Nonequilibrium Capillary Pressure Research Articles

Numerical analysis for two-phase flow with non-equilibrium capillary pressure in anisotropic porous media

Numerical analysis for two-phase flow with non-equilibrium capillary pressure in anisotropic porous media

Nonequilibrium Capillary Pressure of a Miscible Meniscus.

We examine the dynamics of a miscible displacement in a capillary, calculating the nonequilibrium capillary pressure of a moving (and slowly diffusing) miscible meniscus. During the displacement, the capillary pressure varies with time following stretching and smearing of a miscible interface. The capillary pressure remains different from zero for a long time (on a diffusion time scale), slowing the displacement. This capillary pressure is however completely ignored by all theories currently available for practical modeling of miscible displacements in capillaries and porous matrices.

Homogenized Model of Two-Phase Flow with Local Nonequilibrium in Double Porosity Media

We consider two-phase flow in a heterogeneous porous medium with highly permeable fractures and low permeable periodic blocks. The flow in the blocks is assumed to be in local capillary disequilibrium and described by Barenblatt’s relaxation relationships for the relative permeability and capillary pressure. It is shown that the homogenization of such equations leads to a new macroscopic model that includes two kinds of long-memory effects: the mass transfer between the blocks and fractures and the memory caused by the microscopic Barenblatt disequilibrium. We have obtained a general relationship for the double nonequilibrium capillary pressure which represents great interest for applications. Due to the nonlinear coupling and the nonlocality in time, the macroscopic model remains incompletely homogenized in general case. The completely homogenized model was obtained for two different regimes. The first case corresponds to a linearized flow in the blocks. In the second case, we assume a low contrast in the block-fracture permeability. Numerical results for the two-dimensional problem are presented for two test cases to demonstrate the effectiveness of the methodology.

Velocity‐dependent capillary pressure in theory for variably‐saturated liquid infiltration into porous media

Standard theory for liquid infiltration into porous media cannot explain saturation overshoot at an infiltration front. Based on a recent generalization of the Green‐Ampt approach, a new theory for variably‐saturated flow is presented that assumes that capillary pressure does not only depend on liquid content but also on the flow velocity. The Eulerian expression for the nonequilibrium capillary pressure is rotationally invariant and attempts to capture the conjecture that dynamic effects are more pronounced if flow is associated with moving fluid‐fluid interfaces which cause a dynamic contact angle. The new theory correctly predicts how overshoot depends on the downstream and upstream liquid contents as well as on grain size. The theory also yields the hydrostatic pressure distribution in the liquid and allows for liquid contents exceeding the saturated liquid content of the main imbibition curve that may occur for overshoot.

Generalized nonequilibrium capillary relations for two-phase flow through heterogeneous media

For two-phase flow in porous media, the natural medium heterogeneity necessarily gives rise to capillary nonequilibrium effects. The relaxation to the equilibrium is a slow process which should be introduced in macroscopic flow models. Many nonequilibrium models are based on a phenomenological approach. At the same time there exists a rigorous mathematical way to develop the nonequilibrium equations. Its formalism, developed by Bourgeat and Panfilov [Computational Geosciences 2, 191 (1998)], is based on the homogenization of the microscale flow equations over medium heterogeneities. In contrast with the mentioned paper, in which the case of a sufficiently fast relaxation was analyzed, we consider the case of long relaxation, which leads to the appearance of long-term memory on the macroscale. Due to coupling between the nonlinearity and nonlocality in time, the macroscopic model remains, however, incompletely homogenized, in the general case. At the same time, frequently only the relationship for the nonequilibrium capillary pressure is of interest for applications. In the present paper, we obtain such an exact relationship in two different independent forms for the case of long-term memory. This relationship is more general than that obtained by Bourgeat and Panfilov. In addition, we prove the comparison theorem which determines the upper and lower bounds for the macroscopic model. These bounds represent linear flow models, which are completely homogenized. The results obtained are illustrated by numerical simulations.

Upscaling of fractured oil reservoirs using homogenization including non-equilibrium capillary pressure and relative permeability

Recovery from incompletely water-wet fractured reservoirs can be extremely low. A reason for the low recovery is related to wetting issues, whereas the reason for slow recovery can be the non-equilibrium behavior of capillary pressure. One of the non-equilibrium theories is developed by Barenblatt et al. and it modifies both capillary pressure and relative permeabilities. The other theory is developed by Hassanizadeh et al. and it only deals with non-equilibrium effects for capillary pressure. To incorporate non-equilibrium in larger-scale problems, we apply homogenization to derive an upscaled model for fractured reservoirs in which the non-equilibrium effects are included. We formulate a fully implicit three-dimensional upscaled numerical model. Furthermore, we develop a computationally efficient numerical approach to solve the upscaled model. We use simulations to determine the range of delay times and capillary-damping coefficients for which discernable effects occur in terms of oil recovery. It is shown that at low Peclet numbers, i.e., when the residence time of the fluids in the fracture is long with respect to the imbibition time, incorporation of delay times of the order of few months have no significant effect on the oil recovery. However, when the Peclet number is large, the delay times reduce the rate of oil recovery. We discuss for which values of the delay time (Barenblatt) and capillary-damping coefficient (Hassanizadeh), significant delays in oil production occur.

