A STUDY OF THE EFFECT OF SHORT DURATION DISTURBANCES

ON PRESSURE TRANSIENT ANALYSIS

by

WELDON THOMAS PIERSON, B.S. in P.E.

A THESIS

IN

PETROLEUM ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

PETROLEUM ENGINEERING

Approved

Accepted

May, 1989

/r t

^0 ^

Copyright Weldon Thomas Pierson 1989

ACKNOWLEDGEMENTS

The writing of this paper and the completion of two degrees in my

educational process would not have been possible were it not for the

following people. Each and every one of the faculty members in the

department of Petroleum Engineering has contributed to my knowledge and

understanding of the complex field of petroleum engineering which I have

chosen as a career. The members of my committee, friends, and family

are largely responsible for giving me the fortitude leading to this

masters degree.

I would like to thank Dr. Marion Arnold and Dr. Carlon Land for

contributing their time spent on my committee in professional guidance

and review of my research efforts.

Additionally, I would like to thank Professor Duane A. Crawford for

having served on my committee but, more importantly, for presenting the

topic of "Pressure Transient Testing" in such a manner as to raise my

level of interest to the point of investing a year of my time on an

in-depth study within the topic and preparation of this paper.

I owe a special thanks to Tim Hauss for his time spent on answering

questions about his simulation model, suggestions on modifications, and

for making a program copy available to me. The approach taken in this

research would not have been possible without his program.

11

Lastly, it would not be appropriate for me to conclude without

thanking the people who have contributed the most through their love,

support, and encouragement; my parents, my wife Marianne, and my wife's

parents. To each of you I owe an immeasurable amount of gratitude.

Ill

CONTENTS

ACKNOWLEDGEMENTS ii

TABLES V

FIGURES vi

NOMENCLATURE vii

CHAPTER

I. INTRODUCTION AND PURPOSE 1

Research Goal ^ Chapter Summary 5

II. RADIAL SIMULATION MODEL 7

Radial Diffusivity Equation 11 Modelling A Pressure Disturbance 14

III. REVIEW OF CURRENT THEORY 25

Horner's Method for Variable Rates 26 Shut-in and Flow Period Combinations 32 Minimizing Error Caused by A Disturbance 39

IV. RADIAL SIMULATOR METHOD 4 o

Disturbance and Flow-Period Combinations Studied .. 43 Analysis 45 Simulation Results 46 Discounting Afterflow Effects 56

V. CONCLUSIONS 59

REFERENCES 62

APPENDIX A: SEMI-LOG PLOTS FOR THIRTY TEST RUNS 64

APPENDIX B: MODIFIED RADIAL SIMULATOR PROGRAM LISTING 97

IV

TABLES

3-1. Shut-in pressures, shut-in times and superpositioned time

increments used in Guerrero's example [5] 36

4-1. Reservoir parameters held constant for all test runs ... 44

4-2. The slope (m) and percent error (e) associated with various flow-period-to-disturbance ratios 54

FIGURES

2-1. Fundamental Model Assumptions 10

2-2. Comparison between the Ei solution (Ei) and the finite difference (FD) model of Hauss's original radial simulator 15

2-3. Semi-logarithmic representation of initial reservoir pressure distribution 18

2-4. Initial reservoir presssure distribution to a radial distance of 30,000 feet 19

2-5. Initial reservoir pressure distribution to a radial distance of 1,000 feet 20

2-6. Initial reservoir pressure distribution to a radial distance of 100 feet 21

2-7. Initial reservoir pressure distribution to a radial distance of 12 feet oo

3-1. Rate vs. time relationship for a well having produced at a constant rate since time of completion 28

3-2. Rate vs. time relationship for multiple flow periods at constant production rate [ 5 ] 33

3-3. Shut-in pressure versus superpositioned time for flow periods with and without disturbances in Guerrero's example [ 5 ] 37

4-1. Semi-log plot of an uninterrupted buildup test 47

4-2. Semi-log plot of a ten hour disturbance followed by a five hour flow period depicting two distinct semi-log straight lines 50

4-3. Percent error associated with various flow-period-to-disturbance ratios 55

4-4. Characteristic behavior of afterflow on a pressure buildup test [9] 58

vi

NOMENCLATURE

Svmbol - •

^wb cross sectional area of wellbore (tubing)

Ct total compressibility

Cg shrinkage factor

Cs wellbore storage coefficient

h formation thickness

Kc constant (9.667 - 10"^)

' ICQ effective permeability to oil

m slope

M cumulative production

p pressure

Pbh bottomhole pressure

n shut-in pressure

Pw ^

q production rate

q surface production rate

( q )j sandface production rate

r radius

ni block midpoint radius

At incremental time since beginning of buildup test

t cumulative time of production at instant of shutin for buildup test

cgs

cm

atm"-'-

std cc :/cc

En glish

0

ft-

psi~

STB/RB

cc/atm

cm

cc

atm

atm

cm

cm

sec

sec

bbl/psi

ft

atm cm /gm

Darcies md

atm/cycle psi/cycle

STB

psi

psi

atm psi

std cc/sec STB/D

std cc/sec STB/D

std cc/sec STB/D

ft

ft

hours

hour;

vii

Symbol

t_ pseudoproducing time

tcum cumulative time

BQ formation volume factor

C£S_

sec

sec

cc/std cc

English

hours

5 duration of short-term shut-in period sec

T flowtime following shut-in period

Pf fluid density

(J) porosity

sec

gms/cc

fraction

hours

RB/STB

hours

hours

lbs/ft^

fraction

Vlll

CHAPTER 1

INTRODUCTION AND PURPOSE

Pressure transient testing is an valuable tool in the field of

petroleum engineering for characterizing fluid and rock properties of

petroleum reservoirs. Based on known rock and fluid properties, proper

testing procedures can yield important parameters such as permeability,

interwell capacity, and reservoir pressures. In addition, pressure

transient tests are used in the following ways: to determine flood front

positions, to detect faults, to delineate boundaries, to detect and

describe natural and induced fractures, and to evaluate well-to-well

interference. Furthermore, zones of altered permeability may be

detected and the data allow the calculation of the degree of damage or

stimulation near the wellbore (the skin effect), and the areal extent of

such an altered zone [3]. However, even though pressure transient

testing can reveal salient factors, the degree of accuracy of any

calculation is heavily dependent on four considerations. These

considerations include the proper preparation of the well prior to

testing, the events occurring between the end of well preparation and

completion of the test, the length, nature and duration of the test, and

lastly, the correct analysis of the test data. A mistake in any one of

these areas can render the data and subsequent analysis results useless.

Proper conditioning of a well (i.e., stabilized well flow) prior to

testing is one of the foremost requirements to obtain accurate test

1

data. Also known as well stabilization, the need for the procedure is

widely accepted; however, an acceptable length of time for the duration

of the stabilization period varies as there is not a uniform industry

standard at this time. Miller et al. [10] state that stabilization of

the well is always desirable and in some cases a necessity. They

further state that the lack of stabilization is a contradiction of some

of the assumptions made in the derivation of the buildup flow equations.

Although the condition of stabilization may be approximated in many

cases, conventional techniques are not applicable to a well severely

unstable when it is shut in for testing [10]. Thomas [13] noted that it

is imperative that a well be stabilized in order to obtain good data

which is representative of the reservoir system under study.

Opinions vary as to the optimal time span for stabilization of a

well for pressure transient testing. Matthews and Russell [9] suggest

that the well be stabilized at its normal rate for a period of one week.

Furthermore, Thomas [13] suggests that proper stabilization is a

function of the fluid and reservoir properties but generalizes that a

period of ten days should be adequate in most cases. Miller et al. [10]

presented a method for calculating the required length of time to obtain

steady state flow behavior. Their method is dependent on the well

spacing and the ratio of permeability to viscosity. In their example,

stabilization times range from 0.4 hours for a well on 20-acre spacing

and a permeability-viscosity ratio of 100, to 333 hours or approximately

14 days for a well on 160-acre spacing and a ratio of unity [10].

However, a rule-of-thumb taught at the university level and used by a

major oil company specifies that the production or injection rate of a

well identified for testing be rate stabilized within 5% for a period of

three to four times the length of the specific testing period [3]. For

example, a 72-hour test would require a stabilization period lasting

from 216 hours to 288 hours or from nine to 12 days provided the

flowrate is within the +_ 5% range.

