C.J. van Duijn

Effective equations for two-phase flow with trapping on the micro scale

In this paper we consider water-drive for recovering oil from a strongly heterogeneous porous column. The two-phase model uses Corey relative permeabilities and Brooks--Corey capillary pressure. The heterogeneities are perpendicular to the flow and have a periodic structure. This results in one-dimensional flow and a space periodic absolute permeability, reflecting alternating coarse and fine layers. Assuming many---or thin---layers, we use homogenization techniques to derive the effective transport equations. The form of these equations depends critically on the capillary number. The analysis is confirmed by numerical experiments.

Injection strategies to overcome gravity segregation in simultaneous gas and water injection into homogeneous reservoirs

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.

Steam injection into water-saturated porous rock

We formulate conservation laws governing steam injection in a linear porous medium containing water. Heat losses to the outside are neglected. We find a complete and systematic description of all solutions of the Riemann problem for the injection of a mixture of steam and water into a water-saturated porous medium. For ambient pressure, there are three kinds of solutions, depending on injection and reservoir conditions. We show that the solution is unique for each initial data.

Gel Placement in Porous Media: Constant Injection Rate

In this paper we analyse advective transport of polymers, crosslinkers and gel, taking into account non-equilibrium gelation, gel adsorption and crosslinker precipitation. In absence of diffusion/dispersion the resulting model consists of hyperbolic transport-reaction equations. These equations are studied in several steps using mainly analytical techniques. For simple cases, we obtain explicit travelling wave solutions, whereas for more complicated cases we rely on analytical techniques to analyse the problem qualitatively. Finally, a numerical solution for the full system of equations is obtained. The results developed in this study can be used to validate numerical solutions obtained from commercial simulators.

Effective Buckley-Leverett Equations by Homogenization

In this paper we consider water-drive to recover oil from a strongly heterogeneous porous column The two-phase model uses Corey relative permeabilities and Brooks-Corey capillary pressure. The heterogeneities are perpendicular to flow and have a periodic structure. This results in one-dimensional flow and a space periodic absolute permeability, reflecting alternating coarse and fine layers. Assuming many — or thin — layers, we use homogenization techniques to derive the effective transport equations. The form of these equations depend critically on the capillary number. The analysis is confirmed by numerical experiments. This paper summarises the results obtained in [10]

A Numerical Scheme for the Micro Scale Dissolution and Precipitation in Porous Media

In this paper we discuss numerical method for a pore scale model for precipitation and dissolution in porous media. We focus here on the chemistry, which is modeled by a parabolic problem that is coupled through the boundary conditions to an ordinary differential inclusion. A semi-implicit time stepping is combined with a regularization approach to construct a stable and convergent numerical scheme. For dealing with the emerging time discrete nonlinear problems we propose here a simple fixed point iterative procedure.

A monolithic phase-field model of a fluid-driven fracture in a nonlinear poroelastic medium

In this paper, we present a full phase-field model for a fluid-driven fracture in a nonlinear poroelastic medium. The nonlinearity arises in the Biot equations when the permeability depends on porosity. This extends previous work (see Mikelić et al. Phase-field modeling of a fluid-driven fracture in a poroelastic medium. Comput Geosci 2015; 19: 1171–1195), where a fully coupled system is considered for the pressure, displacement, and phase field. For the extended system, we follow a similar approach: we introduce, for a given pressure, an energy functional, from which we derive the equations for the displacement and phase field. We establish the existence of a solution of the incremental problem through convergence of a finite-dimensional Galerkin approximation. Furthermore, we construct the corresponding Lyapunov functional, which is related to the free energy. Computational results are provided that demonstrate the effectiveness of this approach in treating fluid-driven fracture propagation. Specifically, our numerical findings confirm differences with test cases using the linear Biot equations.