Nonequilibrium thermomechanical modeling of liquid drainage/imbibition in trickle beds

AbstractWe extend the macroscopic nonequilibrium thermomechanical multiphase flow theory proposed by Hassanizadeh and Gray for porous media to analyze a set of drainage and imbibition experiments in trickle beds. The nonequilibrium model rests on inclusion of mass and momentum conservations for the gas‐liquid interface, nonequilibrium capillary pressure, Helmholtz free energy gradients in the body supply of momentum for fluid bulk phases and gas‐liquid interface, and mass exchange rates between interface and fluid bulks accounting for production and destruction of gas‐liquid interfacial area. To solve the nonequilibrium model, entropy‐consistent constitutive relationships are derived and calibrated using liquid holdup and bed pressure drop measurements in drainage and imbibition. The model captures very well the decay (drainage), and breakthrough (imbibition) curvatures of liquid holdup and pressure drop kinetics, while model closer inspection allows assessing the role of nonequilibrium capillary pressure and of dynamic interfacial mass exchanges for the production/destruction of interfacial area. © 2011 American Institute of Chemical Engineers AIChE J, 58: 3123–3134, 2012

Incorporation of Dynamic Capillary Pressure into the Green–Ampt Model for Infiltration

The Green–Ampt model describes infiltration of water into soil. A sharp front separates the saturated from the unsaturated zones, and capillary pressure is assumed to remain constant during infiltration. We generalized this model to account for a capillary pressure that depends on the flow velocity. Based on dimensional analysis and physical considerations, we posited a functional form for dynamic capillary pressure and assumed the nonequilibrium capillary pressure to depend on the capillary number in the form of a power law. Our model for dynamic capillary pressure describes measurements of capillary pressure versus Darcy velocity by D.A. Weitz et al. and S.L. Geiger and D.S. Durnford. Moreover, the dimensional analysis allows us to collapse three dynamic capillary pressure curves that Geiger and Durnford measured for sands of different grain size onto one curve. Furthermore our model describes capillary rise experiments performed by T. Tabuchi well. We also derived an implicit analytical solution for the front velocity.

Experimental Studies of Liquid Droplet Coalescence on the Gradient Surface

Visual experiments were conducted to study the behaviors of liquid droplet coalescence on the surface with gradient surface energy. The microscopic contour of the gradient energy surface, which was fabricated on the base of a silicon chip by diffusion-controlled silanization of alkyltrichlorosilanes, was characterized by atom force microscopy. The effects of droplets on the three-phase contact line and the contact angle were obtained. It was observed that the process of droplet coalescence accelerates the droplet speed on the gradient surface. On the high-energy surface, the triple-phase contact line of the coalescing droplet is prone to pinning. The interfacial energy released from coalescence is the main source for promoting the motion. Furthermore, the nonequilibrium capillary pressure gradient inside coalescence droplets is considered to aid in accelerating the coalescence.

Modeling observed saturation overshoot with continuum additions to standard unsaturated theory

In uniform soils that are susceptible to unstable preferential flow, the water saturation exhibits a nonmonotonic profile upon continuous infiltration. This overshoot cannot be described by the conventional Richards equation. Here, solutions to the infiltrations using a popular nonequilibrium extension to the Richards equation are obtained using the traveling wave nature of the saturation profile. Quantitative comparisons are made to recent measurements of saturation overshoot. The nonequilibrium solutions can be made to fit the flux range of the overshoot, but the fit to the tip saturations is fair to poor at best. Also, small changes in porous media size and roughness require large changes in the magnitude of the nonequilibrium term to match the flux range. The results suggest that the nonequilibrium capillary pressure does not include the correct physics that controls the overshoot.

Nonequilibrium capillary pressure and relative permeability curves of porous media

AbstractHistory matching of unsteady immiscible displacement experiments performed on model porous media reveals that nonequilibrium capillary pressure and relative permeability functions are sensitive to the dominating flow pattern. As the capillary number increases, the network displacement pattern changes gradually from invasion percolation cluster to stable displacement. For primary drainage, the capillary pressure and relative permeability curves are increasing functions of the capillary number. The dependence of the relevant macroscopic parameters on the capillary number is consistent with power laws derived from scaling arguments of gradient percolation theory, which describes the growth of the displacement front. The material coefficient of the thermodynamic theory of capillary pressure changes mildly with fluid saturation and is a decreasing function of the capillary number.

Quasi–incompressible Cahn–Hilliard fluids and topological transitions

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.