Comput Geosci (2014) 18:797–815

DOI 10.1007/s10596-014-9427-x

ORIGINAL PAPER

A coarse-scale compositional model

Alireza Iranshahr · Yuguang Chen · Denis V. Voskov

Received: 11 January 2013 / Accepted: 2 June 2014 / Published online: 25 June 2014 © Springer International Publishing Switzerland 2014

Abstract In subsurface flow modeling, compositional simulation is often required to model complex recovery processes, such as gas/CO2 injection. However, compositional simulation on fine-scale geological models is still computationally expensive and even prohibitive. Most existing upscaling techniques focus on black-oil models. In this paper, we present a general framework to upscale two-phase multicomponent flow in compositional simulation. Unlike previous studies, our approach explicitly considers the upscaling of flow and thermodynamics. In the flow part, we introduce a new set of upscaled flow functions that account for the effects of compressibility. This is often ignored in the upscaling of black-oil models. In the upscaling of thermodynamics, we show that the oil and gas phases within a coarse block are not at chemical equilibrium. This non-equilibrium behavior is modeled by upscaled thermodynamic functions, which measure the difference between component fugacities among the oil and gas phases. We apply the approach to various gas injection problems with different compositional

A. Iranshahr · D. V. Voskov

Department of Energy Resources Engineering,

Stanford University, Stanford, CA, USA

A. Iranshahr e-mail: Alireza.Iranshahr@shell.com

D. V. Voskov e-mail: dvoskov@stanford.edu

Y. Chen ()

Chevron Energy Technology Company,

San Ramon, CA, USA e-mail: ychen@chevron.com

Present Address:

A. Iranshahr

Shell Global Solutions (US), Inc., Houston, TX, USA features, permeability heterogeneity, and coarsening ratios.

It is shown that the proposed method accurately reproduces the averaged fine-scale solutions, such as component overall compositions, gas saturation, and density solutions in the compositional flow.

Keywords Upscaling · Compositional simulation ·

Non-equilibrium thermodynamics · Subsurface flow ·

Reservoir simulation 1 Introduction

In subsurface flow modeling, compositional simulation is used to model the complex physics associated with enhanced oil recovery (EOR) processes such as gas injection. Subsurface formations are characterized by heterogeneity over multiple length scales, which can have a strong impact on the flow and transport. To account for the multi-scale features, high-resolution geological descriptions are often generated through geostatistical algorithms by integrating data at different scales. Although computational resources have increased considerably in the recent years (which enable flow simulation of large-scale, multimillion-cell models), compositional simulation on the high-resolution geocellular models is still computationally expensive and even prohibitive. Therefore, coarse-scale compositional simulation is of practical importance in modeling complex EOR processes.

In compositional simulation, thermodynamic phase equilibrium calculations are performed in each gridblock and at every time step to determine the number and compositions of the phases present in the gridblock [14, 15, 39]. Phase properties are then computed from correlations, or equations of state (EoS). Compared to black-oil models, the coarse798 Comput Geosci (2014) 18:797–815 scale modeling of compositional flow presents additional complexities due to the variations in phase compositions, as well as in saturation and velocities.

In coarse-scale flow simulation, upscaling is often applied to coarsen the highly detailed geological models to scales that are suitable for flow simulation while maintaining the impact of important fine-scale features. However, most existing upscaling techniques focus on the coarsening of black-oil models, in which the modeling techniques can be classified into the upscaling of single-phase flow parameters and multiphase flow functions, based on the type of upscaled parameters to be computed. Reviews and comparative studies, e.g., [4, 11, 19, 21], provide detailed discussions on various methods and outstanding issues.

Single-phase flow upscaling, e.g., [9, 18, 40, 41], considers absolute permeability, and it represents the most commonly applied upscaling technique in practice. It is also a prerequisite for any accurate multiphase flow simulation. Multiphase flow upscaling, e.g., [7, 16, 38], involves, in addition, the upscaling of rock-fluid properties (such as phase relative permeabilities) to account for the multiphase flow mechanisms.

For the coarse-scale modeling of both single-phase and multiphase subsurface flow, the principle of computing the upscaled properties is to decompose a large-scale global flow problem into a series of small-scale local flow problems. Therefore, another way to categorize the different upscaling techniques is based on the size of the domain, on which the flow problem is solved, and the upscaled properties are computed. This applies to the upscaling of both single-phase and multiphase flow parameters, ranging from local to global methods. In local upscaling, the flow problems are solved locally around each target coarse block with assumed local boundary conditions (e.g., [18, 31]). Global upscaling, by contrast, considers flow solutions on the entire global domain to compute the upscaled quantities (e.g., [40, 41]). Global methods eliminate the spurious effects introduced by the local boundary conditions but do require some type of global fine-scale flow solutions.

In between, quasi-global approaches, such as local-global upscaling (e.g., [6, 8, 9]), incorporate approximate global flow solutions into the local upscaling calculations, thus improving the accuracy of local methods.

Many issues encountered in the black-oil models, such as the impact of boundary conditions on the upscaling calculations, remain as problems in the coarse-scale modeling of compositional flow. Moreover, the interaction between subgrid permeability heterogeneity and the thermodynamic phase behavior causes additional complexities. Previous studies in the upscaling of compositional simulation fall into two categories. The first set of methods involve the use of the so-called transport coefficients computed for each component in each phase to capture the averaged fine-scale fluxes of each component (e.g., [2, 3, 13]). Upscaled multiphase flow functions (i.e., phase relative permeabilities) were also computed to preserve the fine-scale phase flow rates. The second approach focused on the modeling of nonzero residual oil saturation in the coarse-scale compositional simulation due to the subgrid permeability heterogeneity within a coarse block (e.g., [22, 23]). In those methods, the concept of transport coefficients was also applied to account for the component-dependent flow rates on the fine scale.