Interpretation of Fracturing Pressures (1981)

Abstract This paper presents a basis for interpreting fracture treating pressureswhich permits identification of periods of confined-height extension, uncontrolled height growth, and more importantly, a critical pressure. When atreatment reaches the critical pressure, fracture extension is significantlyreduced, and a pressure (or screen-out) condition or undesired fracture heightcan follow. Example applications for data from five treatments are presentedalong with potential explanations for, and implications of, the criticalpressure. Introduction This paper will be limited to the discussion of hydraulically createdfractures that are in the vertical plane. In addition, if these fractures aredesired to be deeply penetrating, they must have confined or limited heightgrowth. There are two fundamentally different concepts for the propagation of aconstant-height vertical-fracture which lead to very different results. Oneconcept1 is that the fracture width is constant across the height ofthe fracture. This requires the assumption that the formation bed beingfractured is independent of the beds above and below - that is, the beds canslip freely of one another at their boundaries. This assumption leads to theconclusion that the fluid pressure required to extend the fracture decreaseswith time. The other concept, presented by Perkins and Kern,2 assumes thatthere is no, or negligible, slip of boundaries along the horizontal planeswhich confine the fracture height. This assumption leads to the conclusion thatthe fluid pressure required to extend the fracture increases with time. Thisconcept, as refined by Nordgren,3 predicts that for a Newtonianfluid creating a confined height fracture at a constant injection rate thewellbore pressure increases proportionally to time raised to an exponent, that isEquation (1) The larger value is for the assumption of a relatively small fluid lossrate, relative to the injection rate, while the smaller value of the exponentis for a relatively large fluid loss rate. For (1) and all other cases in thispaper, reference to pressure will imply pressure above the fracture closurepressure. Fig. 1 shows the wellbore treating pressure, above the in-situ closurepressure, versus time for three massive treatments. The data shown in thispaper was collected by a tubing/annulus wellbore configuration having nopacker. For this configuration the treatment can be pumped down the tubing orannulus with the surface pressure on the static line giving the bottom-holepressure after a correction for hydrostatic head. The closure pressure wasdetermined by the pump-in/flowback procedure discussed in Ref. 4 and shown onFig. 7 of that reference. The initial portions of the data are not shown inFig. 1 or 2 because they contained periods of significant variations ofinjection rate or fluid viscosity. The figures also show the time at whichproppant was introduced into the wellbore. The data in Fig. 1 is shown on a log-log plot, for which the slope of astraight line defines the exponent e given in (1). Although the figure presentspressure versus time, accumulative-fluid-volume-injected could be substitutedfor time with the same results since a constant injection rate is assumed. Thefigure shows that the slope of the initial portion of each treatment fallswithin the Nordgren bounds given in (1). Therefore, the positive slope portionof each treatment can be interpreted as confined height extension governed bythe no-slip concept of Perkins and Kern. The portion of the treatments whichdeviate from this behavior will be discussed in later sections.