Upscaling of geothermal properties is necessary given the computational cost of numerical simulations. Nevertheless, accurate upscaling of thermo-physical properties of layers combined in simulation grid blocks has been a long-standing challenge. In stratified porous media, non-uniform velocity between layers combined with transverse thermal conduction across layers causes spreading of the thermal front: thermal Taylor dispersion. Neither effect of heterogeneity is accounted for in conventional upscaling. Based on thermal Taylor dispersion, we develop a new upscaling technique for simulation of geothermal processes in stratified formations. In particular, we derive a model for effective longitudinal thermal diffusivity in the direction of flow, α_eff, to represent this phenomenon in two-layer media. α_eff, accounting for differences in velocity and transverse thermal conduction, is much greater than the thermal diffusivity of the rock itself, leading to a remarkably larger effective dispersion. We define a dimensionless number, N_TC, a ratio of times for longitudinal convection to transverse conduction, as an indicator transverse thermal equilibration of the system during cold-water injection. Both N_TC and α_eff equations are verified by a match to numerical solutions for convection/conduction in two-layer systems. We find that for N_TC > 5, thermal dispersion in the system behaves as a single layer with α_eff This suggests a two-layer medium satisfying N_TC > 5 can be combined into a single layer with an effective longitudinal thermal diffusivity α_eff. Compared with conventional approaches by averaging, the α_eff model provides more accurate upscaling of thermal diffusivity and thus more-accurate prediction of cooling-front breakthrough. In stratified geothermal reservoirs with a sequence of layers, upscaling can be conducted in stages, e.g. combining two layers satisfying the N_TC criterion in each stage. The application of the new technique to upscaling geothermal well-log data will be presented in a companion paper.

Foam injection is a promising enhanced-oil-recovery (EOR) technology that significantly improves the sweep efficiency of gas injection. Simulation of foam/oil displacement in reservoirs is an expensive process for conventional simulation because of the strongly nonlinear physics, such as multiphase flow and transport with oil/foam interactions. In this work, an operator-based linearization (OBL) approach, combined with the representation of foam by an implicit-texture (IT) model with two flow regimes, is extended for the simulation of the foam EOR process. The OBL approach improves the efficiency of the highly nonlinear foam-simulation problem by transforming the discretized nonlinear conservation equations into a quasilinear form using state-dependent operators. The state-dependent operators are approximated by discrete representation on a uniform mesh in parameter space. The numerical-simulation results are validated by using three-phase fractional-flow theory for foam/oil flow. Starting with an initial guess depending on the fitting of steady-state experimental data with oil, the OBL foam model is regressed to experimental observations using a gradient-optimization technique. A series of numerical validation studies is performed to investigate the accuracy of the proposed approach. The numerical model shows good agreement with analytical solutions at different conditions and with different foam parameters. With finer grids, the resolution of the simulation is better, but at the cost of more expensive computations. The foam-quality scan is accurately fitted to steady-state experimental data, except in the low-quality regime. In this regime, the used IT foam model cannot capture the upward-tilting pressure gradient (or apparent viscosity) contours. 1D and 3D simulation results clearly demonstrate two stages of foam propagation from inlet to outlet, as seen in the computed-tomography (CT) coreflood experiments: weak foam displaces most of the oil, followed by a propagation of stronger foam at lower oil saturation. OBL is a direct method to reduce nonlinearity in complex physical problems, which can significantly improve computational performance. Taking its accuracy and efficiency into account, the data-driven OBL-based approach could serve as a platform for efficient numerical upscaling to field-scale applications.

We present a CT coreflood study of foam flow with two representative oils: hexadecane C₁₆ (benign to foam) and a mixture of 80 wt% C₁₆ and 20 wt% oleic acid (OA) (very harmful to foam). The purpose is to understand the transient dynamics of foam, both generated in-situ and pre-generated, as a function of oil saturation and type. Foam dynamics with oil (generation and propagation) are quantified through sectional pressure-drop measurements. Dual-energy CT imaging monitors phase saturation distributions during the corefloods. With C₁₆, injection with and without pre-generation of foam exhibits similar transient behavior: strong foam moves quickly from upstream to downstream and creates an oil bank. In contrast, with 20 wt % OA, pre-generation of foam gives very different results from co-injection, suggesting that harmful oils affect foam generation and propagation differently. Without pre-generation, initial strong-foam generation is very difficult even at residual oil saturation about 0.1; the generation finally starts from the outlet (a likely result of the capillary-end effect). This strong-foam state propagates backwards against flow and very slowly. The cause of backward propagation is unclear yet. However, pre-generated foam shows two stages of propagation, both from the inlet to outlet. First, weak foam displaces most of the oil, followed by a propagation of stronger foam at lower oil saturation. Implicit-texture foam models for enhanced oil recovery cannot distinguish the different results between the two types of foam injection with very harmful oils. This is because these models do not distinguish between pre-generation and co-injection of gas and surfactant solution.

