## Fundamentals of Fluid Flow in Porous Media

*Chapter 4*

# Immiscible Displacement

# 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:

Where,

*q _{i}* = linear injection rate,

*x* = length,

*y* = thickness,

*Δρ* = density difference

*k _{x} , k_{y}* = effective permeabilities,

*μ _{o} , μ_{d}* = fluid viscosities,

*σ* = interfacial tension,

*∅* = contact angle,

*g* = gravitational constant,

*R _{a}* = directional permeability constant,

*R _{b}* = mobility ratio,

*R _{c}* = viscous to capillary forces constant,

*R _{d}* = 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:

Where,

*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,

*k _{H}* = permeability of the model in x and y directions,

*k _{v}* = permeability of the model in the z direction,

*μ* = fluid viscosity,

*μ _{av}* = average viscosity

*[(*

*μ*

_{1}+*μ*

_{2}) / 2],*ρ* = density,

*Δρ* = density difference,

*h* = height of rectangular models or diameter of cylindrical models.

Their experiments indicated that the effect of diffusion becomes significant, starting from a sharp interface, when the following is true,

Where,

*D _{e}* = effective diffusion coefficient of fluid = 0.95 x 10

^{-5}cm

^{2}/ s,

*b* = width of rectangular models.

- The effect of gravity on the displacement of a fluid from a horizontal linear model by a fluid of equal viscosity but different density is described by the following equation:

Where,

*U* = total fluid flow,

*x* = distance from outlet to initial position of interface,

*E _{bt}* = breakthrough efficiency (volume of original fluid recovered at the breakthrough divided by the volume of the injected fluid).

- When steady-state circulation is established in a closed vertical model, the advance of the more viscous fluid is given by the equation given below. However, the measured velocity may be influenced by fluid mixing.

- The maximum stabilized zone length in an inclined model is
*2x*_{cr}sin*∝*, where*∝*is the inclination of the x-axis with the horizontal and*x*is the critical distance. The stabilized zone length may exceed the minimum by a substantial amount if the initial interface is sharp._{cr} - When calculating the effect of diffusion, the three-dimensional nature of the motion can be an important factor. This is of particular importance when the height is small compared to the width (presumably always true in radial systems).

Slobad and Howlett (1964) examined the effects of gravity segregation in laboratory studies of miscible displacement in vertical unconsolidated porous media. Fluids of varying viscosity and density were injected into the top of the core and moved at constant rates of twenty-five, fifty or one hundred feet per day using a positive displacement pump. The refractive index was measured to determine the composition of the efflux. The following cases were examined:

*μ*_{1}/*μ*favourable,_{2}*Δρ*favourable*μ*_{1}/*μ*favourable,_{2}*Δρ*unfavourable*μ*_{1}/*μ*unfavourable,_{2}*Δρ*favourable*μ*_{1}/*μ*unfavourable,_{2}*Δρ*unfavourable

They conducted their experiments in a vertical core of unconsolidated sand, measuring four feet in length and two inches in diameter ( *∅* = 0.37 , *k* = 18 Darcies ), and they demonstrated the effects of the four cases on the mixing zone length. They also discussed the effect of other parameters on the mixing zone length, including: molecular diffusion, permeability variations, convective mixing, and fingering and adverse gravity segregation. For their system, they showed the length of the mixing zone can be as short as 0.06 – 0.07 PV under favorable conditions. It was found that gravity will tend to increase the length of the mixing zone under favorable density conditions. In the case of a favorable viscosity ratio (viscosity of displaced fluid less than viscosity of displacing fluid) a favorable density difference was found to reduce the mixing zone length and with sufficient time, allowed gravity segregation to sharpen the zone. When the viscosity ratio was unfavorable the mixing zones were longer due to fingering, while favorable density differences drastically reduced the mixing zone length. Overall, the mixing zone was found to be a function of the relative magnitude of the viscous forces to the gravity forces.

Hiatt (1968) outlined a general fluid displacement equation for two-phase, incompressible vertical flow through porous media and discussed the effect of fluid drive, gravity and capillary pressure. The equations of this report are not mathematically solved and the results are not verified experimentally.

Thompson and Mungan (1969) performed miscible displacement tests on artificially fractured sandstones in order to examine the effect of various reservoir and process parameters on oil recovery. They discussed the importance of sub-vertical fractures, matrix permeability, displacement rate and fracture density in oil recovery and they stated that the magnitude of the fractured permeability, the fracture orientation, the core length and the connate water have little effect on the process.

Sheffield (1969) used matrix techniques and computer algorithms to solve the differential equations for one and two-dimensional reservoir performance predictions with all forces present. He did not provide any experimental support for his results. Kuo *et al*. (1970) gave a theoretical prediction of oil production by gravity drainage for a steam soak process. However, their results were not experimentally verified. Spivak (1974) used a reservoir simulator to examine the effect of gravity segregation in two-phase displacement processes. His work dealt with a wide variety of problems but it also lacked experimental verification.

Dumore and Schols (1974) performed capillary pressure experiments in order to examine whether mercury porosimetry curves can be transformed to drainage capillary pressure curves. They found that with the aid of measured surface tension, permeability and porosity, various fluid combinations could give a unique dimensionless capillary pressure curve. They also found that the mercury-air interfacial tension is time dependent. For both capillary drainage and gravity drainage they observed the following:

**The presence of connate water in the permeable medium led to very low residual oil saturations in the rock-gas-oil systems used by the authors.****The low oil saturations obtained were independent of whether or not the oil phase spreads on water in the presence of gas.****The drainage rate of oil in the presence of connate water and gas is possibly film flow, after the relatively short period in which the main oil production occurs.****The contribution of film flow to oil production in secondary gas caps is thought to be generally negligible in the lifetime of the reservoir. It may play a role, however, when film flow takes place over short distances in very permeable rock. In primary gas caps, very low oil saturation may occur where film flow of oil may have been active for geological periods of time,**

Lefebvre Du Prey (1978) examined gravity and capillary effects on imbibition in porous media. He used a scale-up procedure and expressed his results as functions of the scale-up parameters.

Hagoort (1980) stated that gravity drainage could be a very effective way of production. In his theoretical discussion he covered the importance of relative permeability to oil. Additionally, he performed relative permeability studies in a centrifuge in order to get an exact representation of field conditions.

Joshi and Threlkeld (1985) conducted an experimental study of a thermally aided gravity drainage mechanism using a horizontal well. The laboratory apparatus was a glass bead packed rectangular box with one transparent wall, allowing for flow visualization. For comparison reasons, various combinations of vertical and horizontal wells were tested. Results indicated that the horizontal well pair (both injecting and producing wells being horizontal) yielded maximum oil recovery. Their experimental data was used to justify the theoretical and experimental work of Butler *et al.* (1981) and Butler and Stephens (1981) where steam was used to assist the gravity drainage of heavy oil. The temperature distribution in steam-assisted gravity drainage using horizontal wells was also investigated in this work and by Chung and Butler (1988).

**References**

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