Having properly prepared the selected well for testing, the next

considerations are the events prior to beginning the test itself. For

one reason or another, the well may need to be closed in for a short

period of time before the prescribed test. Common reasons for this

include insufficient stock tank volume, installation or replacement of a

valve, or the attachment of diagnostic equipment and pressure recording

devices [5]. These situations are usually unavoidable and the length of

shut-in time is kept to a minimum. However, a flow line or valve leak

during the stabilization period, automated shutdown due to vibration or

other early warning detectors, power failure, and scheduled maintenance

of compressors or injection stations may require shutting in the well

for a substantially longer period. Human error may also be a cause of

procedural error because field personnel, being unaware of the planned

pressure test, may inadvertently shut in the well. Finally, the testing

company or individuals may shut in the well for a short period as a

means of equipment calibration or "indexing" prior to beginning the

test. However, a short time into the test, a malfunction with the

equipment or calibration setting may require that the test be

interrupted. In the case of an interrupted test, a short flow period

may follow to facilitate any adjustments before the test is continued.

Regardless of the reason, temporarily shutting in a well during the

stabilization period introduces a pressure transient which migrates

through the reservoir. The result of such a pressure surge prior to

transient testing, will be to reduce the degree of stabilization

attained. Consequently, the greater the degree of instability, the

greater will be the deviation from the assumptions embodied in the

formulas used in analyzing the data. Several mathematical studies

indicate that a short pressure disturbance prior to a pressure buildup

test increases the slope of the semi-log straight line and results in

lower calculated values of capacity and permeability than the actual

values for those variables. Although in most cases the duration of the

disturbance is small in relation to the length of the test itself, the

error introduced by using an incorrect slope has been calculated to be a

high as 13% [5,11].

Research Goal

The objective of this research was to quantify the error introduced

through conventional analysis caused by disturbances of short duration

and to investigate the effectiveness of an additional flow period to

reduce or cancel that error. The study was done using a mathematical

reservoir simulation model and the methodology used was two-fold.

First, relatively short disturbances of varying durations were

introduced prior to an otherwise ideal pressure buildup test, and

secondly, the disturbances were followed by flow periods ranging from

one-half to ten times the length of the disturbance.

The approach taken utilized a single-phase radial simulator as a

source of information. The simulator was originally developed to studv

the influence of a varying wellbore storage coefficient on a pressure

drawdown test [6]. However, minor modifications were made which allowed

the simulator to be used for modelling pressure disturbances preceding a

pressure buildup test. The buildup test was simulated with a constant

wellbore storage coefficient. However, both the length of the

disturbance and the length of the flow period following the disturbance

were varied in the 30 test runs made.

The data from these 30 tests were analyzed using accepted

conventional procedures and the results were compared with a pressure

buildup test run without a pre-test disturbance. Additionally, the

results were compared with results from papers by Guerrero [5] and Nisle

[11] dealing with predictions based on the point-source solution of the

radial diffusivity equation.

In contrast to the point-source studies of Guerrero [5] and

Nisle [11], the reservoir simulator is programmed using a finite

difference approximation of the radial diffusivity equation. The

solution of the finite difference equation is an approximation of the

exact solution of the diffusivity equation.

Chapter Summary

Following in Chapter 2 is a description of the radial simulator,

the assumptions used in its development, and a description of the minor

modifications necessary for the simulation of the disturbances.

Chapter 3 is a review of current theory and includes the assumptions and

equations necessary to predict the behavior of a well subjected to a

disturbance prior to the test as outlined by Guerrero and Nisle. The

results of the research are stated in Chapter 4 and compared with the

published forecasts. Any discrepancies are noted, considered, and

discussed before recommendations are stated. Based on the comparisons,

conclusions are stated in Chapter 5 which relate the length of a

disturbance and the length of a flow period following a disturbance to

the error associated with the combination. Plots of all the test runs

are included in the Appendix A. Lastly, Appendix B contains a complete

listing of the simulation model used for studying short-term

disturbances.

CHAPTER 2

RADIAL SIMULATION MODEL

The need to vary both the length of a disturbance and the length of

the flow period following the disturbance in a predetermined structured

manner, made the location and use of actual pressure buildup data

impractical. A total of 30 combinations of the two variables in

addition to the uninterrupted "base line" buildup test was studied.

Extended shut-in periods several times longer than conventional test

periods cannot be achieved in the field without substantial production

losses to the producer or the contacting of reservoir boundaries;

therefore, an alternate means of information acquisition was needed.

A computer program written at Texas Tech University [6] accurately

simulates a pressure drawdown test in a one-well-centered radial system.

Unsteady-state radial flow of a slightly compressible fluid through

porous media is the basis for the simulation model. In developing the

model, Hauss presented a partial differential equation identical to the

radial diffusivity equation of Horner [7] except Hauss included a

production source/sink term. Because of the complex nature of the

radial diffusivity equation presented by Hauss, a solution using a

finite difference equation was substituted for the exact solution from

the radial diffusivity equation. Programming of a solution to the

finite difference equation on a computer enables a user to rapidly solve

for pressure values at nodal position levels; the time required for

8

solution would be prohibitive using a calculator. The partial

differential equation and its finite difference solution were developed

to study the effect of a changing wellbore storage coefficient on the

analysis of a pressure drawdown test. However, the mathematics and

method of solution are equally applicable to pressure buildup tests with

a constant wellbore storage coefficient.

Fundamental to the development of the partial differential equation

is a mass balance of the fluid entering, leaving, and remaining within a

small finite element of porous material and the incorporation of Darcy's

law describing the transfer of mass. For the case of radial flow, Hauss

applied the mass balance to an elemental cylinder. The finite

difference model employs a series of concentric cylinders, the width of

which increases logarithmically with distance from the center. This

comprises the radial flow system under consideration. A mass source or

sink term is included to compensate for fluid being injected into or

fluid being produced from this system [6]. A number of reservoir

parameters were assumed constant. These assumptions have been discussed

and justified in a number of past publications including that of

Hauss [6]. In order to understand the specific case, a sufficient

comprehension of the more general case must first be obtained. The

knowledge gained from the study of a single well in an infinite, single

phase reservoir can be applied towards understanding the complexities of

a multiple-well, multiple-phase reservoir.

Two broad assumptions were made that are common in derivations of

partial differential equations involving petroleum reservoirs. The

first is termed the "Black Oil" assumption meaning that for all times

9

the composition of the produced fluid, or more specifically, the density

of the fluid at standard conditions is constant. This assumption is

correct over the majority of crude gravities and is contradicted only by

very light crude oils of gravity greater than 45* API. Secondly,

Darcy's law must apply within the reservoir, implying the fluid to be in

laminar flow regime. This second assumption applies to virtually all

flow within the reservoir and is in error only for high injection or

production rates near the wellbore and in the presence of extremely high

pressure gradients which may cause turbulent flow [1].

In addition, a constant formation thickness is assumed and

exclusion of reservoir heterogeneities is implied with the assumptions

of constant porosity and permeability. These and some other assumptions

that follow result primarily from the fact that a one-well system

provides data from only one well and those data are therefore the best

available. Additional simplifications on fluid properties include

constant viscosity and constant slight compressibility. This assumption

is reinforced by the fact that the effect of pressure on viscosity is of

the same order of magnitude but opposite in sign to the effect of

pressure on formation volume factor [1]. Thus, the effects tend to

negate each other. Futhermore, concerning restrictions on fluid flow,

non-horizontal flow is not considered, thereby eliminating gravitational

influences of dipping reservoirs. Lastly, fluid flow within the

reservoir is assumed isothermal [6]. The aforementioned assumptions and

the component of the system on which they act are categorized in

Figure 2-1.

