Mathematical models for the repairs of the Kariba Dam

李瀟+谷洋+李保坭

（大連理工大學）

一、Digest

The famous Kariba Dam， a concrete arch dam along the Zambezi River， is now in a dangerous state for the decreasing of its stability and intensity. The ZRA proposed a possible plan suggesting removing the Kariba Dam and replacing it with a series of ten to twenty smaller dams along the Kariba River to improve the situation. Based on this plan， this article presents a math model to determine the optimal location， numbers and height of the smaller dams， and tries to seek balance between security and economical efficiency to make the plan possible.

二、 Notation

（1） Constants

Volume of the Lake is 1.62*1011m3；

·Length of the Lake is 2.2*105m；

·Average depth ofthe Kariba Lake （H0） is 29m；

·Height of the Dam is 128m；

·Width of the mid-stream is 180m；

·width of the down-stream is 6000m.

（2） Variables

三、General assumptions

Considering the difficulty of describing all the details of the basin planning， we ought to simplify the situation by the necessary assumptions below.

·We dont consider the actual shape of the dams except their height and width.

·We assume that there is no soil-erosion phenomenon in this process. What is more， we dont take any energy loss into account.

四、Model establishment

（一） 0-1 programming

Since we are uncertain about the exact number of new dams， we choose to de- termine the existence of a specific dam by 0-1 programming method. We regard the existence as the number 1， while the nonexistence is the number 0. According to the ZRAs requirements， the number of new dams is between 10 and 20. Thus， we can define the 0-1 variables as are shown below：

If the （ i+1） th dam exists， the ith dam will certainly exist. So we get the first con-strained condition，

（二）Math evaluatingindicator

Firstly， we regard this area as a “reservoir”. To evaluate the safety degree of the dams， we can choose the overall volume of all “reservoirs” as an indicator. The bigger the overall volume is， the safer the system is.

As we all know， the volume of a “reservoir” is the multiple of the height and the bottom area. The height of the “reservoir”， is determined by the lower one of the two dams， that is，

The bottom area of this “reservoir” is，

So， the overall volume of all the “reservoir” is，

To match the indicator ， we define the cross sectional area as another indicator of investment ， that is，

五、Conclusion

Many arguments appeared about the safety and profits of the Kariba Dam since its establishment. As the report showed， the water erosion， Alkali Silicate Reaction （ASR） and other issues can result in damages and the collapse of the dam， which may result in a tsunami like wall of water released downstream. For these reasons， We put forward three available methods to prevent catastrophic accidents.

·Rebuilding the dam

·Repairing the dam

·Replacing the dam with small dams

Comparing to the two approaches we mentioned above， establishing a series of small dams may not handle catastrophic floods successfully. However， considering the decreasing flow rate of Zambezi River in recent years， If we can design the small dams properly and manage them efficiently， the new system will have the same capabilities as the Kariba Dam nowadays.

Weighing all the elements we mentioned above， we finally conclude that the best way to get rid of our present dilemma is replacing the big dam with several small dams.

Reference：

[1]"The marooned baboon： Africa's loneliest monkey". BBC. 3 October 2014. Retrieved 3 October 2014.

[2]http：//www.newyorker.com/tech/elements/one-of-africas-biggest -dams-is-falling-Apart

[3]http：//www.newworldencyclopedia.org/entry/Zambezi_ Ri-ver