The effect of microporosity on transport properties in porous media

Highlights

• Our deterministic algorithm adds microporosity into two-scale, 3D unstructured network models.

• Microporosity is added within partially dissolved grains or within pores.

• Off-diagonal complexity can distinguish different types of microporosity added.

• Relative permeability behavior strongly depends on the microporosity type.

• If cementation disconnects larger scale porosity, sometimes neither fluid flows.

Abstract

Sizeable amounts of connected microporosity with various origins can have a profound effect on important petrophysical properties of a porous medium such as (absolute/relative) permeability and capillary pressure relationships. We construct pore-throat networks that incorporate both intergranular porosity and microporosity. The latter originates from two separate mechanisms: partial dissolution of grains and pore fillings (e.g. clay). We then use the reconstructed network models to estimate the medium flow properties. In this work, we develop unique network construction algorithms and simulate capillary pressure–saturation and relative permeability–saturation curves for cases with inhomogeneous distributions of pores and micropores. Furthermore, we provide a modeling framework for variable amounts of cement and connectivity of the intergranular porosity and quantifying the conditions under which microporosity dominates transport properties. In the extreme case of a disconnected inter-granular network due to cementation a range of saturations within which neither fluid phase is capable of flowing emerges. To our knowledge, this is the first flexible pore scale model, from first principles, to successfully approach this behavior observed in tight reservoirs.

Introduction

We propose a new network model for capillary pressure/relative permeability–saturation relationships based on the pore level petrophysical description of porosity in multiple scales. We first clarify the terminology we use in this work. We classify porosity into two categories, namely macroporosity and microporosity. Macroporosity refers to the original intergranular (primary) porosity in typical subsurface porous materials including those intergranular pores that remain open after cementation and compaction. Void spaces that reside inside grains (and act “in parallel” to macropores, e.g. due to dissolution) or inside materials filling the macropores (and act “in series” to the macropores, e.g. porosity within clay) will be referred to as micropores. The parallel and series terminology is conceptual and borrowed from common analogy of (resistance to) flow of fluid in porous media and (resistance to) the flow of electrical current in circuits. The interplay of macroporosity and microporosity on transport properties is difficult to observe experimentally: for instance, a single imaging technique can rarely capture both length scales in desired detail. Herein lies the motivation to develop a deterministic simulation model that sheds light onto transport mechanisms between the two porosity types. All reported lengths in this work are normalized by the grain sizes. Although we analyzed each porosity class separately, both classes co-exist in our porous medium model and are treated simultaneously. Further, while we present the results only based on granular media, the algorithm has enough flexibility to be used on image-based networks (see preliminary results in [1]).

Conventional relationships, such as Archie’s law for resistivity, the Carman–Kozeny permeability estimate and the Brooks–Corey parameterization of relative permeability, were developed for rocks whose pores are mostly intergranular and well connected. The conventional models and theoretical relationships therefore cannot be expected to do well in cases where microporosity dominates the pore space. Such domination is common in many carbonates and tight gas sandstones (TGSS). For instance, most TGSS with porosities less than 10% exhibit steep capillary-pressure curves, low permeability, and high irreducible saturations at drainage and imbibition [2]. The authors in [2] fitted many TGSS experimental curves, but such fits do not necessarily offer qualitative understanding behind the observed behavior. It would thus be instructive to generate the behavior from first principles.

Carbonate rocks are another class of rocks with highly interconnected and heterogeneous distribution of porous and microporous regions [3]. These rocks are therefore difficult to classify [4] and their experimental behavior is hard to explain [5]. This leads us further to contend that, when macroporosity and microporosity are considered separately in a porous medium, the multiphase constitutive relationships are not a simple function of equivalent, separate constitutive relationships.

Network models that consider the single length scale for either pore sizes or connectivity are often inadequate in modeling complex media. Mousavi and Bryant [6] attempted to characterize TGSS by numerically cementing a wide range of sphere packs (for overview of network models, see Section 2), which is a valid approach in modeling conventional sandstones [7]. Nevertheless, the wide range of pore sizes observed, and the high capillary pressures measured, could not be simulated by the conventional, single-scale network model. To circumvent this issue in the absence of the actual TGSS network, the authors [6] used the spatial distribution of a conventional network that was representative of sandstone with a porosity of 5%. They used throat sizes that matched the TGSS distribution from a Green River basin in Wyoming and simulated the drainage curves in an attempt to match the experimental mercury porosimetry data. Neither a qualitative nor a quantitative match ensued.

