q913 re1 w5 lec 20

Reservoir Engineering 1 Course (1st Ed.)

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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

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 5

Water Saturation Profile 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.


Water Saturation Profile 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).


Fractional Flow Curve


Water Saturation Profile 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.


Fluid Distributions at Different Times


Mathematical Derivation DeterminationSome 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.


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:


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:


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


Cumulative Water Injected at BreakthroughAssuming a constant water-injection rate, the

cumulative water injected at breakthrough is calculated from:

Where WiBT = cumulative water injected at breakthrough, bbl


Cumulative Water Injected 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

breakthroughPV = total flood pattern pore volume, bbl


Cumulative Water Injected at Breakthrough (Cont.)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:


Cumulative Water Injected at Breakthrough (Cont.)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


Average Water Saturation at Breakthrough


Average Water Saturation 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.


Tangent from Swi


Average Water Saturation 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.


Average Water Saturation before Breakthrough


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.


Average Water Saturation after Breakthrough


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


Determination of S-W after Breakthrough


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.


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:


Cumulative Water Injected

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, bblEA = areal sweep efficiencyEV = vertical sweep efficiency

5. For a constant injection rate iw, the total time t to inject Winj barrels of water is given by:


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, QoWater production rate, QwSurface water–oil ratio, WORsCumulative oil production, NpRecovery factor, RFCumulative water production, WpCumulative water injected, WinjWater-injection pressure, pinjWater-injection rate, iw


Sample of Performance Curves


Sample of Performance Curves (Cont.)


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

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