Producing–Equipment, Methods and Materials - The Effect of Restricted Fluid Entry on Well Productivity

In the past, other authors.1, 2 have studied the influence of a skin effect on the productivity of a well. This skin effect, expressed by the skin factor S, is considered to be caused by a thin layer of impaired permeability immediately around the wellbore and extending vertically over the whole productive interval penetrated by the well. The skin factor S is defined as follows. Based on this, the impairment in productivity caused by a skin can be expressed by the fractional loss in productivity I, which is the loss in productivity divided by the total unimpaired productivity. For compressible flow in a stabilized well which drains a circular area of radius re, 1 is given by The present paper deals with a different kind of productivity impairment. Consider a well in which part of the productive formation is blocked off completely, either by incomplete penetration or by exclusion of parts of the productive zone by blank casing. In Fig. 1 (A, B and C), three examples are shown. Fig. 1(A) shows the situation where a well only partially penetrates the formation. This often is done to combat the actual or imagined danger of bottom-water coning. Fig. 1(B) shows a well producing from only the central portion of a productive interval. This type of completion is sometimes used where both water and gas coning are a problem. Although the case of a well producing through perforated casing cannot be treated in a manner similar to the previous two cases (where radial flow in the horizontal plane is assumed), Fig. 1(C) shows several intervals open to production and qualitatively describes this case (as will be discussed later). To study the loss in productivity in all these cases, two parameters are introduced which fully determine the impairment. The first is the penetration ratio "b," i.e., the total interval open to fluid entry divided by the total thickness of the productive zone. The second is the ratio hlrw. In this ratio, rw is the wellbore radius. The definition of h is more cumbersome. In Fig. 1(A), h is the thickness of the total productive interval. The streamline configuration for this case of partial penetration is basic to the other two cases considered (Views B and C of Fig. 1). It will be obvious from Fig. i (A) that the flow lines in the uppermost portion of the formation will be essentially horizontal, while those in the lower portion will curve upward toward the well. In Fig. 1(B), with only the middle portion of the zone open to production, the streamline configuration. of the upper half will be an exact mirror image of that in the lower half of the zone. Hence, for the case illustrated in Fig. 1(B), h is defined as one-half the total sand thickness. It follows, then, that in Fig. 1(C) h is one-half the distance between corresponding points in adjacent intervals. (In gun-perforated casing, h would be one-half the distance between perforations.) In a paper by Nisle4 and a paper given by the present authors5, the mathematical theory was developed for the cases under consideration. In the present publication, the emphasis is put on the results of these studies; consequently, the equations derived in Refs. 4 and 5 will be omitted, for the most part. The pressure drop p in a well producing from only a portion of the total formation thickness can, in analogy with Eq. 1, be expressed by Numerical solution of Eq. 5 by use of the IBM 650 leads to the following important conclusions. First, during a short period after starting production (usually on the order of a few minutes), the function p(b) is given by