# Matching of Water Breakthroughs in a Low-Resistivity Oil Reservoir Using Permeability Anisotropy
## 1 Introduction
### Permeability Anisotropy  
Permeability anisotropy is mathematically described as the ratio of vertical to horizontal permeability (Kv/Kh) within a formation. Rock permeability is influenced by the size, shape, and orientation of the grains that make up the reservoir rock. When these grains are not uniformly distributed or have a preferred orientation (anisotropy), they can cause permeability variations measured in different directions. This phenomenon is known as permeability anisotropy [8]. The permeability anisotropy established during deposition may be further modified by burial diagenetic processes including the compaction, dissolution, and cementation of grains. The packing and shape of grains change due to the compaction, leading to higher permeability in the horizontal than in the vertical direction. [PERSON] (1969) indicated that the horizontal permeability (Kh) is higher than the vertical permeability (Kv) for both large and flat rock grains and concluded that generally, vertical permeability is lower than horizontal permeability, especially if the sand grains are small and have irregular shapes [9].  
Moreover, laminar heterogeneities are very common in sandstone reservoirs, resulting in successive layers that may differ in permeability by several orders of magnitude [10]. In such layering structures, the permeability measured parallel to the layers is higher than in the perpendicular direction. However, Kv can be bigger than Kh if measured along fractures [11].  
Anisotropy is also a scale-dependent property. Smaller volumes tend to be isotropic, as can be seen in core plugs. In contrast, at the formation scale, bedding fabric can lead to significant differences between Kv and Kh. In fluvial systems, the arrangement of channel and inter-channel elements can have a substantial effect on anisotropy. For example, well-intercalated channel systems have a higher tendency to be isotropic, whilst the preservation of more discrete channels will exhibit a more anisotropic behavior.  
The permeability anisotropy controls single-phase fluid flow as well as two-phase effective mobility of immiscible phases such as oil-water, and provides a framework to upscale the effects of such structures to field-scale grids for reservoir models [11; 12].  
A more homogeneous permeability distribution has important consequences for the efficiency of fluid flow, and hence, the implication is made that fluid displacement processes commonly applied in the industry (as in a waterflood project) will be more efficient in a sub-vertical direction. In contrast, displacement fronts are expected to be more highly fingered in a horizontal direction, supparallel to bedding or laminae.  
The ratio of vertical to horizontal permeability (Kv/Kh) represents the contrast in permeability between the vertical and horizontal planes within a formation, and it is used in simulation. Therefore, the derivation of relevant Kv/Kh ratios is of great importance to predict and guide hydrocarbon recovery at the time scale of a few decades typical of subsurface exploitation schemes, but also to model secondary migration processes leading to the location and filling of hydrocarbon reservoirs over geologic time.  
In the present study, the effects of permeability anisotropy and relative permeabilities are studied on reservoir simulation results to find an explanation for early water breakthroughs and subsequent high water production in the oil reservoirs. The water breakthroughs from the production wells occurred before the start of water injection in the field; it is therefore believed that a decent volume of unrecovered oil could have been bypassed. Petrophysical and logging data of seven wells were processed using Techlog software and further imported into Petrel software for modelling and simulation. Various reasons for the early water breakthroughs and mismatches in water production were investigated. The maps of the remaining oil are presented.