The accepted large scale equations of motion for two phase flow involve a generalization of Darcy’s law to relative permeabilities including offdiagonal viscous coupling terms [349, 270, 322, 321, 350, 351, 352]. The importance of viscous coupling terms has been recognized relatively late [353, 354, 355, 356, 357]. The equations which are generally believed to describe multiphase flow on the reservoir scale as well as on the laboratory scale may be written as [322, 350]

(6.38) |

(6.39) |

(6.40) |

(6.41) |

where

Eliminating

(6.42) | ||||

(6.43) | ||||

for these two unknowns. Equations (6.42) and (6.43) are coupled nonlinear partial differential equations for the large scale pressure and saturation field of the water phase.

These equations must be complemented with large scale boundary conditions. For core experiments these are typically given by a surface source on one side of the core, a surface sink on the opposite face, and impermeable walls on the other faces. For a reservoir the boundary conditions depend upon the drive configuration and the geological modeling of the reservoir environment, so that Dirichlet as well as von Neumann problems arise in practice [339, 351, 352].

The large scale equations of motion can be cast in dimensionless form using the definitions

(6.44) |

(6.45) |

(6.46) |

(6.47) |

(6.48) |

where as before

(6.49) |

where

(6.50) |

Thus the dimensionless pressure is defined in terms of the inflection
point

The definition (6.48) differs from the traditional analysis [49, 329, 330, 331]. In the traditional analysis the normalized pressure field is defined as

(6.51) |

which immediately gives rise to three problems. Firstly the permeablity is a tensor, and thus a certain nonuniqueness results in anisotropic situations [339]. Secondly equation (6.51) neglects the importance of microscopic wetting and saturation history dependence. The main problem however is that Eq. (6.51) is not based on macroscopic capillary pressures but on Darcy’s law which describes macroscopic viscous pressure effects. On the other hand the normalization (6.48) is free from these problems and it includes macroscopic capillarity in the same way as the microscopic normalization (6.15) includes microscopic capillarity.

With the normalizations introduced above the dimensionless form of the macroscopic two-phase flow equations (6.42),(6.43) becomes

(6.52) | ||||

(6.53) | ||||

In these equations the dimensionless tensor

(6.54) |

plays the role of a macroscopic or large scale capillary number. Similarly the tensor

(6.55) |

corresponds to the macroscopic gravity number.

If the traditional normalization (6.51) is used instead of the normalization (6.48), and isotropy is assumed then the same dimensionless equations are obtained with

(6.56) |

where

For simplicity only the isotropic case will be considered from
now on, i.e. let

The unsteady state or displacement method of measuring relative permeabilities consists of monitoring the production history and pressure drop across the sample during a laboratory displacement process [2, 359, 337]. The relative permeability is obtained as the solution of an inverse problem. The inverse problem consists in matching the measured production history and pressure drop to the solutions of the multiphase flow equations (6.52) and (6.53) using the Buckley-Leverett approximation.

In the present formulation the Buckley-Leverett approximation comprises several independent assumptions. Firstly it is assumed that gravity effects are absent, which amounts to the assumption

(6.57) |

Secondly the viscous coupling terms are neglected, i.e.

(6.58) |

Finally the resulting equations

(6.59) |

(6.60) |

are further simplified by assuming that the term involving

Combining (6.57) with the traditional normalization (6.56) yields the consistency condition

(6.61) |

for the application of Buckley-Leverett theory in the determination
of relative permeabilities. It is now clear from the definition
of the macroscopic gravity number, see Eq. (6.55), that
the consistent use of Buckley-Leverett theory for the unsteady
state measurement of relative permeabilities depends strongly on
the flow regime. This is valid whether or not the capillary pressure
term

The comparison between the macroscopic and the microscopic dimensional analysis is carried out by relating the microscopic and macroscopic velocities and length scales. The macroscopic velocity is taken to be a Darcy velocity defined as (see discussion following equation (5.79))

(6.62) |

where

Using these relations between microscopic and macroscopic length and time scales together with the assumption of isotropy yields

(6.63) |

as the relationship between microscopic and macroscopic capillary numbers. Similarly one obtains

(6.64) |

for the gravity numbers. Taking the quotient gives

(6.65) |

for the macroscopic gravillary number. Note that the ratio

(6.66) | ||||

(6.67) | ||||

(6.68) |

are the macroscopic counterparts of the microscopic numbers defined in equations (6.22), (6.30) and (6.34).

