Q913 re1 w5 lec 19

Reservoir Engineering 1 Course (1st Ed.)

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4.

Recovery Efficiency
Displacement Efficiency
Frontal Displacement
Fractional Flow Equation

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2.
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6.

Water Fractional Flow Curve
Effect of Dip Angle and Injection Rate on Fw
Reservoir Water Cut and the Water–Oil Ratio
Frontal Advance Equation
Capillary Effect
Water Saturation Profile

Parameters Affecting
on the Water Fractional Flow Curve
Therefore, the objective is to select the proper
injection scheme that could possibly reduce the
water fractional flow.
This can be achieved by investigating the effect of
The injected water viscosity,
Formation dip angle, and
Water-injection rate on the water cut.

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Reservoir Engineering 1 Course: W5L19 Fractional Flow / Frontal Advance

5

Fw: Viscosity Effect
Next slide shows the general effect of oil viscosity
on the fractional flow curve for both water-wet and
oil-wet rock systems.
This illustration reveals that regardless of the system
wettability, a higher oil viscosity results in an upward
shift (an increase) in the fractional flow curve.

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Effect of Oil Viscosity on Fw

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Fw: Water Viscosity Effect
Water viscosity:
The apparent effect of the water viscosity on the water
fractional flow is clearly indicated by following Equation.
(Higher injected water viscosities = reduction in fw (i.e.,
a downward shift)).

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Fw: Dip Angle Effect
To study the effect of the formation dip angle α and the
injection rate on the displacement efficiency, consider
the water fractional flow equation as represented by
the Equation.
Assuming a constant injection rate and realizing that
(ρw – ρo) is always positive and in order to isolate the
effect of the dip angle and injection rate on fw, the
Equation is expressed in the following simplified form:
(where the variables X and Y are a collection of different
terms that are all considered positives.)

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Effect of Dip Angle on Fw

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Updip Flow, i.e., sin(α) is Positive
When the water displaces oil updip (i.e., injection
well is located downdip), a more efficient
performance is obtained.
This improvement is due to the fact that the term X
sin(α)/iw will always remain positive, which leads to a
decrease (downward shift) in the fw curve.
A lower water-injection rate iw is desirable since the nominator
1 – [X sin(α)/iw] of the Equation will decrease with a lower
injection rate iw, resulting in an overall downward shift in the
fw curve.

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Downdip Flow, i.e., sin(α) is Negative
When the oil is displaced downdip (i.e., injection
well is located updip), the term X sin(α)/iw will
always remain negative and, therefore, the
numerator of the Equation will be 1+[X sin(α)/iw].
Which causes an increase (upward shift) in the fw curve.
It is beneficial, therefore, when injection wells are
located at the top of the structure to inject the water at
a higher injection rate to improve the displacement
efficiency.

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Downdip Flow, Counterflow
It is interesting to reexamine the Equation when
displacing the oil downdip. Combining the product
X sin(α) as C, it can be written:

The above expression shows that the possibility
exists that the water cut fw could reach a value
greater than unity (fw > 1) if:
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Downdip Flow, Counterflow (Cont.)
This could only occur when displacing the oil
downdip at a low water injection rate iw.
The resulting effect of this possibility is called a
counterflow,
Where the oil phase is moving in a direction opposite to that of
the water (i.e., oil is moving upward and the water downward).
When the water injection wells are located at the top of a tilted
formation, the injection rate must be high to avoid oil migration
to the top of the formation.

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Effect of Injection Rate
for a Horizontal Reservoir
Note that for a horizontal reservoir, i.e., sin(α) = 0,
the injection rate has no effect on the fractional
flow curve.
When the dip angle α is zero, the Equation is
reduced to the following simplified form:

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Water Cut Fw vs. Water–Oil Ratio Wor
In waterflooding calculations, the reservoir water cut fw and the
water–oil ratio WOR are both traditionally expressed in two
different units: bb/bbl and STB/STB.
The interrelationships that exist between these two parameters
are conveniently presented below, where
Qo = oil flow rate, STB/day
qo = oil flow rate, bbl/day
Qw = water flow rate, STB/day
qw = water flow rate, bbl/day
WORs = surface water–oil ratio, STB/STB
WORr = reservoir water–oil ratio, bbl/bbl
fws = surface water cut, STB/STB
fw = reservoir water cut, bbl/bbl
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I) Reservoir Fw – Reservoir WORr
Relationship

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II) Reservoir Fw – Surface WORs
Relationship

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III) Reservoir WORr – Surface WORs
Relationship

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IV) Surface Fws – Surface WORs
V) Surface Fws – Reservoir Fw
IV) Surface Fws – Surface WORs

V) Surface Fws – Reservoir Fw Relationship

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Frontal Advance Equation Concept
The fractional flow equation is used to determine
the water cut fw at any point in the reservoir,
assuming that the water saturation at the point is
known.
The question, however, is how to determine the water
saturation at this particular point.
The answer is to use the frontal advance equation.
The frontal advance equation is designed to determine the
water saturation profile in the reservoir at any give time during
water injection.