10

Formation

Fluid

Transport

constant thickess constant porosity constant permeability

slightly compressible fluid density

slightly compressible fluid viscosity

constant compressibility

Darcy flow isothermal horizontal

Figure 2-1. Fundamental Model Assumptions

11

Radial Diffusivity Equation

A mass balance treatment of fluid within a cylindrical reservoir

element considers the mass rate of flow into the element, mass rate of

flow out of the element, and the source or sink term in the form of

production or injection [6]. The relationship of these terms to the

accumulation or reduction of mass within the element is demonstrated in

Equation 2-1,

mass rate of accumulation mass flow rate in

- mass flow rate out - mass production rate

(2-1)

Defining each of the terms in the equation separately, applying the

assumptions, and several repetitions of the chain rule of calculus,

results in Equation 2-2, which is the radial diffusivity equation,

describing the flow of a slightly compressible fluid in porous media

with a source/sink term included as presented by Hauss, [6]. The term

q is the source/sink term and was developed to have the sign

convention of positive (+) for production and negative (-) for

injection. While Equation 2-2 contains the source/sink term, it does

not consider any wellbore effects.

r„ 3r 1 ^ V dr 'm

qos Bo io (0^0 Ct ^ ap at 2 7C fn Ar h ko

(2-2) o y

The storage or depletion of mass from the wellbore usually

accompanies instantaneous flow-rate changes in a well [6]. Xo better

example of this exists than with the case of pressure buildup testing

12

where wells are subjected to an instantaneous change from a positive

production rate to a no-flow or shut-in condition. While the surface

rate may be zero, reservoir pressures distant from the wellbore and the

slight compressibility of the wellbore fluid permit the reservoir to

continue to produce from the sandface into the wellbore until the weight

of the fluid column is in equilibrium with reservoir pressure. This

period of afterflow into the well results in a dynamic gas-liquid

interface the height of which at any given time is proportional to the

reservoir pressure at the sandface at that time. The theory for

incorporating this afterflow characteristic into the reservoir model is

that proposed by van Everdingen and Hurst [14]. Their work states that

the volume of fluid entering or leaving the wellbore per atmosphere of

pressure is a constant and can be expressed in (cc/atmosphere) at

reservoir conditions. Starting with the statement that the difference

between the surface and sandface flow rates is equivalent to the time

derivative of mass within the wellbore, Hauss arrives at an expression

for the wellbore effect of a changing fluid level.

Cs dPbh los - (qos)s = B 5J- ' (2-3)

where the term Cs is referred to as the wellbore storage coefficient and

is defined as

13

A positive rate differential represents accumulation in the wellbore

while a negative rate differential represents a depletion of fluid from

the wellbore. The wellbore storage effect is combined with Equation 2-2

in the development of the finite difference equation.

In the form of Equation 2-2, an exact solution to the radial

diffusivity equation would be difficult. A numerical technique for

solving a partial differential equation can be developed from a finite

difference equation. The solution to the finite difference equation is

an approximation of the solution to the original partial differential

equation [1].

Finite difference equations have been used to solve problems in

studies including constant rate production of gas wells, reservoir

discontinuities, vertically fractured gas wells, and flow of a slightly

compressible fluid in a composite reservoir [6]. Hauss developed his

particular finite difference scheme based on a system of concentric

elemental cylinders. A consistent, logarithmic increase in the width of

these grid elements provides smaller nodal spacing in areas of

relatively higher pressure gradients near the wellbore. Pressures at

the nodes are approximately a linear function of the logarithm of the

radius during unsteady-state flow. Therefore, a consistent pressure

drop between nodes is accomplished by means of a logarithmic

transformation. Although originally logarithmically spaced prior to the

transformation, equal nodal spacing is present after the transformation

to the linear system [6]. A total of 240 nodes are used in both the

original and modified versions of Hauss's program. After developing the

finite difference equation, Hauss programmed an implicit method for

14

solution on a personal computer in the "QuickBasic" programming

language. The finished program was then compiled into machine language

for the purpose of gaining additonal computational speed. A complete

listing of the computer program as listed by Hauss [6] is included in

his thesis which is shelved at the Texas Tech University library.

After the programming process was completed, the accuracy of the

model was verified. Various runs of the program were analyzed and

compared (see Figures 2-2 a, b) with the Ei and logarithmic

approximations of the radial diffusivity equation as calculated by

Lee [8]. The model shows excellent agreement with these solutions,

validating the theory and mathematics of the finite difference solution

of the partial differential equation. Indeed, the error associated

within five feet of the wellbore is one half of one percent (0.5%) in

some cases [6]. With the availability of the radial reservoir

simulator, only minor programming revisions and the addition of several

time location checkpoints are necessary to model the migration of a

short-duration pressure disturbance into the formation.

Modelling A Pressure Disturbance

Although the program in its original form was useful and

verifiable, the programming sequences were not in place to model the

short duration disturbances of interest in this research. Therefore,

modifications to the program were necessary. One area of modification

was an initial dynamic pressure distribution within the grid nodes prior

to introducing the disturbance. The original drawdown simulator of

Hauss [6] loaded a constant initial reservoir pressure into all of the

nodes used in the calculations. For the purpose of simulating a

15 • =•'

Press. (psia)

Initial Press

.-•-• -•-•-

^. 2720^ Pwf

2650 I I

0.50 600.50 1200.50

rw Radius (ft.)

1800.50

(a)

Press. (psia)

2825 T

2775 ••

2725 ••

Pwf •"

2675

— Ei • FD

..-••

0.50 rw

1.50 2.50 Radius (ft.)

(b)

3.50 4.50

Figure 2-2. Comparison between the Ei solution (Ei) and the finite difference (FD) model of Hauss's original radial simulator. Full reservoir distribution (a) and near wellbore distribution (b) are sncvn [6].

16

pressure buildup test; however, the program needed to have an initial

pressure distribution representative of a production history. Secondly,

the surface flow rates must alternate between a flowing and non-flowing

condition while maintaining the pressure distribution within the

reservoir immediately prior to interrupting its execution. In addition,

the length of the initial shut-in period (the disturbance) and the

length of the flow period following the disturbance must be easily

varied by information supplied from the keyboard and prompted by the

parameter menu portion of the program. Lastly, the period of the

buildup test after the influence of wellbore afterflow and prior to

contact with a boundary must be substantially long to permit the longest

possible period of semi-log-straight-line behavior. Hurst and

van Everdingen observed that a plot of shut-in pressure versus the log

of shut-in time reveals a straight-line portion possessing a slope

proportional to the permeability of the formation [14], The analysis of

this semi-log straight line is one part of a methodology which has

become known as conventional analysis. The degree of influence of the

flow/disturbance combinations on the slope of this semi-log straight

line is the topic under investigation.

The original program simulates a pressure drawdown test and initial

pressure arrays are loaded with an original reservoir pressure value of

3000 psi. In the case of a pressure buildup test however, the well is

assumed to have been on production and thus has already established a

pressure distribution of lowest pressures near the wellbore with a

non-linear pressure progression with increasing distance from the

wellbore. The pressure distribution cannot be arbitrary but must be a

17

true representation of the pressure gradients. To incorporate this

characteristic into the pressure buildup model, the unedited program

version of the pressure drawdown test must be run in its entirety

(over 24,600 hrs of test time) and the nodal pressure distribution saved

to disk storage space for access by the modified program version. The

nodal pressure distribution is read into memory in the early execution

stages of the modified program and loaded as the initial pressure array.

Figure 2-3 displays pressure versus the logarithm of radius from the

wellbore as loaded into the the initial pressure array. Figures 2-4,

2-5, 2-6, 2-7 represent the same pressures on a strictly Cartesian plot

for radial distances of 30,000, 1000, 100, and 12 feet, respectively.

The second major programming consideration involves insertion of

appropriate time checks so that correct program execution takes place

during the correct flow or shut-in period. In other words, the program

must determine when to change the flow rates and reduce the time step

immediately after a rate change while allowing proper time step

sequencing between rate changes. For a well that has been producing at a

constant rate for some period, the well is shut in and the disturbance

initiated at time (t-,). This initial shut-in period lasts until

allowing the well to flow again at time (t2). The flow period ends and

the complete test begins at (t) when the well is once again shut in.

Lastly, the test ends and the well is put back on production at

time (t^). The first shut-in period is equivalent to a pressure surge

during stabilization or the beginning of an interrupted test as the case

may be. The length of the disturbance is then (t^ - ti) (hrs). The

duration of the flow period following the disturbance is t - t^ (hrs)

18

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23

during which any corrections, repairs or recalibrations are carried out.

The length of the complete testing period is t + At (hrs); time

increments and corresponding pressure readings are zeroed at t (hrs).

Menu prompts in the modified version of the program allow for a

wide range of variation in the time span for both the initial shut-in

periods and flow periods. Conversions to cumulative times are automatic

from within the program. Insertion of time checks into the main loop of

the program insures that the surface flow rates are changed at the

designated times. The intial flowrate of the model is 750 (STB/Day).

At times t and t^ the production rate is changed to 0 (STB/Day) and at

time t2 the rate is switched from 0 to 750 (STB/Day). For reasons of

stability, at each toggle point in the program, the time step is divided

by a constant before program execution continues. Thus, smaller time

steps are taken with the new flow rate.

For purposes of this study, boundary effects are not considered.

