Delta time, Equivalent time, Stretched Equivalent time, Super-Equivalent time, Stretched-Super-Equivalent time.

One of the reasons we analyze buildup well test data is to determine the reservoir characteristics. We do this through shape

recognition of the derivative. However, the shape of the derivative varies, depending on the time function used to calculate it,

AND the time function used to plot it. (Yes, these time functions can be different from each other!).

“Derivative” is defined as d(p)/d(ln(time function)). The objective of selecting a

particular time function is to generate a derivative that closely matches the shape

of the drawdown derivative (because the shape of the drawdown derivative is

readily associated with a specific reservoir shape or description). At late times

(which is where reservoir characteristics are determined) the shape can vary

significantly, depending on which of the above time functions is selected.

Case 1: Infinite-acting reservoir:

It is easy to prove theoretically, that for a build-up following a single flow

rate, “Equivalent-time” is the correct time function to use. It always makes

the buildup derivative EXACTLY the same shape as the drawdown

derivative. (In this discussion, replace the word “Equivalent” by “Super-

Equivalent” when a multi-rate flow precedes the build-up).

When “Delta-time” is used, the shapes are the same ONLY WHEN the flow

duration is long. For short flows, the derivative using “Delta-time” has a

downward curve, which can be MISINTERPRETED as a constant pressure

or improved permeability. See Figure 1.

Conclusion: For an infinite acting reservoir, always use “Equivalent time”

or “Stretched-Equivalent-time”. (See definition below).

Volume 2, Issue 1

WHAT TIME IS IT ? By: Louis Mattar, Fekete & Associates, Calgary, Canada

March, 2001

SAUDI ARAMCO

WELL TEST ANALYST

Figure 1— Variation in Derivative signature for an infinite acting reservoir with different time functions.

1

10

100

0.001 0.01 0.1 1 10 100

Drawdown

Equivalent-Time

Delta-Time

The shape of the derivative varies, depending on the time function used to calculate it, AND the time function used to plot it.

A Quarterly Newsletter of the Well Testing Unit, Dhahran

The Well Testing Unit in Dhahran is pleased to bring you this issue of the Well Test Analyst. This newsletter will bring you Well Testing articles that are of local interest showing innovations and advances from well tests in Saudi ARAMCO and around the world.

The current guest article was written by Louis Mattar during his visit to Dhahran to present his course on Practical Well Test Interpretation. It highlights the problem faced in deciding on the correct time function to be used in Pressure Transient Analysis.

Case 2: A well near a boundary:

There does NOT exist a theoretical time function that will make the

buildup and drawdown derivatives look the same. For long flow

durations, all the time functions do a reasonable job of looking like

the drawdown derivative. For short flows, using “Delta-time” to

calculate the derivative is the least appropriate choice. Using the

“Equivalent-time” is somewhat better, but has a big disadvantage, in

that late time is compressed (no matter how long the shut-in, the

maximum value of Equivalent-time is equal to the flow time). This

distorts the late time portion of the derivative to the point where it

does NOT resemble the drawdown at all.

One way of mitigating this distortion is to plot the “Equivalent time”

derivative on the scale of “Delta-time” instead of the scale of

“Equivalent-time” from which it was derived. This results in a

“Stretched-Equivalent-time”, which resembles the shape of the draw-

down derivative more closely, See Figure 2. There has been no

theoretical derivation to explain why this procedure works, or

whether it works in every situation. In a paper submitted for presentation at the SPE conference (fall, 2001), Louis Mattar and

co-workers show that this “stretching” procedure makes the buildup derivative look like the drawdown derivative in every

case that they tested (this did not include closed systems which exhibit pseudo-steady state).

Conclusions:

1. Use Stretch-Equivalent-time both for infinite-acting reservoirs and for wells near boundaries. Of all the time functions, it is

the one that makes the buildup derivative look the most similar to the drawdown one.

2. When evaluating the shape of a buildup derivative, make sure you know what is being plotted and on which time scale.

We hope that this article has been informative. If there is any topic that is of interest, or, that you would like us to address in

future issues, please contact the Well Testing Unit in Dhahran. The contact list is shown below:

S.M. Al-Dawas (Saleh) 874-7811 Manager, RDSD

M.R. Al-Khatib (Mohammed) 872-8263 General Supervisor, AWTD

A.A. Al-Gattan (Abdulaziz) 872-8430 Supervisor, WTU

A.A. Al-Nahdi (Hakim) 872-8437

S.S. Al-Qahtani (Salem) 872-8451

F.M. Al-Thawad (Faisal) 872-8445

R.M. Al-Zayer (Rayed) 872-8448

H.S. Gill (Harmohan) 872-8450

I.A. Al-Samin (Isa) 872-8442

S.A. Bin Akresh (Saud) 872-8439

A.H. Al-Ghamdi (Abdulla) 872-8438

R.H. Al-Obaid (Rashid) 872-8452

S.A. Bubshait (Saleh) 872-8458

A.M. Al-Terairi (Abdullah) 872-8453

P A G E 2

Figure 2 - Variation in Derivative signature for a well near a no flow boundary with different time functions.

1

10

100

0.001 0.01 0.1 1 10 100

Drawdown

Stretched-Equivalent

Equivalent-Time

Delta-Time

V O L U M E 2 , I S S U E 1