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One of the major challenges to maximizing recovery of reserves is that every oil or gas reservoir rock is more or less heterogeneous at all scales (micro, mega, and pore) which leads to disproportionate production and injection outcomes. Generally, the higher the level of reservoir heterogeneity the more difficult it becomes to achieve maximum fluid distribution or conformance. Improving conformance in a non-homogenous material such as a hydrocarbon reservoir inherently means improving flow through lower permeability regions. Ideally, during a conventional well stimulation using a treatment fluid such as acid, we wish to move the fluid through the majority of the rock volume but the physical constraints of fluid flow negatively impact that ideal outcome. Dynamic fluid pulse technology provides for high inertial fluid momentum which improves the flow efficiency of fluids injected into the wellbore, the near wellbore region, and the reservoir. The nature of fluid displacement energy ensures that pulsed fluid will penetrate the matrix proximal to where the tool is placed thus achieving enhanced fluid distribution. Prior to a stimulation operation a dynamic mathematical model associated with fluid pulse technology is employed to generate a precise well program (pumping schedule) to maximize the contact volume of the treatment fluid along the completed interval. Compared with conventional stimulation dynamic fluid pulsation has been demonstrated to bring significant financial benefits to well stimulation without impacting results including: reduced chemical costs; improved post-stimulation sustainability; and, better overall post-stimulation well performance as a greater volume of the completed interval hence matrix is contacted by the treatment fluids.

It has been known since the time of Biot–Gassman theory (Biot, J Acoust Soc Am 28:168–178, 1956, Gassmann, Naturf Ges Zurich 96:1–24, 1951) that additional seismic waves are predicted by a multicomponent theory. It is shown in this article that if the second or third phase is also an elastic medium then multiple p and s waves are predicted. Futhermore, since viscous dissipation no longer appears as an attenuation mechanism and the media are perfectly elastic, these waves propagate without attenuation. As well, these additional elastic waves contain information about the coupling of the elastic solids at the pore scale. Attempts to model such a medium as a single elastic solid causes this additional information to be misinterpreted. In the limit as the shear modulus of one of the solids tends to zero, it is shown that the equations of motion become identical to the equations of motion for a fluid filled porous medium when the viscosity of the fluid becomes zero. In this limit, an additional dilatational wave is predicted, which moves the fluid though the porous matrix much similar to a heart pumping blood through a body. This allows for a connection with studies which have been done on fluid-filled porous media (Spanos, 2002).

Summary A set of macroscopic equations of motion for inhomogeneous and anisotropic porous media is constructed. The porous medium considered consists of an elastic solid with interconnected void spaces filled with a chemically inert viscous fluid. The constituents are assumed homogeneous in their material properties, but the unperturbed porosity is spatially varying and the distributions of pores and interfaces are uneven. The physics at the pore scale, which underpins the approach, is never lost sight of. Although very different in approach from that taken by Biot, a close correspondence to the Biot (1962) theory is established in this paper. The dynamic perturbation in porosity accompanying deformation is treated as kinematically independent of the macroscopic solid displacement field and the macroscopic fluid velocity field. The viscous loss within the pore fluid, which is absent in the Biot approach, is not excluded here. For the most general case, 27 independent macroscopic parameters enter into the macroscopic constitutive equations, not counting the spatial gradient of unperturbed porosity itself, which appears explicitly at various places. Whereas the elastic constants of the constituents are contained within Biot's parameters, here they are factored out. Thus, the parameters are directly linked to the manner in which distributed pores and interfaces control the deformation.

In current lattice gas models particles are constrained to travel along the principal lattice directions between sites, and must advance from one site to another at each time-step. The collision rules are therefore restricted to redistributing momentum along these principal lattice directions. We introduce a computational technique that does not suffer from these limitations. We associate with each particle a continuous momentum which is used to control the motion of the particle on the lattice. This motion is probabilistic, a quasi-random walk, so after many time-steps the particle's average motion is a straight line. The collision rules are no longer constrained by the lattice and can be more realistically implemented; here we use Lorentz invariant binary elastic collisions. We show that the equilibrium distribution of our model is the relativistic Boltzmann distribution which permits us to find the temperature of the lattice gas.