The longest semi-log straight line with no boundary influence is

obtained by incorporating an extremely large radius to the closed

boundary. Although an external radius (re) exceeding 10 ft may seem

extreme, it serves the purpose. If conventional analysis techniques are

to be used, the longest display of semi-log-straight-line behavior

possible is desirable. Therefore, a small wellbore storage coefficient

and long total elapsed time are necessary. With these noted exceptions,

and a few modifications to output lines for screen and printer, the

program as developed and listed by Hauss [6] was used in its entirety.

Neither the form of the partial differential equation nor the finite

difference equation was changed. A complete listing of the reservoir

simulator as modified for simulating short-duration pressure

disturbances is included as Appendix B.

CHAPTER 3

REVIEW OF CURRENT THEORY

Theory dealing with pressure disturbances appeared as early as 1951

in Horner's classic paper entitled "Pressure Build-Up in Wells" [7].

Horner presented an approximate solution to the radial diffusivity

equation based on the point-source solution. Furthermore, he presented

solutions for three well scenarios; a single well in an infinite

reservoir, a well located in an infinite reservoir but near a fault, and

a single well completed in a finite reservoir. Horner also outlined a

method to compensate for the effects of variable production rates prior

to testing. The concept was first used in studying two specific events

in a new well, initial completion and shutin preceded by a constant

production rate from the time of completion. Horner then extended the

method to consider variable rates by superimposing the independent

effects of the individual production rates and arriving at resultant or

combined effect [7].

This technique is called superpositioning and is used widely in

petroleum reservoir engineering for calculating water influx, fault

influence, and such considerations as the effect of other wells in a

reservoir on the production rates and pressures of a specific well.

Nisle [11] published a paper in 1956 which applied Horner's theory to

the case of a single production rate occurring during multiple flow

25

26

periods. Nisle concluded that under conventional analysis techniques, a

flow period ten times the duration of a disturbance prior to testing

'.vQuld reduce the error associated with the disturbance to no greater

than 10% [11]. Horner's method was again used in Guerrero's book [5] to

study the effects of various flow period and shut-in period combinations

on the semi-log -straight-line portion of a pressure buildup plot. Each

of these applications is discussed in detail in following sections of

this thesis. In certain instances, older non-standard symbols have been

replaced by the approved standard symbols as outlined by The Society of

Petroleum Engineers [12]. The equations will be in terms of proper

field units with any exceptions noted as they appear.

Horner's Method for Variable Rates

Horner began with the radial diffusivity equation for the flow of

slightly compressible fluids through porous media,

2 ] 2 ^ T BT k 9t

££ + i i£ = iiii ii • (3-1)

The assumptions included in Equation 3-1 are identical to those used to

develop the reservoir simulation model and include the following

restrictions. The reservoir is assumed homogeneous and horizontal with

a constant formation thickness. The fluid flow is single-phase radial

into the wellbore and obeys Darcy's law. This Darcy flow assumption

implies laminar flow is occurring. Finally, the fluid itself is of

slight compressibility and fluid viscosity varies little over the

pressure range encountered within the reservoir [7]. Horner noted that

27

the exact solution to the radial diffusivity equation is complex,

involving Bessel functions. The equation is diificult to solve and he

therefore presented a solution using the "point source theory." The

solution. Equation 3-2, is exact for an infinite external boundary at

constant pressure and for an internal boundary (i.e., the wellbore)

being a point source.

(3-2)

where: A = 70.6 (a constant)

B = 948.2 (a constant) _ stock tank volume _ 1

° ~~ reservoir volume ~ B,

The zero-wellbore-dimension requirement resulted in the solution's being

termed the "point-source solution" [7]. Equation 3-2 and variations

with other units and boundary conditions are widely accepted as

solutions for the original diffusivity equations for the respective

conditions of applicability. The error introduced by the actual

wellbore dimension is assumed to be of little consequence in the

presence of other generalities such as reservoir homogeneity and single-

phase fluid behavior.

Horner then considered the case of the well's being centered in an

infinite reservoir. The well, having been completed at time (0), has

been producing at a constant rate until being shut in for a transient

test at time (t). The elapsed time since shutin is referenced by delta

t ( At ). This rate-time relationship is depicted in Figure 3-1.

28

I

B«s<nning of pressurefxjildup

test

O ^ ^

0 t + At

Figure 3-1 Rate vs. time relationship for a well having produced at a constant rate since time of completion.

29

Horner accounted for this production history by recognizing that

the influence of the production rate at any time during the test acts

over the time period (t + At), while the influence of the non-production

state acts on the time period At. The combined influence of the two

periods can be calculated from the combination of two independent

applications of the point source solution resulting in

Pw = Po +

70.6 q3c lo Bo

koh 1 Ei ^ 948.2 T^ (|) lo c, ^

V ko (t + At ) J

- Ei ^ 948.2 rj^ (D to c, ^

V ko At y (3-3)

Horner states that for values of (x) less than 0.01, Ei(-x) can be

approximated accurately by the natural logarithmic expression

(Ei (-x) « In X + 0.5772 ...) and shown as follows:

Pw = Po +

70.6 q c lo Bo

koh In

^ 1688 T^(^^o<^x

V ko (t + At)

1688 T^^<^[ioC, ] ^

ko At j J

(3-4)

The equation can be further transformed into the common log (log^g) form

which, after collecting terms, yields

162.6 qsc |io Bo , / t + At Pw = Po T-T log koh At

(3-5)

30

Now consider the more realistic case of a well with multiple

production rates in its history prior to being shut in for transient

testing. The well was initially completed at time t. The well

produced at a constant rate ( Qo ) until tj ; the well then produced at

the new rate ( q ) until t2 . Two additional flow periods followed until

the well was shut in for testing at time ( t ). Following the procedure

above and taking into consideration that the flow rate (q) varies with

each application of the point-source solution, it can be shown that the

variable production is

Pw = Po -162.6 \i^ Bo

koh qo log r 14- At t + At - t.

+ qi log /t + At - t,

Vt + At - t 2 y

+ q2 log / t + At - t2 ^ / t + At - t3

t + At - t3 j + ^ l°g I Xt (3-6)

However, Horner introduced the concept of a corrected time which

would closely approximate the variable production rate. If the last

production rate is sufficiently long (several days), corrected time can

be obtained by dividing the cumulative oil production of the well in STB

by the last established production rate in STB/Day resulting in a time

quantity in days. The result is then multiplied by 24 hours/day placing

the time quantity into hours. This corrected time (also called pseudo­

producing time t or Horner equivalent time) can then replace the

completion time t in Equation 3-5 resulting in Equation 3-7,

31

162.6 q^c^oBo fL, + At \ P- = P° kTh ^^H—A^J

(3-7)

where:

t = i i L . x 2 4 ^ • (3-8) ^ qiast . day

Horner arrived at several significant observations based on the

equations presented above. The first conclusion was that a plot of

shutin pressure (pw) versus the logarithmic term of Equation 3-5

- ( 4 ^

or pw versus the sum of the rate-logarithmic terms

f i + At \ , f t + At - t ^o i°g [ T T A T ^ I T J ^ 1 °§ l t ^ A t - t 2 > '^'•'

of Equation 3-6 will result in a straight line, if the effects of

afterflow into the well are ignored. Secondly, extrapolation of the

line to a At of infinity will yield the original reservoir pressure as

read from the pressure axis. Lastly, the slope of such a semi-log

straight line can be represented by Equation 3-9 for the exact time

represented by Equation 3-6 or Equation 3-10 for corrected time [7].

, 162.6 Ho B„

"" = k„h (3-9) -o

3' 162.6 qsc^ioBo

m = i - ; (3-10)

Current well-test procedures and analyses are based largely on these

assumptions and are included in "conventional analysis."

Horner developed Equation 3-6 for consideration of multiple flow

rates prior to a buildup test. Later publications discussed the

application of Horner's theory to a well which produces at a constant

rate but is subjected to alternating periods of flow followed by shut-in

periods as would be the case when a buildup test is interrupted for any

reason, such as recalibrating recording equipment. Two authors who

explored this situation from a mathematical approach are Guerrero [5]

and Nisle [11],

Shut-in and Flow-Period Combinations

One study of the effect of shut-in and flow period combinations on

pressure buildup testing using Horner's method of superpositioning was

conducted by Guerrero [5]. Guerrero applied superpositioning to the

rate-time relationship depicted in Figure 3-2. Superpositioning is

applied by considering the following rates and the periods over which

they act:

+ q for t + At

- q for t + At - ti

+ q for t + At - t2

- q for At .

Once again, considering Equation (3-3) for a well flowing at a constant

rate prior to shutin,

33

ate

1" o

o a.

o

Duration of s^ort-term stiut-in

4

• z

\

Beginning of pressure-Ouildup

test 1 r

At

0

Figure 3-2. Rate-time relationship for multiple flow periods at constant production rate [5].