In tight gas sandstones, relative permeability curves can have a range of fairly large gas and water saturations in which neither phase holds effective (measurable) flow capacity (sometimes referred to as “permeability jail” [8], defined for rocks with absolute permeability <0.05 mD in the region where both relative permeabilities are less than 0.02). Simulations by Silin et al. [9] approach this type of behavior using a direct simulation of imbibition in imaged TGSS samples characterized by thin microcracks. However, the microporosity connecting to cracks was poorly resolved, and its overall volume and influence are not clear.

We hypothesize that ignoring microporosity, and specifically how it connects to macroporosity in pore scale modeling, is the main reason why existing functional relationships (correlations) in many cases do not match experimental data (either qualitatively or quantitatively). Our objective is to develop a flexible model that can consider a wide range of spatial distributions for macro- and microporosity, as well as their (inter)connectivity, and thus overcome the limitations of available models (see Section 1.3). With the developed model, we can begin to understand and systematically quantify the conditions under which microporosity dominates the transport properties and needs to be explicitly accounted for.

We present an algorithm to geometrically match pore throat networks from two separate scales (and possibly from two different imaging modalities). In these networks, the boundary between the length scales can take an arbitrary shape. For clarity, in this work we focus on construction from model granular media, but a similar algorithm can be devised for imaged porous media (see Section 4.2). Since the resultant pore network is a single entity (i.e. all scales are deterministically included and dealt with simultaneously), we can apply existing network modeling techniques without requiring any additional “bridging physics” to stitch the length scales together. We present results on estimated transport properties from these two-scale media. Before we proceed with the method description (in Section 2), we briefly overview the existing models with similar objectives and their current limitations.

When modeling multi-phase fluid flow, one must account for processes occurring on a broad range of scales. In particular, the detailed structure of the pore space can play a critical role in determining the spatial distribution of fluid phases and ultimately influences the macroscopic flow properties. Different length scales often demand different approaches. A direct simulation within a geometrically detailed medium is very costly. Hence, modeling at the pore scale is often done after mapping the pore space onto a representative network of idealized pores and throats. Fluid displacements are then simulated through discrete events (for overview of network models refer to Section 2.1) on the network. At larger scales, one usually constructs continuum numerical models, in which individual grid blocks implicitly contain sufficient pores so that the system within each grid block evolves smoothly with time. Large (continuum) scale models all require parameters and constitutive relationships that are provided by pore scale models, theory or experiments. Continuum models capable of accounting for two scales – the so-called dual permeability models [10], [11], [12] – have been constructed, but they all require prior knowledge of constitutive relationships. Efforts have also been made in building hybrid methods that couple regions, modeled via pore scale networks, with regions that are treated with continuous approaches [13].

We briefly overview pore scale methods that incorporate microporosity. Wu and coauthors [14] showed the importance of including microscale pores using a direct Lattice–Boltzmann based simulation of drainage capillary pressure in imaged pore spaces. The simulation was first run by omitting the submicron pores in a two-scale siltstone reconstruction, and then iteratively adding stochastically generated submicron pores. A significant difference in capillary pressure threshold and residual saturation in drainage was observed in spite of the fact that both length scales were well connected. The model was based on a single image, did not report on relative permeability, and prohibited modification in the relative spatial position of the microporosity and macroporosity, all of which issues we address in this work. Toumelin [15] on the other hand, studied the effect of microporosity on the electrical properties (Archie’s cementation and saturation exponents m and n respectively) through the random walk method in model carbonates with well-connected intergranular porosity. The authors concluded that the intra-granular porosity (microporosity) added in parallel with the intergranular porosity (macroporosity) did not have much effect on the cementation factor in water wet media (m changed from 2 to 1), but had a noticeable effect on the resistivity index for oil wet rocks. We work with similar models, but include the ability to vary intergranular porosity connectivity as well as the spatial position of microporosity. In this work we focus on fluid flow rather than electrical properties. Finally, Youssef et al. [16] modeled carbonates using the macro- and microporosity information estimated from high resolution micro-tomography images. The macroporous network model was augmented with cuboids that represented microporous regions and were placed in parallel with macroporosity. The physical size (cross-sectional area and length) of the microporous regions was based upon upscaled (effective) parameters and was thus somewhat arbitrary, but the estimated pore volume was preserved. We present microporosity that acts both in parallel and in series with the macroporosity.