Mathematical Basis for the Occurrence of Derivative Time-Shift During Numerical Well Test Simulation Using Large Grid Blocks

Abstract During pressure transient tests, a well is being shut-in at the wellhead; this leads to the occurrence of a compressibility-driven after-flow phenomenon called the wellbore storage effect. This after-flow effect means that flow continues from the sandface into the wellbore long after the well has been shut-in at the wellhead. However, during numerical pressure transient simulation, the well is being shut-in at the sandface, thereby eliminating the wellbore storage effect. Nevertheless, we have observed another kind of compressibility-driven after-flow phenomenon that affects pressure transient simulation; we refer to this phenomenon as grid-block storage effect. Grid block storage is a numerical artifact that occurs during reservoir simulation whereby the simulated fluid storage in a grid block does not accurately represent the actual fluid storage in the reservoir. During simulation, we observed that as the sizes of the grid blocks became larger, it led to the occurrence of a derivative time-shift in the pressure derivative curve. In this paper, we developed a mathematical model that governs the occurrence of this empirical observation. Beginning from first principles, we applied the fundamental equation for fluid compressibility, which accounts for the pressure differences between adjacent simulation grid blocks (i+1 and i). Next, we introduced a grid storativity coefficient (Gs), which accounts for inter-grid fluid storage; and subsequently incorporated the Bourdet's pressure derivative equation and Horner's pressure build-up equation as key mathematical frameworks to back up our concept. Three major investigations were carried out: firstly, the effect of Gs on the derivative plot; secondly, the effect of grid size on Gs; and thirdly, the effect of permeability on the derivative plot. We then performed a hypothetical full-field reservoir simulation across four model scenarios (6,000ft × 6,000ft × 60ft) with constant Δz = 1ft. Grid dimensions were varied: Δx = Δy = 50ft, 100ft, 300ft, and 5ft. All scenarios featured consistent properties: K = 50mD, φ = 0.15, kv/kh = 0.1, μo = 0.594, βo = 1.565, using oil as the fluid. A producer well was placed centrally and operated at 500 bbl/day for 300 hours, then shut-in for 100 hours. Results showed that coarser grids (e.g. 300ft) caused the derivative unit slope line to prevail, while finer grids (e.g. 5ft) eliminated the unit slope line entirely, marking an earlier onset of infinite-acting radial flow (IARF). Major findings from the investigations include: (i) Gs is inversely proportional to IARF derivative stabilization (m), with smaller grids reducing Gs and accelerating IARF onset; (ii) Gs increases with grid size, delaying IARF onset; and (iii) IARF derivative stabilization (m) is inversely proportional to permeability, with lower permeability further delaying IARF onset. Therefore, to evade the derivative time-shift problem, we advise that practitioners adopt a fine-grid simulation.