Plane Poiseuille flow is simulated using a thermodynamic automaton model. This well-understood problem is used to demonstrate theconsistency of the physical predictions obtained from this new lattice gas model with established physical theory. The model presented is a nonrelativistic version of a model constructed by other researchers in which the collision and propagation rules could be used to derive the relativistic Boltzmann equation. The present model has great potential for describing hydrodynamic processes since it does not suffer from many of the limitations of models that rely on differential equations. For example, it can be used to model highly nonlinear processes or flow past complicated boundaries as easily as it can be used for the Poiseuille flow problems presented here. In the present paper, the physical foundations for using this model to study hydrodynamic processes are demonstratedby showing the Boltzmann nature of the gas, the construction of thermodynamic boundary conditions and consistency with physical theory when the temperature dependence of viscosity is modeled. The principal objective of this paper is to establish the physical validity of this model by comparison with well-established theoretical results from equilibrium thermodynamics. PACS No.: 05.10

A thermodynamic automaton model of fluid flow in porous media is presented. The model is a nonrelativistic version of a Lorentz invariant lattice gas model constructed by Udey et al. (1998). In the previous model it was shown that the energy momentum tensor and the relativistic Boltzman equation can be rigorously derived from the collision and propagation rules. In the present paper we demonstrate that this nonrelativistic model can be used to accurately simulate well known results involving single phase flow and diffusion in porous media. The simulation results show that (1) one-phase flow simulations in porous media are consistent with Darcy's law; (2) the apparent diffusion coefficient decreases with a decrease in permeability; (3) small scale heterogeneity does not affect diffusion significantly in the cases considered.

Summary Using the static form of a system of equations for seismic waves (de la Cruz & Spanos 1989), we show how various compressibilities can be calculated in a straightforward manner. The results obtained have many points of contact with those found in the literature. In particular, we verify all identities among drained compressibilities given in, e.g., Zimmerman (1991), thus providing an alternative route towards them. The undrained compressibility is described within the context of this work and its relation to the various drained compressibilities (Gassmann 1951) is verified. For greater experimental flexibility, we introduce a one-parameter family of compressibilities which includes the drained and the undrained compressibilities as members. The family of compressibilities is also used to obtain an expression for the pore-pressure build-up coefficient. In this work we also address the problem of macroscopic shearing. Experiments are proposed for the determination of the macroscopic shear modulus, leading to natural expressions for ‘Young's modulus’ and ‘Poisson's ratio’ for the porous medium under drained conditions. We also establish connections with Biot's (1956a) parameters.

The macroscopic pressure difference between two immiscible, incompressible fluid phases flowing through homogeneous porous media is considered. Starting with the quasi-static motions of two compressible fluids, with zero surface tension, it is possible to construct a complete system of equations in which all parameters are clearly defined by physical experiments. The effect of surface tension is then formally included in the definition of the specific process under consideration. Incorporating these effects into the pressure equations and taking the limit as compressibilities go to zero, the independent pressure equations are shown to yield indeterminate forms. However, the difference of the two pressure equations is found to yield a new process-dependent dynamical equation.

The instability of a plane front between two phases of the same fluid (steam and water) in a porous medium is considered. The configuration is taken to be initially stationary with the more dense phase overlying the less dense phase. The frontal region is assumed sharp, so that macroscopic boundary conditions can be utilized. This assumption precludes the existence of dispersion instabilities. The stabilizing influence of phrase transition as well as the implication of different macroscopic pressure boundary conditions on the stability of the front are discussed and illustrated.

Equilibrium thermodynamics for porous media is considered with special emphasis on its basis in pore-scale thermodynamics. It is shown that porosity, the new purely macroscopic variable, enters the relations on the same footing as mass densities and the strain tensors. Biot’s use of elastic energy potential, which lies at the foundation of his theory of poroelasticity, is examined in light of the results obtained here.

