The plots and equations that quantify the Drill stem well test interpretation for the discovery well 15-9/19-SR are presented here.
The Horner plot for the main build up period
The Horner plot meant to study for the main build up period. The time function is also called Horner time is:
\frac{t_{p}+\triangle t}{\triangle t}
For multi-rate well test we normalize the rates to the final flowing rate and similarly calculate the Horner’s time function.
Since Horner time is based on the radial flow equation, it should only be used for analyzing radial flow. For linear or bilinear flow, linear or bilinear time functions should be used instead for analysis. Horner time is valid only when the reservoir is infinite-acting and the rate prior to shut-in was constant.
Determine k (Permeability)
p_{i}-p_{ws}(t)=162.6\frac{qB\mu}{kh}[\log_{}{\frac{t_{p}+\triangle t}{\triangle t}}]
This equation can be used to calculate k (permeability)
So the shut in bottom hole pressure BHP, pws, should be a straight line function of \frac{t_{p}+\triangle t}{\triangle t} . This line should have the slope of 162.6\frac{qB\mu}{kh}
Image 1: Semi Log Analysis – Horner Plot
Determine p_{i} (Initial Pressure)
p_{ws}=p_{i}-162.6\frac{qB\mu}{kh}\left[\log_{}{\left(\frac{1688\phi\mu c_{t} r_w^2}{k t_{p}}\right)}-0.869s\right]
p_{ws}=p_{i}+m\left[\log_{}{\left(\frac{1688\phi\mu c_{t} r_w^2}{k t_{p}}\right)}-0.869s\right]
At shut in time ,
p_{ws}=p_{i}-m\left[\frac{\left(t_{p}-\triangle t\right)}{\triangle t}\right]
Determine s (Skin factor)
s = 1.151\left[\frac{p_{ws} - p_{wf}}{m}\right]+1.151\log_{}{\left[\frac{1688\phi\mu c_{t}r_w^2}{k\triangle t}\right]}+1.151\log_{}{\left(\frac{t_{p}+\triangle t}{\triangle t}\right)}
Any \triangle t can be used bu the industry practice us to use \triangle t = 1 as long as it is on the straight line. Also assume that the log of \left(\frac{t_{p}+\triangle t}{\triangle t}\right) is negligible. With these assumptions
s = 1.151\left[\frac{p_{1hr} - p_{wf}}{m}\right]-\log_{}{\left[\frac{k}{\phi\mu c_{t}r_w^2}\right]}+3.23
Type Curve analysis
This generally done using commercial software. However, the math is demonstrated here.
We first pick the time period where the radial flow has occurred,
Well bore storage is normally assumed to be constant during a test and, in practice, this assumption is often reasonable. However, there are numerous situations where well bore storage is not constant. This changing well bore storage may be caused by a changing well bore fluid compressibility, by phase redistribution, or by a change in the type of storage from a changing liquid level to a liquid filled well bore.
Based on a random reference point that fits the above graph, select P_{D}, t_{D}, \triangle P , \triangle t .
k h=\frac{141.2q\mu B}{\triangle P}.P_{D}
\phi c_{t}=\frac{0.000264k\triangle t}{\mu r_w^2}.\frac{1}{t_{D}}
c_{D}=\frac{0.000264k\triangle t}{\phi\mu c_{t}r_w^2}.\left[\frac{t}{\frac{t_{D}}{c_{D}}}\right]_{MP}
s = 0.5\ln{C_{D}e^{2s}/C_{D}}
r_{inv}=\sqrt{\frac{kt}{948\phi\mu c_{t}}}