34

Pw = Po + 70-6 q^ ^o B Q

koh Ei

^ 948.2 r; (() Ho Ct ^

V ko (t + At ) J

- Ei ^ 948.2 T^ (D lo c

V ko At y (3-3)

the more general case can be expanded as suggested by Horner. Here,

since the rate is constant, (q) is factored out of the brackets leaving

70.6q3c^ioBo ^ Pw = Po + T—T 1 El koh

^_ 948.2 rj^ (|) Ho c, '

ko ( t + At ) J

- Ei ' ^ 948.2 r ^ (|) Ho Ct '

, ko ( t + At - ti ) > + Ei

V

948.2 r ^ (j) Ho c, ^

ko ( t + At - t2 )

- Ei

r

V

948.2 rJ^<l)HoCt^

ko At J (3-11)

which after collecting terms results in Equation 3-12 below

Pw = Pc 162.6 q^cM-pBc

koh log

t + At \ f t + At - t2

At

\

V t + At - ti ) (3-12)

Guerrero studied a well shut in for three hours followed by flow periods

of three, nine, 21, and 117 hours; and shut in for 12 hours with a

12-hour period flow immediately following the shut-in period and prior

to the test.

Guerrero's example was based on the following information;

Spacing

Production rate since completion

Producing time at constant rate

Oil viscosity

Formation volume factor

= 80 Acre

= 80 STB/Day (constant)

= 10 days

= 1.6 cp

=1.42 RB/STB

The sandface pressures, time increment and columns plotted appear

in Table 3-1, and the plot appears as Figure 3-3. The relationship

describing the slope of the line of the semi-log plot of pressure versus

time is

m = 162.6 qsc^ioBo

koh (3-13)

Now solving Equation 3-12 in terms of capacity, k h, and substituting

the given information, the formula for calculating the permeability-

thickness product (capacity) is

koh =

koh =

koh =

162.6 qsc^ioBo

m

(162.6) (80) (1.6) (1.42)

m 29,554

m

(3-14)

(3-15)

(3-16)

36

Table 3-1. Shut-in pressures, shut-in times and superpositioned time increments used in the example [5],

(1)

2.775

2.84S

2.915

3.030

3.065

3.135

3.158

3.170

3.180

3.185

;2i

.11. Sft

4

7

10 17

20 26 41

55 70 32

(3)

» + At

i» 240 ^ (2)

(2)

61.0

35.3

25.0

15.1

13.0

10.2

6.85

5.36

4.43

3.93

(4)

1.. = 237

1, = 22* (3 + Ai)/

(6-(.a!)

0.700

0.769

0.813

0.370

0.885

0.906

0.936

0.951

0.961

0.966

>5)

'., = 231

1, = 228

'.9 + a»»/

(12 + Al)

0.313

0.842

0.864

0.897

0.906

0.921

0.943

0.955

0.963

0.968

:AI ,7)

!-'= 219

1, = 216

(21 +:il)/

(244-^1)

0.893

0.903

0.912

0.927

0.932

0.940

0.954

0.962

0.968

0.972

1.. = I 23

<, = 120

(1 17 + tl/

(120 + iil

0.976

0.976

0.977

0.978

0.979

0.979

0.981

0.983

0.984

0.985

lat

I- = 223

1, = 216

(1 2 — All/

'2 4-;. ill

a.571

0.613

0.647

0.707

0.727

0.760

0.315

0.348

0.372

0.387

(9»

1, = 237 t, = 234

(3) X ()

42.7 27.1 20.3 13.1

n.5 9.24 6.41

5.10

4.26 3.80

(10)

/ ' I 1, = 231

t, = 228

(3) X (J)

49.6

29.7

21.6

13.5

11.8

9.39

6.46

5.12

i.27

3.30

+ -If

At

(11)

W '*"" M .+..-U = 219

t, =: 216

(3) X (6)

54.5

31.9

22.3

14.0

12.1

9.59

6.53

5.16

4.29

3.82

(12)

' \ ., / 1, = 123

1, = 120

(3) X (7)

59.5

3i.5

24.4

14.3

12.7

9.99

6.72

5.27

4.36

3.87

(131

1.. = 228

1, = 216

(3) X (S)

34.3

21.6

16.2

10.7

9.45

7.75

- 5.58

4.55

3.36

3.49

37

0

a « w 3

-0 c 0

3,300r

3,200^

3.l00t

3.000t

2.900r

2.800t

2.7001 6.0 80 10.0

< l I

300 *Q0 60.0 80.0 100.0 Of ( i l ^ L \ / ' •'^'•i? \

I :k I / \ i -<i i - i , J

Figure 3-3. Shut-in pressure versus superpostioned time for flow periods with and without disturbances [5].

38

From the plot, the slope of the line with no shut-in period before the

test as determined by Guerrero is 112 psi/cycle. Therefore, the

capacity is

m = 112 psi/cycle (3-17)

29,554 - ^ ^^ kh = ,,^ = 264 mdft * (3-18) o 112

The slope of the line representing a three-hour shut-in period followed

by a three-hour flow period is 118 psi/cycle. Calculation of the

capacity corresponding to this slope is shown below:

m = 118 psi/cycle (3-19)

29 554 k h = - f W " ^ ™^^^ ^^'^^^ % Error = ^ ^250^^^ * 100 % = 5.6% . (3-21)

Thus a 5.6 % error was calculated. However, the slope of the line

representing a 12-hour shut-in period followed by a 12-hour flow period

is

m = 127 psi/cycle (3-22)

,. 29,554 o ^ 127 ^ (3-23)

% Error = ^ 233 ' ^ * 100 % = 13.3 % . (3-24)

thereby resulting in a 13.3 % error. The remaining flow periods

following the 3 hour shut-in period were not plotted as they each

resulted in an error less than 5.6 Z.

39

Guerrero [5] concluded from his work that the error introduced by

the short-term shut-in period was a function of two variables. These

are the ratio of the length of the flow period following the short-term

shutin to the length of the shut-in period, and the actual duration of

the disturbance [5]. The fact that a flow period of three times the

disturbance duration produced an error of less than 5.6 % will be

compared to the results of the second investigator, Nisle [11].

Minimizing Error Caused by A Disturbance

Nisle began, as did Guerrero, with the exponential integral (Ei)

solution to the diffusivity equation, which expressed in proper field

units, takes the following form:

Pw = Po + AqscM>o Co koh

Ei

f

V

Br^^<|)HoCt ^

kot (3-2)

J

where: A = 70.6 (a constant)

B

a = 948.2 (a constant) stock tank volume _ J_ reservoir volume B,

He also considered the case of a constant production rate with several

flow periods as was illustrated in Figure 3-2. Separate applications of

the point source solution as developed by Horner, application of the log

approximation, and collection of terms as before yields:

Pw = Po -162.6 q c Ho Bo

koh log

t + At j V t 4- At - t2 (3-25)

Now,

40

let X - t - t2 = flow time foUowing short term shutin

5 = t2 - tj = duration of short term shutin

so t - ti = X + 5

(3-26)

(3-27)

(3-28)

substituting into Equation 3-25 results in

Pw = Po -162.6 q c o Bo r , / At \ f z + d + At

koh log t + At X + At

(3-29)

Observe that

as 5 « X X + At J approaches 1 . (3-30)

The greatest possible error will occur immediately after shutin when

5 + X At = 0 . If one lets the difference between and 1 be less than

X

an acceptable value of 10%, then

5 + X - 1

X

X X

i.1-1 X

^ < 0.1 T

< 0.1

< 0.1

< 0.1

(3-31)

(3-32)

(3-33;

(3-34)

5 < 0.1 X (3-35)

105 < X . (3-36)

Al

Therefore, the length of the flow period needed to bring the error to

within 10% is ten times the length of the short-term shutin period.

Since the period was determined at the moment of greatest error, an

error of less than 10% at that time will be reduced for subsequent

times.

Nisle used an example to demonstrate this principle. The

short-term shutin period lasted 120 minutes and the flow period

following the disturbance lasted 300 minutes; thus, the ratio of the

flow period to the disturbance is 2.5. Nisle then calculated that the

error caused by neglecting the shutin and subsequent flow period prior

to the test is 8%. Nisle concluded his paper by stating that if a flow

period of 2.5 times the short term shutin results in an error of 8%, a

flow period of ten times the disturbance would certainly be tolerable.

Having presented the findings of Guerrero and Nisle, the remaining

step of this research is to compare the results obtained from the

simulation model to those of their papers.

