The Richards equation is a mathematical model for the unsaturated flow through porous media. This paper considers an extension of the Richards equation, where non-equilibrium effects like hysteresis and dynamic capillarity are incorporated in the relationship that relates the water pressure and the saturation. The focus is on travelling wave solutions, for which the existence is investigated first for the model including hysteresis and subsequently for model including dynamic capillarity effect. In particular, such solutions may have non monotonic profiles, which are ruled out when considering standard, equilibrium type models, but have been observed experimentally. The paper ends with numerical experiments confirming the theoretical results. In this sense the original system of partial differential equations is solved by means of an implicit scheme. This relies on an equivalent, mixed formulation of the system. For solving the resulting nonlinear, time-discrete problems, a linear iterative scheme is proposed.
Summary
We extend a model for gravity segregation in steady-state gas/water injection into homogeneous reservoirs for enhanced oil recovery (EOR). A new equation relates the distance gas and water flow together directly to injection pressure, independent of fluid mobilities or injection rate. We consider three additional cases: coinjection of gas and water over only a portion of the formation interval, injection of water above gas over the entire formation interval, and injection of water and gas in separate zones well separated from each other.
If gas and water are injected at fixed total volumetric rates, the horizontal distance to the point of complete segregation is the same, whether gas and water are coinjected over all or any portion of the formation interval. At fixed injection pressure, the deepest penetration of mixed gas and water flow is expected when fluids are injected along the entire formation interval.
At fixed total injection rate, injection of water above gas gives deeper penetration before complete segregation than does coinjection, but again exactly where the two fluids are injected does not affect the distance to the point of segregation. At fixed injection pressure, injection of water above gas is predicted to give deeper penetration before complete segregation. When injection pressure is limited, the best strategy for simultaneous injection of both phases from a vertical well would be to inject gas at the bottom of the reservoir and water over the rest of the reservoir height, with the ratio of the injection intervals adjusted to maximize overall injectivity.
The 2D model applies equally to gas/water flow and to foam, and to injection of water above gas from separate intervals of a vertical well or from two parallel horizontal wells, as long as injection is uniform along each horizontal well. Sample computer simulations for foam injection agree well with the model predictions if numerical dispersion is controlled.
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