The Existence of an Optimum Damkohler Number for Matrix Stimulation of Carbonate Formations
Authors: Fredd C.N. , Tjia R. , Fogler H.S. 
Published: December 2007

Abstract
Effective matrix stimulation treatments rely on the formation of dominant wormhole channels. Several investigators have reported the existence of an optimum injection rate at which dominant wormhole channels are formed and a minimum volume of fluid is consumed. Theories predicting this optimum injection rate have assumed either mass transport or reaction rate limitations. Unfortunately, no fundamental study has included the combined effects. Therefore, it is not possible to describe alternative fluid systems such as chelating agents or weak acids.
Chelating agents such as ethylenediaminetetraacetic acid (EDTA), 1,2-cyclohexanediaminetetraacetic acid (CDTA), and diethylenetriaminepentaacetic acid (DTPA) have been shown to stimulate carbonates at moderate or non-acidic pH values (4 to 13) and at low injection rates where HCl is ineffective. This study extends the investigation of chelating agents of the aminopolycarboxylic acid family as alternative stimulation fluids. Rotating disk experiments have shown that the dissolution of calcite by EDTA, CDTA, and DTPA is not just limited by the rate of reactants transport. Similar results were obtained for the weakly dissociating acetic acid. Therefore, we have derived a generalized description of the dissolution phenomenon that includes the effects of convection, reactant transport, reversible surface reactions, and products transport. Dimensional analysis reveals a common dependence on the generalized Damkohler number for flow and reaction. Neutron radiographs of the wormhole patterns reveal that the structure depends on this generalized Damkohler number. In addition, there exists an optimum generalized Damkohler number at which a single dominant wormhole channel is obtained and the pore volumes to breakthrough is minimized. This optimum Damkohler number occurs at approximately 0.17 for a wide range of fluid/rock systems.

Introduction
The flow and reaction of acids in carbonate porous media results in the formation of highly conductive flow channels or wormholes. The structure of these wormholes is strongly dependent upon the injection rate and the fluid/rock properties. Typical wormhole structures range from face dissolution (or complete dissolution of the core starting from the inlet flow face) at low injection rates to uniform dissolution resulting in ramified wormhole structures at high injection rates. Single dominant wormhole channels are obtained at intermediate injection rates. These dominant wormhole channels represent the most effective mode of stimulation since they minimize the pore volumes of fluid required to obtain a given depth of wormhole penetration.
Several investigators have studied the phenomenon of wormhole formation in a variety of fluid/rock systems and have reported the existence of an optimum injection rate.
Daccord et al. investigated the water/plaster system and reported the optimum injection rate to occur at a Peclet number just above unity. The Peclet number is defined as the ratio of transport by convection to transport by diffusion. A similar dependence on the Peclet number was observed by Mostofizadeh and Economides for the HCl/limestone system. Frick et al. also studied the HCl/limestone system and combined the concepts of fractal geometry and Peclet number. Wang et al. and Huang et al. investigated HCl/carbonate systems and proposed that the optimum injection rate occurred at a transition between reaction rate and fluid-loss limited regimes. Despite mass transport having a major influence on wormhole formation, diffusion plays only a minor role in their theory.
Hoefner and Folger investigated HCl/carbonate systems and found that the phenomenon of wormhole formation is governed by the Damkohler number for flow and reaction. The Damkohler number (Da) is defined as the ratio of the net rate of dissolution by acid to the rate of convective transport of acid. P. 249^