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Reservoir Waterflood Simulation Using Tubular Flow Paths
Adam Smith, SPE, University of Tulsa
April 4, 2014
Executive Summary
The objective of this simulation was to model the progress of a waterflood through a reservoir layers with
simplified dimensions via a series of tubular flow paths; each with different flow rates and pore volumes. The
purpose of this task was to add complexity to previously conducted waterflooding models that assumed that the
front, moving through the reservoir, flowed through a rectangular one dimensional flow path. The simulation used
Microsoft Excel® and Schlumberger’s Eclipse Reservoir Simulator. To reduce the volume of data created when
modeling the entire reservoir, only the first layer was simulated in Excel. The waterflood characteristics found for
this layer were then compared to those modeled by Eclipse when it simulated a waterflood through all the layers
of the reservoir. Assumed in the experiment was fluid incompressibility. The agreement between proceeding
simulated values means that a proof of concept can be established and further modeling of various layers could be
effectively conducted via excel using the tubular flow path method.
The reservoir simulation was conducted using a line drive type well spacing pattern with 1864 ft. between
injector and producer. A Neuman Function as well as a Stream Function in Excel to model dimensionless pressure
differential and flow potential (which helps find % pore volume available to injection). In order to initialize the
Excel simulation injection rates into the first layer at various time steps were assumed to be those found in
Eclipse. A total injection rate of 37,300 was entered into Eclipse when modeling the entire reservoir volume. The
breakthrough time for layer one (QiBT) was found to be 187 days in Excel and 198 days in Eclipse, resulting in a
5.48% error. The cumulative oil produced from layer one at time of abandonment (economic limit) was found to
be 4,874,192 barrels in Excel and 5,401,577 barrels in Eclipse, which resulted in a 9.76% error.
Table 1: Waterflood Characteristics for Significant Reservoir Events – Layer 1 (Eclipse & Excel)
Breakthrough Economic Limit
Excel Eclipse Error Excel Eclipse Error
Time (Days) 187 198 5.48% 765 1035 26.09%
Np (bbl) 2,812,735 2,839,060 0.93% 4,874,192 5,401,577 9.76%
WOR 0.273 0.255 7.33% 19.8 19.5 1.93%
∆P (psi) N/A 4849 - N/A 4358 -
Ed 24.2% 23.7% 2.20% 41.0% 42.1% 2.66%
Ea 76.8% 73.2% 4.92% ~1 ~1 -
Ei N/A 12.2% - N/A N/A -
The Ea, and Ed mentioned above represent efficiencies of planar and volumetric oil production through the
reservoir. Both of these values were found to be extremely similar. Ei, the depth vs. distance cross-sectional
efficiency was only able to be calculated via Eclipse due to the fact that the Eclipse simulation was run for all
layers but the Excel simulation was not.
The proceeding graph shows expected tubular injection volume percentage at different stages of the
waterflood:
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Figure 1: Injection Volume Percentage in Each Tube of Layer 1 (Excel)
Introduction
With the increasing scarcity of economically feasible reservoirs that are able to produce under their own pressure
drive, the enhanced recovery of oil is becoming more and more important in today’s economy. While there are
various methods with which to increase the percentage of oil recovered from a reservoir when compared to the
original oil in place, one of the most efficient is that of the reservoir waterflood. In this method water is injected into
a formation (or group of formations) via an injection well. This water acts similarly to a piston in a gasoline engine.
The water pushes in-situ oil towards a production well due to differences in viscosity, interfacial tension, and
viscosity between the water and the oil.
Because these waterfloods can require a great deal of initial capital in before break even points are met, it
is frugal to conduct reservoir simulations that use known reservoir physical characteristics as well as the
properties of the injection fluid and oil in place. The purpose of this analysis was to conduct and compare the
efficiencies with which oil was recovered from a ten layer reservoir using two different types of simulation
software. The software – Microsoft Excel® and Schlumberger’s Eclipse Reservoir Simulator – was compared at
producer breakthrough and the economic limit for injection.
Procedure
Eclipse
The first step in modeling the waterflood was to code and initiate the simulation using Schlumberger’s
Eclipse Simulator. This involved populating the dimensions of the reservoir volume with a number of cells.
Additionally entered were reservoir characteristics by layer, such as porosity, relative permeability, and
saturations based on capillary pressures. The reservoir simulation was then begun and values for flow rate,
cumulative production (Np), displacement efficiency (Ed), as well as other characteristic quantities could be
obtained. Shown below are images of oil saturation at breakthrough in map view and cross-sectional view found
via Eclipse:
Figure 2: Map View of Oil Saturation in Layer 1 Figure 3: Cross Section of Oil Saturation in All Layers
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
0 1 2 3 4 5 6 7 8 9 10
InjectionVolume
Percentage
Tube #
Initial
Economic Limit
Breakthrough

Adam Smith 3
Excel
The first step towards simulating the tubular-flow waterflood using Excel was constructing a Neuman
Function and Stream Function that modeled differential pressure loss and flow potential. Using these functions as
well as the COUNTIFS statement in Excel, each tube’s pore volume was able to be estimated as well as the
pressure drop at various tube lengths. The functions were then combined to relate dimensionless pressure drop as
a function of injected volume (∆P vs. Qi/Qi-BT).
Figure 4: Neuman Function – Dimensionless Pressure Drop from Injector to Producer (Line Drive Pattern)
Figure 2: Stream Function Showing Reservoir “Tubes” – Pore Volumes from Injector to Producer (Line Drive Pattern)

Adam Smith 4
Figure 5: Layer 1 Dimensionless Pressure from Injector to
Producer
Figure 6: Layer 1 Pore Volume per Tube
Figure 7: Pressure Drop as a Function of Injection Volume for Layer 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0% 50% 100%
DimensionlessPressureDrop
x/L
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
18.0%
20.0%
1 2 3 4 5 6 7 8 9 10
PercentPoreVolume
Tube #

