HOME

TheInfoList



OR:

In
multiphase flow In fluid mechanics, multiphase flow is the simultaneous Fluid dynamics, flow of materials with two or more thermodynamic Phase (matter), phases. Virtually all processing technologies from Cavitation, cavitating pumps and turbines to paper-making ...
in
porous media In materials science, a porous medium or a porous material is a material containing pores (voids). The skeletal portion of the material is often called the "matrix" or "frame". The pores are typically filled with a fluid (liquid or gas). The sk ...
, the relative permeability of a phase is a
dimensionless Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that align with another sy ...
measure of the effective permeability of that phase. It is the ratio of the effective permeability of that phase to the absolute permeability. It can be viewed as an adaptation of Darcy's law to multiphase flow.


Formulation

For two-phase flow in porous media given
steady-state In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p'' ...
conditions, we can write :q_i = -\frac \nabla P_i \qquad \text \quad i=1,2 where q_i is the
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications in physics. For transport phe ...
, \nabla P_i is the
pressure drop Pressure drop (often abbreviated as "dP" or "ΔP") is defined as the difference in total pressure between two points of a fluid carrying network. A pressure drop occurs when frictional forces, caused by the resistance to flow, act on a fluid as i ...
, \mu_i is the
viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
. The subscript i indicates that the parameters are for phase i. k_i is here the phase permeability (i.e., the effective permeability of phase i), as observed through the equation above. Relative permeability, k_, for phase i is then defined from k_i = k_k, as :k_ = k_i / k where k is the permeability of the porous medium in single-phase flow, i.e., the absolute permeability. Relative permeability must be between zero and one. In applications, relative permeability is often represented as a function of water saturation; however, owing to capillary hysteresis one often resorts to a function or curve measured under
drainage Drainage is the natural or artificial removal of a surface's water and sub-surface water from an area with excess water. The internal drainage of most agricultural soils can prevent severe waterlogging (anaerobic conditions that harm root gro ...
and another measured under
imbibition Imbibition is a special type of diffusion that takes place when liquid is absorbed by solids-Colloid, colloids causing an increase in volume. Water surface potential movement takes place along a concentration gradient; some dry materials absorb ...
. Under this approach, the flow of each phase is inhibited by the presence of the other phases. Thus the sum of relative permeabilities over all phases is less than 1. However, apparent relative permeabilities larger than 1 have been obtained since the Darcean approach disregards the viscous coupling effects derived from momentum transfer between the phases (see assumptions below). This coupling could enhance the flow instead of inhibit it. This has been observed in heavy oil petroleum reservoirs when the gas phase flows as bubbles or patches (disconnected).


Modelling assumptions

The above form for Darcy's law is sometimes also called Darcy's extended law, formulated for horizontal, one-dimensional,
immiscible Miscibility () is the property of two chemical substance, substances to mix in all mixing ratio, proportions (that is, to fully dissolution (chemistry), dissolve in each other at any concentration), forming a homogeneity and heterogeneity, homoge ...
multiphase flow in homogeneous and
isotropic In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...
porous media. The interactions between the fluids are neglected, so this model assumes that the solid porous media and the other fluids form a new porous matrix through which a phase can flow, implying that the fluid-fluid interfaces remain static in steady-state flow, which is not true, but this approximation has proven useful anyway. Each of the phase saturations must be larger than the irreducible saturation, and each phase is assumed continuous within the porous medium. Based on data from special core analysis laboratory (SCAL) experiments, simplified models of relative permeability as a function of saturation (e.g. water saturation) can be constructed. This article will focus on an oil-water system.


Saturation scaling

The water saturation S_\mathit is the fraction of the pore volume that is filled with water, and similarly for the oil saturation S_\mathit. Thus, saturations are themselves scaled properties or variables. This gives the constraint : S_\mathit + S_\mathit = 1 \Leftrightarrow S_\mathit = 1 - S_\mathit The model functions or correlations for relative permeabilities in an oil-water system are therefore usually written as functions of only water saturation, and this makes it natural to select water saturation as the horizontal axis in graphical presentations. Let S_\mathit (also denoted S_\mathit and sometimes S_\mathit) be the irreducible (or minimal or connate) water saturation, and let S_\mathit be the residual (minimal) oil saturation after water flooding (imbibition). The flowing water saturation window in a water invasion / injection / imbibition process is bounded by a minimum value S_\mathit and a maximum value S_\mathit = 1 - S_\mathit. In mathematical terms the flowing saturation window is written as : S_\mathit \leq S_\mathit \leq S_\mathit = 1 - S_\mathit By scaling the water saturation to the flowing saturation window, we get a (new or another) normalized water saturation value : S_\mathit = S_\mathit(S_w) = \frac = \frac and a normalized oil saturation value : S_\mathit = 1-S_\mathit = \frac