A set of equations with ‘generalized permeability’ functions has been proposed by de la Cruz and Spanos, Whitaker, and Kalaydjian to describe three-dimensional immiscible two-phase flow. We have employed the zero interfacial tension limit of these equations to model two phase miscible flow with negligible molecular diffusion. A solution to these equations is found; we find the generalized permeabilities to depend upon two empirically determined functions of saturation which we denote asA andB. This solution is also used to analyze how dispersion arises in miscible flow; in particular we show that the dispersion evolves at a constant rate. In turn this permits us to predict and understand the asymmetry and long tailing in breakthrough curves, and the scale and fluid velocity dependence of the longitudinal dispersion coefficient. Finally, we illustrate how an experimental breakthrough curve can be used to infer the saturation dependence of the underlying functionsA andB.

Porosity diffusion is shown to propagate from a seismic source in fluid-saturated porous media. It is observed that this process would cause a travelling pulse of pressure associated with the spreading front of porosity and would result in variations in fluid levels in the earth's crust following a seismic event. The interaction between the associated fluid fluxes and seismicity is considered. It is shown that this process can cause secondary fractures, aftershocks and can lead to the interaction of earthquakes.

The mode conversions which occur during the reflection and transmission of seismic waves at the boundaries of porous media are analysed. It is shown how the energy partitioned to the various modes depends on the incident angle and on the physical properties of the fluid and solid components on each side of the boundary. The boundary conditions used here predict the occurrence of bright and dark spots as are currently observed in seismic studies of heavy oil reservoirs. They also give rise to a class of pseudo interface waves which propagate in a direction almost parallel to the surface and which become true interface waves in the limiting case where the porous media degenerate to elastic solids. When thermomechanical coupling is an important attenaution mechanism in one of the media it is also observed to have a substantial effect on the mode conversions which occur at the boundary.

In general, existing porous media descriptions are thermodynamically incomplete or improperly constituted. This incompleteness arises because of two widely accepted but incorrect assumptions : • Porosity is a scalar invariant (rather than a thermodynamic state variable) • The energy state for a representative elementary volume can be described by a unique energy function These two assumptions lead to flawed thermodynamical statements for porous media containing multiphase fluids, with a number of consequences that hinder theoretical developments in thermomechanics, and impair development of practical applications. The article seeks to fully express the equilibrium thermodynamics relationships for porous media, based on a first-principles derivation.

Abstract Laboratory tests initiated in January 1997 demonstrated clearly that periodic, large-amplitude, low-frequency strain excitation of porous media leads to large flow enhancements. Based on these results, a new liquid flow enhancement technology for reservoirs was formulated, and a successful full-scale field experiment was executed in early 1999. Other field projects in 1999 through 2001 waterfloods in heavy oil cold production wells with sand influx confirmed the expectation that pressure pulsing, properly executed, increases oil production rate at low cost. The first trial showed that periodic application of large amplitude, liquid-phase pressure pulses increased oil production rates, decreased water-oil ratio, and increased the percentage of sand produced, even without large-scale injection. Though experience to date is in heavy oil, the process is general and will work in all porous media that have interconnected pore space. Furthermore, the method works in single-phase and two-phase liquid saturated cases, although the presence of large amounts of free gas is detrimental. Based on the field and laboratory work, and considering the nature of the physical processes, it appears likely that pressure pulsing will also help reduce coning and viscous fingering instabilities, help overcome capillary blockages, and result in more total oil recovery over time. Introduction In the oil industry, progress in production technologies is most commonly based on empirical discoveries, and only later followed by attempts to develop a consistent physical theory to explain, analyse, and predict field behaviour. This is the case for all processes, such as CHOPS (Cold Heavy Oil Production with Sand), SAGD (Steam Assisted Gravity Drainage), and so on. Often, a fully rigorous physical theory remains elusive (e.g., for oil-gas-water-sand slurry flow in CHOPS production), and practice is refined through empirical models, physical reasoning and trial and error. However, the results obtained from laboratory and field research and development in pressure pulsing over the past five years were initially predicted from a new, more rigorous physical theory for porous media flow. The new theory(1) is a more complete system of equations for dynamic behaviour of interconnected multiphase porous media (matrix-liquid, including matrix-water-oil systems). It was developed by considering all relevant pore-scale physical processes (micro scale), followed by rigorous volume averaging to scale the physics up to the scale of a representative elementary volume (REV) that can statistically represent mesoscopic behaviour (cm scale). The theory is therefore consistent with the laws of thermodynamics in component phases at the pore scale, and leads to large-scale thermodynamic relationships (equations) in which porosity is found to play a fundamental thermodynamic role, similar to that of temperature in single-component systems. In other words, porosity must be treated as a basic thermodynamic variable in porous media(2). Furthermore, induced dynamic variations in porosity are responsible for the observed flow rate enhancement effect. The physical theory also dictates the experiments required to determine the parameters that arise in the equations(3). A slow dynamic wave called the porosity dilation (or porosity diffusion) wave is predicted(4).

