The first step in the formulation of acid-fracturing fluids is to specify an optimal range for each of the fluid properties. To do this, a new procedure is developed for the optimization of hydraulic acid-fracturing treatments. This procedure first predicts the optimum fracture geometry, namely, the geometry which maximizes the stimulation (or productivity) ratio for a given acid volume and then identifies the fracture fluid properties needed to create this geometry.

By utilizing the calculus of variations, an analytical solution is found for the optimum fracture geometry. This solution has several important features. First, the optimum dimensionless fracture shape (conductivity profile or acid-spending profile) in undamaged reservoirs is invariant with respect to both acid volume and formation properties; and, furthermore, this shape is nearly uniform (i.e., constant conductivity) over the entire length of the fracture. Second, the optimum length is a function of a single dimensionless variable. Third, the optimum shape (acid-spending profile) in damaged reservoirs is nearly the same as in undamaged reservoirs and essentially independent of the extent of damage, so long as the fracture extends beyond the damaged zone.

By combining the variational solution for optimum fracture geometry with existing analytical models of the acid reaction and fracture growth processes, it is possible to define ranges of fluid properties which will yield a treatment closely approximating the optimum one. The primary conclusions reached by this method are that the Peclet number for viscoεified acids must be large, approximately 4.0, to achieve the optimal shape; and that optimal fluid-loss coefficients are greater than those previously thought desirable. In fact, it appears that fluid loss except in highly fissured or vuggy carbonate formations is not a critical design variable. Furthermore, very low injection rates or relatively high diffusion coefficients may require unacceptably high acid viscosities.

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