Linear Darcy Law Research Articles

Experimental Analysis of Core Crushing and Core Movementin RTM and SRIM Foam Cored Composite Parts

In liquid composite molding, SRIM can typically achieve a higher pressure than the RTM process. When molding a foam cored composite part at high flow rates using an SRIM system, the following related effects can be observed: high molding pressures, core shifting, core crushing, variable permeability and reinforcement compaction. Because of the variable permeability in such conditions, the linear Darcy law does not apply for flow simulation. Finally, all this can easily result in dry spot defects. To study the various problems occurring in the molding of foam cored composite parts, two molds equipped with pressure transducers were used. The first one was a picture frame square plaque mold. To ease the measurement of reinforcement compaction and core crushing, flat panels were molded with reinforcement on one side only. The second part was a quasi-rectangular foam cored beam with braided reinforcement all around the foam core. In this paper, it is refered as the Small Core part. With this part, reinforcement compaction, core crushing and core shifting was measured. The picture frame plaque mold was used with the RTM and SRIM processing equipment. Continuous strand mat with a constant fiber volume fraction was used with polyurethane foam of variable density. For the RTM process, Vinyl Ester resin was injected at constant pressure while for SRIM, Polyurethane resin was injected at constant flow rate. Parts were cut to measure laminate thickness and core deformation. For the Small Core mold, only the SRIM system was used with a constant flow rate. In this case the core materials used were Polyurethane of various densities. All parts were braided with ±45 and axial (0°) glass filaments. For some parts, locators were placed on the core surface to prevent its shifting. Short shots were used to observe the resin flow progression. For the picture frame experiments, the RTM process gave no significant core deformation. With the SRIM system, results showed a non-negligible core deformation at some foam densities. The SRIM system was also used with the Small Core tool. The results revealed the dominance of the core shifting and fabric compaction over core crushing. The use of locators placed on the core surface led to a better flow progression and nearly eliminated the core shifting, resulting in a part without defects. The flow simulation software RTMFLOT developed in our laboratory and based on a linear Darcy Law was used with these complex cases. The simulation results gave a good approximation of the gate pressure during filling.

The dam problem for linear Darcy's law and nonlinear leaky boundary conditions

Starting from the model proposed in [15], we study a stationary flow of fluid, governed by a linear Darcy's law, through an inhomogeneous porous medium with leaky boundary conditions. We establish properties of solutions and regularity of the free boundary. Finally, we prove a comparison result from which we deduce some uniqueness results.

Mathematical analysis of the influence of power-law fluid rheology on a capillary diffusion zone

Non-Newtonian fluids are used in current oil recovery processes. These fluids do not satisfy the linear Darcy Law for flow through porous media. A generalization is needed to model the flow processes involved. Furthermore, when two immiscible fluids are present in a porous medium, capillary pressure will cause a transition zone to develop between them. This transition zone may lead to early breakthrough of water into an oil well. In this paper, we study the effect of the non-Newtonian behaviour of fluids on a capillary transition zone. A general framework is set up for modelling processes involving two-phase flow of non-Newtonian, immiscible and incompressible fluids in a porous medium. The equations are applied to the one-dimensional diffusion process of power-law fluids. The model allows for general capillary pressure and relative permeability functions. The mathematical model consists of a degenerate diffusion equation, giving rise to a free boundary formulation. The free boundaries represent the endpoint of the diffusion zone. Qualitative properties, and some analytical solutions, can be obtained for the saturation profile. Two numerical methods are presented. One is applicable if the rheology of both fluids is modelled with equal powers. The other is applicable to the general situation. Both methods make use of the qualitative, analytical properties of the solutions, which clearly improved the results obtained by standard methods. One-dimensional results can be used to interpret the general multi-dimensional flow behaviour of non-Newtonian fluids when capillarity is considered. They can also be used to test numerical algorithms developed for multi-dimensional displacement processes.

INSTABILITY OF A SHARP INTERFACE IN POLYMER FLOODING

Non‐Newtonian fluids are used in current oil recovery processes. These fluids do not satisfy the linear Darcy law for flow through porous media. To model the recovery processes, a generalization of Darcy's law is used. A numerical method, developed originally for salt and fresh groundwater flow, has been adapted to incorporate the generalized Darcy law. We use it to model the two‐phase, two‐dimensional flow of immiscible fluids in a porous medium. In particular it will be applied to investigate the stability of the fluid/fluid interface. The results verify the theoretically predicted critical velocity above which the displacement of oil by polymer flooding becomes unstable, leading to low recovery.

