q913 re1 w5 lec 19

Reservoir Engineering 1 Course (1st Ed.)

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http://www.about.me/AlamiNia

1. Recovery Efficiency

2. Displacement Efficiency

3. Frontal Displacement

4. Fractional Flow Equation

1. Water Fractional Flow Curve

2. Effect of Dip Angle and Injection Rate on Fw

3. Reservoir Water Cut and the Water–Oil Ratio

4. Frontal Advance Equation

5. Capillary Effect

6. Water Saturation Profile

Parameters Affecting on the Water Fractional Flow CurveTherefore, 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.

2013 H. AlamiNia 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.


Effect of Oil Viscosity on Fw


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)).


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.)


Effect of Dip Angle on Fw


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.


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.


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:


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.


Effect of Injection Rate for a Horizontal ReservoirNote 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:


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


I) Reservoir Fw – Reservoir WORr Relationship


II) Reservoir Fw – Surface WORs Relationship


III) Reservoir WORr – Surface WORs Relationship


IV) Surface Fws – Surface WORsV) Surface Fws – Reservoir FwIV) Surface Fws – Surface WORs

V) Surface Fws – Reservoir Fw Relationship


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.


Frontal Advance Equation

Buckley and Leverett (1942) presented what is recognized as the basic equation for describing two-phase, 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


Water Flow through a Linear Differential ElementConsider a differential element of porous media,

having a differential length dx, an area A, and a porosity φ.


Frontal Advance Equation 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


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/dayA = cross-sectional area, ft2qt = total flow rate (oil + water), bbl/day(dfw/dSw)Sw = slope of the fw vs. Sw curve at Sw


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.


Total Distance Any Specified Sw Travels during tTo 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 water-injection rate, the cumulative water injected is given by:

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


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.


Mathematical Difficulty in the EquationNext 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:


Fw Curve with Its Saturation Derivative Curve


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:


Effect of the Capillary Term on the Fw Curve


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)


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:


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 reservoir-flooded zone with this range of saturations the stabilized zone.


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.


Water Saturation Profile as a Function of Distance and TimeFigure

illustrates the concept of the stabilized zone.


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.


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

Education

water flow rate

water fractional flow

oil flow rate

downdip flow

effect of injection

updip flow

water injection wells

fw reservoir water