# Presentation: 657811

## The Nuts and Bolts ofFalloff Testing

**Ken Johnson**
**Environmental Engineer**
**(214) 665-8473**
**johnson.ken-e@epa.gov**
** **
**Susie Lopez**
**Engineer**
**(214) 665-7198**
**lopez.susan@epa.gov**

**Sponsored by EPA Region 6**

- March 5, 2003

**Notes:**
Ken Johnson has a BS in Chemical Engineering from Texas A&M and a Master in Civil Engineering with an Environmental Engineering option from the University of Texas at Arlington.  Ken has worked in the Region 6 Groundwater/UIC Section reviewing no migration petition submittals; injection well pressure transient tests; and production logs for the past 4 years.  Prior to the EPA, Ken worked over 3 years at the TNRCC Region 4 Office in Arlington as an Industrial and Hazardous Waste Field Inspector.  Ken also has extensive injection well experience, having worked over 13 years in the oil and gas industry and served as one of the company’s experts in falloff test analysis.  Ken is a Registered Professional Engineer in Texas and holds an industrial and hazardous waste inspector certification from the TNRCC.

Susie Lopez has a BS in Mechanical Engineering from Oklahoma State University and is a registered Professional Engineer in the State of Oklahoma.  Susie has worked with EPA’s Ground Water/UIC Section for the past eleven years where she has been involved with the processing of no migration petitions and the review of falloff tests for Class I hazardous waste injection wells.  Prior to the EPA, Susie worked with falloff tests for ten years in the oil and gas industry in the production, reservoir, and reservoir simulation engineering aspects of the industry.

## What’s the Point of a Falloff Test?

**Satisfy regulations**
**Measure reservoir pressures**
**Obtain reservoir parameters**
**Provide data for AOR calculations**

- 40 CFR

- Part 148

- 40 CFR

- Part 146

**Notes:**
1.  Both the nonhazardous and hazardous regulations in 40 CFR Part 146 have monitoring requirements that state the Director shall require monitoring of the pressure buildup in the injection zone annually, including at a minimum, a shut down of the well for a time sufficient to conduct a valid observation of the pressure falloff curve.§146.13 (Non-haz) / §146.68 (Haz)
     Though the regulations may not directly require a falloff test be conducted for Class II wells, the Director can request additional testing to ensure protection of the USDW.

2. Injection and static reservoir pressure measurements can determine if an endangerment or no migration problem exists

3. Transmissibility value, kh/, provides a permeability value for use in the UIC permit and no migration petition demonstrations.

4. These reservoir parameters are used to estimate the maximum pressure buildup in the reservoir for the AOR or ZEI calculations and to support the modeling parameters in a no migration petition.

## Characterize injection interval

Identify reservoir anomalies

Evaluate completion conditions

Identify completion anomalies

**Identify reservoir anomalies**
**Evaluate completion conditions**
**Identify completion anomalies**

**What’s the Point of a Falloff Test?**

**Notes:**
A falloff test can be used to characterize the nature of the injection zone such as indicating if the interval is homogeneous or naturally fractured.
The test may identify reservoir anomalies such as a faults, pinchouts, or boundary from late time data.
The wellbore skin value evaluates the completion condition of the well by indicating if the well is damaged or stimulated, i.e., acidized or frac’d.
Tests can also identify completion anomalies such as partial penetration for a well with wellbore fill or only completed in a portion of a thick sand.

## What Is a Falloff Test?

**Rate, q (gpm)**

**0**

**INJECTING**

**SHUT IN**

- t

**Time, t**

- tp

**Bottom-Hole**
** ****Pressure, P**

**Time, t**

**Falloff start**

**Falloff pressure decline**

**P****wf**

**-100**

**Notes:**
A falloff test is:
Part of pressure transient theory that involves shutting in an injection well and measuring the pressure falloff
Equivalent to a pressure buildup test in a producing well
Analyzed using the same pressure transient analysis techniques used for buildup and drawdown tests.
A replay of the injection period but is less noisy because there is no fluid going by the pressure gauge.

## Effect of Injection and Falloff

**P****wf**

**Pressure, P**

**P****i**

**INJECTION**

**FALLOFF**

******t**

******t=0**

**Pressure recovery at injection well from initial pressure transient due to ceasing injection**

**Pressure transient**
**from injection well**
**continues at the well **

**Time, t**

**t****p**

**Effect of Injection and Falloff**

**Notes:**
Initiating injection creates a pressure transient at the well 
Ceasing injection creates another pressure transient at the well
Remember:  RATE CHANGES CAUSE PRESSURE TRANSIENTS

## Pressure Transients

**Rate changes create pressure transients**
**Simplify the pressure transients**
**Do not shut-in two wells simultaneously**
**Do not change the rate in two wells simultaneously**

**Notes:**
Rate changes create pressure transients in the reservoir.
It only makes sense to minimize the pressure transients in the reservoir during the test.
   Keep the injection rate constant
Do not shut-in two wells simultaneously
Do not change the rate in two wells simultaneously, e.g., shutting in the test well and increasing the rate in an offset well
Always take a good look at an operators falloff testing procedures to confirm multiple pressure transients will not be initiated in the reservoir that could cause a test to be unanalyzable

## Falloff Test Planning

**Notes:**
Successful welltests require a lot of planning
Several unanalyzable welltests could be prevented with proper planning.

## General Planning

**Most problems are avoidable**
**Preplanning**
**Review procedures**

**Notes:**
Most problems encountered are within the operator’s control and are avoidable such as:
1.  Allowing adequate time for both injection and falloff periods to reach radial flow
  Injecting at a constant rate during the injection period preceding the falloff
Reservoir considerations include an understanding about the type of reservoir you are testing.  
	Is it a sandstone or naturally fractured?
	Is there only one injection interval or several completed in the well?
	If the previous welltest was storage dominated, can the wellbore damage be reduced with stimulation prior to the test?
Carefully review procedures to identify any pitfalls before they occur.

## Injection well constraints
Type of completion
Downhole condition
Wellhead configuration
Pressure gauge installation
Shut-in valve

**Type of completion**
**Downhole condition**
**Wellhead configuration**
**Pressure gauge installation**
**Shut-in valve**

**Operational Considerations**

**Notes:**
Injection well constraints:
Review the wellbore schematic included with the procedures – if one is not available, get one.
What type of completion does the well have?  Perforated, gravel packed, or open hole?
Look at the construction of the well.
	Will the downhole condition of the well impact the gauge depth or use of a downhole gauge?
	Is there a liner or junk in the hole?
Look to see the wellbore fill depth tagged from the previous RAT.  
	Wellbore fill can cause partial penetration effects requiring additional time to reach radial flow
How is the wellhead configured?
	Is there a crown valve installed so the well won’t have to be shut-in to install the pressure gauge?
	Is the shut-in valve located near the wellhead?  Shut-in the well near the wellhead to minimize the portion of the test dominated by wellbore hydraulics instead of the reservoir.

## Operational Considerations

**Surface facility constraints**
**Adequate injection fluid **
**Adequate waste storage**
**Offset well considerations**

**Notes:**
Is there adequate injection fluid to maintain a constant injection rate prior to the falloff?
What type of fluid is going to be used for the test?
Plant waste, brine brought in from offsite or a combination of both?  
If there is brine brought in, where will it be stored? Frac tanks? 
Is there room for the number of frac tanks needed on the well’s location?
Is there adequate waste storage for the duration of the falloff test?
Tests are often ended prematurely because of waste storage issues
Offset well considerations
Are there offset wells completed and operating in the same injection interval?
If an offset well is shut-in prior to and during the test, additional waste storage capabilities must be available
If an offset well is not shut-in, a constant injection rate must be maintained both prior to and during the falloff test

Remember:  rate changes cause pressure transients in the reservoir
Confirm that diverting waste from the test well does not impact an offset well’s injection rate

## Recordkeeping:
Maintain an accurate record of injection rates
Obtain viscosity measurements

**Maintain an accurate record of injection rates**
**Obtain viscosity measurements**

**Operational Considerations**

**Rule of thumb:  At a**** bare minimum****, maintain injection rate data equivalent to twice the length of the falloff**

**Notes:**
Maintain an accurate record of injection rates
An adequate rate metering system is a must.  If there are rate fluctuations, try to account with them through superposition
The operator should maintain rates on:
Injection well - prior to shut-in
Offset wells - prior to and during the test
It is also recommended to get viscosity measurements of the injectate to confirm the consistency of the waste injected.
This is more critical if waste is coming from several different tanks or the test involves a combination of plant waste and brine.

Rule of Thumb:  At a bare minimum, maintain injection rate data equivalent to the length of the falloff

## Instrumentation

**Pressure gauges**
**Use two**
**Calibration**
**Types of pressure gauges**
**Mechanical**
**Electronic **
**Surface readout (SRO)**
**Surface gauge**

**Notes:**
Pressure gauges
Use two, one serves as a backup.  This backup gauge does not have to be an identical type
Pressure span of the gauge should not grossly exceed expected test pressures
Calibration:  The wireline company may haul the same gauge around for years without taking it back to the manufacturer for calibration. Ask to see a copy of the vendor calibration sheet.

