On Optimizing an Acid Treatment
Authors: Closmann P.J. 
Published: March 2010

Abstract
A theoretical approach has been used to develop a method for optimizing an acid treatment of a formation. The technique is based on an experimental determination of the permeability increase with porosity due to reaction with the acid. The calculation applies mostly to matrix acidizing and is more uncertain when wormholes are present

Introduction
In view of recent studies in the theory and practice of acidizing a formation matrix to increase reservoir production(1,2,3,4,5), it seems useful to outline an approximate theoretical approach which optimizes the effect of acid on the rock by determining a preferred distribution of permeability. That a suitable choice of radius of penetration and permeability around the wellbore may give favorable results was suggested by Rowan(6) a number of years ago. The approach in this Note determines the permeability distribution around the well likely to yield the maximum improvement in productivity for a given amount of acid injected into a uniform formation. The method considers the idealized case in which some definable relationship between the permeability and rock porosity attained during acidizing can be obtained experimentally. This is possible in those cases where acid penetrates the Rock matrix in a relatively uniform manner. These would include sandstones and sometimes carbonates under carefully controlled conditions(7). For carbonates, however, the tendency to develop wormholes during acid injection makes such a relationship more uncertain(7). However, in such cases, if a relation between permeability and porosity can be obtained the following analysis would apply.
The mathematical technique which permits specifying the manner of distribution of the porosity and, hence, of the acid depends on the Calculus of Variation(8). Throughout the Note it is recognized that the theory is idealized and would ultimately depend on the development of practical field techniques for its application.

Theory
It is sufficient for present purposes to consider the simple ease of steady-state radial flow of a single phase incompressible fluid and a permeability which varies radially but not vertically. For this case Darcy's law may be applied to radial flow to give the following for the overall pressure drop:
Equation (1) (Available in full paper)
The permeability k(r) is allowed to vary with radial distance from the wellbore and is determined by the local porosity. The productivity index is given by
Equation (2) (Available in full paper)
Define dimensionless variables
Equation (3a,b,c) (Available in full paper)
Equation (3d,e,f) (Available in full paper)
where t = treated radius and substitute into Equation (2) obtaining
Equation (4) (Available in full paper)
We assume that the productivity JD will have its maximum value when kD(rD) is chosen to make the integral in Equation (4) a minimum.
A constraint on the choice of kD(rD) is provided by the requirement that a given amount of acid will react with a fixed amount of rock. The local value for permeability will then be determined by the local porosity, i.e., kD = kD [ΦD(rD)]. This method depends upon the possibility of obtaining a relationship of permeability to porosity by laboratory tests on the rock of interest.