Analysis of Buildup Curves Obtained After Well Treatment

Two new unit functions, based on the continuity equation of unsteady-state flow of slightly compressible fluids in permeable media, give the pressure decrease in a well produced at unit rate after its effective radius has been considerably enlarged by acidizing or when the wellbore is intersected by an induced, vertical fracture. Introduction Many papers have appeared that contain detailed analyses of the pressure data in wells from the moment the well is shut in until such data form the straight-line relationship, described by Horner,1 between pressure increases and the log of ?t/(t+?t). When formations are not very permeable, it may take some time to gather sufficient pressure data. Attempts have been made to limit the duration of buildup tests and to extract information on the characteristics of the formation and the efficiency of well completion from data recorded in the first hours of the shut-in period. Details are well presented in Refs. 2 and 3, which also contain an extensive bibliography of articles of interest. The relation between pressure and time is a function of the producing rates and the characteristics of the formation and its fluid content, which can cover a considerable range as shown by the following data. Depending on the formation, the permeability, k, can have a value anywhere between 1 and 1,000 md; the porosity, f, varies from 5 to 35 percent; the fluids can have a viscosity, µ, between 0.02 for gas and 5 cp for oil. The compressibility, c, can be somewhere between 5×10−1 atm−1 for undersaturated fluids and 10−2 atm−1 for gas, whereas the hole diameter can be as small as 4 in. and as large as 10 in. (radius, rw, 5 to 12.5 cm). To simplify computational work, these units were combined into dimensionless groups in Ref. 4 so that only a few fundamental curves are required to analyze flow problems in wells and reservoirs. The following conversion factors are used in these analyses. Dimensionless time:Equation 1 Dimensionless rate:Equation 2 Dimensionless cumulative production:Equation 3 As it is difficult to measure, with any precision, pressure increases occurring in less than 1 minute after shut-in, 1 minute is about the shortest time interval to be considered. The following table gives the values assigned to the factors in Eq. 1; they convert 60 seconds to 62 dimensionless time units.k=0.001 darcyt=60 secondsf=0.12µ=0.02 cpc=0.005 atm−1rw=9 cm