Generally, you get what you pay for -
There are several types of pressure gauges that range in price to use
Electronic gauges tend to have a higher resolution than the mechanical gauges
Downhole surface readout (SRO) gauge enables tracking of pressures in real time, but are more expensive than a downhole memory gauge
Surface gauge may be impacted by ambient temperature (sunrise to sunset)

## Pressure Gauge Selection

**Selection criteria**
**Wastestream**
**Well goes on a vacuum**
**Wellbore configuration**
**Pressure change at the end of the test**
**Accuracy and resolution**

**Notes:**
Lots of different factors need to be considered when selecting a pressure gauge:
The wastestream may prevent the use of a downhole gauge.
Surface gauges are insufficient if the well goes on a vacuum.
Junk or casing patch in the hole may prevent the use of a downhole gauge.
The pressure change at the end of the test and the accuracy and resolution of the gauge are dependent of each other.   The gauge resolution must be sufficient to measure the pressure changes anticipated at the end of the test.
Ideally, the maximum test pressure should be at least 50% of the gauge pressure limit.

## Example: Pressure Gauge Selection

**What pressure gauge is necessary to obtain a good falloff for the following well?**
**Operating surface pressure: 500 psia**
**Injection interval: 5000’**
**Specific gravity of injectate: 1.05**
**Past falloff tests have indicated a higher permeability reservoir of 500 md**
**Injection well goes on a vacuum toward the end of the test**
**Expected rate of pressure change during radial flow portion of the test is 0.5 psi/hr**

## Example: Pressure Gauge Selection

**Calculate the flowing bottomhole pressure**
**500 psi+(0.433 psi/ft)(1.05)(5000) = 2773 psi (neglect tubing friction)**
**Pick a downhole pressure gauge type and range**
**2000 psi gauge is too low**
**5000 and 10,000 psi gauges may both work**
**Resolution levels:**
**Mechanical gauge - 0.05% of full range**
**Electronic gauge - 0.0002% of full range**
**Mechanical gauge:**
**5000(0.0005) = 2.5 psi****	****     10,000(0.0005)= 5 psi**
**Electronic gauge:**
**5000(0.000002)=.01 psi    10,000(0.000002)=.02 psi**

**Notes:**
In this example, the mechanical gauges do not provide enough resolution for the 0.5 psi/hr anticipated pressure change at the end of the test
Both the 5000 and 10,000 psi electronic gauges provide adequate resolution
The 5000 psi electronic gauge is a better pick because more of the full range of the gauge is used

## Falloff Test Design

**Questions that must be addressed:**
**How long must we inject?**
**How long do we shut-in?**
**What if we want to look for a boundary?**
**Radial flow is the basis for all pressure transient calculations**** **
**Confirm that the test reaches radial flow during both the injection and falloff periods**

**Notes:**
1.  Besides the practical aspects of planning a falloff, there is also a theoretical side to falloff test planning
2.  Some preliminary assumptions and calculations are needed to answer these theoretical questions
 If possible, simulate the falloff test using the assumed parameters
Falloff Test Design considerations must address these questions:
The radial flow portion of the test is the basis for all pressure transient calculations
The ultimate goal of the test design is to reach radial flow during both the injection and falloff portions of the test because the falloff is a replay of the injection.  If the injection period did not reach radial flow, neither will the falloff.

## Falloff Test Design

**The radial flow period follows the wellbore storage and transition periods**
**Wellbore storage: Initial portion of the test governed by wellbore hydraulics**
**Transition period: Time period between identifiable flow regimes**
**Radial Flow: Pressure response is only controlled by reservoir conditions**

**Notes:**
When is a test in radial flow?
The radial flow period typically follows a wellbore storage and transition period
Wellbore storage is the initial portion of the test when the pressure response at the well is governed by wellbore hydraulics instead of the reservoir.
A transition period is the time period between identifiable flow regimes.
During radial flow, the pressure response is only controlled by reservoir conditions

## Falloff Test Design

**Falloff is a replay of the injection period**
**Both the injection period and falloff must reach radial flow**
**Calculate the time to reach radial flow**
**Different calculations for the injectivity and falloff portions of the test**

**Notes:**
The falloff is a replay of the injection portion of the test so the injection period controls what is seen on the falloff
Therefore, it is necessary to calculate the time to reach radial flow in both the injection and falloff periods
The equations used to calculated the time to reach radial flow are different for the injectivity and falloff portions of the test

## Time to Radial Flow Calculation

**Wellbore storage coefficient, C in bbl/psi**
**Fluid filled well:**
**Well on a vacuum:**

**Falling fluid level in the wellbore so that the well goes on a vacuum at the surface**

**Based on fluid filled wellbore so that pressure is maintained at the surface throughout the duration of the test**

**Notes:**
Empirical, back of the napkin type, equations can be used to calculate the time to radial flow, though simulating the test is best.
To calculate the time to reach radial flow, first estimate the wellbore storage coefficient, C:
The calculation for C is different for wells with positive surface pressure and wells on a vacuum
Vw= Total wellbore volume, bbls
cw=fluid compressibility, psi-1
Vu=Wellbore volume per unit length, bbls/ft

## Time to Radial Flow Calculation

**Small C:  The well is connected with the reservoir within a short timeframe if the skin factor is not excessively large**
**Large C:  A longer transition time is needed for the well to display a reservoir governed response**

**Notes:**
These empirically derived equations can be used with limitations:
If C is small, the well is hooked up with the reservoir within a short timeframe if the skin factor is not excessively large
If C is large, a longer transition time is needed for the well to respond to changes in the reservoir
Some carbonate reservoirs contain vugs which cause larger C values
C can be minimized by a downhole shut-in
High skin also prolongs wellbore storage

## Time to Radial Flow Calculation

**Calculate the time to reach radial flow for an injectivity test:**
**Calculate the time to reach radial flow during the falloff test:**
**Note the skin factor,s, influences the falloff more than the injection period**

## Example Radial Flow Calculation

**What injection and falloff timeframes are necessary to reach radial flow given the following injection well conditions?**
**Assumptions:**
**Well maintains a positive wellhead pressure**

**Parameters:**
_**Reservoir**_**	****	**_**Wellbore**_
**h=120 ft****	****     7” tubing (6.456” ID)**
**k=50 md****	****     9 5/8” casing (8.921” ID)**
**s=15****	****	****     Packer depth: 4000’**
******=.5 cp****	****     Top of the injection interval:  4300’**
**c****w****=3e-6 psi****-1**

**Notes:**
High speed example:

## Example Radial Flow Calculation

**Calculate wellbore volume, V****w****:**
**tubing volume + casing volume below packer**
**Calculate wellbore storage coefficient, C **
**C=V****w****c****w**
**	****Note: assume the wellbore storage coefficient is the same for both the injection and falloff periods**

**Notes:**
Value of C is small because the well starts fluid filled and the compressibility and storage coefficients are small.

## Example Radial Flow Calculation

**Calculate minimum time to reach radial flow during the injection period, t****radial flow**
**Note: The test should not only reach radial flow, but also sustain a timeframe sufficient for analysis of the radial flow period**

**Notes:**
Time to radial flow occurs quickly because of the small C and small skin of 15.
If a well is on a vacuum, C could be 2 orders of magnitude larger and the resulting time to radial flow would be much greater.

Note: The test should not only reach radial flow, but also sustain a timeframe sufficient for analysis of the radial flow period

## Example Radial Flow Calculation

**Calculate minimum time to reach radial flow during the falloff, t****radial flow**
**Use with caution!**
**This equation tends to blow up in large permeability reservoirs or wells with high skin factors**

**Notes:**
Time to radial flow is still short, but notice the falloff time is 4 times the time to radial flow calculated for the injection period
Use with caution!  This equation tends to blow up in large permeability reservoirs or wells with high skin factors

## Additional Test Design Criteria

**Decide on the test objectives**
**Completion evaluation**
**Determining the distance to a fault**
**Seeing “x” distance into the reservoir**
**	****Note: Equations for transient test design are discussed in detail in SPE 17088 provided in the reference portion of this presentation**

**Notes:**
Define Test Objectives:
Completion evaluation:  Test must reach radial flow to calculate the skin factor which indicates the condition of the well
Determining the distance to a fault
Seeing “x” distance into the reservoir.  Use radius of investigation to calculate time.
Note: Equations for transient test design are discussed in detail in SPE 17088 provided in the reference portion of this presentation

## Additional Test Design Criteria

**Type of test:**
**Falloff**
**Multi-rate**
**Interference test**
**Simulate the test**
**Review earlier test data if available**

**Notes:**
Determine the type of test needed to produce analyzable results
Falloff, multi-rate, or interference test
The best approach is to simulate the test using estimated parameters
It is easy to conduct sensitivity cases to evaluate the effects of varying reservoir parameters in simulated tests 
Review earlier test data if available
If the test wasn’t good the previous year, find out if changes have been made to the procedure or well for the operator to predict the results will be different this year.

