# Matching of Water Breakthroughs in a Low-Resistivity Oil Reservoir Using Permeability Anisotropy
## 1 Introduction
### Difficulties in Reservoir Simulation of Matter Fields  
The revision of existing data accumulated over a long period of production from mature oil fields is required for the evaluation of remaining oil reserves to plan future operations such as the drilling of new wells, water injection, pump settings, as well as the repair and maintenance of existing wells. Reservoir simulation is used to model and reproduce the original state in the field using history matching to predict future well and field performances. A good history match serves as a means for verifying the accuracy of the dynamic model to adequately present the field performances and their appropriateness for the predictions of future field characteristics such as oil production, water production, and pressure. The static and dynamic models undergo a series of revisions to obtain a match between the measured field data and the calculated data from the model. However, a prolonged production history with field operations overlapping in time and space complicates the simulation. Additionally, production from multiple reservoirs can occur simultaneously, which could lead to uncertainties regarding the individual performances of the reservoirs and wells. The original data are often missed, which makes it impossible to compare with the base case that occurred before the start of production. New wells are often drilled in the areas already under previous water injections. In this regard, the following wireline logging cannot provide reliable data for the water saturation. All of the above-mentioned issues contribute to inconsistency and a lack of robust evidence supporting the explanations of observed anomalies. In addition, the use of different parameters during history matching by various investigators often requires rescaling. The segmentation and isolation of various formation intervals can also be required through the introduction of baffles or barriers vertically and horizontally. As a result, models and predictions vary largely from investigator to investigator, leading to contradictory conclusions.  
Early water breakthrough during water injection may lead to the bypassing of oil reserves, especially in heterogeneous formations. Other sources of water breakthroughs such as those resulting from borehole cementing integrity issues, water encroachments along the high permeable streaks, fractures, or underlying aquifers of unclear sources and paths make the history matching especially complicated. In the zone of active aquifers and dependent on the structure of the traps, tilted OWC can be formed. This can lead to early water breakthroughs in the areas of the uplifted plane of OWC [6].  
Fluid flows within the formation are controlled by the mobility ratio and permeability anisotropy. The mobility ratio (\(M\)), which describes the stability of a displacement process, is defined as the ratio of the mobility (\(\lambda\)) of the displacing fluid to the mobility of the displaced fluid:  
\[M=\frac{\lambda_{water}}{\lambda_{oil}}=\frac{K_{water}}{K_{oil}}\cdot\frac{ \mu_{oil}}{\mu_{water}} \tag{1}\]  
where \(Kr\) is the relative permeability, \(\mu\) (cP) is the viscosity, crude oil is the displaced fluid, and water is the displacing fluid.  
Efficient piston-like frontal displacement of oil occurs at \(M\leq 1\). A significant contrast between relative permeabilities and/or viscosities creates a high mobility ratio indicative of an unstable displacement front and fingering [7]. Notwithstanding, reservoir engineers widely use relative permeability curves as a parameter in history matching because of the inherent uncertainties in these flow curves. By varying relative permeability curves, reservoir engineers can adjust the fluid flow behavior in the model to match the observed production data. Therefore, uncertainty in relative permeability is an important consideration in history matching and reservoir engineering.