Dimensionless Straight‐Type Lines for Aquifer Tests

Nonlinear flow-type straight-line expressions are derived on the basis of exponential flow law instead of the linear Darcy law. A procedure of hypothesis testing for the flow-regime linearity is explained step-by-step. It is concluded that any straight line for late time-drawdown data on semilogarithmic paper does not necessarily guarantee the accurate application of the classical Jacob methods but confirms that the flow is radial. A dimensionless time-drawdown plot is proposed for the identification of flow regime. If the dimensionless time-drawdown plot for the late time-drawdown data fits a straight line with the same slope as the jacob straight line, then the underlying flow regime is linear. Otherwise the flow regime is nonlinear. Once the flow regime is nonlinear, the necessary equations are given for the estimation of parameters such as the storability, turbulence exponent, and transmissivity. Finally, the application of the methodology has been performed for some field data available in the literature.

Macroscopic laws for immiscible two‐phase flow in porous media: Results From numerical experiments

Flow through porous media may be described at either of two length scales. At the scale of a single pore, fluids flow according to the Navier‐Stokes equations and the appropriate boundary conditions. At a larger, volume‐averaged scale, the flow is usually thought to obey a linear Darcy law relating flow rates to pressure gradients and body forces via phenomenological permeability coefficients. Aside from the value of the permeability coefficient, the slow flow of a single fluid in a porous medium is well‐understood within this framework. The situation is considerably different, however, for the simultaneous flow of two or more fluids: not only are the phenomenological coefficients poorly understood, but the form of the macroscopic laws themselves is subject to question. I describe a numerical study of immiscible two‐phase flow in an idealized two‐dimensional porous medium constructed at the pore scale. Results show that the macroscopic flow is a nonlinear function of the applied forces for sufficiently low levels of forcing, but linear thereafter. The crossover, which is not predicted by conventional models, occurs when viscous forces begin to dominate capillary forces; i.e., at a sufficiently high capillary number. In the linear regime, the flow may be described by the linear phenomenological law ui = ΣjLijfj, where the flow rate ui of the ith fluid is related to the force fj applied to the jth fluid by the matrix of phenomenological coefficients Lij which depends on the relative concentrations of the two fluids. The diagonal terms are proportional to quantities commonly referred to as “relative permeabilities.” The cross terms represent viscous coupling between the two fluids; they are conventionally assumed to be negligible and require special experimental procedures to observe in a laboratory. In contrast, in this numerical study the cross terms are straightforward to measure and are found to be of significant size. The cross terms are additionally observed to be approximately equal, which is the behavior predicted by Onsager's reciprocity theorem. However, persistent transient effects can render the reciprocity unobservable. The numerical study is performed with a discrete numerical model of the molecular dynamics of immiscible mixtures called the immiscible lattice gas. The immiscible lattice gas models both the Navier‐Stokes equations and surface tension. Numerical tests presented here additionally provide quantitative validation of the method's ability to simulate wetting phenomena and the effects of capillary pressure. Whereas the numerical study of the linear phenomenological laws utilizes a highly simplified porous medium with one pore and two throats, numerical examples of wetting and nonwetting invasion experiments in a geometrically complex 2‐D porous medium are also provided.

Influence of Bed Size on the Flow Characteristics and Porosity of Randomly Packed Beds of Spheres

Experiments were performed to investigate the flow characteristics and porosity of randomly packed beds of glass spheres for conditions where the bed size cannot be regarded as infinitely large compared with the sphere size. The operating conditions of the flow experiments extended over a Reynolds number range for which the flow was governed by the Forchheimer extension of the linear Darcy law. The influence of the bed bounding walls on the permeability, on the coefficient of the Forchheimer inertia term, and on the porosity, was studied by using beds of rectangular cross section. It was found that the permeability was not apparently influenced by the bounding walls when the equivalent diameter of the bed was greater than 12 times the sphere diameter, whereas the coefficient of the inertia term was affected by the presence of the walls for bed equivalent diameters as high as 40 sphere diameters. The porosity of the beds was not influenced by the bed size for values of the bed equivalent diameter greater than 15 sphere diameters. When the large-bed porosity value was used with the Carman-Kozeny relationship for the permeability as a function of sphere diameter, an excellent representation of accumulated experimental data was attained.