## Falloff Test Design

**What if no falloff data is available?**
**Review the historical well pressure and rate data **
**Look for “pressure falloff” periods when the well was shut-in**
**This information **_***may***_** provide some information that can be used to design the falloff test**

## Data Needed To Analyze a Falloff

**Time and pressure data**
**Rate history prior to the falloff**
**Basic reservoir and fluid information**
**Wellbore and completion data**

**Notes:**
Time and pressure data:
Surface or bottomhole pressure
Impacted by the gauge type - “Get what you pay for”
Rate history prior to the falloff
Include rate history of offset injection or production wells completed into the same interval
Basic reservoir and fluid information
Viscosity, porosity, compressibility, thickness
Wellbore and completion data
Wellbore radius, rw
Perforated, gravel packed, open hole

## Time and Pressure Data

**Record sufficient pressure data**
**Consider recording more frequently earlier in test**
**Consider plotting data while test is in progress to monitor the test**

**Notes:**
Record sufficient pressure data to analyze
Consider recording more frequently earlier in test
More frequent data with an electronic gauge generally provides a better quality derivative curve
Consider plotting data while test is in progress to monitor the test

Operators with waste storage issues may have to end the test prematurely.  Result from poor planning.

## Reservoir Parameters

**net thickness (h)**
**well log and cross-sections**
**permeability (k)**
**core data and previous well tests**
**porosity (********)**
**well log or core data**
**viscosity of reservoir fluid (********f****)**
**direct measurement or correlations**
**total system compressibility (c****t****)**
**correlations, core measurement, or well tests**

## Injectate Fluid

**viscosity of waste (********w****)**
**direct measurement or correlation**
**specific gravity (s.g.)**
**direct measurement**
**rate (q)**
**direct measurement**

**Rule of thumb: No q, no k**

**Notes:**
No q – no k

## “Quick” Falloff Planning Checklist

**Wellbore construction - depths, dimensions, configuration, obstructions, fill depth**
**Injectivity period – constant rate if possible, record rate history, sufficient test duration, waste storage capacity**
**Falloff period – time and pressure data, rate history, sufficient test duration, waste storage capacity**

## Checklist (cont.)

**Instrumentation – resolution, surface vs. bottomhole gauges, backup gauge**
**General reservoir and waste information – h, ********, c****t****, ********f****, ********waste**
**Area geology – boundaries, net thickness trends, sandstone or carbonate formation**

## Pressure Transient Theory Overview

**P-T theory correlates pressures and rates as a function of time**
**P-T theory is the basis for many types of well tests**
**Used in petroleum engineering, groundwater hydrology, solution mining, waste disposal, and geothermal projects**

## Involves working the problem backwards:
From the measured pressure response, determine the reservoir parameters
Start at the wellbore
Work out to the reservoir boundaries

**From the measured pressure response, determine the reservoir parameters**
**Start at the wellbore**
**Work out to the reservoir boundaries**

**Pressure Transient Theory**

## Pressure Transient Theory

**Start with what you know:**
**Well and completion history**
**Geology**
**Test conditions**
**Pressure responses show dominant features called flow regimes**

## P-T Theory Applied to Falloffs

**Falloff testing is part of P-T theory**
**Falloff tests are analyzed in terms of flow models **
**Flow models are solutions to the flow equations**

## P-T Theory Applied to Falloffs (cont.)

**P-T Theory Applied to Falloffs (cont.)**

**The starting point is a partial differential equation (PDE) **
**The PDE is solved for a variety of boundary conditions**
**The solution allow calculation of pressure or rate as a function of time and distance**

## Partial Differential Equation (PDE)

**Partial Differential Equation (PDE)**

**Notes:**
This equation assumes an infinite, homogeneous, isotropic reservoir with a slightly compressible fluid 
ct, k, , , are independent of P

## What’s the Point of the PDE?

**Why do we need all these equations and assumptions?**
**Provide an injection well behavior model**
**Provide a method for reservoir parameter evaluation**
**Only work during radial flow**

## How Do We Solve the PDE?

**Assume conditions to solve the PDE and obtain a model**
**Typical constraints:**
**At the well**
**Finite wellbore radius**
**Constant rate injection**
**Away from the well**
**Infinite-acting **
**Uniform reservoir properties and initial pressure**

## Solution to the PDE

**The exact solution to the PDE is in terms of cumbersome Bessel functions**
**Fortunately an approximate solution based on the exponential integral (Ei) gives almost identical results:**
**where:**

## Simplifying the PDE Solution

**Ei functions:**
** ****tabulated and easy to use**
** ****valid until boundary effects occur**
** ****give the pressure in the reservoir as a    function of both time and distance from the well center**
**simplified with a log approximation:**
**This leads us to our flow model for falloff analysis:**

## Simplifying the PDE Solution

**Simplifying the PDE Solution**

**Notes:**
Note the use of dimensionless variables, PD, tD, and rD

## Predicting Injection Well  Pressure Using the PDE Solution

**Example: Estimate the pressure of an injection well located in an infinite acting reservoir with no skin (s=0).  The well has injected 100 gpm for 2 days.  Other reservoir data are:**

** ****P****i ****= 2000 psi**
** ****k = 200 md**
** ******** ****= 0.6 cp**
** ******** ****= 30 %**

** ****h**** ****= 50 ft**
** ****B****w**** = 1 rvb/stb**
** ****c****t**** = 6e-6 psi****-1**
** ****r****w**** = 0.4 ft**

## Example (cont.)

- First, let’s calculate the dimensionless variables:
- rD, tD, and PD

- Since we’re calculating the pressure
- at the well r = rw and rD = 1

## Example (cont.)

- Now look up PD on the graph or calculate PD from
- the following equation:

- From Figure C.2 in SPE Monograph 5:  at tD= 14.65x106 and rD=1

## Example (cont.)

- At tD= 1.465x107 and rD=1, PD= 8.5   (Figure C.2 in SPE Monograph 5)

## Example (cont.)

- Now calculate the pressure increase at the well:

- (a pressure increase of 251 psi)

## What happens if the injection reservoir isn’t infinite?

**Not infinite if limited by a fault or pinchout**
**Represent limits as virtual barriers using “image” wells**
**A linear PDE  means the Ei solutions can be added to consider pressure changes from multiple wells**

## How to Account for Boundary Effects

**Add the real injector and image well to account for the boundary**
**1 injector with 1 boundary requires 1 image well**
**Image wells are more complex with multiple boundaries**

**Notes:**
Adding means summing the pressure contribution of each injector
Image wells may have to be mirrored due to the interactions with boundaries

## Boundary Effects (cont.)

**Boundary Effects (cont.)**

## What happens if the pre-falloff injection rate varies?

**Again, the PDE is linear**
**Each rate change creates a new pressure response to be added to the previous response**
**Account for each rate change by using an image well at the same location**

**Notes:**
Use an image well at the same location as the injector with a time delay.  Sum the image well pressure contributions.

## Superposition

**Superposition is the method of accounting for the effects of rate changes on a single point in the reservoir from anywhere and anytime in the reservoir including at the point itself using the PDE solution**

- Image well contribution

**Notes:**
In dimensionless terms for any point in time, t, the following equation results:

## Superposition (cont.)

- q1

- Shut-in

- tp

- t

- t

**Superposition (cont.)**

- Pressure, P

- Rate, q

- q2

- Pstatic

- Pwf1

- Pwf2

- 0

- Pressure recovery
-  from q1 to q2

- Pressure recovery
-  from q1 to SI

- Pressure recovery
-  from q2 to SI

## “Kitchen Sink” Solution to the PDE

**If we were to account for all wells and potential boundaries (image wells) in a reservoir, the pressure change at any point could be given by:**

**This is essentially what an analytical reservoir simulator does!**

## PDE Solution At The Injector

**The PDE can give the pressure at any reservoir location**
**At the wellbore, r****D**** =1, so:**

**Notes:**
Use of dimensionless variables can be useful for AOR calculations
Note:  This leads directly to the semilog plot

## Semilog Plot

**Applies **_**only**_** during radial flow!**
**Write PDE solution as a straight line equation with a slope and intercept:**

**Where ****m ****is the semilog plot**** slope:**

**Notes:**
By grouping the slope and intercept terms together, the solution to the PDE can be written in the following form which is the basis for a semilog plot

## Finding the Semilog Slope, m

- 0.01         0.1         1.0         10.0        100.0

- P1

- P2

- Elapsed time, hrs

- log(t1)

- log(t2)

- if t2 / t1=10 (one log cycle), then log (t2 / t1) = 1 and the slope is P2-P1

- Pressure

## The Many Faces of the Semilog Plot

**4 semilog plots typically used:**
**Miller Dyes Hutchinson (MDH) Plot**
**Pressure vs log ********t**
**Horner Plot**
**Pressure vs log (t****p****+ ********t)/ ********t**
**Agarwal Time Plot**
**Superposition Time Plot**

**Notes:**
There are four different semilog plots typically used in pressure transient and falloff test analysis
MDH
Horner
Agarwal uses equivalent time- Pressure vs log equivalent time
Pressure vs log superposition time function
Pressure/rate vs log superposition time function
Superposition

## Miller Dyes Hutchinson (MDH) Plot

**Applies to wells that reach  pseudo-steady state during injection**
**Plot pressure vs log ********t**
**Means response from the well has encountered all limits around it **
**Only applies to **_**very**_** long injection periods at a constant rate**

**Notes:**
t is the elapsed shut-in time of the falloff
Pseudo-steady state means the response from the well has encountered all the boundaries around the well

