in a bounded domain, 1. One dimensional simulation." Water Resouces Research, Vol. 15, No. 3, pp. 521-528.

Smith, L. and Freeze, R.A. (1979b). "Stochastic analysis of steady state groundwater flow in a bounded domain 2. Two dimensional simulations." Water Resources Research, Vol. 15, No. 6, pp. 1543-1559.