Darcy

Abstract: We study the gas flow processes in ultra-tight porous media in which the matrix pore network is composed of nanometre- to micrometre-size pores. We formulate a pressure-dependent permeability function, referred to as the apparent permeability function (APF), assuming that Knudsen diffusion and slip flow (the Klinkenberg effect) are the main contributors to the overall flow in porous media. The APF predicts that in nanometre-size pores, gas permeability values are as much as 10 times greater than results obtained by continuum hydrodynamics predictions, and with increasing pore size (i.e. of the order of the micrometre), gas permeability converges to continuum hydrodynamics values. In addition, the APF predicts that an increase in the fractal dimension of the pore surface leads to a decrease in Knudsen diffusion. Using the homogenization method, a rigorous analysis is performed to examine whether the APF is preserved throughout the process of upscaling from local scale to large scale. We use the well-known pulse-decay experiment to estimate the main parameter of the APF, which is Darcy permeability. Our newly derived late-transient analytical solution and the late-transient numerical solution consistently match the pressure decay data and yield approximately the same estimated value for Darcy permeability at the typical core-sample initial pressure range and pressure difference. Other parameters of the APF may be determined from independent laboratory experiments; however, a pulse-decay experiment can be used to estimate the unknown parameters of the APF if multiple tests are performed and/or the parameters are strictly constrained by upper and lower bounds.

Abstract: (1957). DARCY'S LAW AND THE FIELD EQUATIONS OF THE FLOW OF UNDERGROUND FLUIDS. International Association of Scientific Hydrology. Bulletin: Vol. 2, No. 1, pp. 23-59.

Abstract: Darcy's law q = KJ(q = vector of specific discharge; J = — grad E = hydraulic gradient; E — energy of flow; K — hydraulic conductivity of porous medium) is historically reviewed as well as its numerous extensions. The author starts from the Navier-Stokes hydrodynamic equations of viscous flow, and using statistical methods (space averaging) shows that: The head (or energy) used in Darcy law is different from that used in viscous flow; which recommends large piezometric intakes. The formula J = aq + bqq + c∂q/∂t is obtained (a, b, c = coefficients in isotropic media). Usually the third term may be neglected. At low values of Reynolds numbers, Darcy's law is obtained with a = 1/K in the Kozeny-Carman form. It is found that a = the hydraulic resistivity is more important than K. At larger values of Reynolds number, the Forchheimer formula is obtained, with b depending on the soil texture (grain diameter d) and porosity, not on temperature nor viscosity. The Laplace equation is replaced by a non linear Poisson-type equation with K = f(J). Darcy's law then becomes physically meaningless. A minimum principle is found: Darcy flow is such that the kinetic energy or the dissipation becomes minimum. The application of the Navier-Stokes equations to rocks with seams or cracks gives Darcy's law in its tensorial form.