## Horner Plot

**Plot pressure vs. log (t****p****+********t)/********t**
**Used only for a falloff preceded by a constant rate injection period**
**Calculate injecting time, t****p****= V****p****/q (hours)**
**Where V****p****= injection volume since last pressure equalization**
**V****p ****is often taken as cumulative injection volume since completion**
**Caution: Horner time can result in significant analysis errors if the injection rate varies prior to the falloff**

## Agarwal Time Plot

**Plot pressure vs log equivalent time, ********t****e**
******t****e ****= log(t****p**** ********t)/(t****p****+********t)**
**Where t****p**** is as defined for a Horner plot**
**Similar to Horner plot**** **
**Time function scales the falloff to make it look like an injectivity test**

## Superposition Time

**Accounts for variable rate conditions prior to a falloff test**
**Most rigorous semilog analysis method**
**Requires operator to track rate history**

**Rule of thumb:  ****At a bare minimum****, maintain injection rate data equivalent to twice the length of the falloff**

## Calculating Superposition Time Function

**Superposition time function:**
**Can be written several ways – below is for a drawdown or injectivity test:**
**Pressure function is modified also:**
**	**

## Which Time Function Do I Use?

**Depends on available information and software:**
**If no rate history, use Horner**
**If no rate history or cumulative injection total, use MDH**
**If you have rate history equal to or exceeding the falloff test length, use superposition**
**Horner or MDH plots can be generated in a spreadsheet**
**Superposition is usually done with welltest software**

## Which Time Function Do I Use?

**Rules of thumb:**
**Use MDH time only for **_**very**_** long injection times (e.g., injector at pseudo-steady state)**
**Use Horner time when you lack rate history or software capability to compute the superposition function**
**Superposition is the **_**preferred**_** method if a rate history is available**

## Which Time Function Do I Use?

**Horner **_**may**_** substitute for superposition if:**
**The rate lasts long enough to reach the injection reservoir limits (pseudo-steady state)**
**The rate prior to shut-in lasts twice as long as the previous rate**
**At a minimum, the rate prior to shut-in lasts as long as the falloff period**
**Horner is a single rate superposition case**

## One Falloff Test Plotted with Three Semilog Methods

**MDH Plot**

**Horner Plot**

**Superposition Plot**

**k= 1878 md**
**s = 57**

**k= 2789 md**
**s = 88.6**

**k = 1895 md**
**s = 57.7**

## Other Uses of a Semilog Plot

**Calculate radius of investigation, r****i**
**Completion evaluation, skin factor, s**
**Skin pressure drop, ********P****skin**
**False extrapolated pressure, P*******

## Radius of Investigation

**Distance a pressure transient has moved into a formation following a rate change in a well**** ****(Well Testing by  Lee)**
**Use appropriate time to calculate radius of investigation, r****i**
**For a falloff time shorter than the injection period, use t****e**** or the length of the injection period preceding the falloff to calculate r****i**

## Radius of Investigation

**There are numerous equations that exist to calculate r****i**** in feet**
**They are all square root equations, but each has its own coefficient that results in slightly different results ****(OGJ, Van Poollen, 1964)**
**Square root equation based on cylindrical geometry**
**	****From SPE Monograph 1: (Eq 11.2) and Well Testing, Lee (Eq. 1.47)**

## Skin Factor

**The skin factor, s, is included in the PDE**
**Wellbore skin is the measurement of damage near the wellbore (completion condition)**
**The ****skin factor**** is calculated by the following equation:**

**Notes:**
The equation deviates from the usual equation for skin in that the factor,
 tp/(tp+delta t), where delta t = 1 hr, appears in the log term. 
Delta t is usually assumed to be negligible, however, for short injection periods this term could be significant

## Skin Factor

**Wellbore skin is quantified by the skin factor, s **
**“****+” positive value - a damaged completion**
**Magnitude is dictated by the transmissibility of the formation**
**“****-” negative value - a stimulated completion**
**- 4 to - 6 generally indicates a hydraulic fracture**
**-1 to - 3 typical acid stimulation results in a sandstone reservoir**
**Negative results in a larger effective wellbore**

## Effective Wellbore Radius Concept

**Ties the skin factor into an effective wellbore radius (wellbore apparent radius, r****wa****)**
**r****wa****= r****w****e****-s**
**A negative skin results in a larger wellbore radius and therefore a lower injection pressure**

## Effective Wellbore Radius

**Example:  A well with a radius of 5.5” had a skin of +5 prior to stimulation and –2 following the acid job.  What was the effective wellbore radius before and after stimulation?**
**r****wa****= r****w****e****-s**
**A little bit of skin makes a big impact on the effective wellbore radius**

**Before**

**After**

## Pressure Profile with Skin Effect

**r****w**

**Wellbore**

**Damaged**
**Zone**

**P****static**

******P****skin**** = Pressure drop across skin**

**P****wf**

**Pressure**

**Distance**

## Completion Evaluation

**The assumption that skin exists as a thin sheath is not always valid**
**Not a serious problem in the interpretation of the falloff test**
**Impacts the calculation of correcting the injection pressure prior to shut-in**
**Note the term t****p****/(t****p****+********t), where ********t = 1 hr, appears in the log term and this term is assumed to be 1**
**For short injection periods this term could be significant (DSTs)**

## Completion Evaluation

**Wellbore skin**
**Increases the time needed to reach radial flow in a falloff**
**Creates a pressure change immediately around the wellbore**
**Can be a flow enhancement or impediment**

## Completion Evaluation

**Too high a skin may require excessively long injection and falloff periods to establish radial flow**
**The larger the skin, the more of the falloff pressure drop is due to the skin**

## Skin Pressure Drop

**Skin factor is converted to a pressure loss using the skin pressure drop equation**
**Quantifies what portion of the total pressure drop in a falloff is due to formation damage**

**Where,**
**P****skin**** = pressure due to skin, psi**
**m = slope of the Horner plot, psi/cycle**
**s = skin factor, dimensionless       **

**Notes:**
Dependent on the last injection pressure prior to shut-in so an accurate recording of this pressure point is important

## Corrected Injection Pressure

**Calculate the injection pressure with the skin effects removed**
**P****corrected**** is injection pressure based on  pressure loss through the formation only**

**Where:**
**P****corrected**** = adjusted bottomhole pressure, psi**
**P****inj****= measure injection pressure at ********t = 0, psi**
**P****skin**** = pressure due to skin, psi**

## False Extrapolated Pressure

**False Extrapolated Pressure, P*********, is the pressure obtained from the semilog time of 1**
**For a new well in an infinite acting reservoir, it represents initial reservoir pressure**

## False Extrapolated Pressure

**For existing wells, it must be adjusted to P, average reservoir pressure**
**Requires assumption of reservoir size, shape, injection time, and well position within the shape**
**For long injection times, P********* will differ significantly from P**
**P***** ****to P conversions are based on 1 well reservoirs, simple geometry**
**We don’t recommend using P*******
**Use the final measured shut-in pressures, if well reaches radial flow, for cone of influence calculations**

## Semilog Plot Usage Summary

**A semilog plot is used to evaluate the radial flow portion of the well test**
**Reservoir ****transmissibility ****and ****skin factor**** are obtained from the slope of the **_**semilog straight line **_**during radial flow**
**Superposition is used for rate variations**

**Notes:**
The key to the semilog plot:  Semi-log plots are not useful if the test does not reach radial flow.

## Identifying Flow Regimes 

**Notes:**
Flow regimes are characterized by mathematical relationships between pressure, rate, and time.
Flow regimes are a visualization of what goes on in the reservoir

## Identifying Flow Regimes

**Create a master**** diagnostic ****plot, the ****log-log plot**
**Log-log plot contains two curves**
**Individual flow regimes:**
**Characteristic shape**
**Sequential order**
**Specific separation**
**Critical flow regime - ****radial flow**

**Notes:**
The first plot used to identify flow regimes is the log-log plot.
The log-log plot is a master diagnostic plot that contains two curves, a pressure curve and a derivative curve
The log-log plot identifies the various stages and flow regimes present in a falloff test
Individual flow regimes have characteristic slopes and a sequential order on the log-log plot with the critical flow regime being radial flow.  
The radial flow portion of the log-log plot is identified and then the corresponding timeframe on the semilog plot is used for the calculations
Flow regimes are characterized by specific slopes and trends for P and P', as well as specific separation between the two curves
The radial flow portion is the critical flow regime because it is the portion of the test that the calculations are based

## Example Log-log Plot  

**Pressure**
**Data**

**Radial Flow**

**Semilog Pressure**
**Derivative Function**

**Transition period**

**Unit slope during**
**wellbore storage**

**Derivative flattens**

**Wellbore Storage Period**

**Example Log-log Plot**

**Notes:**
Pressure curve - red
Derivative curve - blue
Wellbore storage period:  Pressure and derivative curves overlay on a unit slope
Radial flow:  Derivative flattens
Notice the pressure curve flattens prior to radial flow so the pressure curve is not a good indicator of radial flow

## Log-log Plot Pressure Functions

**	****Rate variations prior to falloff test determine how the pressure function is to be plotted**
**Constant rate - Plot pressure**
**Variable rate - Normalize pressure**

**Notes:**
Just like the time function, rate variations prior to the shut-in of the well determine how pressure is plotted on the Y axis
Constant rate - Plot pressure
Variable rate - Normalize pressure data or normalize time function using the rates. P/q term

## Log-log Plot Time Functions

**Rate variations prior to shut-in dictate the log-log plot time function:**
**Use if the injection rate is constant **_**and**_** the injection period preceding the falloff is significantly longer than the falloff**** **
**Elapsed time, ********t**

**Notes:**
As with the semilog plot, rate variations prior to shut-in also dictate the log-log plot time function to use on the log-log plot:
Time function is plotted on the x-axis of the log-log plot
Elapsed time, t  (Real time)
	Calculate as: telapsed = tshut in -  teach test data point
	Use if the injection rate is constant and the injection period preceding the falloff is significantly longer than the falloff 
Time function is similar to the time function for the MDH semilog plot.

