The Modeling of Velocity Enhancement in Polymer Flooding

In single-phase polymer flooding experiments it has repeatedly been observed that the average velocity of the polymer molecules is higher than the average velocity of the water molecules. This effect is incorporated in many conventional Enhanced Oil Recovery (EOR) simulators by the introduction of a constant velocity enhancement factor. In this paper we show that, in absence of dispersion, a constant enhancement factor in the mathematical model for two-phase polymer flow (Buckley--Leverett displacement) leads to ill-posedness of the model equations. We propose a saturation dependent enhancement factor, derived from a model based on percolation concepts, for which this problem does not occur.     

The Riemann Solution for the Injection of Steam and Nitrogen in a Porous Medium

We solve the model for the flow of nitrogen, vapor, and water in a porous medium, neglecting compressibility, heat conductivity, and capillary effects. Our choice of injection conditions is determined by the application to clean up polluted sites. We study all mathematical structures, such as rarefaction, shock waves, and their bifurcations; we also develop a systematic method to find fundamental solutions for thermal compositional flows in porous media. In addition, we unexpectedly find a rarefaction evaporation wave which has not been previously reported in any other study.     

Recovery of Light Oil by Medium Temperature Oxidation

We study one aspect of combustion in porous media for the recovery of light oil. We assume that there is a temperature range above low temperature combustion where oxygen is added to the aliphatic oils to form oxygenated compounds and below the temperature where cracking and coke formation occurs. In the intermediate range oil is combusted to form small combustion products like water, CO $_2$ , or CO. We call this medium temperature oxidation (MTO). Our simplified model considers light oil recovery when it is displaced by air at medium pressures in linear geometry, for the case when water is absent. The resulting MTO combustion displaces all the oil. There are adjacent vaporization and combustion zones, traveling with the same speed. The MTO reaction is assumed to be slow, so that vaporization is much faster. The solution of the model equations leads to a thermal wave upstream, a MTO wave in the middle and a cold isothermal Buckley–Leverett gas displacement process downstream. One of the unexpected results is that vaporization occurs upstream of the combustion zone. In the initial period the recovery curve is typical of gas displacement, but after a critical amount of air has been injected the cumulative oil recovery increases linearly until all oil has been recovered. In our model, the oil recovery is independent of reaction rate parameters, but the recovery is much faster than for gas displacement. Finally, the recovery is slower for higher boiling point and higher oil viscosity, but faster at higher injection pressure. We give a simple engineering procedure to compute recovery curves for a variety of different conditions.     

Negative Saturation Approach for Non-Isothermal Compositional Two-Phase Flow Simulations

This article deals with developing a solution approach, called the non-isothermal negative saturation (NegSat) solution approach. The NegSat solution approach solves efficiently any non-isothermal compositional flow problem that involves phase disappearance, phase appearance, and phase transition. The advantage of the solution approach is that it circumvents using different equations for single-phase and two-phase regions and the ensuing unstable procedure. This paper shows that the NegSat solution approach can also be used for non-isothermal systems. The NegSat solution approach can be implemented efficiently in numerical simulators to tackle modeling issues for mixed CO₂–water injection in geothermal reservoirs, thermal recovery processes, and for multicontact miscible and immiscible gas injection in oil reservoirs. We illustrate the approach by way of example to cold mixed CO₂–water injection in a 1D geothermal reservoir. The solution is compared with an analytical solution obtained with the wave-curve method (method of characteristics) and shows excellent agreement. A complete set of simulations is carried out, which identifies six bifurcations. The two main bifurcations are (1) when the most downstream compositional wave is replaced by a compositional shock and (2) when an extra Buckley–Leverett rarefaction appears. The plot of the useful energy (exergy) versus the CO₂ storage capacity shows a Z-shape. The top horizontal part represents a branch of high exergy recovery/relatively lower storage capacity, whereas the bottom part represents a branch of lower exergy recovery/higher storage capacity.     

