Foam front propagation in anisotropic oil reservoirs

Modelling foam improved oil recovery: towards a formulation of pressure-driven growth with flow reversal

The pressure-driven growth model that describes the two-dimensional (2-D) propagation of a foam through an oil reservoir is considered as a model for surfactant-alternating-gas improved oil recovery. The model assumes a region of low mobility, finely textured foam at the foam front where injected gas meets liquid. The net pressure driving the foam is assumed to reduce suddenly at a specific time. Parts of the foam front, deep down near the bottom of the front, must then backtrack, reversing their flow direction. Equations for one-dimensional fractional flow, underlying 2-D pressure-driven growth, are solved via the method of characteristics. In a diagram of position versus time, the backtracking front has a complex double fan structure, with two distinct characteristic fans interacting. One of these characteristic fans is a reflection of a fan already present in forward flow mode. The second fan however only appears upon flow reversal. Both fans contribute to the flow's Darcy pressure drop, the balance of the pressure drop shifting over time from the first fan to the second. The implications for 2-D pressure-driven growth are that the foam front has even lower mobility in reverse flow mode than it had in the original forward flow case.

Breakdown of similarity solutions: a perturbation approach for front propagation during foam-improved oil recovery

The pressure-driven growth model has been employed to study a propagating foam front in the foam-improved oil recovery process. A first-order solution of the model proves the existence of a concave corner on the front, which initially migrates downwards at a well defined speed that differs from the speed of front material points. At later times, however, it remains unclear how the concave corner moves and interacts with points on the front either side of it, specifically whether material points are extracted from the corner or consumed by it. To address these questions, a second-order solution is proposed, perturbing the aforementioned first-order solution. However, the perturbation is challenging to develop, owing to the nature of the first-order solution, which is a similarity solution that exhibits strong spatio-temporal non-uniformities. The second-order solution indicates that the corner’s vertical velocity component decreases as the front migrates and that points initially extracted from the front are subsequently consumed by it. Overall, the perturbation approach developed herein demonstrates how early-time similarity solutions exhibiting strong spatio-temporal non-uniformities break down as time proceeds.

Foam–liquid front motion in Eulerian coordinates

A mathematical model formulated as a system of Hamilton–Jacobi equations describes implicitly the propagation of a foam–liquid front in an oil reservoir, as the zero-level set of the solution variable. The conceptual model is based on the ‘pressure-driven growth’ model in Lagrangian coordinates. The Eulerian mathematical model is solved numerically, where the marching is done via a finite volume scheme with an upwind flux. Periodic reinitialization ensures a more accurate implicit representation of the front. The numerical level set contour values are initially formed to coincide with an early time asymptotic analytical solution of the pressure-driven growth model. Via the simulation of the Eulerian numerical model, numerical data are obtained from which graphical representations are generated for the location of the propagating front, the angle that the front normal makes with respect to the horizontal and the front curvature, all of which are compared with the Lagrangian model predictions. By making this comparison, it is possible to confirm the existence of a concavity in the front shape at small times, which physically corresponds to an abrupt reorientation of the front over a limited length scale.

Pressure-driven growth in strongly heterogeneous systems

The pressure-driven growth model for advance of a foam front through an oil reservoir during foam improved oil recovery is considered: specifically the limit of strong heterogeneity in the reservoir permeability is treated, such that permeability variation with depth more than outweighs the tendency of the net pressure driving the front to decay with depth. This means that the fastest moving part of the front is not at the top of the solution domain, but rather somewhere in the interior. Moreover the location of the foam front on the top boundary of the system can no longer be specified as a boundary condition, but instead must be determined as part of the solution of the problem. Numerical solutions obtained from the pressure-driven growth model under these circumstances are compared with approximate analytic solutions. An early-time approximate solution is found to break down remarkably quickly (far more quickly than breakdown would occur in the analogous homogeneous system). Numerical solutions agree much better with local quasi-static solutions centred about local maxima in the front shape, each local maximum corresponding to a depth within the reservoir at which a high permeability stratum is found. These individual local solutions meet together at sharp concave corners to cover the entire depth of the foam front. As time continues to progress however, the system evolves towards a long-time, global quasi-static solution, corresponding to the fastest moving of the aforementioned local maxima. Additional key features of the predicted front shapes are elucidated. The foam front is found to meet the top boundary obliquely despite an established convention in pressure-driven growth that the front and top boundary should meet at right angles. In addition, at each sharp concave corner, discontinuous jumps are predicted in the path length that material points travel to reach either side of the corner. Moreover the long-time, global quasi-static solution is found to admit smooth concavities, as opposed to the aforementioned sharp concave corners, which only tend to be prominent earlier on.