Abstract When technology based on new science is started, there can be a lot of skepticism, which is a healthy reaction. The skepticism encountered by PE-TECH as pressure pulsing is gradually introduced has been partly overcome, but only in the Canadian heavy oil industry. Here are some typical remarks we have encountered over the last three years, accompanied by our responses. "This is not predicted by Darcy's Law." This is quite correct since Darcy's Law is a static law and cannot handle inertial effects. For example, look at how Darcy's Law is fudged to handle turbulent flow. "Well, it works in heavy oil, but it won't in light oil." Or, "Our reservoirs are different." Physics is physics; it will work in all liquid-saturated systems, but will have to be optimized in individual cases. "We don't need this because we use horizontal wells." It can be used in many different configurations, and can help horizontal wells just as it does vertical and inclined wells. "It's not the pulsing, it's a relative permeability effect (or permeability increase or viscosity decrease, etc.)." Nope, we've proven otherwise, although in the case of highly viscous oils of large molecular weight, there may be an additive effect of viscosity reduction. "It won't work in consolidated rocks." We're confident of field success; limited laboratory tests indicate that it does indeed work, but we have a lot more testing to do. "Sounds like Cold Fusion to me." Frankly, we were pretty startled ourselves at how large the effect is, but you can do the experiments yourself. "Come back when you have some real field data." We have. We're here. History Pressure pulsing is an emerging technology. Its roots go back several decades; as a rigorous theory, it goes back about 15 years. Russian engineers noticed decades ago that large earthquakes often caused changes in oil well behaviour, usually a short-term rate increase. This was variously ascribed to compaction, shaking loose of particles blocking pore throats, or changes in permeability, viscosity, and capillary entry pressure. However, attempts to use seismic excitation have, to our knowledge, met with failure in China, Canada, and the United States. Senior engineers from western oil companies have examined claims that mechanical vibrations are being used successfully, they appear unconvinced, even after site visits. Also, the numerous articles in the Russian literature, mainly by geophysicists, tend to be mathematicallyopaque, unfathomable in terms of physical processes, based on debatable premises, or without sufficient clear and unequivocal information to allow evaluation. Nevertheless, published Russian data appear convincing, and R&D programs are active in Alberta, USA, and other areas, albeit without much apparent success. We believe we know why: seismic excitation is the wrong type of impulse, of insufficient amplitude, and applied at the wrong place. Pressure pulsing experiments in the laboratory started only in January 1997; in the field, it was first tried in June 1998. PETECH conducted the first commercial applications in workover mode in September 1998, and in full field-wide rate enhancement mode in June 1999.

High-amplitude, low-frequency pressure pulsing was developed as a new oil well workover technique, and has been used in > 40 heavy oil wells in Alberta and Saskatchewan during the period October 1998 to October 1999, with a good success ratio. The method is particularly effective in initiation sand production and in cases where mechanical skin arising from blocked perforations, immobilized fines and asphaltenes are responsible for a high skin factor. Perforation blockage, restriction of near-wellbore flow paths through fines accumulation, pore throat precipitation of asphaltenes, and sand re-compaction around the well are all thought to be responsible for rapid production rate declines in many wells. In heavy oil wells, chemical treatments alone are rarely useful, but large physical perturbations of the material around the wellbore have been found to be effective in reestablishing production. A downhole device was developed to apply large periodic pressure pulses to the perforation face. This device also permits injection of fluid at measured rates into the nearwellbore region while executing prolonged (9–12 hour) high-amplitude perturbation. The action loosens mechanical skin, allowing the source of flow blockage to be physically removed through remolding and liquefaction of the sand around the wellbore. This liquefied zone is produced when the well is placed back on production, and in most cases the well is rejuvenated. In some cases, wells that never produced economic rates of oil because sand influx could not be initiated were turned into reasonable producers after a pressure pulse treatment. The method can also be used to deliberately introduce treatment chemicals while reducing the negative effects of channeling and fingering, as the front of treatment fluid ingress is well dispersed by the pressure pulsing.

