New Solution for Anisotropic Formation Damage Due to Produced Water Re-injection
Authors: Salehi Mojarad R. , Settari A. 
Published: March 2010

Abstract
Despite the economics and environmental benefits of PWRI (Produced Water Re-Injection) projects, the permeability reduction due to deposited particles is a persistent problem. Various models for permeability damage calculation are available. Most of them are based on one-dimensional laboratory parameters and have not considered the anisotropy of the media. In previous attempts at anisotropy modelling, the initial anisotropy of the media has been considered. However, when it comes to the damage intensity calculation, isotropic parameters have been used for the entire media. As a result, the relative damage in these models is isotropic.
In this paper, a robust approach for anisotropic permeability impairment is developed based on micromechanical considerations. The damage mechanics is coupled with numerical flow code. The model formulation has been successfully tested in 1D flow against the core flood tests from the Masila Block onshore Yemen. Then, the damage model has been extended to 3D using pseudo directional parameters to capture the anisotropy. A dynamic anisotropic mathematical formulation for damage intensity has been derived, implemented in a 3D numerical code and successfully tested on a field case study. The new model exhibits the expected anisotropy of damage.

Introduction
Anisotropic formation damage has been studied by many researchers, especially for horizontal wells. However, in all previous work it has been assumed that the damage intensity in different directions (defined as a ratio of the damaged permeability to the original permeability) would remain the same in all directions(1, 2). This assumption (sometimes clearly stated, sometimes not) means that, although the damage profile and front would be dissimilar in different directions (sometime considered to be anisotropic damage), the ratio of the damaged permeabilities in the horizontal and vertical direction kh/kv (called "anisotropy ratio") stays constant. The common result of this assumption for a horizontal well is to have an elliptical cross-sectional damage profile, which will grow with time proportionally in both directions.
The hypothesis for the mechanics of real anisotropic damage is obtained by starting with an elliptical cross-section for the damage profile. The following phenomenon is plausible. Since we have a higher velocity (and volume) of damaging water flowing in the direction of higher permeability (horizontal), we would continue to have a deeper invasion in this direction compared to the direction of lower permeability. Therefore, the anisotropy persists as the damage grows. However, we would expect the intensity of damage to be higher in the vertical direction because of lower initial permeability and, therefore, smaller pore throats. Hence, the anisotropy ratio (kh/kv) is expected to increase with time, rather than to stay constant.
To be able to model the anisotropy phenomenon, consider the velocity-based damage model(3) (VDM). First, we note that in isotropic permeability, it was proven that the use of the vector of velocity would create the correct solution for a multidimensional case(4). In an anisotropic situation, the original model produces a constant anisotropy ratio. We therefore need to have different initial damage coefficients a for different directions and they need to be changed with time in a different fashion.