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Dynamic Data Analysis – v5.12.01 - © KAPPA 1988-2017
Chapte
r 4 – R ate Transient Analysis (RTA)- p155/743
In case of RTA deconvolution, the target is the measured production data while the input are
features, which are functions of pressure and time. Mathematically, the features can be written
as:
x
(i)
=
[
∑(∆P
(j)
− ∆P
(j−1)
)
i−1
j=1
∑(∆P
(j)
− ∆P
(j−1)
)
i−1
j=1
log(t
(i)
− t
(j)
)
∑(∆P
(j)
− ∆P
(j−1)
)
i−1
j=1
(t
(i)
− t
(j)
)
∑(∆P
(j)
− ∆P
(j−1)
)
i−1
j=1
√(t
(i)
− t
(j)
)
]
where x
(i)
above contains four features, defined at time i. In the above example, the first
feature represents rate as a superposition of pressure drop changes, the second feature
represents infinite acting radial flow, and the third represents wellbore storage and pseudo-
steady state flow while the fourth feature represents linear flow.
In this approach, the flow rate q
(i)
at time i is represented to be a linear combination of the
features x
(i)
:
q
(i)
= θ
T
x
(i)
+ ε
(i)
, i = 1,2,3, … , n
where θ is a N
f
dimensional vector with unknown constants (N
f
being the number of features
employed), ε captures the measurement error and other discrepancies and n is the number of
flow rate and pressure measurements.
In matrix form, this can be represented as:
= θ
T
+ ε
where q is a n-dimensional vector and X is a n x N
f
matrix.
Assuming that the mean of the error is zero, the exercise then becomes that of minimizing the
mean-square error function:
J(θ) =
1
2
∑(θ
T
x
(i)
− q
(i)
)
2
n
i=1
Since this is a linear system of equations, the unknown θ can be solved directly by:
θ = (X
T
X)
−1
X
T
q
4.E.6.c
Model Regularization
One danger in data mining methods is to over fit the data, which degraded the predictive
capability of the model. Ridge regression (a type of model regularization technique) is widely
used to address the overfitting issue by reducing the prediction variance. Instead of minimizing
the function J(θ) above, ridge regression minimizes: