The PI of a horizontal well in an anisotropic medium is calculated during a process of partial acidization, in which damage removal occurs over only a fraction of the well length. Two numerical simulators are used in this work: an acidizing simulator, which calculates the permeability distribution around the wellbore; and a reservoir simulator, which calculates the PI of the well. By applying mathematical transformations, it is possible to reduce the acidizing model from a 2D to a 1D problem.

By comparing treatment results, it is possible to select the optimal volume of acid, injection rate, and fraction of the length to be acidized for each well. The same procedure also may be applied for vertical wells both with and without anisotropy.

Simulation results indicate that the application of a partial acidizing strategy reduces the total amount of acid required for significant improvement in well productivity, substantially reducing the financial, operational, and environmental risks involved in the treatment of a horizontal well. In most cases, the optimal injection rate for sandstone acidizing is the maximum rate that does not fracture the formation.

Matrix acidizing of vertical wells is a reasonably well understood technique and generally is modeled with a radial geometry. In recent years, however, the wide use of horizontal wells has required that the standard procedure for matrix acidizing be adapted to a new environment. Two major factors must be considered when designing an acid treatment for a horizontal well. First, the area exposed to the formation is much larger because of the extended length of the wells. Second, the normally large contrast between the horizontal and vertical permeabilities must be taken into account. Applying the conventional design procedure to a horizontal well would result in the use of huge amounts of acid and extremely long operations. Corrosion problems would be impossible to avoid with currently known corrosion inhibitors. It is very important that a new technique that reduces the total volume of acid used be developed to acidize horizontal wells.

The main objective of this work is to use numerical simulators to evaluate the feasibility of partially acidizing horizontal wells. **Fig. 1** illustrates this concept; the arrows show the relative flow of fluids into the well after acidizing is complete. If only a fraction of the well length is treated, an acceptable productivity increase can be obtained without injecting an excessive amount of acid. Such issues as the optimal volume of acid and optimal injection rate are addressed. Two criteria have been proposed for selecting the injection rate: the maximum possible injection rate^{1,2} and an optimal rate that minimizes the acid volume.^{3,4}

Partial acidizing means the removal of formation damage in only a few equally spaced intervals of the well, while the other segments remain untreated. Should a commonly used slotted-liner completion or a less usual openhole completion be applied, all portions of the well would contribute to the production of fluids. This concept is similar to that of partially perforating a long horizontal well introduced by Goode and Wilkinson.^{5,6} It has been shown that the perforated length of a horizontal well can be considerably reduced without substantial loss of well performance in many cases. The same kind of results may be expected for the problem of partially acidizing a horizontal well.

The acidizing simulator used in this work was UTACID,^{7,8} which considers the geometry of the flow to be radial. The problem of injecting acid in an anisotropic medium was solved by using conformal mapping to transform the 2D problem into a simpler 1D one and solving the diffusivity equation.^{9} The acid flux was derived in the new coordinates and introduced into the mass-balance equations. The output of UTACID is the permeability distribution around the wellbore and the average permeability of the treated zone to be used by the reservoir simulator UTCHEM^{10–12} in the calculation of the PI of the horizontal well.

To simulate matrix acidizing of vertical wells, we define dimensionless variables for radial geometry.^{4} The radial-flow hypothesis implies that the permeabilities normal to the well axis are equal. However, when the well is horizontal, it is not possible to maintain that assumption because the horizontal permeability is normally larger than the vertical permeability. A new procedure for solving the problem is developed in this work, with the goal of keeping the geometry 1D so that the existing acidizing simulator, UTACID, could be adapted to the new environment.

Determining the pressure field is the first step toward matrix acidizing of anisotropic formations. Peaceman^{9} found a solution for that problem, presented schematically in **Fig. 2.** The 2D steady-state diffusivity equation for an anisotropic reservoir is given by

Equation 1

where *p=p _{w}* at

*r=r*. The solution of Eq. 1 starts with the following variable transformation:

_{w}Equations 2a and 2b

Eq. 1 becomes the following Laplace equation:

Equations 3a and 3b

Eqs. 3a and 3b may not be solved directly because of the nature of the boundary condition. A second step necessary to the solution of the problem is the use of conformal mapping of the *u-v* space:^{9}

Equation 4