Ohen and Civan (1991, 1992, 1993), skin factor varies over time and can be predicted by means of a formation damage model.Ohen and Civan (1992) for prediction of the skin factor associated with formation damage resulting from fines migration and clay swelling effects in the nearwellbore formation.

## Model-Assisted Analysis of the Near-Wellbore Permeability Alteration using Pressure Transient Data

The modeling and parameter estimation methods for determination of near-wellbore permeability alteration from pressure transient analysis data by Olarewaju (1990) are presented here. Olarewaju (1990) considered a reservoir system, composed of two concentric zones, denoted as zones 1 and 2 in this article. Zone 1 is located near the wellbore and its permeability has been altered by formation damage or stimulation processes. For example, zone 1 includes the near-wellbore formation, in which permeability impairment occurs by mud fluid and particle invasion and the mud cake formed over the sand face during drilling. Zone 2 represents the undamaged formation located beyond zone 1. The permeabilities of zones 1 and 2 are denoted by K\ and K2 and the radius of zone 1 of the skin effect region is r}. The external drainage radius of zone 2 is r2. The objective is to estimate the values of K{, K2, and TJ using build-up pressure test data, such as by Olarewaju (1990) from a reservoir in which the permeability of a near wellbore formation has been enhanced by acid stimulation. Ultimately, this information will be used to determine the skin factor as a measure of the effectiveness of the acid treatment.

For this purpose, Olarewaju (1990) developed a simplified mathematical model by considering

1 a slightly compressible single phase fluid,

2 constant thick-horizontal reservoir,

3 a constant rate producing well, and

4 a reservoir, with no-flow boundaries at the top, bottom, and external drainage radius.

The dimensionless partial differential equations of the Olarewaju (1990) model are given as following:

subject to the following conditions of solution:

Initial conditions (uniform initial pressure):

The dimensionless variables and/or parameters are defined as following:

In these equations, the indices 1 and 2 denote zones 1 and 2; rw, re, and r represent the wellbore and drainage radii and radial distance from the center of the well, respectively; t is time, B is the formation volume factor, Pt and P} denote the initial reservoir and zone 1 radius pressures, |i is fluid viscosity, ɸ, K, and h represent the formation porosity, permeability, and thickness; ct is the total compressibility, and a = 0.0002637 and β = 0.007082 are some constant factors resulting from conversion from Darcy to field units. The skin factor is calculated by

Eqs. 22-26 through 32 can be solved by an appropriate numerical method, such as by the finite difference method. However, Olarewaju (1990) obtained an analytical solution for the wellbore fluid pressure in the terms of the modified Bessel series / and K0, in the Laplace domain, as:

and then inverted it numerically using the Stephfest algorithm. Readers interested in details are referred to Olarewaju (1990). Olarewaju (1990) presents the pressure and its derivative curves for different parameter values generated by the above model. Olarewaju (1990) also presented an application of this model to acidized well pressure build-up test data. Applying an automatic parameter estimation technique, Olarewaju (1990) obtained an excellent history match of data as shown in this article. For this purpose, Olarewaju (1990) used r w = l f t , A = 10 ft, ɸ = 0.10, Pi = 4,000 psia, q = 8.27 STB/D, B = l.2l RB/STB, \i = 1 cp, and ct = 9.8 x 1CT6 psr1. Olarewaju (1990) began the history matching process by the initial estimates of K{ = 1 md, K2 = 0.1 md, and r} = 5 ft and obtained the best match with K{ =9.82md, K2 =0.05md, and r{ =51 ft. Consequently, the skin factor was calculated as s = -5.29 using Eq. 22-40. However, Olarewaju (1990) warns that the solution is not unique because an infinite number of combinations of K1, K2, and r1, may yield the same skin factor value.

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