Three major new technologies developed in the past decade in the heavy oil regions of Alberta and Saskatchewan are long, shallow horizontal wells, Steam-Assisted Gravity Drainage (SAGD), and Cold Production (CP). These three technologies are briefly reviewed. They can be combined in a general approach to heavy oil reservoir development that will give the advantages of reasonable early oil production rates, continued long-term production, and excellent long-term recovery. A combination of vertical wells and horizontal wells are used to achieve early CP, which is then gradually replaced by long-term SAGD production. As the processes progress, vertical CP wells will decline in production rate; when they are no longer economical to produce, these wells are converted to other purposes. They can be used as inert gas injectors, control wells, low-rate thermal injection or production wells, or even sand disposal wells, as long as they help maintain the reservoir at the appropriate pressure for SAGD. The proper balance of pressures through injection or production to maintain the long-term stability of processes in the gravity-dominated flow regime is critical to SAGD success. Given price fluctuations and long-term projections, economic factors related to heavy oil must be addressed. It is appropriate to point out that within 10-15 years, the current conditions of low prices and over-supply of conventional crude oil will be replaced by a world-wide shortfall in oil supply, which should bring a permanent increase in oil prices. This suggests that corporations should be making substantial investments in heavy oil at the present time.

Two phase countercurrent steady-state flow through permeable media in one dimension is discussed. For steady-state countercurrent flow in water wet porous media, a saturation profile is predicted with the water saturation decreasing in the direction that the water phase is flowing. The de la Cruz and Spanos equations predict that the Muskat relative permeability curves for countercurrent flow will be less than the Muskat relative permeability curves for steady-state cocurrent flow. This result has immediate implications regarding the use of external drive techniques to determine relative permeabilities based on the Buckley-Leverett theory and Muskat's equations. These equations and current experimental evidence involving countercurrent flow indicate that Muskat's equations do not adequately describe the multiphase flow of immiscible fluids.

A perturbation of an immiscible displacement process causes relative motion of the two fluids involved. At the macroscopic scale such relative motions are considered to propagate throughout the porous medium in the form of fluid waves. A description of these waves is given on surfaces of constant saturation in a similar fashion to the description of a surface wave propagating on the interface between two fluids. In the porous medium, however, the wave propagation is not restricted to the surface of constant saturation and as a result one obtains a wave equation that is both dissipative and diffusive.A stability analysis is also considered for the immiscible displacement process. Here, a characteristic time for instability to occur can be calculated when the inertial terms are included in the equations of motion. Also a generalization of the wave equations and stability criteria are considered for an inhomogeneous porous medium.

The instability of the plane interface between two uniform, superposed, and streaming fluids through porous media is considered. The configuration is taken to be bottom-heavy. In the absence of surface tension, perturbations transverse to the direction of streaming are found to be unaffected by the presence of streaming if perturbation in the direction of streaming are ignored, whereas for perturbations in all other directions there exists instability for a certain wavenumber range. The surface tension is able to suppress this Kelvin–Helmholtz instability for small wavelength perturbations and the medium porosity reduces the stability range given in terms of a difference in streaming velocities. For the top-heavy configurations, the surface tension stabilizes a certain wavenumber range.

Exact solutions can be found for steady fluid flow under constant shear, even if the stress–strain rate relation is nonlinear and shear heating connects the material properties to thermal behaviour. We present such solutions for a Newtonian material in which viscosity decreases exponentially with temperature, and for two empirical equations valid for high temperature creep. The onset of melting limits the range in which these solutions are applicable. If we assume that the region of the low velocity zone for shear waves is close to melting and that drag on this region by plates appears to a first approximation as a constant stress, we can predict surprisingly reasonable values for the plate velocity with respect to the mantle. The low viscosity of the zone becomes a consequence of melt induced by shear heating. Such melt would also explain a low Q and a reduction in shear velocity. A final solution is then given for an inhomogeneous material whose viscosity increases with depth. This can be interpreted as a material whose melting point increases with depth at a faster rate than the temperature of the material.