CHAPTER 4

RADIAL SIMULATOR METHOD

The method of using simulation results as a source of information

for analysis is a different approach than Guerrero and Nisle used in

their papers. For Guerrero's calculations, ten pressure-versus-time

data pairs were used as the basis for all calculations [5]. Nisle used

only nine data pairs for his calculations [11]. Although the equations

are based on Horner's superposition principle, the end result is that

the disturbances and after-disturbance flow periods are compensated for

through a plotting technique. The radial simulation model on the other

hand, used rigorous numerical analysis techniques. A pressure pulsation

introduced into the mathematical framework of the program will be

present throughout the remainder of the operations and calculations.

As stated previously in Chapter 2, under the subheading

"Modifications," several minor modifications to the original simulation

program were necessary to model the disturbances. The length of the

pressure pulses as well as the after-disturbance flow periods had to be

selectively varied. To accomplish this, various checks were installed

of the cumulative elapsed time in the program. Each of these checks was

determined from the input information and corresponded to the time at

which an instantaneous flowrate change will occur. When the cumulative

time matched the predetermined time, the surface flow rate was changed

42

43

to zero or 750 STB/day whichever the case may be. Associated with each

instantaneous rate change is the need for a time-step reduction to

maintain numerical stability. Immediately following a surface-rate

change, the time step is divided by a constant which decreases the size

of the time steps. The time steps then progressively increase until a

point is reached where another rate change occurs or program execution

is stopped.

The end result is a single-phase radial simulator which has the

capabilities of combining virtually any combination of shut-in and flow

period lengths with a pressure buildup test. The final buildup program

version has been compiled into machine language by the Quickbasic

compiling option for two reasons. First, the execution time is reduced

substantially when a compiled version of the program is run.

Additionally, compiling in this manner allows the program to run on any

IBM compatible personal computer regardless of the particular

programming languages (i.e., Basic, Quickbasic, Fortran, etc.) available

for that specific machine. However, because of the number of loops,

matrix operations, and mathematical computations required for each run

of the simulator, a mathematical co-processor, high clock speed, and

advanced central processing unit are advantageous.

Disturbance and Flow-Period Combinations Studied

The most critical requirement when programming the model was to

determine the times at which to change the flowrate. To isolate the

flowrate variable, all other reservoir parameters, such as thickness and

porosity, were held constant. Table 4-1 lists the various parameters

input into the simulator which were held constant for all test runs.

44

Table 4.1. Reservoir parameters held constant for all test runs.

Parameter Value

Porosity

Permeability

Original reservoir pressure

Formation thickness

Oil compressibility

Oil viscosity

Oil Formation volume factor

Wellbore radius

Initial production rate

Wellbore storage coefficient

20%

8.0 md

3000 psia

91 ft

7*10"^ psi"^

1.2 cp

1.01 RB/STB

0.5 ft

750 STB/Day

4.6628 - 10"^ bbl/psi

•40

The length of time a well would be shut in for maintenance, such as

to replace a valve or to install diagnostic equipment, should not amount

to more than a few hours in most cases. Should the well be shut in

longer than ten to 12 hours during the stabilization period of a well at

constant rate, any planned pressure transient test most likely would be

rescheduled.

Based on these considerations, disturbances of two, four, six,

eight, and ten hours in duration were selected for study. Furthermore,

since Guerrero's findings indicate that error attributed to a pressure

disturbance is a function of the ratio of the flow period to the

disturbance and the length of the disturbance itself, flow periods of

one-half, one, two, four, seven, and ten times the length of the

pressure disturbance were simulated.

In addition to the buildup tests which include a short-term shut-in

period prior to testing, a full length uninterrupted buildup test was

also simulated to act as the baseline for comparison. The analysis of

this full buildup test serves two important functions.

First, an uninterrupted test which after analysis yields the known

formation permeability will verify that the simulation model is

executing correctly. In addition, the value of the slope as read off

the semi-log plot and the corresponding calculated permeability value

will be the accepted values used in calculating the error associated

with the various disturbance-flow-period pairs.

Analysis

Each program run generates a total of 600 computed data sets. The

sets include pressure, cumulative time since the beginning of program

46

execution, and sandface flow rate. Approximately 30 points from each

test run were selected for purposes of plotting a graph of semi-log

pressure versus time. The cumulative time readings are adjusted by

subtracting the cumulative time at final shutin from each of the values

for the thirty selected points. Thus, at the time of final shutin the

corrected data sets possess the correct incremental time relationship

needed for analysis. Pressure values are plotted on (x-y) two

dimensional plot with the time or x-axis being logarithmic and the

pressure axis being Cartesian. Since the simulator equivalent of

producing time prior to initial shut-in period is over 24,000 hours from

tp + At the full length run of the pressure drawdown simulator, plotting

would result in extremely high numbers. However, the shut-in pressure

may be plotted versus the logarithm of At and the semi-log slope of

this plot will be the same as that of a Horner plot for the same data.

This plot of incremental time since shut in versus pressure is called a

Miller-Dyes-Hutchinson (MDH) plot [3]. The associated straight-line

slope may then be used in the capacity equation

162.6 q^c^oBo ^ ^ koh = ' ( -1) ^ m

in the standard manner for calculation of capacity and ultimately,

permeability.

Simulation Results

The plot of the uninterrupted build-up test, Figure 4-1, displays a

semi-log straight-line trend for the majority of the plotted test. The

in CD h-Q.

U CD • o a> Q.

. 1 — >

c c ID

D

I;

(eisd) ajnss5J(j

48

slope of the the straight-line portion is 200 psi/cycle. The degree of

accuracy to which the pressure scale can be read is + 5 psi/cycle. If

the given reservoir information of Table 4-1 is substituted into

Equation 4-1, and the equation solved for permeability, the result is

, (162.6) (750) (1.2) (1.01) K = 7^7; (4-2)

m (91) 1624.21 , ,

ko = . ^ j ^ . (4-3)

Substituting the slope of 200 psi/cycle results in a calculation of

k' = i^^Hi (4-4) ^ 2 0 0

ko = 8.12 md (4-5)

for the reservoir permeability. If Equation 4-1 is solved for the ideal

slope, the resulting equation is

162.6 qsc^oBo " = k^T ' ^^-6)

substituting the given information yields the following:

_ (162.6) (750) (1.2) (1.01)

""" (8.00) (91) " "'

mjdeai = 203.03 psi/cycle . (4-8)

^9

The percent error in permeability associated with the incorrect slope is

Of J: (203.03 - 200.0)

"" ^ ' = (200.0) " '00^" ( - )

or

% Error = 1.51% . (4-10)

This error of 1.5% can be attributed to the manual selection of the best

fit line or to the degree to which the slope can be measured.

Considering the slope has an uncertainty of + 5 psi/cycle, the error

associated with a difference of 3.03 psi/cycle is acceptable.

The remaining 30 plots were analyzed in precisely the same manner

as described above. Early in the analysis, it became evident that there

were two distinct straight lines which one might consider as the

serai-log straight line portion for each of the combinations tested.

Figure 4-2 illustrates this phenomenon. Therefore, both slopes were

determined in those cases where two slopes were present. Errors

associated with the various disturbance-flow-period combinations range

from a maximum of just under 24% to a minimum of 0. However, for

convenience 200 psi/cycle (rather than 203 psi/cycle) was used as the

accepted slope so an additional 1.5% can be added to the upper end of

the errors making the maximum approximately 25%.

Two specific trends were noted from the analysis of the 31 plots.

The first of these trends deals with the accepted slope of

200 psi/cycle. One distinct advantage the simulator has over the use of

field data, if it were available, is that the simulator can run for a

period of time, much longer than would be feasible for testing an actual

well. In fact, it is not uncommon for the buildup tests from the

50

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simulator to have shut-in periods lasting thousands of hours, a testing

period unrealistic for actual tests. Every run of the pressure buildup

simulator eventually had a straight line portion with a slope of

200 psi/cycle. However, the time for this slope to appear is often

hundreds of hours into the buildup test. The lower the ratio of the

flow period to the disturbance, the longer the test must be run to

demonstrate the accepted slope of 200 psi/cycle. Although this trend is

present in all the disturbance lengths (i.e., two, four, six, eight, ten

hours), discrepancies exist. A test which has a flow period ten times

the length of the disturbance may show the accepted slope within 13

hours whereas a test with a flow period seven times the length of the

disturbance may display the accepted slope in ten hours. Additionally,

in the lowest ratios of flow period length to disturbance length, a test

with a 1:2 (0.5) ratio and a 2:1 ratio may exhibit the accepted slope at

appproximately the same time. However, the overall trend still exists

with the correct slope being displayed sooner in the high ratios and the

accepted slope being displayed later with the lower ratios.

