Impact of Vertical Grid Refinement on Water Injectivity in Vertical Wells Using a Least Resistance Principle

Summary An analytical expression for the bottomhole pressure (BHP) difference between a vertical injection and a production well is derived based on a balance between the external and internal chemical potential and the principle of least flow resistance. This expression is used to explain and interpret the impact of vertical grid refinement and varying permeability distributions of the fine gridblocks on simulated BHP differences and injectivity indices. The two potentials are manifested as the BHP difference and the viscous pressure loss given by Darcy’s law, respectively. The expression is, furthermore, developed by dividing the total water injection rate into flow paths, which involves the sum of the relationship between distance traveled, effective permeability, and volumetric flow rate for each path. It therefore enables physical justifications for the BHP and injectivity index (II) variations, as it is based on fundamental laws and principles. That is, it explains why and, in principle, how a total constant volume rate injected is distributed into a set of paths based on grid properties and nature’s tendency to minimize the external work density supplied, i.e., the BHP difference. The expression is first used to explain the decrease in BHP difference and corresponding increase in II upon refining one of the layers in the original grid in the vertical direction. The decrease is explained as an effect of reduced crossflow distances in the vertical direction when water is flowing in the finer compared to the coarser grid. Shorter crossflow distances require less supply of external work density and thereby lower BHP differences. It is, furthermore, used to explain BHP differences when populating refined gridblocks with different permeability distributions when one layer is divided into two and 20 layers. All the distributions have equal flow capacities. The BHP difference between the wells increased, corresponding to a decrease in the II for the two-layer case when the permeability contrast between the layers increased. It is explained as an effect of the reduced total flow capacity of the two composite refined layers as one of them becomes almost impermeable. The reduced total flow capacity consequently forces more water to crossflow out of the refined composite region, which requires a higher BHP difference. The same trend is likewise observed for the refined 20-layer cases, where the BHP difference increases, with a corresponding decrease in the II, as the contrast between the maximum and minimum permeability layers grows. The analytical expression has only been used for a limited set of refinement cases in the vertical direction herein, so further testing on other flow systems should be performed. Grid refinements laterally are also expected to contribute significantly to BHP variations, as the decrease in crossflow distance upon implementing grid refinement conventionally is even larger than in the vertical direction highlighted here. It should, from a more practical engineering point of view, additionally also be tested in reservoirs containing more than one fluid phase and under transient flow conditions, although accuracy will decrease the more the fluid properties of the second phase deviate from water and the flow departs from stationary conditions. The expression highlights nature’s holistic approach as the complete flow distribution changes upon altering permeability values locally. It may, therefore, have implications for how upscaling should be performed and support procedures conventionally used when determining flow fields through heterogeneous geological formations.