An interesting way of rewriting these relationships arises from
interpreting the permeability as an effective microscopic
cross sectional area of flow, combined with the Leverett

(6.69) |

denote a microscopic length which is characteristic for the pore space transport properties. Then equations (6.63), (6.64) and (6.65) may be rewritten as

(6.70) | ||||

(6.71) | ||||

(6.72) |

where

The present section gives order of magnitude estimates for the relative importance of capillary, viscous and gravity effects at different scales in representative categories of porous media. These estimates illustrate the usefulness of the macroscopic dimensionless ratios for the problem of upscaling.

Three types of porous media are considered: high permeability
unconsolidated sand, intermediate permeability sandstone
and low permeability limestone. Representative values for

Quantity | Sand | Sandstone | Limestone |
---|---|---|---|

0.36 | 0.22 | 0.20 | |

10000 mD | 400 mD | 3 mD | |

2000 Pa |

Quantity | Definition | Unconsolidated Sand | Sandstone | Limestone | |||
---|---|---|---|---|---|---|---|

Laboratory | Reservoir | Laboratory | Reservoir | Laboratory | Reservoir | ||

To estimate the dimensionless numbers the same microscopic velocity

The first row in Table VII can be used to check the consistency of the Buckley-Leverett approximation with the traditional normalization. The consistency condition (Eq. (6.61)) is violated for unconsolidated sand and sandstones. Such a conclusion, of course, assumes that the values given in Table VI are representative for these media.

The fifth row in Table VII gives the ratio between macroscopic
and microscopic capillary numbers which according to Eq. (6.63)
is length scale dependent. The last row in Table VII compares this
ratio to the typical critical capillary number

As a consequence one expects differences between residual oil saturation

The values of the dimensionless numbers in Table VII allow an assessment
of the relative importance of the different forces for a displacement.
To illustrate this consider the values

Sand | Sandstone | Limestone | |||
---|---|---|---|---|---|

pore scale | |||||

traditional analysis | |||||

large scale | [47] | laboratory scale | |||

[48] | field scale |

Obviously, the relative importance of the different forces may change depending on the type of medium, the characteristic fluid velocities and the length scale. Perhaps this explains part of the general difficulty of scaling up from the laboratory to the reservoir scale for immiscible displacement.

The characteristic macroscopic velocities, length scales and kinematic viscosities defined respectively in equations (6.66), (6.67) and (6.68) are intrinsic physical characteristics of the porous media and the fluid displacement processes. These characteristics can be useful in applications such as estimating the width of a gravitational segregation front, the energy input required to mobilize residual oil or gravitational relaxation times.

The macroscopic gravillary number

Similarly, the macroscopic capillary number defines an intrinsic
specific action (or energy input)

The gravitational relaxation time is the time needed to return to gravitational equilibrium after its disturbance. This may be defined from the balance of gravitational forces versus the combined effect of viscous and capillary forces. Analogous to equation (6.35) for the microscopic case the dimensionless ratio becomes

(6.73) | ||||

which defines the gravitational relaxation time

(6.74) |

Estimated values are given in Table IX. They correspond to gravitational
relaxation times of roughly

Another interesting intrinsic number arises from comparing the strength of macroscopic capillary forces versus the combined effect of viscous and gravity forces

(6.75) | ||||

where

(6.76) |

is an intrinsic system specific characteristic flow rate.
The estimates for

Quantity | Sand | Sandstone | Limestone |
---|---|---|---|

In summary, the dimensional analysis of the upscaling problem
for two phase immiscible displacement suggests to normalize the
macroscopic pressure field in a way which differs from the
traditional normalization.
This gives rise to a macroscopic capillary number