cumulatively by the straight line a a, its capacity must be such that it will hold not only the 11 % surplus of the same year, but that, on June 10th, when this surplus has been used to satisfy the demand, it will still contain the water c d—19%—stored from a previous year; otherwise between June 10th and August 31st the reservoir will be empty and only the dry weather flow of the stream will be available for supply. In short, if the reservoir is to equalize the whole flow of this year, it must have a capacity equal to the greatest deficiency c d of the cumulative flow below the cumulative demand, plus the greatest excess e f of the cumulative flow over the cumulative demand. This capacity is represented by the height of the line a′a′ (drawn parallel to a a from the point of maximum surplus f) vertically above the point of greatest deficiency c, and equal, on the vertical scale, to the difference between the height c = 48% and g = 78% or 30% of the stream-flow during the driest year. A reservoir so proportioned to the stream-flow with a proper addition to avoid drawing off the bottom water, would probably be safe in Great Britain in any year for a uniform demand equal to the cumulative stream-flow; or, if it failed, that failure would be of very short duration, and would probably only occur once in 50 years.
Fig. 3.
It may be at first sight objected that a case is assumed in which there is no overflow before the reservoir begins to fall, and therefore no such loss as generally occurs from that cause. This is true, but it is only so because we have made our reservoir large enough to contain in addition to its stock of 19% at the beginning of the year, all the surplus water that passes during the earlier months in this driest year with its least favourable time-distribution of flow. Experience shows, in fact, that if a different distribution of the assumed rainfall occurs, that distribution will not try the reservoir more severely while the hitherto assumed uniform rate of demand is maintained. But, as above stated, the time-distribution of demand is never quite uniform. The particular drought shown on the diagram is the result of an exceptionally early deficiency of rainfall which, in conjunction with the variation of demand shown by the dotted line b b, is the most trying condition. The reservoir begins to fall at the end of February, and continues to do so with few and short exceptions until the end of August, and it so happens that about the end of August this dotted line, b b representing actual cumulative demand, crosses the straight line a a of uniform demand, so that the excess of demand, represented by the slope from June to September, is balanced by the deficiency of demand, represented by the flatter slope in the first five months, except as regards the small quantity b e near the end of February, which, not having been drawn off during January and February, must overflow before the end of February. To avoid this loss the 11% is in this case to be increased by the small quantity b e determined by examination of the variation of the actual from a constant demand. After the reservoir begins to fall—in this case at the end of February—no ordinary change in the variation of demand can affect the question, subject of course to the cumulative demand not exceeding the reservoir yield for the assumed year of minimum rainfall. In assuming a demand at the beginning of the year below the mean, resulting in an overflow equal in this case to b e at the end of February and increasing our reservoir to meet it, we assume also that some additional supply to that reservoir beyond the 11% of the stream-flow from the driest year can be obtained from the previous year. In relation to this supply from the previous year the most trying assumption is that the rainfall of that year, together with that of the driest year, will be the rainfall of the two driest consecutive years. We have already seen that while the rainfall of the driest of 50 years is about 63% of the mean, that of the driest two consecutive years is about 75% of the mean. It follows, therefore, that the year immediately preceding the driest cannot have a rainfall less than about 87% of the mean. As the loss by evaporation is a deduction lying between a constant figure and a direct proportional to the rainfall, we should err on the safe side in assuming the flow in the second driest year to be increased proportionally to the rainfall, or by the difference between 63 and 87 equal to 24% of the mean of 50 years. This 24% of the 50 years’ mean flow is 38% of the driest year’s flow in fig. 3, and is therefore much more than sufficient to ensure the reservoir beginning the driest year with a stock equal to the greatest deficiency—19%—of the cumulative flow of that year beyond the cumulative demand.
But in determining the capacity of reservoirs intended to yield a supply of water equal to the mean flow of two, three or more years, the error, though on the safe side, caused by assuming the evaporation to be proportional to the rainfall, is too great to be neglected. The evaporation slightly increases as the rainfall increases, but at nothing like so high a rate. Having determined this evaporation for the second driest consecutive year and deducted it from the rainfall—which, as above stated, cannot be less than 87% of the mean of 50 years—we may, as shown on fig. 3, extend our cumulative diagram of demand and flow into the reservoir from one to two years.
The whole diagram shows, by the greater gradient of the unbroken straight lines, the greater demand which can be satisfied by the enlargement of the reservoir to the extent necessary to equalize the flow of the two driest consecutive years. The new capacity is either c h or c′ h′, whichever, in the particular case under investigation, is the greater. In the illustration the c′ h′ is a little greater, measuring 4712% of the flow of the driest year. In the same way we may group in a single diagram any number of consecutive driest years, and either ascertain the reservoir capacity necessary for a given uniform yield (represented cumulatively by a straight line corresponding with a′ a′ , but drawn over all the years instead of one), or conversely, having set up a vertical from the most trying point in the line of cumulative flow (c or c′ in fig. 3—representing, in percentage of the total annual flow of the driest year, the capacity of reservoir which it may be convenient to provide) we may draw a straight line a′′′ a′′′ of uniform yield from the head of that vertical to the previous point of maximum excess of cumulative flow. The line a′′ a′′ drawn from zero parallel to the first line, produced to the boundaries of the diagram, will cut the vertical at the end of the first year at the percentage of the driest year’s flow which may be safely drawn continuously from the reservoir throughout the two years. It is to be observed that any irregularity in the rate of supply from the reservoir may occur between the critical periods of maximum excess of cumulative flow and maximum deficiency