## Log-log Plot Time Functions

**Agarwal equivalent time, t****e**
**Calculate as:  **
**Use if the injection period is short**
**Superposition time function**
**Use if the injection rate varied**
**Most rigorous time function**

**Notes:**
Equivalent time, te: 
	Also referred to as Agarwal equivalent time
	Calculate as a Horner time function and should be used if the injection period is short
In this equation, the injection volume since that last stabilization period on the well divided by the injection rate is used to calculate the injecting time, tP: 
    tp= Vp/q (hours)
	     Where,	Vp= injection volume since last pressure equalization
			Vp is often mistaken as cumulative injection volume since completion
Superposition Time Function should be used if the injection rate varied prior to shut-in for the falloff and the rate history is available.
The superposition time function is the most rigorous time function – some type of computer software is needed to calculate

## Pressure Derivative Function

**Magnifies small changes in pressure trends **
**Good recording device critical**
**Independent of skin**
**Popular since 1983**

**Notes:**
The derivative function is graphed on the log-log plot 
The main use of the derivative is to magnify small changes in pressure trends (slope) of the semilog plot to help identify:
Flow regimes
Boundary effects
Layering
Natural fractures (dual porosity)
Derivatives amplify reservoir signatures and noise so the use of a good pressure recording device is critical
The derivative for a specific flow regimes is independent of the skin factor, while the pressure is not
Use of derivative curves have been around since 1983 and are not a new technology

## Pressure Derivative Function

**Combines a semilog plot with a log-log plot**
**Calculates a running slope of the MDH, Horner, or superposition semilog plots**
**The logarithmic derivative is defined by:**

**Notes:**
The derivative combines a semilog plot with a log-log plot
The derivative is the running slope of the MDH, Horner, or superposition semilog plots of pressure vs log delta t
The derivative function is simply the slope of the semilog plot which is the change in pressure over the change of log delta t

## Pressure Derivative Function

**Recent type curves make use of the derivative by matching both the pressure and derivative simultaneously**
**A test can show several flow regimes with “late time” responses correlating to distances farther from the wellbore**

**Notes:**
Plotted on a log-log plot, flow regimes are characterized by specific slopes and trends for P and P', as well as specific separation between P and P' 

Use of the pressure and derivative curves provide a better type curve match since two curves are matched instead of just one.  The derivative curve also offers more shape to match than the pressure curve.

## Example:  For a well in an infinite acting reservoir with radial flow

	so that

The constant derivative value plots as a “flat spot” on the log-log plot

**	****so that**
**The constant derivative value plots as a ****“flat spot”**** on the log-log plot**

**Pressure Derivative Function**

- constant value

**Notes:**
When the reservoir is in radial flow, the logarithmic derivative plots at a constant value for all times
The constant derivative value plots as a “flat spot” on the log-log plot
If you use dimensionless variables, the derivative calculates to be 0.5.  Dimensionless variables are used for some type curves.
Remember:  During radial flow P’ plots as a constant value, that is flat
For wellbore storage:
	The log-log plot of the derivative will plot the same slope as the pressure curve, and both will have a unit slope
For Linear Flow:
	Both the derivative and pressure curves will have the same slope, but the derivative is lower than the pressure curve, usually about a third of a log cycle lower

## Pressure Derivative Function

**Usually based on the slope of the semilog pressure curve**
**Can can be calculated based on other plots:**
**Cartesian**
**Square root of time:**
**Quarter root of time:**
**1/square root of time:**

**Notes:**
Derivative function is usually based on the slope of the semilog plot
However, the derivative can also be based on other specialized plots to make identification of a specific flow regime easier to identify

## What Flow Regimes Are Active?

**Examine what might happen in and near the wellbore to determine early time behavior**
**Examine the reservoir geology, logs, etc., to determine late time behavior**

**Notes:**
When trying to identify flow regimes, recognize the test is first going to see what is happening in or near the well.  If the test is not dominated by wellbore hydrualics, i.e. wellbore storage, the test will then observe the reservoir responses away from the well
Always use common sense during the preplanning of the welltest
1. Well completion history – check wellbore fill
The completion condition of the well will dictate the early time behavior
2. The test must get outside the influences of the well prior to and into the reservoir during the lat time behavior of the test.  If you want to specifically check for a reservoir anomaly away from the well, these are things that must be considered when planning the test.
Single fault
Multiple boundaries
Layering
Natural Fractures (dual porosity)

## Wellbore Storage

**Occurs during the early portion of the test**
**Caused by shut-in of the well being located at the surface rather than at the sandface**
**After flow -  fluid continues to fall down the well after well is shut-in**
**Location of shut-in valve away from the well prolongs wellbore storage **

**Notes:**
Welbore storage is caused by the well being shut-in at the surface instead of at the sandface.
The length of the wellbore storage period can be impacted by the location of the shut-in valve.  Always shut-in the well near the wellhead.

## Wellbore Storage

**Pressure responses are governed by wellbore conditions not the reservoir**
**High wellbore skin or low permeability reservoir may prolong the duration of the wellbore storage period**
**A wellbore storage dominated test is unanalyzable**

**Notes:**
High wellbore skin or low permeability reservoir results in a slower transfer of fluid from the well to the formation, therefore, extending the duration of the wellbore storage period

If the welltest does not reach radial flow, the test is unanalyzable

## Wellbore Storage Log-log Plot

**Identifying characteristics: Pressure and derivative curves overlay on a unit slope line during wellbore storage**

**Wellbore Storage Log-log Plot**

**Notes:**
Identifying characteristics of wellbore storage
Unit slope on the log-log plot for both the pressure and derivative curves
Caused by shut-in of the well being located at the surface rather than at the sandface so there is after flow - fluid continues down well after well is shut-in
Wellbore storage can be impacted by where the shut-in valve is located.
During wellbore storage, pressure responses are governed by wellbore conditions and are not representative of reservoir behavior
High wellbore skin or low permeability reservoir results in a slower transfer of fluid from the well to the formation extending the duration of the wellbore storage period
A welltest dominated by storage is unanalyzable

## Radial Flow

**The **_**critical**_** flow regime from which all analysis calculations are performed**
**Used to derive key reservoir parameters and completion conditions**
**Radial flow characterized by a straight line on the semilog plot**
**Characterized by a flattening of the derivative curve on log-log plot**

**Notes:**
Radial flow is the ultimate goal of the Falloff Test

## Radial Flow

**A test needs to get to radial flow to get valid results**
**May be able to obtain a minimum permeability value using the derivative curve on the log-log plot if well does not reach radial flow**
**Try type curve matching if no radial flow**

**Rule of thumb:  Leave the well shut-in for an additional 1/3 log cycle after reaching radial flow to have an adequate radial flow period to evaluate**

**Notes:**
In tests where the derivative did not reach a plateau (i.e. radial flow), some pressure transient computer software packages can estimate a transmissibility from either the log-log plot derivative, by taking an antilog, or using the semilog plot slope.  The transmissibility obtained at this point in the test is a minimum because the derivative has not reached its minimum value.  The derivative reaches its minimum value at the radial flow plateau, resulting in a smaller slope value and, consequently, a larger transmissibility.

