Fundamentals of Fluid Flow in Porous Media
Vertical and Volumetric Sweep Efficiencies: Review of Gravity Related Oil Recovery Studies
There are a number of studies in the literature that deal with the effect of gravity forces on displacement behavior. Emphasis has been given to gravity drainage, i.e. the downward self – propulsion of oil in the reservoir rock (Lewis, 1944). This phenomenon was studied extensively from 1940 to 1965; results from the first gravity drainage study were presented by Stahl et al. (1943). In this study, air was used to displace various liquids from a column containing Wilcox Sand. The results showed the dependence of liquid saturation on column height at both equilibrium and dynamic conditions. They also showed that the drainage rates were temperature dependent. Lewis (1944) gave an extensive review of the general aspects of gravity drainage and discussed conditions that favor gravity drainage. He also reported results from some field studies. Terwilliger et al. (1951) performed gravity drainage experiments on silica sand using brine and gas. The main difference between the work of Terwilliger et al. (1951) and that of Stahl et al. (1943) is that Terwilliger et al. (1951) conducted experiments with continuous production of the wetting phase. Stahl et al. (1943) however, halted the experiment momentarily for sampling. As a result, the work of Terwilliger et al. (1951) yielded more typical Buckley-Leverett plots. They applied the Buckley-Leverett approach in order to model their displacement tests and succeeded in matching their experimental results to this model.
Marx (1956) describes a method for predicting the complete gravity drainage characteristics of arbitrary, long columns from centrifuge drainage measurements on reconstituted core samples. Marx claims that oil residuals corresponding to hundreds of years of normal gravity depletion can be obtained in a few hours on the centrifuge. The flow rate due to gravity at any stage of the depletion process may be determined from the time correlation obtained in this study.
Craig et al. (1957) used scaled reservoir models to study gravity segregation in frontal drives. The scaling criteria were as follows:
qi = linear injection rate,
x = length,
y = thickness,
Δρ = density difference
kx , ky = effective permeabilities,
μo , μd = fluid viscosities,
σ = interfacial tension,
∅ = contact angle,
g = gravitational constant,
Ra = directional permeability constant,
Rb = mobility ratio,
Rc = viscous to capillary forces constant,
Rd = viscous to gravity forces constant.
The tests included unconsolidated and consolidated media in either a five spot pattern or linear pattern and variable length. Unconsolidated stratified systems were also used. In these studies, the experiments were discontinued after breakthrough of the injected fluid. From this Craig et al. (1957) concluded:
- For linear gas or water injection operation in flat formations of uniform rock texture, segregation of the fluids, due to gravity effects, can result in oil recoveries at breakthrough as low as twenty percent of those otherwise expected.
- In five spot injection operations in flat uniform systems, the oil recoveries at breakthrough can be as low as forty percent of those predicted by methods that assume negligible gravity effects.
- In secondary recovery operations in stratified rock formations, the oil recovery at breakthrough may be affected to a greater degree by fluid segregation due to variations in rock properties, rather than by gravity effects.
- The magnitude of segregation of the fluids due to gravity is influenced by the average injection rate, rather than day-to-day or week-to-week variations.
Hovanessian and Fayers (1961) used a finite difference approach to solve the one-dimensional displacement equation for a homogeneous porous medium, including the effects of gravity and capillary forces. The authors state that the inclusion of capillary effects eliminates the triple valued Buckley-Leverett saturation profiles.
Templeton et al. (1962) examined the gravity counter-flow segregation in a closed system. They completed immiscible displacement studies in glass beads and tried to calculate relative permeabilities by applying Darcy’s law. Moreover, they observed saturation distributions as a function of time with a method similar to the one of Terwilliger et al. (1951). They concluded that Darcy’s equations, if modified for the separate phases, are generally valid for counter-flow due to density differences.
Cook (1962) performed a mathematical analysis of gravity segregation process during natural depletion conditions. He developed and solved the differential equations for two types of flows: 1) A “distributed flow” where vertical permeability in the reservoir is zero and fluids flow only in the dip direction while uniformly distributed over a hypothetical sand thickness, and 2) A “segregated flow” where vertical permeability is sufficient to permit gas to segregate against the sand top before proceeding up dip.
Gardner et al. (1962) performed a series of experiments on gravity segregation of miscible fluids in linear models with both rectangular and circular cross-sections. They concluded that:
- A sharp interface between fluids with different densities and different viscosities moving in a closed horizontal linear model tends to become straight and tilts in accordance with the following equation:
x = distance of leading edge of the viscous fluid from initial position of the interface,
t = time measured from the instant when the interface is perpendicular to the length of the model,
kH = permeability of the model in x and y directions,
kv = permeability of the model in the z direction,
μ = fluid viscosity,
μav = average viscosity