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Buckley and Leverett (1942) presented what is
recognized as the basic equation for describing twophase, immiscible displacement in a linear system.
The equation is derived based on developing a material
balance for the displacing fluid as it flows through any
given element in the porous media:
Volume entering the element – Volume leaving the
element
= change in fluid volume

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Water Flow through
a Linear Differential Element
Consider a differential element of porous media,
having a differential length dx, an area A, and a
porosity φ.

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Demonstration
During a differential time period dt, the total
volume of water entering the element is given by:
Volume of water entering the element = qt fw dt

The volume of water leaving the element has a
differentially smaller water cut (fw – dfw) and is
given by:
Volume of water leaving the element = qt (fw – dfw) dt

Subtracting the above two expressions gives the
accumulation of the water volume within the
element in terms of the differential changes of the
saturation dfw:
qt fw dt – qt (fw – dfw) dt = Aϕ (dx) (dSw)/5.615
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Velocity of Any Specified Value of Sw
Simplifying:
qt dfw dt = A (dx) (dSw)/5.615
Separating the variables gives:

Where
(υ) Sw = velocity of any specified value of Sw, ft/day
A = cross-sectional area, ft2
qt = total flow rate (oil + water), bbl/day
(dfw/dSw)Sw = slope of the fw vs. Sw curve at Sw
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Velocity of Any Specified Value of Sw
(Cont.)
The above relationship suggests that the velocity of
any specific water saturation Sw is directly
proportional to the value of the slope of the fw vs.
Sw curve, evaluated at Sw.
Note that for two-phase flow, the total flow rate qt
is essentially equal to the injection rate iw, or:

Where
iw = water injection rate, bbl/day.
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Total Distance Any Specified Sw
Travels during t
To calculate the total distance any specified water saturation will
travel during a total time t, the Equation must be integrated:

Above Equation can also be expressed in terms of total volume of
water injected by recognizing that under a constant waterinjection rate, the cumulative water injected is given by:
Where

 iw = water injection rate, bbl/day
 Winj = cumulative water injected, bbl
 t = time, day
 (x)Sw = distance from the injection for any given saturation Sw, ft

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Total Distance Any Specified Sw
Travels during t (Cont.)
Above Equation also suggests that the position of
any value of water saturation Sw at given
cumulative water injected Winj is proportional to
the slope (dfw/dSw) for this particular Sw.
At any given time t, the water saturation profile can
be plotted by simply determining the slope of the
fw curve at each selected saturation and calculating
the position of Sw from above Equation.
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Mathematical Difficulty
in the Equation
Next slide shows the typical S shape of the fw curve
and its derivative curve.
However, a mathematical difficulty arises when
using the derivative curve to construct the water
saturation profile at any given time.
Suppose we want to calculate the positions of two
different saturations (shown in the Figure as
saturations A and B) after Winj barrels of water
have been injected in the reservoir. Applying the
Equation gives:
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Fw Curve with Its Saturation Derivative
Curve

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Capillary Pressure Gradient Term
Figure indicates that both derivatives are identical,
i.e., (dfw/dSw) A = (dfw/dSw) B, which implies that
multiple water saturations can coexist at the same
position—but this is physically impossible.
Buckley and Leverett (1942) recognized the physical
impossibility of such a condition.
They pointed out that this apparent problem is due to
the neglect of the capillary pressure gradient term in the
fractional flow equation. This capillary term is given by:

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Effect of the
Capillary Term on the Fw Curve

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Graphical Relationship Correction
Including the above capillary term when
constructing the fractional flow curve would
produce a graphical relationship that is
characterized by the following two segments of
lines, as shown in previous slide:
A straight line segment with a constant slope of
(dfw/dSw)Swf from Swc to Swf
A concaving curve with decreasing slopes from Swf to (1
– Sor)

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Velocity at Lower Values of Sw
Terwilliger et al. (1951) found that at the lower
range of water saturations between Swc and Swf, all
saturations move at the same velocity as a function
of time and distance.

We can also conclude that all saturations in this
particular range will travel the same distance x at
any particular time, as given below:

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Stabilized Vs. Nonstabilized Zone
The result is that the water saturation profile will
maintain a constant shape over the range of
saturations between Swc and Swf with time.
Terwilliger and his coauthors termed the reservoirflooded zone with this range of saturations the
stabilized zone.

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Stabilized vs. Nonstabilized Zone
(Cont.)
They define the stabilized zone as that particular
saturation interval (i.e., Swc to Swf) where all points
of saturation travel at the same velocity.
The authors also identified another saturation zone
between Swf and (1 – Sor), where the velocity of
any water saturation is variable.
They termed this zone the nonstabilized zone.

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Water Saturation Profile
as a Function of Distance and Time
Figure
illustrates the
concept of the
stabilized
zone.

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Water Saturation at the Front
Experimental core flood data show that the actual
water saturation profile during water flooding is
similar to that of previous Figure.
There is a distinct front, or shock front, at which the
water saturation abruptly increases from Swc to Swf.
Behind the flood front there is a gradual increase in
saturations from Swf up to the maximum value of 1 –
Sor.

Therefore, the saturation Swf is called the water
saturation at the front or, alternatively, the water
saturation of the stabilized zone.
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1. Ahmed, T. (2006). Reservoir engineering
handbook (Gulf Professional Publishing). Ch14

1. Welge Analysis
2. Breakthrough
3. Average Water Saturation

Q913 re1 w5 lec 19

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