More conclusive evidence is present with the second trend. After

the characteristic swan neck or S-shape of early time effects is seen on

the MDH plots, the first apparent semi-log straight line is present.

This earlier "false" line is present in each and every test plot

involving a disturbance and possessing two distinct straight lines.

This first apparent straight line has a slope which is less than that of

the accepted value. As seen in Appendix A, the slope increases as the

ratio of the flow period to the disturbance length increases. Using the

four-hour disturbance series as an example, for a two-hour flow period

52

after the disturbance, a ratio of 1:2 or 0.5, the first semi-log

straight line has a slope of 163.8 psi/cycle while a flow period of 4

hours giving a 1:1 ratio results in a premature slope of 172.6

psi/cycle. For a value of m equal to 163.8 psi/cycle,

m = 163.8 psi/cycle (4-11)

1624.2

^ = T63T ^'-'-^ ko = 9.916 md . (4-13)

For a value of m equal to 172.6 p s i / c y c l e ,

m = 172.6 psi/cycle (4-14)

ko = ^ ^ 2 T ^'-''^

ko = 9.41 md . (4-16)

In addition, a ratio of 2:1 results in a slope of 184 psi/cycle and a

ratio of 4:1 showed the first and second straight lines meet and form

one line with the accepted slope of 200 psi/cycle. The error associated

with these false slopes are given in percent as follows: 19.3, 14.9,

9.3, and 1.5%, respectively. All five of the disturbance lengths tested

gave similar results. In addition, the errors associated with each

ratio regardless of the actual length of the diturbance, are all less

than 6%. The longer the flow period following a disturbance, the lower

the error associated with the first apparent semi-log straight line from

the MDH plot. Moreover, at a ratio of 4:1 the associated error drops

below 5% regardless of the actual length of the disturbance. Although

53

this trend contradicts the findings of Guerrero [5], the fact that

analyzing the uninterrupted buildup test results in the calculation of

the correct, known permeability and the fact that the trend is repeated

for every disturbance length tested would tend to reinforce the validity

of the results. Table 4-2 lists the apparent slopes of the first

semi-log straight line and the error associated with using those slopes

in a permeability calculation for each of the disturbance-flow period

combinations tested. Figure 4-3 is a graphical representation of

Table 4-2. Note the errors for ratios of 4:1 and greater are less than

5%.

These findings are not in agreement with those of Guerrero and

Nisle. Guerrero suggested that the error caused by a pressure

disturbance was a function of the ratio of the flow period following the

disturbance to the length of the disturbance and also the length of the

disturbance itself. While a three-hour disturbance followed by a

three-hour flow period, a 1:1 ratio, resulted in an error of 5.6 %, a

12-hour disturbance followed by a 12-hour flow period, also a 1:1 ratio,

resulted in an error of 13.3%. The degree of error associated with the

disturbance appeared to be independent of the fact that both were 1:1

ratios and dependent on the actual duration of the disturbance. Nisle

suggested that for the error to be less than 10%, the flow period must

be ten times the length of the disturbance or a ratio of 10:1. It

should be noted however, that Nisle's example of a two-hour shut-in

period followed by a five-hour flow period, giving a ratio of 2.5:1,

resulted in an error of 8% in the capacity calculation. Thus, a

disturbance less than that used by Guerrero (two hours as compared to

54

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56

three hours) in Nisle's example followed by a longer flow period (2.5

times the disturbance as compared to one times) had an associated error

greater than that of Guerrero's example. One would expect that a

shorter disturbance followed by a longer flow period than that used by

Guerrero would result in a smaller error than the 5.6% given in his

example. Furthermore, in Guerrero's example flow periods of three times

the length of the three-hour disturbance produced an error less than

5.6% and were not plotted in Figure 3-3. Both of the examples

demonstrated therefore, that errors of less than 10% are possible with

flow periods of as little as two and one-half times the length of the

disturbance. Recall however, that the suggested flow period-to-

disturbance ratio suggested by Nisle was ten for an error less than 10%.

Thus, an inconsistency exists between the results of the two authors

[5,11].

Discounting Afterflow Effects

Although the first conclusion one might reach is that the first

apparent line is still experiencing influence from afterflow, this

possibilty would seem unlikely based on current understanding of

afterflow influence. Miller describes the influence of afterflow on the

semi-log straight line slope used for permeability determination as

'follows:

However, an earlier apparent straight line section which is not accounted for by the buildup equations may be evident in this [semi-log] plot. The earlier straight line section will have a greater slope than that to be used in the permeability equation. . . .it was found that the most obvious deviation [from conditions specified in the derivation of the equations] was the

3/

fact that the buildup relations were derived for a well shut in at the sandface. [10, pg. 94]

Figure 4-4 demonstrates the effects of wellbore storage on the earlier

stages of a pressure buildup test. Note the high slopes within the

S-shaped portion of the graph [9].

The fact that the slope of the first apparent line of the simulator

has a slope less than that of the accepted 200 psi/cycle is not at all

like the behavior described by Miller [10] and shown by Matthews and

Russell [9]. In the absence of any current explanation of this early

slope behavior, the result can only be attributed to the

disturbance-flow period combinations.

58

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CHAPTER 5

CONCLUSIONS

The modelling of pressure disturbances preceding a pressure

buildup test, by means of a reservoir simulation model, has given some

very unexpected but interesting results. The contradiction of these

results with those of Guerrero [5] and Nisle [11] suggests the need

for further research and clarification. Several important conclusions

can be made based on the results of the'simulation study.

1. Modelling of a short duration pressure disturbance can be

adequately achieved through the use of radial simulation models

describing single-phase fluid flow of a slightly compressible

liquid through porous media.

2. A short-term shut-in period during the rate stabilization period

prior to a pressure transient test, does in fact transmit a

disturbance throughout the reservoir.

3. The influence of this short-term pressure disturbance is evident

in the analysis of semi-log pressure versus time plots using

conventional and accepted methods.

4. Two semi-log straight lines will be present for a well subjected

to a short term shut-in period prior to transient testing. The

first of these lines will possess a slope less than that needed

for proper permeability calculations. The second line vill have

59

60

the correct slope, but may not appear in all tests because of the

long shutin time necessary for this second straight line to

appear.

5. The effect of a short duration disturbance can be reduced or

negated if the disturbance is followed by a flow period prior to

beginning a subsequent pressure buildup test. The longer the

flow period, the closer the first apparent semi-log straight line

slope is to the correct slope needed for analysis. If the flow

period is four times the length of the disturbance or greater,

the two lines become one, with the resulting slope being equal to

or very near the correct slope.

6. If the length of the flow period is at least two times the length

of the short-terra disturbance, the error associated with using

the first (which may be the only) apparent semi-log straight line

slope for permeability calculations will be reduced to 10% or

less. Furthermore, a flow period of at least four times the

length of the disturbance will reduce the error to 5% or less.

7. The error introduced by using the first apparent semi-log

straight line slope is a function of the ratio of the duration of

a flow period following the disturbance to the duration of the

disturbance, and is independent of the actual length of the

short-term shut-in period.

8. The findings of this research are not in agreement with those

published by Guerrero [5] and Nisle [11] based on the Horner

method of superposition. Additional research is needed to

confirm these findings.

61

9. A comparison of the results from the radial simulation model of

pressure disturbances with those of actual field data or the

exact solutions of the radial diffusivity equation by means of

Bessel functions should be made in an effort to clarify the

effect of a pressure disturbance and flow period on pressure

buildup testing.

REFERENCES

1. Arnold, Marion D.: Department of Petroleum Engineering, Texas Tech University (1988), Reservoir Simulation I (Class Notes).

2. Craft, B. C , and Hawkins, M. F.: Applied Petroleum Reservoir Engineering, Prentice-Hall Inc., Englewood Cliffs, N.J. (1959).

3. Crawford, D. A.: Department of Petroleum Engineering, Texas Tech University (1987), Pressure Transient Testing (Class Notes).

4. Crawford, D. A.: Department of Petroleum Engineering, Texas Tech University (1987), Private Communication.

5. Guerrero, E. T. : Practical Reservoir Engineering, The Petroleum Publishing Co., Tulsa, Oklahoma (1968), 141-143.

6. Hauss, William T.: A Numerical Simulation Study On The Characteristics Of A Variable Wellbore Storage Pressure Transient Response, Masters Thesis In Petroleum Engineering, Texas Tech University (May 1988).