## Example: Well in a Channel

**Well observes linear flow after reaching the channel boundaries**

- Radial Flow

- Linear Flow

- Semilog derivative plot

**Notes:**
Two derivative curves are plotted along with the pressure curve:
The pressure curve is shown in red
The blue curve is based on the semilog plot and the magenta curve is based on the square root of time plot.
When the reservoir is in radial flow and infinite acting, the logarithmic derivative of the semilog plot (blue curve) plots at a constant value for all times, that is, the constant semilog derivative value plots as a “flat spot” on the log-log plot
For wellbore storage, both the pressure and semilog derivative plots have a unit slope and overlay each other 
For linear flow, a pressure curve and semilog derivative curve will have the same slope, but the derivative curve will be lower, generally about a third of a log cycle lower.
The derivative based on the square root of time is flat when the well is in linear flow.  This illustrates the advantages of being capable of plotting the derivative on various time functions

Radial flow – pressure derivative plots as a constant value or as a flat spot in the curve
Wellbore storage – pressure and pressure derivative plot as a unit slope
Linear flow – pressure derivative and pressure plot have the same slope, but the derivative is lower 

## Typical Log-log Plot Signatures

- P

- t

- Log t

- Log P
- Log P'

- 1

- 1

- 1

**WellboreStorage**

- P

- Log t

- Log t

- Log P
- Log P'

**RadialFlow**

**slope = m**

**P & P’ overlay**

**P****'**

**P**

**P****'**** ****= dP/d(log t)**

## Example SemiLog Plot

- Straight line during radial flow period

## Typical Log-Log Plot Signature

- P

- Log t

- Log P
- Log P'

- 1

- 2

**LinearFlow**

**slope = m****'**

- 1

- 2

**P**

**P****'**

**P****'**** ****= dP/d(log t)**

**Notes:**
Here’s an example of a linear flow regime.
Linear flow regimes may result from: Flow in channel / Parallel faults / Highly conductive hydraulic fracture
Log-log plot:
Half slope on both the pressure and derivative curves
Derivative curve approximately 1/3 of a log cycle lower than the pressure curve
Square root time plot:  Straight line

## Log-log Plot Dominated bySpherical Flow

**Log-log Plot Dominated by****Spherical Flow**

**Partial Penetration characterized by a negative 1/2 slope line**

**Notes:**
Spherical Flow:  
Indicates a partial penetration effect.  This is very common with wells having a lot of wellbore fill
Characterized by a derivative negative half slope on the log-log plot
The welltest in this example does not get to radial flow and is therefore not analyzable

## Hydraulic Fracture Log-log Plot

**Hydraulic Fracture Log-log Plot**

**Derivative drop due to constant pressure**

**Half slope on both curves – linear flow**

**Notes:**
Flow regimes:
Wellbore Storage
Fracture Linear Flow
Usually hidden by wellbore storage
Bilinear Flow
Result of simultaneous linear flows in the fracture and from the formation into the fracture
Log-log plot:  Quarter slope on both the pressure and derivative curves
Quarter root plot: Straight line
Psuedo-Linear Flow
Follows bilinear flow after a transition period
Log-log plot: Half slope of the derivative curve
Square root time plot: straight line
Formation Linear Flow
Linear flow from formation into fractures
Log-log plot:  Half slope on both the pressure and derivative plots
Square root time plot:  Straight line

## Hydraulic Fracture Response

**1⁄4 slope trend**

**1⁄2 slope trend**

**Pseudo-radial flow**

**Pressure response**

**Derivative**
**Response**

**Notes:**
Flow regimes:
Bilinear Flow
Result of simultaneous linear flows in the fracture and from the formation into the fracture
Log-log plot:  Quarter slope on both the pressure and derivative curves
Quarter root plot: Straight line
Psuedo-Linear Flow
Follows bilinear flow after a transition period
Log-log plot: Half slope of the derivative curve
Square root time plot: straight line
Formation Linear Flow
Linear flow from formation into fractures
Log-log plot:  Half slope on both the pressure and derivative plots
Square root time plot:  Straight line

## Naturally Fractured Rock

- Dual Porosity Log-log Plot

**Fracture system will be observed first on the falloff followed by the total system (fractures + tight matrix rock)**
**Complex falloff analysis involved**
**Falloff derivative trough indicates the level of  communication between fractures and matrix rock**

## Layered Reservoirs

- Homogeneous behavior of the total system

- Crossflow

- Homogeneous behavior of the higher permeability layer

- Layered System with Crossflow

- Figures taken from Harts Petroleum Engr Intl, Feb 1998

## Layered Reservoirs

- Commingled

- Figures taken from Harts Petroleum Engr Intl, Feb 1998

- Layered system response

- Homogeneous system response

- Homogeneous behavior
- Both layers infinite acting

- High perm layer bounded
- Low perm layers infinite acting

- Psuedo-steadystate flow

## Layered Reservoirs

**Analysis of a layered reservoir is complex**
**Different boundaries in each layer**
**Falloff objective for UIC purposes is to get a total transmissibility from the whole reservoir system**

## Pressure Derivative Flow Regime Patterns

**	**** **_**Flow Regime**_**            **_**Derivative Pattern**_**	****              **
**Wellbore Storage ....... Unit slope**
**Radial Flow ............... Flat plateau**
**Linear Flow ............... Half slope**
**Bilinear Flow ............. Quarter slope**
**Partial Penetration ..... Negative half slope**
**Layering ................... Derivative trough**
**Dual Porosity ............ Derivative trough**
**Boundaries ............... Upswing followed by plateau**
**Constant Pressure ..... Sharp derivative plunge**

## Log-log Plot Summary

**Logarithmic derivative combines the slope trend of the semilog plot with the log-log plot to magnify flow regime patterns**
**The derivative trend determines what portion of the test can be used to evaluate the semilog straight line**
**Various flow regimes show up on the derivative plot with specific patterns**

## Falloff Test Evaluation Procedure

**Data acquisition:**
**Well information **
**Reservoir and injectate fluid parameters**
**Reservoir thickness**
**Rate histories**
**Time sync injection rate data with pressure data**

**Notes:**
Data acquisition:
Well information obtained from the well schematic
Well radius, rw
Type of completion
Reservoir and injectate fluid parameters
Porosity,  (well log or core data)
Compressibility, ct (correlations, core measurement, or well test)
Viscosity, f and w  (direct measurement or correlations)
Estimate of reservoir thickness, h
Review flow profile surveys or slug chases from MIT
Well log and cross-sections
Rate histories
Test well prior to the test
Constant or variable
Offset wells prior to and during the test
Constant or variable
Time sync injection rate data with pressure data

## Falloff Evaluation Procedure

**Prepare a Cartesian plot of pressure and temperature versus time**
**Confirm stabilization of pressure prior to shut-in**
**Look for anomalous data**
**Did pressure change reach the resolution of the gauge?**

**Notes:**
Prepare a Cartesian plot of pressure and temperature versus time
Confirm stabilization of pressure and temperature measurements prior to shut-in
Look for anomalous data
Missing data
Pressure rise or jump in data
Fluctuations in temperature can impact the pressure measurement
Determine if the pressure change reached the resolution of the gauge
If the test has not reached radial flow, a Cartesian plot can indicate if continuing the test can provide additional data given the resolution of the pressure gauge used for the test.

## Falloff Evaluation Procedure

**Prepare a log-log plot of the pressure and the derivative**
**Use appropriate time scale**
**Identify the radial flow period**
**Flattening of the derivative curve**
**If there is no radial flow period, resort to type curve matching**

## Falloff Evaluation Procedure

**Make a semilog plot**
**Use the appropriate time function**
**Horner or Superposition time**
**Draw a straight line of best fit through the points located within the equivalent time interval where radial flow is indicated by the derivative curve on the log-log plot**
**Determine the slope m and P****1hr**** from the semilog straight line**

**Notes:**
Calculate reservoir and completion parameters
transmissibility, kh/
skin factor, s
radius of investigation, ri, based on Agarwal equivalent time, te
Check results using type curves (optional)

## Falloff Evaluation Procedure

**Calculate reservoir and completion parameters**
**transmissibility, kh/******
**skin factor, s**
**radius of investigation, r****i****, based on Agarwal equivalent time, t****e**
**Check results using type curves (optional)**

**Notes:**
Use a common sense check of the values calculated for s and kh/ 
Are these parameters what would be expected of the completed reservoir and well condition?
Skin will be used to correct injection pressure for skin effects.
The distance into the reservoir observed in the test is based on the ri
Should any boundaries have been observed?

## Gulf Coast Falloff Test Example

**Well Parameters:**
** ****r****w****= .4 ft**
** ****cased hole perforated completion **
**6020’- 6040’**
**6055’- 6150’**
**6196’- 6220’**
** ****Depth to fill: 6121’**
**Gauge depth:  6100’**
**Panex 2525 SRO**

**Notes:**
Based on this limited data, what potential issues might you already be looking for?
Layering based on the three sets of perforations
Partial penetration, since the lower set of perforations is completely covered by wellbore fill

## Example (cont.)

**Reservoir Parameters:**
**Reservoir thickness, h:  200’**
**Average porosity, ********: 28%**
**Total compressibility, c****t****: 5.7e****-6**** psi****-1**
**Formation Fluid Properties**
**Viscosity, ********f****: 0.6 cp **

## Example (cont.)

**Temperature**

**Pressure**

**Rate**

**Well shut-in**

**End of test**

**Several rate fluctuations prior to shut-in**

**Notes:**
Notice the slight incline in the pressure just prior to shut-in.  The pressure in this well had not stabilized prior to shut-in for the falloff test.
Several different rate changes occurred prior to shut-in so use a superposition time function to analyze this test.  This requires the previous rate history that is available for this well.

