In preparing a well for pumping, observations are often made of the fluidlevel in the well bore or bottom-hole pressures at various times beforeequilibrium has set in. From a qualitative point of view one may immediatelyinfer that if the rise in the bottom-hole pressure or fluid level is rapid thepermeability of the sand about the well bore is large, and converse conclusionmay be drawn if the rise is slow. However, a more quantitative estimate willgive additional information of value, as will be shown in this paper.
To derive such an estimate we may proceed as follows. Let h be the fluidheight, above the sand face, at time t, and let "Yo be the average densityof the fluid that enters the bore. The bottom-hole pressure, for a fluid heighth will therefore be p = "Yogh, neglecting, of course, the friction drop inthe well bore, while the fluid is rising, as may be justifiably done for mostpumping wells.
If a is the free area of the open-flow string-assumed to be uniform-the rate ofproduction from the sand during the rise of fluid will be:
ah a ap
Q = a- = - - = f(p)
at "Yog dt
where the last part of the equation merely indicates that, in general, theproduction rate Q must be considered as a function of the back pressure, and isdetermined by it alone. In fact, equation 1 shows that if the rate of rise ofthe pressure or fluid level is determined as a function of the height of rise, or p, for a number of values of the latter, this functional relation between Qand p will be immediately given.
Conversely, if we know this functional relation, the rate of increase of p or hmay be predicted, or, by observing the rate of rise of p or h some of theconstants of the sand may be determined.