Endpoints

Let K_\mathit be oil relative permeability, and let K_\mathit be water relative permeability. There are two ways of scaling phase permeability (i.e. effective permeability of the phase). If we scale phase permeability w.r.t. absolute water permeability (i.e. S_\mathit = 1), we get an endpoint parameter for both oil and water relative permeability. If we scale phase permeability w.r.t. oil permeability with irreducible water saturation present, K_\mathit endpoint is one, and we are left with only the K_\mathit endpoint parameter. In order to satisfy both options in the mathematical model, it is common to use two endpoint symbols in the model for two-phase relative permeability. The endpoints / endpoint parameters of oil and water relative permeabilities are :\begin K_\mathit(S_\mathit) = K_\mathit && \text && K_\mathit(S_\mathit) = K_\mathit \end These symbols have their merits and limits. The symbol K_\mathit emphasize that it represents the top point of K_\mathit. It occurs at irreducible water saturation, and it is the largest value of K_\mathit that can occur for initial water saturation. The competing endpoint symbol K_\mathit occurs in imbibition flow in oil-gas systems. If the permeability basis is oil with irreducible water present, then K_\mathit = 1. The symbol K_\mathit emphasizes that it is occurring at the residual oil saturation. An alternative symbol to K_\mathit is K_\mathit^o which emphasizes that the reference permeability is oil permeability with irreducible water S_\mathit present. The oil and water relative permeability models are then written as :\begin K_\mathit = K_\mathit \cdot K_\mathit(S_\mathit) && \text && K_\mathit = K_\mathit \cdot K_\mathit(S_\mathit) \end The functions K_\mathit and K_\mathit are called normalised relative permeabilities or shape functions for oil and water, respectively. The endpoint parameters K_\mathit and K_\mathit (which is a simplification of K_\mathit) are physical properties that are obtained either before or together with the optimization of shape parameters present in the shape functions. There are often many symbols in articles that discuss relative permeability models and modelling. A number of busy core analysts, reservoir engineers and scientists often skip using tedious and time-consuming subscripts, and write e.g. Krow instead of K_\mathit or k_\mathit or krow or oil relative permeability. A variety of symbols are therefore to be expected, and accepted as long as they are explained or defined. The effects that slip or no-slip boundary conditions in pore flow have on endpoint parameters, are discussed by Berg et alios.


Corey-model

An often used approximation of relative permeability is the Corey correlation which is a
power law In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a relative change in the other quantity proportional to the ...
in saturation. The Corey correlations of the relative permeability for oil and water are then :K_\mathit(S_) = K(1-S_\mathit)^ : K_\mathit(S_) = KS_\mathit^ If the permeability basis is normal oil with irreducible water present, then K_\mathit = 1. The empirical parameters N_\mathit and N_\mathit are called curve shape parameters or simply shape parameters, and they can be obtained from measured data either by analytical interpretation of measured data, or by optimization using a core flow numerical simulator to match the experiment (often called history matching). N_\mathit = N_\mathit = 2 is sometimes appropriate. The physical properties K_\mathit and K_\mathit are obtained either before or together with the optimizing of N_\mathit and N_\mathit. In case of gas-water system or gas-oil system there are Corey correlations similar to the oil-water relative permeabilities correlations shown above.


LET-model

The Corey-correlation or Corey model has only one degree of freedom for the shape of each relative permeability curve, the shape parameter N. The LET-correlation adds more degrees of freedom in order to accommodate the shape of relative permeability curves in SCAL experiments and in 3D reservoir models that are adjusted to match historic production. These adjustments frequently includes relative permeability curves and endpoints. The LET-type approximation is described by 3 parameters L, E, T. The correlation for water and oil relative permeability with water injection is thus :K_\mathit=\frac and :K_\mathit=\frac written using the same S_w normalization as for Corey. Only S_\mathit, S_\mathit, K_\mathit, and K_\mathit have direct physical meaning, while the parameters ''L'', ''E'' and ''T'' are empirical. The parameter ''L'' describes the lower part of the curve, and by similarity and experience the ''L''-values are comparable to the appropriate Corey parameter. The parameter ''T'' describes the upper part (or the top part) of the curve in a similar way that the ''L''-parameter describes the lower part of the curve. The parameter ''E'' describes the position of the slope (or the elevation) of the curve. A value of one is a neutral value, and the position of the slope is governed by the ''L''- and ''T''-parameters. Increasing the value of the ''E''-parameter pushes the slope towards the high end of the curve. Decreasing the value of the ''E''-parameter pushes the slope towards the lower end of the curve. Experience using the LET correlation indicates the following reasonable ranges for the parameters ''L'', ''E'', and ''T'': ''L'' ≥ 0.1, ''E'' > 0 and ''T'' ≥ 0.1. In case of gas-water system or gas-oil system there are LET correlations similar to the oil-water relative permeabilities correlations shown above.


Evaluations

After Morris Muskat et alios established the concept of relative permeability in late 1930'ies, the number of correlations, i.e. models, for relative permeability has steadily increased. This creates a need for evaluation of the most common correlations at the current time. Two of the latest (per 2019) and most thorough evaluations are done by Moghadasi et alios and by Sakhaei et alios. Moghadasi et alios evaluated Corey, Chierici and LET correlations for oil/water relative permeability using a sophisticated method that takes into account the number of uncertain model parameters. They found that LET, with the largest number (three) of uncertain parameters, was clearly the best one for both oil and water relative permeability. Sakhaei et alios evaluated 10 common and widely used relative permeability correlations for gas/oil and gas/condensate systems, and found that LET showed best agreement with experimental values for both gas and oil/condensate relative permeability.


See also

* TEM-function *
Permeability (earth sciences) In fluid mechanics, materials science and Earth sciences, the permeability of porous media (often, a rock or soil) is a measure of the ability for fluids (gas or liquid) to flow through the media; it is commonly symbolized as ''k''. Fluids c ...
* Capillary pressure *
Imbibition Imbibition is a special type of diffusion that takes place when liquid is absorbed by solids-Colloid, colloids causing an increase in volume. Water surface potential movement takes place along a concentration gradient; some dry materials absorb ...
*
Drainage Drainage is the natural or artificial removal of a surface's water and sub-surface water from an area with excess water. The internal drainage of most agricultural soils can prevent severe waterlogging (anaerobic conditions that harm root gro ...
* Buckley–Leverett equation


References


External links


Relative Permeability Curves
{{DEFAULTSORT:Relative Permeability Fluid dynamics Porous media Physical quantities