Objectives

The objective of the task is to compare each shut-in with a reference one. The result of the comparison is the consistency between the two shut-ins along with permeability and skin changes between the reference and the current shut-in.

The consistency value is a measure of the distance on the loglog plot between the reference shut-in and the current shut-in, after it has been translated to provide a best fit to the reference shut-in. A perfect fit between the curves results in a consistency of 1. As the fit becomes worse, the consistency value decreases.

In addition the task creates loglog plots with derivatives comparison for each shut-in. The plots allow an engineer to evaluate the quality of the curves matching.

Theory

About the curves translation and permeability multiplier

From type curve matching theory it is known that actual data curves (both dp and p’) have exactly the same shape on loglog plot as the dimensionless theoretical solution provided the data correspond to the model. The dimensional and dimensionless curves are shifted by constants relative to each other. The ordinates are shifted by the amount of $\log \frac{2\pi kh}{qB\mu }$ and the abscissas are shifted by the amount of $\log \frac{k}{\phi \mu c_{\mathit{t}}{r_w}^2}$.

From the above it is evident that if we have two situations with different permeabilities, $k_1$ and $k_2$, and other parameters being equal, the corresponding log curves will be shifted diagonally relatively to each other.

If $k_2=multiplier ·k_1$, then the curves are shifted by the amount of $\log \left(multiplier\right)$ both horizontally and vertically.

About the distance in regression cost function and consistency

The goal of the regression in the user task is to find a permeability multiplier that provides the best overlay of the curve with the reference one. The best overlay is equivalent to the minimum distance and maximum consistency between the two curves.

There are two possible options in the user task, viz. vertical distance consistency check and diagonal distance consistency check.

The vertical distance consistency check is based on the classical formula of vertical distance (Lp norm with p=0.5 to account for outliers): $Vertical distance ={\left[\frac{1}{N}\sum _{\mathrm{i}=1}^{\mathrm{N}}\sqrt{\left|{\mathrm{y}}_{\mathrm{ref} \mathrm{i} }- {\mathrm{y}}_{\mathrm{i} }\right|}\right]}^2$

where:

This approach works well in cases where two curves have similar shapes. However, if the curves are different due to noisy data or different flow regimes in early or/and late data the above approach can fail, often due to the inability to consistently define a $t_{min}$.

The alternative approach uses a diagonal distance that is consistent with the diagonal shift of a curve. Outliers, characterized as either noise or different flow regimes/behaviors, are accounted for in two manners.  First, weights associated with each point in the loglog curve are calculated by comparing the gradients of the loglog curves at the intersection with the reference curve in the diagonal direction (see below, right). Similar gradients will have higher weights, as determined by the dot-product. Second, an iterative weighted least squares approach performs multiple regressions and employs a special arctan cost function which removes the effect of outliers. This approach does not require the definition of $t_{min}$.

Vertical distance

Diagonal distance (diagonal means slope 1 on loglog)

Consistency is a function of the distance after translation. The smaller the distance the better the consistency. Identical curves would result in a value of 1. Lower values imply a greater distance and hence less consistency.

Calculating skin change

When the best fit is obtained the skin change is calculated by the following formula: $\Delta S = (\Delta p - \Delta p_{ref}) / \Delta p\text{'}ref/ 2$

The pressure values are calculated at $t_{min}$ for vertical distance approach, at an input time for the diagonal distance approach.

Input

Output

Loglog plots with reference shut-in derivative, current shut-in derivative and best fit (shifted) current shut-in derivative.

User Task dialog

Output channels (no reference document option)

Loglog comparison, where reference shut-in (highlighted in green), current shut-in (highlighted in blue), and best-fit current shut-in (highlighted in red) are all normalized by the reference rate.