Q913 re1 w5 lec 20

Reservoir Engineering 1 Course (1st Ed.)

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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

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

Welge Analysis Concept
Welge (1952) showed that by drawing a straight
line from Swc (or from Swi if it is different from
Swc) tangent to the fractional flow curve, the
saturation value at the tangent point is equivalent
to that at the front Swf.
The coordinate of the point of tangency represents
also the value of the water cut at the leading edge
of the water front fwf.

2013 H. AlamiNia

Reservoir Engineering 1 Course: W5L20 Welge Analysis / Breakthrough

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at Any Given Time
From the above discussion, the water saturation
profile at any given time t1 can be easily developed
as follows:
Step 1. Ignoring the capillary pressure term, construct
the fractional flow curve, i.e., fw vs. Sw.
Step 2. Draw a straight-line tangent from Swi to the
curve.
Step 3. Identify the point of tangency and read off the
values of Swf and fwf.
Step 4. Calculate graphically the slope of the tangent as
(dfw/dSw)Swf.
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at Any Given Time (Cont.)
Step 5. Calculate the distance of the leading edge of the
water front from the injection well by using following
Equation:

Step 6. Select several values for water saturation Sw
greater than Swf and
determine (dfw/dSw) Sw by graphically drawing a tangent to
the fw curve at each selected water saturation
• (as shown in next Figure).

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Fractional Flow Curve

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at Any Given Time (Cont.)
Step 7. Calculate the distance from the injection well to
each selected saturation by applying following Equation:

Step 8. Establish the water saturation profile after t1
days by plotting results obtained in step 7.
Step 9. Select a new time t2 and repeat steps 5 through
7 to generate a family of water saturation profiles as
shown schematically in next Figure.

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Fluid Distributions at Different Times

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Mathematical Derivation
Determination
Some erratic values of (dfw/dSw)Sw might result
when determining the slope graphically at different
saturations.
A better way is to determine the derivative
mathematically by recognizing that the relative
permeability ratio (kro/krw) can be expressed by:

Notice that the slope b in the above expression has a negative
value.

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Mathematical Derivation
Determination (Cont.)

The derivative of (dfw/dSw)Sw may be obtained
mathematically by differentiating the above
equation with respect to Sw to give:

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Breakthrough Determination
The water front (leading edge) will eventually reach
the production well and water breakthrough
occurs.
At water breakthrough, the leading edge of the water
front would have traveled exactly the entire distance
between the two wells.

Therefore, to determine the time to breakthrough,
tBT, simply set (x)Swf equal to the distance between
the injector and producer L and solve for the time:

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Time to Breakthrough
Note that the pore volume (PV) is given by:
Combining the above two expressions and solving
for the time to breakthrough tBT gives:

Where tBT = time to breakthrough, day
PV = total flood pattern pore volume, bbl
L = distance between the injector and producer, ft
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Cumulative Water Injected
at Breakthrough
Assuming a constant water-injection rate, the
cumulative water injected at breakthrough is
calculated from:

Where WiBT = cumulative water injected at
breakthrough, bbl

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at Breakthrough (Cont.)
It is convenient to express the cumulative water
injected in terms of pore volumes injected, i.e., dividing
Winj by the reservoir total pore volume.
Conventionally, Qi refers to the total pore volumes of water
injected.

Qi at breakthrough is:

Where
QiBT = cumulative pore volumes of water injected at
breakthrough
PV = total flood pattern pore volume, bbl
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A further discussion is needed to better understand
the significance of the Buckley and Leverett (1942)
frontal advance theory.
Cumulative water injected at breakthrough, is given
by:
If the tangent to the fractional flow curve is
extrapolated to fw = 1 with a corresponding water
saturation of S*w, then the slope of the tangent can
be calculated numerically as:
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Combining the above two expressions gives:
The above equation suggests that the water saturation
value denoted as S*w must be the average water
saturation at breakthrough, or:
Where S-wBT= average water saturation in the
reservoir at breakthrough
PV = flood pattern pore volume, bbl
WiBT = cumulative water injected at breakthrough, bbl
Swi = initial water saturation
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Average Water Saturation
at Breakthrough

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at BT Considerations
Two important points must be considered when
determining S-wBT:
1. When drawing the tangent, the line must be
originated from the initial water saturation Swi if it is
different from the connate water saturation Swc, as
shown in next slide.

2. When considering the areal sweep efficiency EA
and vertical sweep efficiency EV, the Equation
should be expressed as:
Where EABT and EVBT are the areal and vertical
sweep efficiencies at breakthrough.
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Tangent from Swi

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in the Swept Area
 Note that the average water saturation in the
swept area would remain constant with a value of
until breakthrough occurs, as illustrated in following
Figure.
At the time of breakthrough, the flood front
saturation Swf reaches the producing well and the
water cut increases suddenly from zero to fwf.
At breakthrough, Swf and fwf are designated and fwBT.

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before Breakthrough

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Sw and Fw after Breakthrough
After breakthrough, the water saturation and the
water cut at the producing well gradually increase
with continuous injection of water, as shown in
following Figure.
Traditionally, the produced well is designated as
well 2 and, therefore, the water saturation and
water cut at the producing well are denoted as Sw2
and fw2, respectively.

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after Breakthrough

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Welge Analysis
Welge (1952) illustrated that when the water
saturation at the producing well reaches any
assumed value Sw2 after breakthrough, the
fractional flow curve can be used to determine:
Producing water cut fw2
Average water saturation in the reservoir w2
Cumulative water injected in pore volumes, i.e., Qi

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Determination of S-W
after Breakthrough

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Welge Analysis (Cont.)
As shown in previous Figure, the author pointed
out that drawing a tangent to the fractional flow
curve at any assumed value of Sw2 greater than Swf
has the following properties:
1. The value of the fractional flow at the point of
tangency corresponds to the well producing water cut
fw2, as expressed in bbl/bbl.

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Cumulative Pore Volumes
of Water Injected
2. The saturation at which the tangent intersects fw = 1
is the average water saturation w2 in the swept area.
Mathematically, the average water saturation is
determined from:

3. The reciprocal of the slope of the tangent is defined as
the cumulative pore volumes of water injected Qi at the
time when the water saturation reaches Sw2 at the
producing well, or:

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4. The cumulative water injected when the water
saturation at the producing well reaches Sw2 is given by:

Where:
Winj = cumulative water injected, bbl
(PV) = pattern pore volume, bbl
EA = areal sweep efficiency
EV = vertical sweep efficiency

5. For a constant injection rate iw, the total time t
to inject Winj barrels of water is given by:
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Oil Recovery Calculations
The main objective of performing oil recovery calculations is
to generate a set of performance curves under a specific
water-injection scenario.
A set of performance curves is defined as the graphical
presentation of the time-related oil recovery calculations in
terms of:
 Oil production rate, Qo
 Water production rate, Qw
 Surface water–oil ratio, WORs
 Cumulative oil production, Np
 Recovery factor, RF
 Cumulative water production, Wp
 Cumulative water injected, Winj
 Water-injection pressure, pinj
 Water-injection rate, iw

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Sample of Performance Curves

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Sample of Performance Curves (Cont.)

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1. Ahmed, T. (2006). Reservoir engineering
handbook (Gulf Professional Publishing). Ch14

Q913 re1 w5 lec 20

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