The concept of improving oil recovery through polymer flooding is analysed. It is shown that while the injection of a polymer solution improves reservoir conformance, this beneficial effect ceases as soon as one attempts to push the polymer solution with water. Once water injection begins, the water quickly passes through the polymer creating a path along which all future injected water flows. Thus, the volume of the polymer slug is important to the process and an efficient recovery would require that the vast majority of the reservoir be flooded by polymer. It is also shown that the concept of grading a polymer slug to match the mobilities of the fluids at the leading and trailing edges of a polymer slug does not work in a petroleum reservoir. While this process can supply some additional stability to the slug, it is shown that for the purposes of enhanced oil recovery this additional stability is not great enough to be of any practical use. It is found that in this case the instability has simply been hidden in the interior of the slug and causes the same sort of instability to occur as was the case for the uniform slug.

A statistical theory for the construction of the equations of viscous displacement in a porous medium is considered. This yields a continuum theory for immiscible displacement which can be applied to either a homogeneous or inhomogeneous porous medium. The relative motions of the fluid are considered in terms of the motion of surfaces of constant saturation which are smoothed surfaces at the macroscopic scale considered. The boundary conditions and initial conditions at the injection boundary are considered as well as the boundary conditions and breakthrough conditions at the recovery boundary and the side boundary conditions. The inertial terms are included in the equations and shown to be of importance in describing these initial conditions and the breakthrough conditions.

The stability of viscous displacement in a porous medium is considered in terms of the stability of fluid wave propagation in the medium. The concept of a critical wavelength is used to describe the criteria for the onset of fingering and the transition from the range for which Darcy's equation applies to the conditions under which the Brinkman equation must be used.

A complete set of equations to describe low‐frequency seismic wave phenomena in fluid‐filled porous media is presented. The approach is based on the mathematics of volume‐averaging, aided by order‐of‐magnitude and physical arguments. The results are immediately utilizable by practicing seismologists. Our equations and those of Biot (1956a) are found to be largely consistent in form, and we suggest how Biot’s parameters may be defined in terms of basic physical parameters. The theory predicts two dilatational waves and two rotational waves. Under certain conditions these behave differently than would be expected on the basis of Biot’s theory.

Boundary conditions for fluid-filled porous media are derived in a form independent of the choice of equations of motion. A definition of the boundary surface, together with the equation of continuity for a porous medium, leads to some of the boundary conditions. The others are suggested by the physical meaning of the various stress tensors and the porosities and the concept of the alignment. For problems of seismic transmission and reflection it is demonstrated that these boundary conditions are complete. When one of the media degenerates to an elastic solid or a fluid continuum, the boundary conditions are specialized accordingly.

Following secondary recovery processes in conventional light oil reservoirs, more than half the original oil in place may remain trapped as a discontinuous phase. During the previous recovery processes these oil ganglia have been pinched off by capillary forces and remain immobile while the continuous phase which surrounds them is able to flow freely. Furthermore if a portion of this oil is mobilized in a tertiary recovery process the conditions required to apply Darcy's equation to the flow of either phase are violated. These are also problems which are encountered during in-situ recovery techniques in tar sands where the mobilization of the heavy oil occurs as a discontinuous phase. In this paper the relevant flow equations are derived. Also a parameter is deduced which directly determines the criterion for mobilization.

We present a first order stability analysis of a steam-water front moving through a porous medium. The interplay between gravity, mobility and phase change are investigated. Several new phasical effects are observed in this dynamic study (e.g. oscillatory solutions, multiple solutions, etc.) and a number of specific limiting solutions are also considered. On presente une analyse de stabilite de premier ordre pour un front vapeur-eau qui se deplace a travers un milieu poreux. On etudie les effets interactifs entre la gratite, la mobilite et le changement de phase; plusieurs nouveaux effets physiques ont ete observes dans cette etude dynamique (par exemple: des solutions oscillatoires, des solutions multiples, etc.) et l'on a aussi examine un certain nombre de solutions limites specifiques.

