A new kind of percolation problem is described which differs from ordinary percolation theory in that it automatically finds the critical points of the system. The model is motivated by the problem of one fluid displacing another from a porous medium under the action of capillary forces, but in principle it may be applied to any kind of invasion process which proceeds along a path of least resistance. The name invasion percolation is proposed for this new process. Similarities to, and differences from, ordinary percolation theory are discussed.

We report on molecular-dynamics simulations of the low-Reynolds-number flow of Lennard-Jones fluids through a channel. Application of a pressure gradient to a single fluid produces Poiseuille flow with a no-slip boundary condition and Taylor-Aris hydrodynamic dispersion. For an immiscible twofluid system we find a (predictable) static contact angle and, when accelerated, velocity-dependent advancing-and receding-contact angles. The approximate local velocity field is obtained, in which the no-slip condition appears to break down near the contact line.PACS numbers: 47. 15.6f, 51.10.+y, 61.20.Ja Although the low-Reynolds-number flow of continuum Newtonian Auids has been successfully described by the Stokes equations for over a century, there remain a number of unsettled questions concerning the appropriate boundary conditions at solid surface. " and fluid interfaces. For example, overwhelming phenomenological evidence supports the "no-slip" condition of zero fluid velocity at a solid boundary, ' yet there is no compelling theoretical argument for why this should be the case. A problem of principle arises when a meniscus separating two immiscible fluids moves along a solid surface. When the Stokes equations are combined with the usual boundary conditions, the viscous dissipation diverges logarithmically at the contact line. This singularity indicates that the problem is not properly formulated, but at present the cure is not known. Various proposals have been advanced to yield a finite result, e.g. , a finite "slip length, " an appeal to surface roughness, nontrivial interfacial shapes near the solid, and precursor films, but no consensus exists.In these and related problems, the macroscopic flow description must be augmented with knowledge of the microscopic physics of the boundary region between the fluids. To this end, we have carried out molecular dynamics (MD) simulations ' of viscous fluid Aows past solid boundaries. We have studied systems consisting of 1536 molecules (per fluid) confined to a region of size 40-100 A, over times up to 10 sec, the computations requiring hours of central processing unit time on a Cray XMP-12. Our results indicate that even such a small system behaves in almost all respects like a continuum fluid in motion. The starting point is a standard molecular-dynamics code in which each pair of molecules interacts through a Lennard-Jones potential r r VLs(r) =4e 0 0 +rBV cut off at r =2. 5a, where BV is chosen so that the force vanishes at the cutoff. For numerical illustration, we will refer to parameters appropriate for liquid argon, where the distance and energy scales are o 3.4 A. and e/ka 120 K, the natural time unit is z=o(m/e)' 2. 16 X10 ' sec, and the molecular mass is m 40 a.u.Newton's law is numerically integrated with a fifth-order predictor-corrector scheme, with a time step of 0.005z. The molecules are initialized on an fcc lattice whose spacing is chosen to obtain the desired density, with initial velocities randomly assigned subject to a fixed temperature. The substance ...

