T E A M 6 .

BUCKLEY-LEVERETT.

KEY DEFINITIONS.

• DISPLACEMENT EFFICIENCY.

• IMMISCIBLE DISPLACEMENT. (PHYSICAL ASSUMPTIONS).

• FRACTIONAL FLOW EQUATION.

DISPLACEMENT EFFICIENCY.

IT’S DEFINED LIKE:

IT HAS SOME VARIATIONS BETWEEN 0 Y 1.

IF 1 WE HAVEN’T OIL SATURATION IN POROSITY MEDIA.

THEORY.

• EFFICIENCY IS LIMITED BY RESIDUAL OIL SATURATION.

During the sweep of a reservoir, the displacement efficiency coincide with the

recovery efficiency (Er), if hypothetically the injected fluid contacted all

oil reservoir.

ASSUPTIONS RELATED TO BUCKLEY-LEVERETT EQUATIONS.

• BIPHASIC FLOW THROUGH THE RESERVOIR IN AN IMBIBITION SYSTEM.• THERE ARE NO SOURCES OR SINKS IN POROUS

MEDIA.• INCOMPRENSIBLE FLOW Qw=Qo• LINEAR FLOW.• POROSITY AND PERMEABILITY ARE CONSTANT.• Be neglected THE CAPILLARY PRESSURE

GRADIENT IN THE FLOW DIRECTION INTO THE RESERVOIR.

ANOTHER IMPORTANT CALCULATION.

• FRACTIONAL FLOW.

• FLUIDS ARE CONSIDERED AS INCOMPREHENSIBLE, THAT’S WHY THESE ARE ADDED.

• In fluid dynamics, the Buckley–Leverett equation is a transport equation used to model two-phase flow in porous media. The Buckley–Leverett equation or the Buckley–Leverett displacement can be interpreted as a way of incorporating the microscopic effects due to capillary pressure in two-phase flow into Darcy's law.

FLUID DYNAMICS

• The Buckley-Leverett equation is based on the principle of conservation of mass for linear flow of a fluid (water or gas) through a reservoir at constant total flow rate.

• To illustrate the derivation of the Buckley-Leverett equation, the case of water displacing oil is used.

• Note that the same equation can be developed representing the case for gas displacing oil.

BASINS

• Consider a volume element of a linear reservoir model shown in Figure. Let the thickness of the element be represented as Δx and located at a distance, x, from the inlet face of the linear model.

• A volumetric balance in terms of the water phase (assuming density of water is constant) for the element of the reservoir model can be written as:

• Expressed algebraically as:

fw = fraction of water in flow streamqt = constant total flow rate,Δt = time interval, daysf = porosity, fraction A = cross-sectional area of flow, ft2Sw = water saturation in the element.

Re-arranging Equation (2) gives:

Taking limits as ∆t → 0 and ∆x →0 yields the continuity equation:

The fractional flow of water is a function of water saturation only if fluid properties and total flow rate are constant. By application of chain rule, fw = fw (Sw), can be expressed as:

(3)

(4)

(5)

Substituting Eq. (4) into Eq. (5) and re-arranging gives:

Equation (6) gives water saturation as a function of time at a given location.

(6)

For any displacement, the distribution of water saturation is a function of both location and time. This is represented as:

(7)

(8)

Since the focus is on a fixed water saturation.

Then equation (8) becomes:

By re-arrangement:

Now substituting Eq. (11) into Eq. (6) gives:

(9)

(10)

(11)

(12)

Total flow rate is assumed to be constant, then fractional flow of water is independentof time. Hence:

• By re-arrangement and substituting becomes to the Buckley-Leverett equation:

Also called the frontal advance equation. Integration of Equation yields a useful form of the Buckley-Leverett equation:

x = distance travelled by a fixed saturation in time t, feetqt = total flow rate t = time interval, daysf = porosity, fractionA = cross-sectional area of flow, ft2(dfw/dSw)Sw = slope of the fractional flow curve at Sw.

• The equation can be used to calculate the distribution of water saturation as a function of time in a linear reservoir under water injection or aquifer influx.

FRACTIONAL FLOW THEORY ON OIL

RECOVERY CALCULATIONS

HOMOGENEOUS RESERVOIRS BUCKLEY & LEVERETT´S

METHOD

A barrel of water injected is equal to one barrel of oil displaced or produced, assuming a steady state. Thus requires the assessment of the recovered oil after the emergence of the front (breakthrough).

After the irruption into the producing wellX2=L

If we make

Distribution of water saturation: 1) in the irruption and 2) after the irruption,

on a linear injection

• At the precise moment of the irruption, Swbt = water saturation in the irruption (breakthrough) = Swf front reaches the production well and reservoir water production increases abruptly from zero to fwbt .

• This confirms the existence of the shock.

DIMENSIONLESS OIL PRODUCTION IN THE IRRUPTION (BREAKTHROUGH)

• After the irruption (breakthrough), we will produce water and oil together.

• At this time we will evaluate the oil recovery.

Substracting Swc from both sides of the equation:

Recovered oil is replaced in the porous medium by the injected water. That is equal to the average water saturation in the formation less initial water saturation Sw

m

GRAPHICAL METHOD

•For each value of Sw>Swbt is drawn tangent to the fractional flow curve.•This tangent intersects the horizontal line fw=1 at the required point Sw

m.

• The oil recovered plotted as a function of the injected water.

• If we knew the injected water flow qi, we could estimate the oil recovered vs. time.

• In this way we calculate the recoverable oil of a linear, one-dimensional porous medium.

IN THE "BREAKTHROUGH"

The oil recovered when the water breaks into the producing well is still equal to the injected water. Then, remains valid Eq 39. It is useful to express it in dimensionless form.

N pD =WiD WiD is the volume of water injected dimensionless respect to pore volume (number of pore volumes of water injected)

This procedure is applied to calculate the time of "breakthrough“

injected water flow on qiD is dimensionless volume porary.

AFTER THE "BREAKTHROUGH"

Is valid for times after the "breakthrough". But the values of Wid and [dfw/ dSw] change with time WiD

The oil recovered is plotted against the water injected in known When injected water flow qi, we could estimate the oil recovered vs. time. This time

is calculated from the equation after the "breakthrough".

Thereby calculating the recoverable oil and a porous linear

unidimensional. With this value of Np is estimated theoretical displacement efficiency

CONCLUSIONS.WE THINK THIS IS AN IMPORTANT METHOD TO PREDICT THE BEHAIVOR OF THE FLUIDS INTO THE RESERVOIR, BECAUSE IT ASSUMES SOME IMPORTANT FACTORS THAT WE’LL PROBABLY NEED IN A SIMULATION OR SOMETHING LIKE THAT.

THANK’S A LOT