Single-phase fluid flow permeability

Single-phase fluid flow in porous media is a well-studied case in the literature. It is important not only for its application, but the characterization of the porous medium itself is also dependent on the study of a single-phase flow. The parameters normally needed are porosity, areal porosity, tortuosity, and permeability. For flow of a constant viscosity Newtonian fluid in a rigid isotropic porous medium, the volume averaged equations can be reduced to the following the continuity equation,... 

Physical systems are modelled by postulating a relationship between three objects - the properties, 95, the state and the auxiliary data, ipa- The auxiliary data and the properties together are referred to as input. The state is sometimes called the output. The properties characterise the unchanging aspects of the system the state characterises the aspects of the system that respond to different selections of the auxiliary data. The auxiliary data corresponds to those aspects of the system that are under human, or other, control. An example is that of single phase fluid flow in a porous medium the properties are the permeability, the state is the pressure and the flux. The auxiliary data are the boundary conditions imposed on the flow system. In a time dependent problem, the auxiliary data will also include the initial conditions. 

We then discussed the modeling for single-fluid phase flow in porous media. In particular, the shear factor and permeability model of Liu et al. (32) is discussed in detail. The bounding wall effects are presented. This section completed the modeling requirements for single-phase incompressible flow in porous media. We showed how to solve the governing equations for flow in porous media and an approximate solution of the pressure drop for an incompressible flow through a cylindrically bounded porous bed was constructed. 

Of the numerous macroscopic parameters used to quantify porous media, those gaiiung widest acceptance in the hterature for describing the flow of single phase fluids are voidage, speciflc surface, permeability and tortuosity. Their values can often be inferred from experiments on the streamline flow of single phase Newtonian fluids. 

To fix ideas, we will formulate the transformed problem for a single-phase, eompressible flow with a eonstant horizontal permeability kh in the areal (x,y) eoordinates, and a variable permeability kv(z) in the layered, vertieal z direetion. The matrix porosity z) may vary with z. The fluid viseosity p is assumed to be eonstant. The governing equation then takes the following form,... 

The rheological description of foam flow in porous media has been treated in different ways. one approach has been to use the single-phase fluid viscosities to calculate relative permeabilities to each fluid on the basis of experimental measurements of flow rates and pressure drop in foam flow through a porous medium. 

The pore geometry described in the above section plays a dominant role in the fluid transport through the media. For example, Katz and Thompson [64] reported a strong correlation between permeability and the size of the pore throat determined from Hg intrusion experiments. This is often understood in terms of a capillary model for porous media in which the main contribution to the single phase flow is the smallest restriction in the pore network, i.e., the pore throat. On the other hand, understanding multiphase flow in porous media requires a more complete picture of the pore network, including pore body and pore throat. For example, in a capillary model, complete displacement of both phases can be achieved. However, in real porous media, one finds that displacement of one or both phases can be hindered, giving rise to the concept of residue saturation. In the production of crude oil, this often dictates the fraction of oil that will not flow. 

For single-phase flow, when the rock is completely saturated with one fluid (or gas) only, the permeability k (= absolute permeability), is a rock property and a constant, irrespective of the nature of the fluid flowing through the pores. For two-phase (water and hydrocarbon) flow, each fluid has its own effective permeability that is dependent on the saturation of the rock to the fluid. The greater the hydrocarbon saturation, the greater the effective permeability of the... 

Before commencing two-phase flow operations, single-phase permeability determinations were made at several flow rates, using brine in the water-wet sample, and Soltrol-oil in the samples that had been treated with Dri-Film. In every case, first the drainage relative permeability curves were determined. The first steady-state point was obtained typically at a wetting fluid/non-wetting fluid flow rate ratio of about 10. The filter velocities used... 

Formation testers are measurement instruments that retrieve reservoir fluid samples from wells during pauses in drilling operations. Various practical questions arise. A type of reverse invasion problem appears how long must pumps be operated in order to obtain true formation fluids and not mud filtrate contaminants How do pump power requirements vary in permeable versus tight zones Can measured pressure transients be interpreted for rock characteristics like permeability and anisotropy Different answers are obtained depending on the fluid model assumed. Later in this book, we will consider constant density, immiscible, two-phase flows with and without mudcake effects. For now we assume transient, compressible, single-phase flow, but within this framework, we formulate and solve a very general problem. 

If gas is used as non-reactive fluid, at low gas pressures the mean free path of gas molecules gets the order of the pore dimensions. Then gas molecules have a finite velocity at the pore wall, but for liquids, a zero velocity at the wall is assumed. The gas slippage effect increases the flow rate and causes an overestimated permeability. Klinkenberg correction uses measurements at different pressures and an extrapolation for a (theoretical) infinite pressure (Cosentino, 2001). It results in the Klinkenberg corrected permeability , which is independent of the type of gas, and approximately the same as for a single phase liquid. Forchheimer effect ... 

The creeping flow of a single fluid phase through a rigid permeable medium is modeled with the continuity equation and Darcy s Law ... 

Foam with large and less stable bubbles is less likely to flow as a single fluid. Mast and Fried deduced that foam is propogated inside a porous medium by the breaking and reforming of foam bubbles. The gas flows as a discontinuous phase while toe liquid is transported as a free phase via toe film network. Nahid proposed that toe gas flow could be treated according to Darcy s law if a correction factor for the gas permeability is used. 

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