Summary
We derive an effective continuum model to describe the nucleation and subsequent growth of a gas phase from a supersaturated liquid in a porous medium, driven by heat transfer. The evolution of the gas results from the reduction of the system pressure at a constant rate. The model addresses two stages before the onset of bulk gas flow, nucleation, and gas-phase growth. The problem arises in internal steam drive - for example, of the type recently discussed in blowdown experiments in carbonate rocks (Dehghani et al.,1 Dehghani and Kamath2).
Important quantities, such as the fraction of pores that host activated sites, the deviation from thermodynamic equilibrium, the maximum supersaturation, and the critical gas saturation depend crucially on the nucleation characteristics of the medium. We use heterogeneous nucleation models in the form of pre-existing gas, trapped in hydrophobic cavities, to investigate the nucleation behavior. Using scaling analysis and a simpler analytical model, we show that the relevant quantities during nucleation can be expressed in terms of a simple combination of dimensionless parameters, which include rate effects. The subsequent evolution of the gas phase and the approach to the critical gas saturation are also described using numerical and analytical models.
The theory predicts that the maximum supersaturation in the system is a weakly increasing function of the decline rate. This function depends sensitively on the probability density function of the nucleation cavity sizes. It also predicts that the final nucleation fraction, and thus the critical gas saturation, is a power law of the decline rate. The theory for both nucleation and phase growth is then compared with available experimental data.
Introduction
The liquid-to-gas phase change in a porous medium and the subsequent growth of the gas phase is encountered in a plethora of applications driven by mass or heat transfer. Typical examples include solution gas-drive for the recovery of oil from oil reservoirs, boiling in porous media, thermal methods for oil recovery, nuclear waste disposal, soil remediation, and others. In this paper, we examine the gas-phase growth from a supersaturated, slightly compressible liquid in a porous medium, driven by heat transfer and controlled by the application of a constant-rate decline of the system pressure. A characteristic example of such a process occurs during cyclic steaming for the recovery of oil from low-permeability reservoirs through hydraulic or natural fractures (Dehghani et al.1). During injection and soaking, steam condenses in the fracture, and hot water imbibes into the matrix. During production, the pressure of the system constantly declines, and when it falls sufficiently below the vapor pressure, it results in the appearance of steam in the matrix (in-situ boiling). The in-situ production and subsequent growth of the steam phase inside the matrix are of interest because they result in expelling additional oil.
Dehghani et al.1 conducted a series of core experiments in order to study the effect of in-situ steam drive on fluid displacement in porous media. Subsequently, Dehghani and Kamath2 conducted experiments in a vuggy carbonate core using a recombined oil to study the contribution of the various recovery mechanisms (thermal expansion, thermally enhanced solution gas-drive, dry distillation, and in-situ steam drive) in steam injection, followed by pressure reduction.
While of interest both from theoretical and applied viewpoints, a more fundamental understanding of the basic aspects has not been obtained to our knowledge. It is the objective of this paper to bridge this gap by providing a model of the nucleation and gasphase growth periods. Internal steam drive has many similarities with the process of solution gas-drive. They both describe the evolution of a gas phase caused by the increase of the supersaturation of the system through a relatively slow pressure decline. Nucleation and subsequent phase growth play a key role in both processes. In a recent publication (Tsimpanogiannis and Yortsos3), we developed a comprehensive effective continuum model to model solution gas-drive under various conditions. In this paper, we extend that approach to the specific problem of internal steam drive.
As discussed in Tsimpanogiannis and Yortsos,3 our effective continuum model is best suited for the early part of the process, in which nucleation and the early stages of bubble growth are dominant. The latter two, particularly the nucleation sequence, are the main areas of interest of this paper as well. We focus on the effect of the nucleation characteristics on the maximum supersaturation, the nucleation fraction, and the critical gas saturation, and provide an analysis of the effect of various parameters, such as pressure decline rate, on these quantities. Results for the gas-phase growth, following the conclusion of nucleation, are also presented for completeness.
Solution gas-drive involves a binary system and is controlled by mass transfer. On the other hand, internal steam drive is fundamentally a single-component system, controlled by heat transfer. Of course, in the more general case, oil will be present in the pore space in addition to water. The presence of oil can affect the nucleation process, depending on which phase is wetting the porous medium, and can prevent a component having access to cavities on the pore walls where nucleation occurs. It also can affect the growth process by restricting the motion and the coalescence of the gas bubbles, thus delaying the reaching of the critical gas saturation. The effects of the combined presence of oil and water on the solution gas-drive process were examined extensively, using visualization experiments in glass micromodels by Hawes et al.,4 Mackay et al.,5 and Bora et al.,6 and with core experiments by Kortekaas and van Poelgeest.7 Dehghani et al.1 and Dehghani and Kamath2 examined the effects of the combined presence of oil and gas on internal steam drive processes. Dehgani et al.1 also discussed in detail the effect of hot-water flash tests on oil recovery for various initial oil saturations. However, our emphasis will be on single-component systems.
The paper is organized as follows: First, we formulate the problem, closely following Tsimpanogiannis and Yortsos.3 A scaling analysis of the resulting equation allows us to recast the problem in a more useful form, to be used for direct predictions. The numerical results are analyzed. It turns out that for their interpretation, a simplified model of the nucleation and growth periods can be developed. We use this simpler model to obtain expressions for the maximum supersaturation as a function of geometric, thermodynamic, and process parameters. This allows us to obtain useful relations for the dependence of the final nucleation fraction (and the critical gas saturation) on process parameters. The theoretical predictions are then compared against experimental results.