Understanding the Non-Gaussian Nature of Linear Reactive Solute Transport in 1D and 2D

In the present study, we examine non-Gaussian spreading of solutes subject to advection, dispersion and kinetic sorption (adsorption/desorption). We start considering the behavior of a single particle and apply a random walk to describe advection/dispersion plus a Markov chain to describe kinetic sorption. We show in a rigorous way that this model leads to a set of differential equations. For this combination of stochastic processes, such a derivation is new. Then, to illustrate the mechanism that leads to non-Gaussian spreading, we analyze this set of equations at first leaving out the Gaussian dispersion term (microdispersion). The set of equations now transforms to the telegrapher’s equation. Characteristic for this system is a longitudinal spreading that becomes Gaussian only in the longtime limit. We refer to this as kinetics-induced spreading. When the microdispersion process is included back again, the characteristics of the telegraph equations are still present. Now, two spreading phenomena are active, the Gaussian microdispersive spreading plus the kinetics-induced non-Gaussian spreading. In the long run, the latter becomes Gaussian as well. Another non-Gaussian feature shows itself in the 2D situation. Here, the lateral spread and the longitudinal displacement are no longer independent, as should be the case for a 2D Gaussian spreading process. In a displacing plume, this interdependence is displayed as a ‘tailing’ effect. We also analyze marginal and conditional moments, which confirm this result. With respect to effective properties (velocity and dispersion), we conclude that effective parameters can be defined properly only for large times (asymptotic times). In the two-dimensional case, it appears that the transverse spreading depends on the longitudinal coordinate. This results in ‘cigar-shaped’ contours.     

Analysis of Nitrogen and Steam Injection in a Porous Medium with Water

We formulate conservation laws governing steam and nitrogen injection in a one-dimensional porous medium containing water. Compressibility, heat conductivity and capillarity are neglected. We study the condensation front and shock waves arising in the flow. We find that there are four possible types of solutions for the initial and boundary conditions of interest. We describe a simple construction in the temperature saturation plane that determines the complete solution for the given conditions. Applications of the theory developed here are in clean up of soil contaminated with nonaqueous phase liquids. We show that a substantial cold gaseous zone develops in all solutions of practical interest, thus counteracting downward migration of the pollutant.     

Pressure dependence of the contact angle in a CO2-H2O-coal system

Carbon dioxide injection into coal layers serves the dual purpose to enhance coal bed methane production (ECBM) and to store CO2. The efficiency of this process is expected to be much higher if water is the non-wetting phase in the coal-water-gas system. Therefore, contact angles in the coal-water-CO2 system have been measured using the captive bubble technique in the pressure range between atmospheric pressure and 141 bar at a temperature of 45 degrees C. At atmospheric pressure the contact angle of a shrinking CO2 droplet increases with time, but stays below 90 degrees . At higher pressures (>2.6 bar) the contact angle increases beyond 90 degrees . The pressure dependence of the contact can be represented by theta=(111 degrees +/-10.5 degrees )+(0.17+/-0.14)P [bar]. The exceptional behavior at atmospheric pressure is possibly related to the stability of water patches on the coal surface. It is concluded that water is the non-wetting phase in this coal-water-CO2 system.     

Diffusive Effects on Recovery of Light Oil by Medium Temperature Oxidation

Volatile oil recovery by means of air injection is studied as a method to improve recovery from low permeable reservoirs. We consider the case in which the oil is directly combusted into small products, for which we use the term medium temperature oil combustion. The two-phase model considers evaporation, condensation and reaction with oxygen. In the absence of thermal, molecular and capillary diffusion, the relevant transport equations can be solved analytically. The solution consists of three waves, i.e., a thermal wave, a medium temperature oxidation (MTO) wave and a saturation wave separated by constant state regions. A striking feature is that evaporation occurs upstream of the combustion reaction in the MTO wave. The purpose of this paper is to show the effect of diffusion mechanisms on the MTO process. We used a finite element package (COMSOL) to obtain a numerical solution; the package uses fifth-order Lagrangian base functions, combined with a central difference scheme. This makes it possible to model situations at realistic diffusion coefficients. The qualitative behavior of the numerical solution is similar to the analytical solution. Molecular diffusion lowers the temperature of the MTO wave, but creates a small peak near the vaporization region. The effect of thermal diffusion smoothes the thermal wave and widens the MTO region. Capillary diffusion increases the temperature in the upstream part of the MTO region and decreases the efficiency of oil recovery. At increasing capillary diffusion the recovery by gas displacement gradually becomes higher, leaving less oil to be recovered by combustion. Consequently, the analytical solution with no diffusion and numerical solutions at a high capillary diffusion coefficient become different. Therefore high numerical diffusion, significant in numerical simulations especially in coarse gridded simulations, may conceal the importance of combustion in recovering oil.