Adam Smith 5
Because the Stream Function showed a horizontal line of symmetry, it was assumed that the pressure drop
as a function of injected volume showed symmetry as well. This is the reason that only five tubes are illustrated
on the above graph (Fig 7). This graph was subsequently used to estimate the weight that each differential
distance of tube dropped injection pressure as the waterflood preceded to the production well. The following
values were estimated graphically:
Table 2: Fractional Pressure Drop Based on Fraction of Injection Volume
After the Eclipse portion of the simulation was complete as well as the Neuman and Stream Function,
injection rate per day into the top-most layer was retrieved from Eclipse and input into Excel. It was assumed that
all tubes took the exact same amount of injected fluid at time = 0. These injection values were then used in order
to calculate apparent viscosity (µapp) for each differential section of each tube independently as well as the
injected volume as a fraction of the break through volume necessary to break through at the outlet of each tube.
These values were correlated to a list of watercut values that were calculated in a previous simulation. The
calculated viscosities were multiplied by the weights that each differential section carried (due to the pressure
drop associated with each of them) and were summed. The summations were used as divisor in the following
equation which related viscosity to flow rate,in which qi is the injection rate into each tube, qref is the injection
rate into the reference tube (tube 1 was used in this simulation), µref is the apparent viscosity of the reference tube
(again tube 1), and µi is the apparent viscosity in each tube. (All q’s were in bbl’s and µ’s were in cP)
Equation 1: Flow Rate Fraction Related to Fractional Viscosity
𝑞 𝑞
𝑞 𝑞𝑞𝑞
=
µ 𝑞𝑞𝑞
µ 𝑞
Therefore the amount of water injected into each tube at a various time was changed and a new time step
was begun. An overall watercut was calculated by summing the product of injection rate and watercut and
dividing by the total injection rate:
Equation 2: Total Watercut
𝑞 𝑞−𝑞𝑞𝑞𝑞𝑞 =
∑ 𝑞 𝑞𝑞𝑞𝑞 ∗ 𝑞 𝑞−𝑞𝑞𝑞𝑞
𝑞 𝑞𝑞𝑞𝑞𝑞
Breakthrough at the production well was assumed to occur when the first increase in total watercut
occurred and economic limit occurred when total watercut became greater than 95%.
The volumetric and arealrecovery efficiency able to be calculated using Excel were then found using the
following equations:
Qi/QiBT 0 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1
FractiondP(psi) 0.28 0.1 0.05 0.02333 0.02333 0.02333 0.02333 0.02333 0.02333 0.05 0.1 0.28

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Equation 3: Volumetric Recovery Efficiency
𝑞 𝑞 = 𝑞 𝑞 𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞
+
(∆𝑞) ∗ (1 − 𝑞 𝑞)
(1 − 𝑞 𝑞 𝑞
) ∗ 𝑞𝑞 𝑞𝑞𝑞𝑞𝑞
Equation 4: Areal Sweep Efficiency
𝑞 𝑞 =
∑ [min (1,
𝑞 𝑞,𝑞
𝑞 𝑞𝑞𝑞
)] ∗ 𝑞𝑞 𝑞
𝑞=𝑞𝑞𝑞𝑞 10
𝑞=𝑞𝑞𝑞𝑞 1
min (1,
𝑞 𝑞,𝑞
𝑞 𝑞𝑞𝑞
) ∗ ∑ 𝑞𝑞 𝑞
𝑞=𝑞𝑞𝑞𝑞10
𝑞=𝑞𝑞𝑞𝑞1
Where 𝑞 𝑞 𝑞
is the layer’s initial water saturation and 𝑞𝑞 𝑞𝑞𝑞𝑞𝑞 is total layer volume in bbl’s in Equation 4. In
Equation 5 the minimum number between 1 and the injected volume fraction is taken. Additionally, in Equation 5
𝑞𝑞 𝑞 is the pore volume of each individual tube.
Finally, Np – the cumulative oil production (bbl) – was found via the following correlation:
Equation 5: Cumulative Oil Production (Np)
𝑞 𝑞 = 𝑞 𝑞0
+ 𝑞 𝑞 ∗ (1 − 𝑞 𝑞)
Where 𝑞 𝑞0
is the previously recorded cumulative production volume, 𝑞 𝑞 is the injection volume for the time
period (bbl), and 𝑞 𝑞 is the total watercut.
Excel & Eclipse
In order to compare Excel and Eclipse, values for watercut were evaluated in Eclipse and breakthrough
and economic limit times were recorded. Also recorded were values of cumulative production, water oil ratio,
pressure differential between injector and producer, and the various production efficiencies. The efficiencies were
found using a computational digitizer that evaluated saturated areas of Figures 2 & 3, comparing them to the total
areas of the illustrations. These values were then set next to corresponding Excel values and percent error was
found.
Conclusions
Through the comparison of the significant values and percentages found via Excel and Eclipse, it can be said that
Excel is effective software in simulating the injection of water through a series of reservoir “tubes”. Technically,
this method could be reused in more complex reservoirs with a greater number of tube systems; however, this
method takes a greater deal of time than a simulator such as Eclipse which doesn’t require the manual
manipulation of equations. It can also be said that one should be weary of the estimation of the time to economic
limit using Excel simulation, as it does not seem to agree well with the well regarded Eclipse simulator.

Adam Smith 7
References
Rose, Stephen C., Buckwalter, John F., Woodhall, Robert J., 1989. The Design Engineering Aspects of
Waterflooding. Book SPE. Vol. 11

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