A new capillary number (N _ca ) definition is proposed for 2-D etched micromodels. We derive the new definition from a force balance on a nonwetting ganglion trapped by capillarity. It incorporates the impact of pore microstructure on mobilization. The geometrical factors introduced can be estimated directly from image analysis of the pore network etched in the micromodel, without conducting flow experiments. The improved fit of the new N _ca to published data supports its validity. The new definition yields a consistent trend in the capillary-desaturation curve. The conventional N _ca definitions proposed for porous rock give a large scatter in the capillary-desaturation curve for data in micromodels. This is due to the different type of flow in micromodels, as 2-D networks, relative to 3-D geological porous media. In particular, permeability is dominated by channel depth in micromodels with shallow depth of etching, and generally, there is no simultaneous multiphase flow under capillary-dominated conditions. Applying the conventional definitions to results in micromodels may lead to misleading conclusions for fluid transport in geological formations.

Foam flow in porous media without oil shows two regimes depending on foam quality (gas fractional flow). Complexity and limited data on foam/oil interactions in porous media greatly restrict understanding of foam in contact with oil. Distinguishing which regimes are affected by oil is key to modeling the effect of oil on foam. We report steady-state corefloods to investigate the effect of oil on foam through its effect on the two flow regimes. We fit the parameters of a widely used local-equilibrium (LE) foam model to data for concurrent foam/oil flow. This research provides a practical approach and initial data for simulating foam enhanced oil recovery (EOR) in the presence of oil. To ensure steady state, oil is coinjected with foam at a fixed ratio of oil (U _o ) to water (U _w ) superficial velocities in a Bentheimer Sandstone core. Model oils used here consist of a composition of hexadecane, which is benign to foam stability, and oleic acid (OA), which can destroy foam. Varying the concentration of OA in the model oil allows one to examine the effect of oil composition on steady-state foam flow. Experimental results show that oil affects both high- and low-quality regimes, with the high-quality regime being more sensitive to oil. In particular, oil increases the limiting water saturation (S _w ) in the high-quality regime and also reduces gas-mobility reduction in the low-quality regime. Unevenly spaced !p contours in the high-quality regime suggest either strongly shear-thinning behavior or an increasingly destabilizing effect of oil. In some cases, the pressure gradient (!p) in the low-quality regime decreases with increasing U _w at fixed gas superficial velocity (U _g ), either with or without oil. This might reflect either an effect of oil, if oil is present, or easier flow of bubbles under wetter conditions. Increasing the OA concentration extends the high-quality regime to lower foam qualities, indicating more difficulty in stabilizing foam. Thus, oil composition plays as significant a role as oil saturation (S _o ). A model fit assuming a fixed S _w and including shear thinning in the low-quality regime does not represent the two regimes when the oil effect is strong enough. In such cases, fitting S _w to each !p contour and excluding shear thinning in the low-quality regime yield a better match to these data. The dependency of S _w on S _o is not yet clear because of the absence of oil-saturation data in this study. Furthermore, none of the current foam-simulation models captures the upward-tilting !p contours in the low-quality regime.

Foam flow in porous media without oil shows two regimes, depending on foam quality (gas fractional flow). Complexity and limited data on foam-oil interactions in porous media greatly restrict understanding of foam in contact with oil. Distinguishing which regimes are affected by oil is key to modelling the effect of oil on foam. We report steady-state corefloods to investigate the effect of oil on foam through its effect on the two flow regimes. We fit parameters of the widely used STARS foam model to data for foam-oil concurrent flow. This research provides a practical approach and initial data for simulating foam EOR in the presence of oil. To ensure steady state, oil is co-injected with foam at a fixed ratio of oil (Uo) to water (Uw) superficial velocities in a Bentheimer sandstone core. Model oils used here consist of two components: hexadecane, which is benign to foam stability, and oleic acid, which can destroy foam. Varying the concentration of oleic acid in the model oil allows one to examine the effect of oil composition on steady-state foam flow. Experimental results show that oil impacts both high- and low-quality regimes, with the high-quality regime more vulnerable to oil. In particular, oil increases the limiting water saturation (Sw^∗) in the high-quality regime and also lessens gas mobility reduction in the low-quality regime. The high-quality regime is strongly shear-thinning in the presence of oil. Pressure gradient (p) in the low-quality regime, in some cases, decreases with increasing Uw at fixed gas superficial velocity (Ug), either with or without oil. This may reflect either an effect of oil, if oil is present, or easier flow of bubbles under wetter conditions. Increasing oleic acid concentration extends the high-quality regime to lower foam qualities, indicating more difficulty in stabilizing foam. Thus oil composition plays as significant a role as oil saturation. A model fit assuming a fixed Sw∗ and including shear-thinning in the low-quality regime doesn't represent each regime when the oil effect is strong enough. In such cases, fitting Sw∗ to each p contour and excluding shear-thinning in the low-quality regime yields a better match to data. The dependency of Sw∗ on oil saturation is not yet clear owing to absence of oil-saturation data in this study. Furthermore, none of the current foam simulation models can capture the upward-tilting p contours in the low-quality regime.