# A New Algorithm Model Based on Extended Kalman Filter for Predicting Inter-Well Connectivity
## 4 Production Performance of Single Well Pair
### Fluid Migration  
In Figure 4, the liquid production rate consists of oil and water production rates. Significant water injection and liquid production rate increases were observed at certain moments (January 2024, January 2027, and January 2031). Especially in 2027, the injection volume soars to over 4 m\({}^{3}\)/day. This phenomenon is closely related to the production system of the assumed single-well pair system. In 2024, 2027, and 2031, the BHP of injector increases from 750 bar to 780 bar, 800 bar, and 760 bar, respectively. Therefore, the increase in pressure directly promotes the increase in injection volume and liquid production, and the maximum production pressure differential corresponds to the maximum liquid production.  
Further examination of the relationship between the water injection rate and liquid production rate suggests that the water injection rate increases simultaneously with the increase in injection pressure. In contrast, the production rate increases slowly after the pressure impulses are applied and decreases slowly after the removal of pressure impulses. In a complex reservoir system, the water injection from the injector can be treated as the input signal while the production is deemed the output signal. This delayed response in fluid production observed in Figure 4 is essentially a reflection of the attenuation of the reservoir system to pressure signals. In general, for a fixed well spacing, the presence of low permeability zones, faults, and the high viscosity of reservoir fluids will all exert a large dissipative effect on the injection signals, which leads to a delay in the fluid production.  
In Figure 4, the water cut profile is also plotted against the water injection and liquid production curves. Water cut refers to the mass percentage of water in a producing well, thus showing an increasing trend with production. The initial water cut is zero until 1 November 2024, followed by a rapid increase to 0.7 within one year, and then the water cut slowly increases to 0.92 in the following simulation. The sudden response of the water cut profile facilitates locating the water breakthrough time of the producer.  
Figure 4: Variation in water injection rate, liquid production rate, and producer water cut affected by artificially applied pressure impulses.  
Figure 5 illustrates the variation in oil saturation in the reservoir at three consecutive time points, combined with the water cut, which helps ascertain the time of water breakthrough. The modeled water breakthrough happens on November 2024, when a connected water flow pathway is formed in the reservoir, allowing water to flow more efficiently to the producer along the model diagonal. This observation is consistent with the time-dependent water cut profile in Figure 4.  
In Figure 5, the water cone has not yet reached the producer on 1 August 2024; on 1 November 2024, the water cone has just contacted the producer; and on 1 February 2024, the water cone has been rippling through the producer for some time. The simulation also suggests that for the oil beyond the water cone, it is almost impossible to be displaced without well pattern thickening or well groups conversion. The formation of advantageous water flooding channels is one of the main factors affecting the efficiency of oil recovery [41; 42].