The viscous surface conditions between two fluids are considered for fluid displacement in a homogeneous porous medium. The term viscous surface is defined here as a mathematical abstraction used to approximate the macroscopic shape of the boundary layer between two fixed saturations of displacing fluid by continuum theory. In the limit as the saturations approach each other, one then obtains a convergence of the viscous surface to a constant saturation contour. This yields a mathematical description of immiscible displacement in porous media which contains a theory of viscous fingering.

Abstract The Buckley-Leverett theory is discussed in light of the recently proposed flow equations involving "generalized relative permeabilities". In particular it is argued that because capillary pressure was ignored in the Buckley-Leverett analysis, internal consistency requires that the parameters be corrected for the removal of interfacial tension. The results indicate a tendency toward removing the inflection point all the fractional flow curve, which is the mathematical source of an unphysical saturation profile. Introduction Any attempt to describe the flow of multiple phases through a porous medium must address the complex problem of incorporating interfacial phenomena from the pore scale into a larger-scale description of the dynamics. In particular, for water-oil displacements, motions of the fluid-fluid interfaces, the formation of oil ganglia and the wetting characteristics of the medium introduces "capillarity" into the dynamics. The importance of these processes in oil recovery mechanisms has been the subject of a great deal of research (e.g. Melrose and Brandner, 1974; Stegemeier, 1977; Mohanty et al., 1980; Takamura and Chow, 1983). However, difficulties in incorporating this information into larger-scale descriptions of the flow dynamics have been a stumbling block in attempts to describe the dynamics of multiphase flow. Furthermore, pore size distribution and pore geometry have been shown to have a large effect on the trapping of oil ganglia (cf. Chatzis et al. 1983, Wardlaw, 1982) and thus residual oil saturations. The description of multiphase flow which has been generally accepted is based on an analogy with single-phase flow. Proposed by Muskat (1946, 1953) over thirty years ago, it assumes that multiphase flow can be adequately described by flow equations of the same form as Darcy's equation. Permeability is replaced by parameters called relative permeabilities which are taken as functions of fluid saturation. Thus all the dynamic interfacial phenomena which occur at the pore scale during multiphase flow are to be handled by this single modified term in each of the flow equations. A more rigorous understanding of the dynamics of multiphase flow, in terms of pore scale phenomena, could not be considered at that stage since Darcy's equation itself was only understood as an empirical relationship. A number of researchers subsequently speculated that Darcy's equation was illustrating the average behaviour of the well understood flow dynamics at the pore scale (i.e. the Navier-Stokes equation) and the effects of the complex boundary between the fluid and the solid medium (e.g. Hubbard, 1956; Hall, 1956; Scheidegger. 1974). The main obstacle in these analyses to a clear formulation of the problem, was how to relate the averaged effect of the differential equations at the pore scale to the Darcy description which expressed the problem as a differential equation of the averaged quantities. A breakthrough came when Whittaker (1969) and Slattery (1969) proved what has come to be known as the Whittaker-Slattery averaging theorem. This mathematical result expresses the volume average of the spatial derivative of a microscopic quantity in terms of the derivative of the volume averaged quantity plus a surface integral over the interfaces between the fluid and solid.

The explicit form of the metric corresponding to two Kerr-Newman sources in equilibrium under the balancing of their electromagnetic and gravitation forces is derived using the method of Israel, Wilson, and Perj\'es. The sources have arbitrary masses and arbitrary but parallel angular momenta. The condition for equilibrium without struts and magnetic poles is discussed.

The procedure of volume averaging is applied to the problem of seismic propagation in a fluid-filled porous medium. The interplay of temperature variation and the mechanical motion is taken into account to first order. A complete set of equations is obtained, which carry, in addition to permeability, two mechanical parameters and one thermal parameter. When thermomechanical coupling is ignored, a subset of the equations reduces to those of Biot (1956a), allowing a correspondence between the parameters to be established.