Abstract:

Characterization of formation heterogeneity is the first step for constructing geological models that are used in simulating subsurface flow migration, such as CO2 geological storage and oil/gas recovery. Technologies such as seismic survey, well logging, and core analysis reveal the formation heterogeneity at different scales. Due to the computational limitations on model construction and flow simulation, the gridblock size of a geological model is usually much larger than a core. The common industry practice is to assign core-scale properties directly to a geological model gridblock. The sub-grid- and sub-core-scale heterogeneities are neglected. The objective of this work is two-fold. First, we demonstrate that these small-scale heterogeneities can influence large-scale CO2 migration during geological storage. Significant modeling error may occur if these heterogeneities are not accounted for properly. Second, we improve the nonlinear convergence performance of numerical simulation, which is a crucial tool for translating small-scale physics in large-scale modeling and for uncertainty quantifications during the scale-up.

Recent advancements in experimental techniques allow quantifying sub-core-scale heterogeneities in a high resolution. Based on observations of heterogeneity distributions in natural core samples, we perform simulations to study the influence of sub-core-scale heterogeneities on large-scale CO2 migration during geological storage. We observe that even the heterogeneities at millimeter scale (the scale of a Representative Elementary Volume) can affect large-scale buoyancy-driven CO2 migration. For the representative examples we study, ignoring small-scale heterogeneities can lead to an overestimation of the migration speed by a factor of two. To analyze the cause of such overestimation, we introduce a dimensionless heterogeneity factor to characterize the level of heterogeneities. The influence on CO2 migration is quantified with respect to a variety of heterogeneity factors, correlation lengths, and fluid viscosity ratios for isotropic and anisotropic media. Our findings suggest that sub-core-scale heterogeneities need to be considered in core analysis. In addition, relative-permeability curves measured from core-flood experiments under high flow-rate conditions (so as to eliminate capillary end-effects) should not be directly used in modeling low-flow-rate CO2 migration if sub-core-scale heterogeneities are present.

The ability to characterize sub-core-scale heterogeneities provides statistical descriptions of the missing information between a core and a geological model gridblock. Knowledge of depositional environment and geostatistics can be applied to quantify the uncertainty associated with the translation between the two scales. Flow simulations are often needed, not only for quantifying the uncertainty of the flow response, but also for computing the effective (upscaled) properties that will be used in large-scale modeling. These steps rely heavily on the computational performance of numerical simulators.

Unfortunately, nonlinear convergence problems in simulations can lead to unacceptably large computational time and are often the main impediment to performing scale-translation analysis and general-purpose reservoir simulation. To understand the cause of convergence failure, we analyze the nonlinearity of the discrete transport (mass conservation) equation for immiscible, incompressible, two-phase flow in porous media in the presence of viscous, buoyancy, and capillary forces. Although simulation problems are multi-dimensional with large numbers of gridblocks and variables, we find that the essence of the nonlinear behavior can be understood by studying the discretized (numerical) flux function for the interface between two gridblocks. The numerical flux is expressed in terms of the saturations of the two gridblocks. Discontinuities in the first derivative of the flux function (referred to as kinks) and inflection lines are identified as the cause of convergence difficulty. These critical features (kinks and inflections) change the curvature of the numerical flux function abruptly, and can lead to overshoots, oscillations, or divergence in Newton iterations.

Based on our understanding of the nonlinearity, a nonlinear solver is developed, referred to as the Numerical Trust Region (NTR) solver. The solver is able to guide the Newton iterations safely and efficiently through the different saturation `trust-regions` delineated by the kinks and inflections. Specifically, overshoots and oscillations that often lead to convergence failure are avoided. Numerical examples demonstrate that our NTR solver has superior convergence performance compared with existing methods. In particular, convergence is achieved for a wide range of timestep sizes and Courant-Friedrichs-Lewy (CFL) numbers spanning several orders of magnitude. In addition, a discretization scheme is proposed for handling heterogeneities in capillary-pressure-saturation relationship. The scheme leads to easier nonlinear convergence compared with the standard Single-point Phase-based Upstream weighting scheme, especially when used together with our NTR solver.

This study shows that sub-core-scale heterogeneities can influence large-scale flow migration in porous geological formations. Any assumption that ignores such small-scale heterogeneities in reservoir characterization should be validated, at least, by flow simulations. Although it may seem that the translation between small and large scales is mainly a physical problem, its numerical challenges are as severe as the physics. Our improved numerical strategy will benefit not only the scale-translation analysis but also general-purpose reservoir simulation.

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Copyright 2014, Boxiao Li: Please note that the reports and theses are copyright to their original authors. Authors have given written permission for their work to be made available here. Readers who download reports from this site should honor the copyright of the original authors and may not copy or distribute the work further without the permission of the author, Boxiao Li.