Lift forces acting on particles play a central role in many cases, such as sediment transport, proppant transport in fractured reservoirs, removal of drill cuttings in horizontal drill holes and cleaning of particles from surfaces. We study the problem of lift using 2D direct numerical simulations and experimental data. The lift-off of single particles and many particles in horizontal flows follow laws of similarity, power laws, which may be obtained by plotting simulation data on log-log plots. Data from slot experiments for fractured reservoirs is processed (for the first time) on log-log plots. Power laws with a parameter dependent power emerge as in the case of Richardson-Zaki correlations for bed expansion by drag.

Summary Since the introduction of the G-function derivative analysis, prefrac diagnostic injection tests have become a valuable and commonly used technique. Unfortunately, the technique is frequently misapplied or misinterpreted, leading to confusion and misdiagnosis of fracturing parameters. This paper presents a consistent method of analysis of the G-function, its derivatives, and its relationship to other diagnostic techniques including square-root(time) and log(?pwf)-log(?t) plots and their appropriate diagnostic derivatives. Four field test examples are given for the most common diagnostic curve signatures. These show how multiple analysis methods can be applied to consistently interpret closure pressure and time, as well as pre- and post-closure flow regimes and reservoir properties from the test data. The cases include normal constant-area and constant permeability leakoff, pressure dependent fissure leakoff, fracture tip extension, and variable fracture storage. In some cases conventionally accepted analysis methods, such as the Sqrt(time) plot, can lead to misleading interpretations. A single consistent approach to analysis is described for each case. The example cases can be used to build a foundation for consistent and less ambiguous analysis of any complex fracture injection/falloff test. Introduction Prefrac diagnostic injection test analysis provides critical input data for fracture design models, and reservoir characterization data used to predict post-fracture production. An accurate post-stimulation production forecast is necessary for economic optimization of the fracture treatment design. Reliable results require an accurate and consistent interpretation of the test data. In many cases closure is mistakenly identified through misapplication of one or more analysis techniques. In general, a single unique closure event will satisfy all diagnostic plots or methods. All available analysis methods should be used in concert to arrive at a consistent interpretation of fracture closure. Relationship of the pre-closure analysis to after-closure analysis results must also be consistent. To correctly perform the after-closure analysis the transient flow regime must be correctly identified. Flow regime identification has been a consistent problem in many analyses. There remains no consensus regarding methods to identify reservoir transient flow regimes after fracture closure. The method presented here is not universally accepted but appears to fit the generally assumed model for leakoff used in most fracture simulators. Four examples are presented to show the application of multiple diagnostic analysis methods. The first illustrates the expected behavior of normal fracture closure dominated by matrix leakoff with a constant fracture surface area after shut-in. The second example shows pressure dependent leakoff (PDL) in a reservoir with pressure-variable permeability or flow capacity, usually caused by natural or induced secondary fractures or fissures. The third example shows fracture tip extension after shut-in. These cases generally show definable fracture closure. The fourth example shows what has been commonly identified as fracture height recession during closure, but which can also indicate variable storage in a transverse fracture system.

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The limitations of Darcy's Law to a relatively small velocity region have long been recognized. A commonly accepted approach has been to use Forchheimer's equation, and its inertial flow parameter (ß), as an extension of Darcy's Law beyond the linear flow region. This "trans-Darcy" flow is especially important in hydraulic fracture conductivity calculations, where flow velocities in the proppant pack are much higher than in the surrounding reservoir. The computation and presentation of ß values as functions of closure stress and permeability has become an important consideration in proppant selection and fracture treatment design. New experimental data, under very high-rate flow conditions, has shown conclusively that Forchheimer's equation, like Darcy's Law, has a limited range of applicability. At high potential gradients the flow rate cannot be predicted from Darcy or Forchheimer equations. These data also show that ß is not a single-valued function of permeability, as has been expected, but is as much a function of Reynolds Number as the apparent Darcy permeability. This leads to different values of ß for the same proppant, depending on the range of flow rates used for the measurement. This paper presents a single new equation that describes the relationship between rate and potential gradient for porous media flow over the entire range of Reynolds Number. The equation simplifies to both the Forchheimer and Darcy equations under their governing assumptions. The equation can be used to determine the correct theoretical ß value and to demonstrate the limits of applicability of ß and the Forchheimer equation. A new method for describing porous media flow using different coefficients, and the relationship of these coefficients to physical parameters, is presented. The development of the complete porous media flow model is supported by extensive laboratory data on various proppant packs. Applicability of Darcy's Law In 1856 Henry Philibert Gaspard Darcy published the results of a series of experiments on water flow through a sand-packed column at various pressure differentials.1 His apparatus consisted of a 0.35 m diameter sand pack with 38% porosity. The height of the sand column varied from 0.58 meters to 1.7 meters and the imposed water head varied from 1 to 14 meters. From the observations of flow rate resulting from these various experiments he concluded that flow rate varied in proportion to the imposed head and inversely to the height of the sand pack. The flow rate could be related to the imposed head through a linear proportionality involving a constant, k, where "k is a coefficient dependent on the permeability of the (sand) layer." This linear proportionality is expressed in Darcy's Law which is given by Equation 1. In the equation the imposed potential gradient is ?P/?L, the fluid viscosity is µ, and the superficial velocity is v. As defined, a permeability of one darcy results in a flow of one cm/sec for a fluid of one centipoise under a gradient of one atmosphere per centimeter.Equation 1 The calculated apparent permeability values from Darcy's five experimental data sets are shown in Figure 1 as a function of the quantity ?v/µ, defined here as the pseudo-Reynolds number (Rp). The parameter Rp is not a true Reynolds Number because it is not dimensionless, but is missing a characteristic length giving it dimensions of 1/L. The plot shows that even Darcy's original observations do not show an absolutely constant permeability, but an apparent decrease in flow capacity with increasing velocity. Darcy mentions that there were experimental difficulties at the higher flow rates in his tests because his water source was a hospital. The maximum flow rate he could achieve was unstable because of other people in the hospital occasionally opening and closing other water taps.

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractThe effects of natural fissure opening, or pressure dependent leakoff, on the pressure behavior observed during fracturing are significant. Previous work has suggested that this behavior can be identified from pressure diagnostic plots during pumping, or from pressure falloff analysis. However, these techniques lead to ambiguous conclusions regarding the magnitude, and even existence, of pressure dependent leakoff. This paper presents a method of pressure falloff analysis which removes this ambiguity and allows an accurate determination of the magnitude of pressure dependent leakoff. The effects of fracture tip extension and recession, height recession, and transient flow in the fracture are also identified. The effect of pressure dependent fracture compliance, which has not been previously published, is also described.