We use computational fluid dynamics to explore the creeping flow of power-law fluids through isotropic porous media. We find that the flow pattern is primarily controlled by the geometry of the porous structure rather than by the nonlinear effects in the rheology of the fluid. We further highlight a macroscale transition between a Newtonian and a non-Newtonian regime, which is the signature of a coupling between the viscosity of the fluid and the structure of the porous medium. These complex features of the flow can be condensed into an effective length scale, which defines both the non-Newtonian transition and the Newtonian permeability.

Summary The saturation distribution after unstable waterflooding into highly viscous oil may have a decisive effect on the efficiency of tertiary polymer flooding, in particular because of hysteresis effects associated with oil banking. In this work, we model waterflood and tertiary polymer-flood experiments performed on Bentheimer sandstone slabs with heavy oils of approximately 2,000 and 7,000 cp, and compare the numerical results with experimental production, pressure, and X-ray data. The unstable waterfloods are initially simulated in two dimensions with our parallel in-house research reservoir simulator (IHRRS) using a high-resolution discretization. In agreement with existing literature, we find that Darcy-type simulations dependent on steady-state relative permeabilities—inferred here from a 3D quasistatic pore-network model (PNM)—cannot predict the measured waterflood data. Even qualitatively, the viscous-fingering patterns are not reproduced. An adaptive dynamic PNM is then applied on a 2D pore network constructed from the statistics of the 3D network. If the fingering patterns simulated with this 2D PNM are qualitatively in good agreement with the experimental data, a quantitative match still cannot be obtained because of the limitations of 2D modeling. Although 3D dynamic PNMs at the slab scale would currently lead to prohibitively high computational cost, they have the potential to address the deficiencies of continuum models at highly unfavorable viscosity ratio. For the tertiary polymer floods characterized by a much more favorable mobility ratio, Darcy-type modeling is applied, and history matching is conducted from the end of the waterfloods. We find that unless hysteresis caused by oil banking is accounted for in the relative permeability model, it is not possible to reconcile the experimental data sets. This hysteresis phenomenon, associated with oil invasion into previously established water channels, explains the rapid propagation of the oil bank. For the considered experiments, a simultaneous history match of good quality is obtained with the production and pressure data, and the simulated 2D saturation maps are in reasonable agreement with X-ray data. This paper addresses the challenges in modeling highly unstable waterflooding, using both a conventional Darcy-type simulator and adaptive dynamic PNM, by comparing the simulated results with experimental data including saturation maps. It also highlights the important role of relative permeability hysteresis in the tertiary recovery of viscous oils by polymer injection.

Polymer flooding in very viscous oil has been gaining interest since its efficiency has been field proven. Multiple laboratory investigations have evidenced that the incremental oil recovered by the tertiary process increases considerably the recovery reached thanks to water flooding. However, such tertiary injection is made all the more complex that it is preceded by unstable displacement of oil by water. Therefore a better understanding of the physics is needed, in order to better predict and optimize the viscous oil reserves associated with tertiary polymer flooding. This work presents the interpretation of three similar tertiary polymer flood experiments carried out at the Centre for Integrated Petroleum Research (CIPR, Norway). Each experiment consisted in a water flood followed by a polymer flood. They involved the same Bentheimer outcrop sandstone, 2000 cP oil, 70 cP polymer solution, 2D slab geometry, but different slab lengths (2 slabs are 30cmx30cm, 1 slab is 30cmx90cm). Saturation evolution was monitored by X-ray. On the one hand, provided simple simulation assumptions, the three water floods under study could be history matched (production, pressure). Similar ratios between water and oil relative permeabilities were found, although the water flood relative permeabilities, matched with non Corey-type curves, reflected an important variability. On the other hand, the tertiary polymer floods were found challenging to match consistently. In particular, using classic history matching approaches, the history matching of the long slab experiment could not be reconciled with that of short slab experiments. Simulations were initialized with saturation maps obtained at the end of the water floods. None of the tested approaches enabled us to match consistently the short and long slab experiments together, unless a hysteresis model was implemented. Indeed, a memory effect was observed experimentally from the quantitative analysis of X-ray saturation maps and interpreted as a hysteresis phenomenon. This simple model, with two additional matching parameters, is then further validated by the comparison of 2D simulations with measured in situ saturations.

Motivated by the problem of gravity segregation in an inclined porous layer, we present a theoretical analysis of interface evolution between two immiscible fluids of unequal density and mobility, both in two and three dimensions. Applying perturbation theory to the appropriately scaled problem, we derive the governing equations for the pressure and interface height to leading order, obtained in the limit of a thin gravity tongue and a slightly dipping bed. According to the zeroth-order approximation, the pressure profile perpendicular to the bed is in equilibrium, a widely accepted assumption for this class of problems. We show that for the inclined bed two-dimensional problem, in the reference frame moving with the mean gravity-induced advection velocity, the interface motion is dictated by a degenerate parabolic equation, different from those previously published. In this case, the late-time behaviour of the gravity tongue can be derived analytically through a formal expansion of both the solution and its two moving boundaries. In three dimensions, using a moving coordinate along the dip direction, we obtain an elliptic–parabolic system of partial differential equations where the fluid pressure and interface height are the two dependent variables. Although analytical results are not available for this case, the evolution of the gravity tongue can be investigated by numerical computations in only two spatial dimensions. The solution features are identified for different combinations of dimensionless parameters, showing their respective influence on the shape and motion of the interface.

The Microemulsion phase behavior model based on oleic-aqueous-surfactant pseudo-phase equilibrium, commonly used in chemical flooding simulators, is coupled to Gas-Oil-Water phase equilibrium in our new four-fluid-phase, fully implicit In-House Research Reservoir Simulator (IHRRS). The method consists in splitting the equilibrium in two stages, where all the components other than surfactant are equilibrated first (e.g. using a black-oil, K-value or equation of state model), and the resulting Gas, Oil and Water phases are then lumped into pseudo-phases to be equilibrated using the Microemulsion model. This subdivision in stages is conceptual, and at each converged time-step the four phases (Gas, Oil, Water and Microemulsion, when simultaneously present) will be in equilibrium with each other.The fluid properties (such as densities, viscosities and interfacial tensions) and rock-fluid properties (such as relative permeabilities), required in the transport equations, are evaluated with models from well-known industrial or academic simulators. Surfactant flooding being usually implemented as a tertiary recovery mechanism, on fields for which complete models that we do not wish to modify already exist, particular care is devoted to ensuring continuity of the physics at the onset of surfactant injection.Our code is validated against a reference academic chemical flooding simulator, on 1D corefloods where the original hydrocarbons in place form a dead-Oil phase, possibly with free dry-Gas. Some numerical aspects of our implementation such as numerical dispersion versus time-step size and nonlinear convergence performance are also discussed. As an application example of our code where it is necessary to account for four phases in equilibrium, we consider a scenario where the chemical flood is preceded by a vaporizing Gas drive.