## Log-log Plot

**Wellbore storage**

**Radial flow**

**Spherical flow**

## Semilog Plot

**Test results:**
**Permeability, k:  780 md**
**Skin factor, s:  52**
**Semilog slope, m: -10.21 psi/cycle**
**P****1hr**** = 2861.7 psi**
**P********* = 2831 psi**

**Semilog straight line**

- Radial Flow

**Notes:**
Positive skin factor as expected.

## Type Curves

**Graphs of P****d**** vs. t****d**** for various solutions to the PDE**
**Provide a “picture” of the PDE for a certain set of boundary conditions**
**Work when the specialized plots do not readily identify flow regimes**

**Notes:**
Type curves:
Graph dimensionless variables, Pd vs. td for various solutions to the P-T PDE
Provide a “picture” of what a solution to the PDE looks like for a certain set of boundary conditions
Determined from either analytical or numerical solutions
Cover a wide range of parameter combinations and work even when specialized plots do not readily identify flow regimes
Applied to field data analysis by a process called “type curve matching”
Generally based on drawdowns/injectivity tests
May require plotting test data with specialized time functions to use correctly
May provide a welltest analysis when specialized plots do not identify a radial flow regime

## Type Curves

**Applied to field data analysis by a process called “type curve matching”**
**Generally based on drawdowns/injectivity**
**May require plotting test data with specialized time functions to use correctly**

## Example: Homogeneous Reservoir Type Curves

## Type Curve Match

**Simulated test results**

**Spherical flow: - 1⁄2 slope**

**Notes:**
Here is a type curve match to our previous example

## Effects of Key Falloff Variables

## Key Falloff Variables

**Length of injection time**
**Injection rate**
**Length of shut-in (falloff) period**
**Wellbore skin**
**Wellbore storage coefficient**

## Log-Log Plot

**Pressure Data**

**Radial Flow Period**

**Derivative**

**Wellbore Storage Period**

**Transition Period**

**Log-Log Plot**

## Effect of Injection Time

**Length of injection period controls the radius of investigation of the falloff test**
**Falloff is a “replay” of the preceding injection period**
**Falloff  period cannot see any further out into the reservoir than the injection period did**
**Injection period should be long enough to establish radial flow**

## Injection Time

**Increase injection time to observe presence of faults or boundary effects**
**Calculate minimum time needed to reach a certain distance away from the injection well**

## Simulated Injection Periods - Same Properties, Varying Duration

**Simulated Injection Periods - Same Properties, Varying Duration**

- 4 hours injection

- 8 hours injection

- 24 hours injection

- Does not reach radial flow

- Barely reaches radial flow

- Well developed radial flow

## Log-log Plots for Injection Periods ofVarying Length

**Log-log Plots for Injection Periods of****Varying Length**

- 4 hours of injection
- 8 hours shut-in

- 8 hours of injection
- 8 hours shut-in

- 24 hours of injection
- 8 hours shut-in

## Summary of Injection Time Effects

**When injection time is shorter than the falloff, it compresses the falloff response on log-log plot**
**Longer injection time extends the falloff response**
**When injection time is very long relative to the falloff time, it has little effect on the falloff response**

## Effects of Injection Rate

**Rate determines the magnitude of pressure rise during the injection period and the amount of pressure falloff during shut-in period**
**Too small a rate can minimize the degree of pressure change measured during a falloff test**
**Rate limit during a test may be constrained by permit limits, formation transmissibility, skin factor, or waste storage capacity**

## Injection Rate Effects

**Injection rate preceding the test may be limited by the UIC permit and no migration petition requirements or operational considerations including:**
**available injectate capacity**
**pumping capacity**
**surface pressure or rate limitations**

## Effect of Increasing Rate on Falloff Test Response

**Effect of Increasing Rate on Falloff Test Response**

- Log-log plots
- look similar

- m=2.9 psi/cycle

- m=8.6 psi/cycle

- m=17.2 psi/cycle

- 60 gpm

- 150 gpm

- 300 gpm

## Summary of Injection Rate Effects

**Higher rate increases the amount of pressure buildup during injection resulting in:**
**Greater total falloff pressure change**
**Larger slope of the semilog plot during radial flow**
**Increased semilog slope enables a more reliable measurement of radial flow**

## Effect of Shut-in Time

**Too little shut-in time prevents the falloff from reaching radial flow, making it unanalyzable**
**Shut-in time exceeding the injection period length is compressed when plotted with the  proper time function on the log-log plot**

## Shut-in Time

**Falloff data should be plotted with an appropriate time function on a log-log plot to account for the effects of the injection period on the shut-in time**
**Increase falloff time to observe presence of faults and boundary effects if preceding injection period was long enough to encounter them **

## Comparison of Shut-in Times for Identical Injection Conditions

**I**
**n**
**c**
**r**
**e**
**a**
**s**
**i**
**n**
**g**
**t**
**i**
**m**
**e**

**Comparison of Shut-in Times for Identical Injection Conditions**

- 4 hr shut-in

- 8 hr shut-in

- 24 hr shut-in

- Does not reach radial flow

- Barely reaches radial flow

- Well developed radial flow

## Summary of Shut-in Time Effects

**Too short a shut-in time results in no radial flow**
**Shut-in time may be dictated by the preceding injection time**
**Falloff is a replay of the injection**
**Wellbore storage, skin, and need to observe a boundary may increase the required shut-in time**

## Effects of Wellbore Storage and Skin Factor

**A positive skin factor increases the time to reach radial flow**
**A negative skin reduces the time to reach radial flow**
**Large wellbore storage coefficient increases time to reach radial flow**
**Caused by well going on a vacuum, formation vugs, presence of fracture or large wellbore tubular dimensions**

## Comparison of Skin Effect for Identical Falloff Conditions

**I**
**n**
**c**
**r**
**e**
**a**
**s**
**i**
**n**
**g**
**s**
**k**
**i**
**n**

**Comparison of Skin Effect for Identical Falloff Conditions**

**s=0**

**s=50**

**s=250**

**Well developed**
**radial flow**

**Less developed**
**radial flow**

**Minimal**
**radial flow**

## Boundary Effects

## What Can I Learn About Boundaries from a Falloff Test? 

**Derivative response indicates the type and number of boundaries**
**If radial flow develops before the boundary effects, then the distance to the boundary can be calculated**

**Notes:**
Derivative response shape indicates the type and number of boundaries
1 fault causes the semilog slope to double
2 perpendicular faults cause the slope to quadruple if fully developed
Derivative response shape can provide the position of the well relative to the boundaries.  If radial flow develops before the boundary effects, then the distance to the boundary can be calculated

## How Long Does It Take To See A Boundary?

**Time to reach a boundary can be calculated from the radius of investigation equation:**
**Where L****boundary**** is the distance in feet to the boundary**
** ****t****boundary**** is in hours**

## How Long Does It Take To See A Boundary?

**For a boundary to show up on a falloff, it must first be encountered during the injection period**
**Additional falloff time is required to observe a fully developed boundary on the test past the time needed to just reach the boundary**

**Rule of thumb:  Allow at least 5 times the length of time it took to see the boundary to see it ****fully**** developed on a log-log plot**

## Example:  Well Located Near 2 Faults

**An injection well injects at 2000 bpd for 10,000 hours and then is shut-in for 240 hours**
**The well is located in the corner of a fault block**
**The reservoir is a high permeability sandstone**

**Injection Well**

**Fault 1**

**Fault 2**

**Fault Distances: 1000’ and 2000’**

## What Does the Falloff Look Like with Boundary Effects?

**Wellbore Storage**

**RadialFlow**

**Start of boundary effects**

**Effects of**
**both faults**

## Type Curve Analysis of Falloff with Boundary Effects

**k= 507 md**
**s = 10**
**2 faults @ 90****0 ****angle**
**Boundary Distances:**
**     ****1955’& 995’**

## Falloff with Boundary Effects Semilog Plot

**m****2****= 21.8 psi/cycle**

**m****1****= 7 psi/cycle**

**m****2**** indicates more than 1 boundary**

## Summary of Boundary Effects on a Falloff Test

**Use the log-log plot as “master test picture” to see response patterns**
**Look for slope changes in pressure and pressure derivative trends **
**Inner boundary conditions such as wellbore storage, partial penetration, and hydraulic fractures typically observed first**
**Outer boundary effects show up after radial flow occurs if you’re lucky!**

**Notes:**
Use the log-log plot as “master test picture” to see response patterns
Look for slope changes in pressure and pressure derivative trends to identify boundary effects
Inner boundary conditions such as wellbore storage, partial penetration, and hydraulic fractures are typically observed first
Outer boundary effects are usually observed after radial flow occurs (flat derivative trend)

## Typical Outer Boundary Patterns

**Infinite acting**
**No outer boundary**
**Only radial flow is observed on log-log plot**
**Composite reservoir**
**Derivative can swing up or down and re-plateau**
**Constant pressure boundary**
**Derivative plunges sharply**

**Notes:**
Infinite acting – no outer boundary observed and only radial flow is observed on log-log plot
Composite reservoir (change in transmissibility (kh/u) or a mobility change (change in permeability/viscosity ratio, k/u) ) - derivative can swing up or down and re-plateau
Constant pressure boundary– derivative plunges sharply
No flow boundary – derivative upswing followed by a plateau – multiple boundaries cause variations in shape and degree of the upswing
Pseudosteady state – all boundaries around the well reached – derivative swings up to a unit slope – injector is in a closed shape

## Typical Outer Boundary Patterns

**No flow boundaries**
**Derivative upswing followed by a plateau**
**Multiple boundaries additional degrees of the upswing**
**Pseudo-steady state**
**all boundaries reached**
**closed reservoir**
**derivative swings up to a unit slope**

