The principle of superposition states that the total pressure drop at any point in a reservoir is the sum of the pressure drops caused by all wells in the reservoir. This allows modeling of multiple wells by treating each as a separate pressure component that can be summed. Superposition can also model bounded reservoirs using "image wells" to represent boundaries. For variable flow rates, each rate change is modeled as a separate well component based on the rate difference and time since the change. Horner developed an approximation that calculates an equivalent pseudoproducing time to simplify modeling of variable rate wells.

1. Petro 450
Class 4
Principle of Superposition
The Ei function the most useful of the approximations assumes a single
well with a constant flow rate starting at time zero. The application of
the principle of superposition can remove some of these restrictions.
It is basically states that the total pressure drop at any point in the
reservoir is the sum of the pressure drops caused by all the wells in the
reservoir.
The pressure drop in well A is the sum of the pressure drops caused by
all three wells.
𝑝𝑖 − 𝑝 𝑤𝑓 𝑡𝑜𝑡𝑎𝑙 𝐴
= 𝑝𝑖 − 𝑝 𝐴 + 𝑝𝑖 − 𝑝 𝐵 + 𝑝𝑖 − 𝑝 𝐶
In terms of the flow equation

2. 𝑝𝑖 − 𝑝 𝑤𝑓 𝑡𝑜𝑡𝑎𝑙 𝐴
= −70.6
𝑞 𝐴 𝐵𝜇
𝑘ℎ
𝑙𝑛
1688∅𝜇𝑐𝑡 𝑟 𝑤𝐴
2
𝑘𝑡
− 2𝑠𝐴
− 70.6
𝑞 𝐴 𝐵𝜇
𝑘ℎ
𝐸𝑖
−948∅𝜇𝑐𝑡 𝑟𝐴𝐵
2
𝑘𝑡
− 70.6
𝑞 𝐴 𝐵𝜇
𝑘ℎ
𝐸𝑖
−948∅𝜇𝑐𝑡 𝑟𝐴𝐶
2
𝑘𝑡
Using this method we can model any number of wells in an infinite
reservoir. This is the basis of pulse or interference testing.
The next application is to model wells in a bounded reservoir.

3. A well with a no flow boundary (sealing fault) a distant L from the well
can be modeled by having an image well 2L from the well. The
pressure gradient is zero at the no flow boundary, which means there is
no flow.
𝑝𝑖 − 𝑝 𝑤𝑓 =
−70.6
𝑞𝐵𝜇
𝑘ℎ
𝑙𝑛
1688∅𝜇 𝑐 𝑡 𝑟 𝑤
2
𝑘𝑡
− 2𝑠 − 70.6
𝑞𝐵𝜇
𝑘ℎ
𝐸𝑖
−948∅𝜇 𝑐 𝑡2𝐿2
𝑘𝑡
This method can be used to model wells 1) pressure distribution for a
well between 2 boundaries intersecting at 90 degrees, 2) well between
parallel boundaries, 3) wells in various locations surrounded on all sides
by boundaries. This last one is used to calculate the average reservoir
pressure.
Superposition is used to get around the assumption of a constant rate.
The build up test has a constant q of 0, but a constant rate before the
shut in is required.

4. For multiple flow rates, each change of rate can be modeled by having
a well for each of the rates.
So for the first rate q1 starts at time to
∆𝑝1 = 𝑝𝑖 − 𝑝 𝑤𝑓 1
− 70.6
𝑞1 𝐵𝜇
𝑘ℎ
𝑙𝑛
1688∅𝜇 𝑐 𝑡 𝑟 𝑤
2
𝑘𝑡
− 2𝑠
For the second rate
∆𝑝2 = 𝑝𝑖 − 𝑝 𝑤𝑓 2
= −70.6
𝑞2−𝑞1 𝐵𝜇
𝑘ℎ
𝐸𝑖
−948∅𝜇 𝑐 𝑡 𝑟 𝑤
2
𝑘 𝑡−𝑡1
− 2𝑠
For the third rate
∆𝑝3 = 𝑝𝑖 − 𝑝 𝑤𝑓 2
= −70.6
𝑞3−𝑞2 𝐵𝜇
𝑘ℎ
𝐸𝑖
−948∅𝜇 𝑐 𝑡 𝑟 𝑤
2
𝑘 𝑡 −𝑡2
− 2𝑠
𝑝𝑖 − 𝑝 𝑤𝑓 = ∆𝑝 1 + ∆𝑝 2 + ∆𝑝 3

5. As you can see if there is a large number of rate changes, which is very
likely in producing wells, these calculations can be very tedious.
Horner came up with an approximation that can be used for variable
rate wells. A pseudoproducing time is calculated by using the last flow
rate and the total production of the well.
𝑡 𝑝 = 𝑁
𝑞𝑙𝑎𝑠𝑡 𝑖𝑛 ℎ𝑜𝑢𝑟𝑠
So the flow equation becomes
𝑝𝑖 − 𝑝 𝑤𝑓 = −70.6
𝑞 𝑙𝑎𝑠𝑡 𝐵𝜇
𝑘ℎ
𝐸𝑖
−948∅𝜇 𝑐 𝑡 𝑟 𝑤
2
𝑘𝑡 𝑝