Summary. Borehole measurements of the nuclear magnetic resonance (NMR) properties of rocks have been of interest for many years, especially for estimating permeability. This paper presents laboratory measurements of the NMR properties of water-saturated rocks and shows that permeability can be estimated well with expressions of the form T, where T1 is the relaxation time constant of the longitudinal nuclear magnetization of hydrogen nuclei. Different methods of representing the laboratory- measured T1 curves are shown, including a new one called the stretched- exponential representation. An improved method for estimating T1 parameters from borehole measurements that can be used with either old or new representations is presented. Introduction In this paper, we pursue permeability estimation from borehole NMR* longitudinal relaxation (T1) measurements. Many previous workers have demonstrated the potential of NMR for this application; however, we make closer and more consistent connections between the components linking permeability and borehole NMR than have previously been published. These components are addressed in the three parts that follow,1. Stretched-exponential representation of laboratory T1 measurements. We present the results of laboratory NMR measurement on approximately 60 water-saturated rocks. We introduce a new representation for the NMR curve, called the stretched-exponential representation, that has the practical advantage of having fewer pa-rameters than the classical two- and three-exponential representations of NMR measurements. Such representations are important in reducing the measurement to a few parameters that can be correlated to properties of practical importance.2. Estimation of permeability from laboratory measurements. We use the data base of 60 rocks from Part 1 to find the best estimator of permeability from NMR T1 measurements. An important result is that permeability is estimated better by T1 than by Seevers classic estimator .3. Extraction of NMR T1 parameters from borehole NMR measurements. To apply the correlations of Part 2 to borehole data, we introduce a new method of extracting the important T, parameters from downhole NMR T1 measurements: in this method, called "global fitting," a model is fitted simultaneously to the set of free induction decay (FID) waveforms collected for different polarizing times during a station measurement. We exhibit two suitable models. Both have the advantage of accommodating some complexities observed in borehole waveforms and verified in a corresponding laboratory measurement. In particular, the observed decay time of the FID waveforms decreases as the polarizing time decreases. This paper concentrates on the NMR property T1 and does not investigate the parameter called free fluid index (FFI). The reasons for this emphasis are two-fold. First, T1 is a more complete measurement, and thus gives a better picture of the potential of NMR in permeability estimation. Second, FFI is specifically a low-field measurement, which is much less convenient to measure in the laboratory. Borehole T1 data can he obtained with existing commercial nuclear magnetic log (NMLTM) equipment by making stationary measurements.A key issue in this paper is compact representation-finding ways to describe accurately the observed behavior with only a small number of parameters. Representation is an issue in Part 1, dealing with laboratory T1 measurements, because a complete curve must be described. Part 2 shows that all the representations used here allow equally good permeability estimation. In Part 3, dealing with borehole T1 data. representation is important because of the need to work around measurement dead-time, and because borehole measurements in practice have a lower signal-to-noise ratio than laboratory measurements. Throughout, compactness of a representation is weighed against its ability to fit the measurements and its appropriateness for estimating permeability. Part 1-Stretched-Exponential Representation of Laboratory T1 Measurements Laboratory Technique. We measured porosity, permeability, and NMR T1 properties on water-saturated sandstone samples from five oilfield wells in different parts of the world, plus a number of quarried sandstone samples. Samples were cut to Hassler collar size2.0 cm [0.78 in.] in diameter and approximately 4 cm [ 1.57 in.] long; the samples were cored parallel to any visible bedding planes in the original rocks. Sample porosities were determined by Archimedes' methodi.e., measuring dry sample weight. saturated weight, and buoyant weight of the water-saturated sample. Permeabilities to water were measured end-to-end on the samples encased in a Hassler collar, at room temperature, with a collar pressure in the neighborhood of 414 kPa [60 psi]. Because our measured permeabilities are thus for single-phase parallel-to-bedding flow, the final output of our permeability estimators will be for the same quantity. Laboratory NMR measurements were made using an IBM/Bruker PC10. The PC10 is a desk-top permanent magnet instrument that makes pulsed measurements of proton resonance at 10 MHz [10(6) cycles/sec). Samples for NMR measurement were surface-dried and then wrapped in Saran(TM) wrap held in place by rubber bands to reduce evaporation during measurement; these wrapping materials contributed a negligible signal for the water volumes of our samples. Before measurement, samples were allowed to equilibrate to the magnet temperature, which is thermostatically maintained at 40 degrees C [104 degrees F]. The fundamental NMR property to be measured is the time evolution of proton magnetization along the direction of the applied magnetic field. This behavior of the "longitudinal" magnetization is called T1. SPEFE P. 622^

Molecular dynamics techniques are used to study the microscopic aspects of several slow viscous flows past a solid wall, where both fluid and wall have a molecular structure. Systems of several thousand molecules are found to exhibit reasonable continuum behavior, albeit with significant thermal fluctuations. In Couette and Poiseuille flow of liquids it is found that the no-slip boundary condition arises naturally as a consequence of molecular roughness, and that the velocity and stress fields agree with the solutions of the Stokes equations. At lower densities slip appears, which can be incorporated into a flow-independent slip-length boundary condition. The trajectories of individual molecules in Poiseuille flow are examined, and it is also found that their average behavior is given by Taylor–Aris hydrodynamic dispersion. An immiscible two-fluid system is simulated by a species-dependent intermolecular interaction. A static meniscus is observed whose contact angle agrees with simple estimates and, when motion occurs, velocity-dependent advancing and receding angles are observed. The local velocity field near a moving contact line shows a breakdown of the no-slip condition and, up to substantial statistical fluctuations, is consistent with earlier predictions of Dussan [AIChE J. 23, 131 (1977)].

We consider capillary displacement of immiscible fluids in porous media in the limit of vanishing flow rate. The motion is represented as a stepwise Monte Carlo process on a finite two-dimensional random lattice, where at each step the fluid interface moves through the lattice link where the displacing force is largest. The displacement process exhibits considerable fingering and trapping of displaced phase at all length scales, leading to high residual retention of the displaced phase. Many features of our results are well described by percolation-theory concepts. In particular, we find a residual volume fraction of displaced phase which depends strongly on the sample size, but weakly or not at all on the co-ordination number and microscopic-size distribution of the lattice elements.

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mance of our system given in terms of the mean effective noise energy I) ^0.034&T r compares favorably with the best results reported in higher-frequency bands, we have inevitably arrived at a rather large limit for the GR energy spectrum density F(v 0 ) at 145 Hz given by Eq. (14).We are grateful to Mr. Kenji Tsunemoto and Mr. Masa-Katsu Fujimoto for their contributions during various phases of the experiment. We thank Mr. Y. Kamiya and Mr. K. Tsubono for their cooperation. . 3 Details of our GR detectors will be published elsewhere. 4 On the assumption that the resonance width v/Q of the antenna cross section o (0,