## Infinite Acting Reservoir – No Boundary

**Derivative plateau for radial**
**flow**

**Derivative hump size **
**increases with skin factor**

**Wellbore**
**storage**

## Boundary Effects from Sealing Faults – Derivative Patterns

**3 faults in U shape**

**2 parallel faults**

**2 perpendicular faults**

**1 fault**

## Boundary Effects from a Composite Reservoir – Derivative Patterns

**Mobility increase away**
**from the well**

**Mobility decrease away from the well**

## Is It a Real Boundary?

**Check area geology **
**Type of injectate **
**Both the injection and falloff have to last long enough to encounter it**
**Most pressure transient tests are too short to see boundaries**

**Notes:**
Check area geology for boundaries and possible permeability and net thickness changes to justify the test response
A composite reservoir can give a similar signature to a conventional boundary
The type of injectate may also impact the test – waste acids, viscous wastes
To observe a boundary in the welltest, both the injection and falloff have to last long enough to encounter it
Most pressure transient tests are too short to see boundaries

## Example:  Hydraulic Fracture Type Curves

## Log-log Plot Examples 

## A Gallery of Falloff Log-log Plots

**Radial flow with boundary effects**
**Falloff with a single fault**
**Falloff in a hydraulically fractured well**
**Falloff in a composite reservoir**
**Falloff with skin damage**
**Falloff after stimulation**
**Falloff with spherical flow**
**Simulated pseudosteady state effects**

## Radial Flow Followed by Boundary Effects

**Wellbore Storage Period**

**Transition to**
**radial  flow**

**Radial Flow Period**

**Boundary Effects**

**Radial Flow Followed by Boundary Effects**

## Falloff with a Single Fault

**Falloff with a Single Fault**

## Falloff with a Hydraulic Fracture

**Falloff with a Hydraulic Fracture**

**Derivative drop due to constant pressure**

**Half slope on both curves – linear flow**

## Falloff in a Composite Reservoir 

**Falloff in a Composite Reservoir**** **

## Falloff with Skin Damage

**k = 4265 md**
**s = 392**

## Falloff with Negative Skin

**k = 99 md**
**s = -1**

**Radial Flow**

## Falloff Dominated bySpherical Flow

**Partial Penetration**
**characterized by a negative 1/2 slope line**

## Simulated Falloff with Pseudo-steady State Effects

## Other Types of Pressure Transient Tests 

## Other Types of Pressure Transient Tests

**Injectivity Test**
**Record pressure, time, and rate data from the start of an injection period following a stabilization period**
**Pros**
**Don’t have to shut in well**
**Generally maintain surface pressure so less wellbore storage**
**Less impact from skin**

## Other Types of Tests

**Cons**
**Noisy data due to fluid velocity by pressure gauge**
**Rate may fluctuate so an accurate history is important**

## Other Types of Tests

**Multi-rate Injection Test**
**Record pressure, time, and rate data through at least two injection periods**
**Pros**
**Can be run with either a decrease or an increase in injection rate**
**Minimizes wellbore storage especially with a rate increase**
**Provides two sets of time, pressure, and rate data for analysis**
**Decreasing the rate provides a partial falloff without shutting in the well**

## Other Types of Tests

**Cons**
**Noisy data due to fluid velocity by gauge**
**1****st**** rate period needs to reach radial flow**

## Other Types of Tests

**Interference Test**
**Use two wells: signal and observer**
**Signal well undergoes a rate change which causes pressure change at the observer**
**Measure the pressure change over time at the observer well and analyze with an Ei type curve or, if radial flow is reached, a semilog plot**

## Other Types of Tests

**Pros**
**Yields transmissibility and porosity-compressibility product between wells**
**May give analyzable results when falloff doesn’t work**
**Cons**
**Generally involves a small pressure change of 5 psi or less so accurate surface or bottomhole gauges are needed**
**Observable pressure change decreases as the distance between the two wells increases**

## Other Types of Tests

**Cons (cont.)**
**Complex analysis if more than two injectors are active**
**Need knowledge of pressure trend at the observer well**
**Test rate should be constant at the signal well**

## Other Types of Tests

**Pulse Test**
**Similar to interference except rate changes at observer well are repeated several times**
**Pros**
**Multiple data sets to analyze**
**Verify communication between wells more than one time**
**Cons**
**Difficult to analyze without welltest software – Monograph 5 methodology**
**Requires more time and planning and careful control of the signal well rate**

## Designing an Interference Test

**For both interference and pulse tests, the best  design approach is to use a well test simulator**
**Interference tests can designed using the Ei type curve**
**Design information needed:**
**Distance between signal and observer wells**
**Desired pressure change to measure**
**Desired injection rate **
**Estimates of c****t****, ********, ********, k, h, r****w******

## Interference Test Design Example

**Two injection wells are located 500’ apart.  Both wells have been shut in over 1 month**
**An interference test is planned with an injection rate of 3000 bpd (87.5 gpm)**
**k = 50 md, h = 100’,  ******** = 20%, ********f**** = 1 cp, c****t ****= 6x10****-6**** psi****-1****, r****w****= 0.3 ft**
**How long will the test need to run to see a 3 psi change at the observer?  **

## Interference Design Example

**Ei Type Curve: from Figure C.2 in SPE Monograph 5**

## Interference Design Example

**Calculate P****D ****and r****D **** from equations listed in PDE discussion**
**Find t****D****/r****D****2**** from corresponding P****D****  value on Ei type curve**
**Calculate t****D **** and solve for t****interference**** **
**Results:  **
**P****D****= 0.0354, r****D****=  1666.7   **
**t****D****/r****D****2**** = 0.15**
**t****D****= 416,666.7**
**t****interference****= 3.4 hours**

## Interference Test Example

**An interference test is conducted between two injection wells at a Gulf Coast area facility.  **
**Reservoir conditions:**
**h=55’,   ********=28%, c****t****=6x10****-6**** psi****-1****, r****w****=0.25 ft**
**Well Data:**
**q = 120 gpm**
**wells are 150’ apart**

## Interference Test Example:Log-log Plot at Observer Well

- Radial flow

## “Real World” Interference Type Curve Match

**Match Results:**
**k = 4225 md**
**********c****t**** = 4.015x10****-6**** psi****-1**

## How Do Falloff Results Impact Area of Review

**The transmissibility obtained from the falloff and the solution from the PDE can be used to project the pressure increase due to injection**
**The PDE solution can also be used to estimate the location of the cone of influence**
**Both the pressure projection and cone of influence location estimate can be set up in a spreadsheet**

## Example Cone of Influence Estimate

## How is Fracture Pressure Determined?

**Typically estimated from  fracture gradient correlations (e.g. Hubbert and Willis, Eaton)**
**Can be determined from a step-rate test**
**Fracture pressure varies with depth, lithology, and geographical region**

## What is a Step-Rate Test?

**Series of constant rate injection steps of equal time duration**
**Each step can be analyzed as a pressure transient test (injectivity test)**

## Step-Rate Test Rate Sequencing

**q, gpm**

- q1

- q3

- q2

- q4

- q5

- q6

- q7

- q8

**Elapsed test time, t (hrs)**

**Total test time for all steps**

**Each rate step is maintained at a constant rate of equal duration**

## Step-Rate Test Pressure Behavior

**Injection pressure (psi)**

**Time (hours)**

**Time**
**Step **
**Size**

- t

- t

- t

- t

- t

- t

- t

## Step Rate Tests Analysis

**Data is analyzed using log-log and linear plots**
**Use the linear plot to estimate fracture pressure (also called the formation parting pressure)**
**Use the log-log plot to verify that fracturing occurs and estimate kh/u and skin**

## Step Rate Test Analysis: Linear Plot

**Fracture or formation parting pressure**

**Injection pressure (psi)**

**Each point is the **_**final **_**injection pressure at each rate step**

**Injection rate (bpd)**

## Example Step Rate Test 

**1****st**** series of step step rate tests**

**Falloff test**

** ****Rates**

**Pressures**

**2****nd**** series of step rate tests**

**2****nd**** Falloff test**

## Log-Log Plot of a Rate Step

**Noisy derivative, but suggests radial flow trend – no fracture signature**

**Analysis of 12****th**** Step in 1****st**** Rate Series**

## Example Step Rate Linear Plot

**No slope decrease – no fracture indicated**

## Other Uses of Injection Rate and Pressure Data

**Monitor injection well behavior**
**Data readily available in Class I wells**
**Hall plot**
**Linear plot**
**x-axis:  cumulative injected water, bbls**
**y-axis:  ********(********BHP*********t), psi-day**
**Can be used to identify fractures**

## Hall Plot

**Cumulative injected water (bbl)**

**Cumulative**** (********P*********t), psi-days**

**m****Hall**** = [141.2*B*u*ln(r****e****/r****wa****)]/(k*h)**

**Wellbore plugging**

**Fracturing near the well**

**Fracture Extension**

**Radial flow**

## Hall Plot Analysis

**Straight-line slope gives transmissibility:**
**Slope changes indicate well conditions**
**Decrease in slope indicates fracturing (skin decrease)**
**Increase in slope indicates well plugging (skin increase)**
**Straight line indicates radial flow**

## Hall Plot Example

## Hall Plot Limitations

**Type of pressure function used impacts the slope of the data plotted**
**Cannot determine kh/******** and s independently from a single slope**
**Pressure data is dependent on gauge quality and can be noisy**