Abstract
The theoretical analysis of the acid-fracturing process for turbulent-flow conditions has been process for turbulent-flow conditions has been reconsidered taking fluid losses into account. For a simple fracture model and an idealized acidizing process, the acid concentration in the fracture process, the acid concentration in the fracture during acid injection and the fracture width have been determined as functions of time and place for three loss conditions:no fluid loss,fluid loss proportional to time, andfluid loss proportional to the square root of time.
proportional to the square root of time. From the results of the analysis, it is concluded that even under the unfavorable conditions of turbulent flow in the fracture and fluid loss, acid penetration is, in general, not a limiting factor in penetration is, in general, not a limiting factor in the application of the acid-fracturing process. However, it will not be possible to predict the productivity increase resulting from a given productivity increase resulting from a given treatment until more experimental data on the conductivity of etched fractures and on certain aspects of the reaction kinetics have been gathered.
Introduction
Acid-fracturing treatments are frequently applied to improve well productivity in limestone formations. In this process, hydrochloric acid is injected into a hydraulically induced fracture, which extends diametrically from the wellbore into the formation. During injection, the limestone faces of the fracture are dissolved. As a result, acid is consumed and its concentration decreases in the direction of flow. The width of the fracture increases, and the fracture faces may become irregularly etched as a result of the natural anisotropies of the formation. The etching pattern produced may contribute to an improvement in fracture conductivity after the fracture is allowed to close. The extent of this etching into the fracture and its final fluid conductivity determine the increase in productivity. Barron et al. have presented an empirical formulation of the acid-fracturing process for laminar flow conditions without fluid loss. When a theoretical description given by Prins et al. concerning the heat-transfer in laminar flow between parallel plates, is applied to the acid-fracturing parallel plates, is applied to the acid-fracturing process, the acid concentration in a fracture for process, the acid concentration in a fracture for steady-state laminar flow can be exactly described, provided that the fracture width is constant and no provided that the fracture width is constant and no fluid loss occurs. A comparison of the acid concentrations calculated from the empirical reaction-rate data of Barron with those theoretically derived according to Prins shows that these values are of the same order of magnitude and can be made equal for acceptable values of the diffusion rate only in the range of low velocities. Judging from the experimental set-up of Barron, we believe that for higher velocities the entrance transition length for fully developed laminar flow should be longer. For this reason, no agreement in the higher velocity ranges can be expected. This view is supported by Williams et al., who compared theoretically derived reaction rates in the heterogeneous calcium-carbonate /hydrochloric-acid system with those of Barron et al., who also conclude that entry effects may be responsible for the discrepancies in the higher velocity range. Nierode and Williams determined a kinetic model for the heterogeneous reaction of hydrochloric acid with limestone. The reaction order and rate constant used in their model were obtained from experiments. On the basis of this model, they derived an acid-fracturing design for laminar flow conditions including fluid loss. In the study described below, the acid-fracturing process has been reconsidered for turbulent-flow process has been reconsidered for turbulent-flow conditions in which both fluid loss and change in fracture width have been taken into account. We feel that the study provides a more realistic description of the process for both a vertical and horizontal fracture and that it may be used as a bask for designing acid-fracturing treatments.
MATHEMATICAL DESCRIPTION OF THE ACIDIZING PROCESS FOR A RECTILINEAR FRACTURE
A vertical rectangular fracture (rectilinear) with initially plan-parallel and flat fracture faces was adopted as a fracture model.
SPEJ
P. 239
Net-exergetic, hydraulic and thermal optimization of coaxial heat exchangers using fixed flow conditions instead of fixed flow rates
