Analytical Modeling of How Capillary End Effects Impact Time Scale and Transient Data During Steady State Relative Permeability Measurements

Abstract

Abstract The role of capillary end effects on steady state relative permeability experiments is investigated focusing on transient measurements. Analytical solutions are derived for saturation distributions, production, pressure drop and time scales for two important cases: (1) When capillary forces are negligible, Buckley-Leverett principles yield the impact of changing injected flow fraction F or total rate Q. (2) When capillary end effects are significant, but the end effect profile is not reaching the inlet we explore increasing Q. The latter is the main novel contribution. The steady state saturation profile is only compressed by higher Q at fixed F. This is assumed to hold also transiently to facilitate analytical solutions. The outlet flux follows from the changes in outlet saturation gradient with time, whereby mass balance results in the requested solutions. In absence of end effects, it takes finite time to reach steady state when F or Q is changed (except for single phase injection). Capillary end effects cause the needed time to be infinite since the solution at late time is exponential. Practically most is produced sooner and a time scale follows from the analytical solutions to reach a specified fraction of production. The injected pore volumes (time) as function of average saturation has a linear term and a logarithmic term. If high Q is applied or it is early time, production is linear with time. The onset of the exponential regime depends on the conditions. Pressure drop increases instantly when Q is increased, but declines afterwards with time (characteristic of end effects) since the end effect is compressed reducing flow resistance. The evolution in pressure drop follows exactly the same profile shape as average saturation. This is the first analytical solution demonstrating how production and pressure drop evolves in response to rate during relative permeability experiments in presence of end effects. Preliminary investigations are presented in this conference paper while more detailed investigations of the model and more theoretical results will be presented in an upcoming journal paper.