7. Horner, D. R. : "Pressure Buildup In Wells," Proceedings Third World Petroleum Congress - Section III (1951), 503-521.

8. Lee, John: Well Testing, First Printing, Society Of Petroleum Engineers Of AIME, Dallas (1982).

9. Matthews, C. S., and Russell, D. G.: Pressure Buildup And Flow Tests In Wells, Monograph One, Society Of Petroleum Engineers Of AIME, Dallas (1967).

10. Miller, C. C , Dyes, A. B., and Hutchinson, C. A., Jr.: "The Estimation Of Permeability And Reservoir Pressure From Bottom Hole Pressure Buildup Characteristics," Petroleum Transactions Of AIME, Vol. 189 (1950), 91-104.

11. Nisle, Robert G.: "The Effect Of A Short Term Shutin On A Subsequent Pressure Buildup Test On An Oil Well," Petroleum Transactions Of AIME, Vol. 207 (1956), 320-321.

12. Society Of Petroleum Engineers: "SPE Symbols Standard," Journal Of Petroleum Technology (1984), 2278-2332.

13. Thomas, G. B.: "Analysis Of Pressure Buildup Data," Petroleum Transactions Of AIME, Vol. 198 (1953), 126-128.

62

63

14. Hurst, W., and van Everdingen, A. F.: "The Application Of Laplace Transformation To Flow Problems," AIME (1949), 186, 305-324.

APPENDIX A

SEMI-LOG PLOTS FOR THIRTY TEST RUNS

64

65

Following are the semi-log plots of pressure versus incremental

time for the thirty test runs involving pressure disturbances. In each

case the first number appearing in the plot title refers to the length

in hours of the disturbance or short-term shutin period. The second

number in the parenthesis corresponds to the length of the flow period

immediately following the disturbance prior to beginning the pressure

buildup test. This appendix contains the following plots:

Length of disturbance (Hours)

2

2

2

2

2

4

4

4

4

4

4

6

6

6

6

6

Length of flow period (Hours)

1

2

4

8

14

20

2

4

8

16

28

40

3

6

12

24

42

66

Length of disturbance (Hours)

8

8

8

8

8

8

10

10

10

10

10

10

Length of flow period (Hours)

60

4

8

16

32

56

80

5

10

20

40

70

100

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APPENDIX B

MODIFIED RADIAL SIMULATOR PROGRAM LISTING

97

98

90 LPRINT CHR$(27): "&12A": LPRINT CHR$(27); "illL": LPRINT CHR$(27); "iieoP' LPRINT CHR$(27): "S^sOC

* 1-PHASE, RADIAL, SLIGHTLY COMPRESSIBLE FLUID FLOW SIMULATION * * •i

*

* MODEL. SINGLE AND CHANGING WEUBORE STORAGE INCLUDED. * PROGRAMMED BY WILLIAM T. HAUSS FOR A MASTESfS * THESIS IN PRESURE TRANSIENT ANALYSIS. * MODIFIED BY WELDON PIERSON WITH ASSISSTANCE FROM TIM HAUSS

95 1000 REM 1010 REM 1020 REM 1030 REM L040 REM 1045 REM 1050 REM 1060 CLS : CLEAR : KEY OFF 1070 DEFDBL A-H, 0-Z 1080 DEF SEG - &H40 1090 POKE &H17, PEEK(&H17) OR 64: REM ** SET CAPS LOCK ON ** 1100 DIMQT(255), BETA(255). GAMMA(255), A(255), B(255), QSFPRT(610) 1110 DIM D(255), R(255), RPLUS(255). DT(50). CSTERM(255), SL(610) 1120 DIM PNEW(255), POLD(255), PWB(610). TWB(610), C(255), ZL(610) 1130 REM 1140 REM **** INITIALIZING DEFAULT DATA **** 1150 QSURF - 0: QT(1) - 0: TSM - 1.6: ITER - 20: M - 240: PI - 3000: H - 91: PHI -0007 1160 Vise - 1.2: BTAW - 1.01: K - 8: RWC - .5: DU - .06: N - 30: DT(1) - 2 : AS - "NO'

CMP

B$

.00

•• TS

1170 C$ - "NO": D$ - "TRUE": DEPTH - 10000: CF - .000003: DENS - 62.4: TBGID - 4: T$ 1175 REM !!!!!!!!!!!!!!!!! HERE IS INTERRUPT DEFAULT DATA !!!!!!!!!!!!!!!!!!!!!!!! 1176 LFSI - 2 * 3600: LTIFP - 4 * 3600 1180 CUMTIME - 0: SCROLL - 11 1190 FLAG - 0: FLAG3 - 0: FLAGS - 0:. FLAG7 - 0: FLAG9 - 0 1200 JJ - 1: REM STARTING POINT FOR Pwf / Time PRIMARY STORAGE ARRAY 1210 PIE - 4 * ATN(l)

1220 ATBG - PIE * TBGID * TBGID / 576: REM• ANNULUS/TBG ARE ISOLATED WITH A PKR. 1230 CLOG - 2.302585094y/ 1240 REM 1250 REM **** CALCULATING TIME STEP ARRAY & LENGTH OF SIMULATION RUN **** 1260 REM 1270 LENGTH - DT(1) 1280 FOR I - 2 TO N 1290 DT(I) - DT(I - 1) * TSM: REM LOADING TIME STEP ARRAY 1300 TIME - ITER * DT(I) 1310 LENGTH - LENGTH + TIME: REM SUMMING LENGTH OF SIMULATION RUN 1320 NEXT I 1325 IF FLAG3 - 1 THEN GOTO 3350 1330 TWBS2 - LENGTH / 2: REM DEFAULT 2WBS IS APPROX. 1/2 SIM. RUN 1340 TWBS2XX - TWBS2: REM DUMMY STORAGE 1350 FLAG - 0

25: PRINT "PROGRAM DATA INITIALIZATION"

"PDD"

1360 COLOR 3: LOCATE 1 L370 PRINT STRING$(80. 1380 COLOR 7 1390 PRINT USING " A) 1400 PRINT USING " B) 1410 PRINT USING " C) 1420 PRINT USING " D) 1430 PRINT USING " E) 1440 PRINT USING " F) 1450 PRINT USING " G) 1460 PRINT USING " H) 1470 TDX - .0002637 * K 1480 PRINT USING " I) 0

RESERVOIR PRESSURE - #.###•# psia"; PI RESERVOIR PERMEABILITY - ####.# md."; K RESERVOIR THICKNESS - ###.# ft."; H WELLBORE RADIUS - #.## ft."; RWC FLUID VISCOSITY - //#.# cp."; VISC FORMATION VOLUME FACTOR - 4.M RB/STB"; BTAW SYSTEM COMPRESSIBILITY - ##.# 1/psi"; CMP POROSITY - #.### (fraction)"; PHI (PHI * VISC * CMP * RWC * RWC) FIRST TIME STEP - iHHhU sec. cD - 'hhhMir: DT(1) TDX * DT(1) / 360

99

1490 PRINT USING " J) 1500 PRINT USING " K) 1510 PRINT USING " L) 1520 PRINT USING " M) 1530 TL - LENGTH / 3600 1540 PRINT USING " N) X 1550 PRINT USING " 1560 PRINT USING " 1570 PRINT USING " 1580 PRINT 1590 COLOR 4: PRINT 1600 LOCATE 16, 68: 1610 LOCATE 21 1620 COLOR 3 1630 PRINT " 1640 LOCATE 25 1650 COLOR 7: LOCATE 23, 5 1660 1$ - INKEY$: IF 1$ - " N GOTO 1710 ELSE IF 1$ - "C ) THEN GOTO 1740 ELSE GOTO 1670

NODAL SPACING MULTIPLIER - tt. ihhHr : DU ITERATIONS PER TI.ME STEP - '/».'» *"; ITER TIME STEP GEOMETRIC MULTIPLIER - 4.'kihi" \ TSM -WMBER OF TIMES TO USE TIME-STEP MULTIPLIER - ."*.*«"; N

END SIMULATION TIME - //y/,#y>#.## hrs., tD - *Ht • ^iHhhM^ TL. TL

11:

0) P) Q)

PRINT PRINT

NODES USED IN COMPUTATIONS - iHHr: M EXTENT OF RESERVOIR, (re) - 4,Mi^,Mi^.-H ft."; RWC * EXPf.' * DU) INITIAL FLOW RATE. + PROD., - INJECT. - iHHHh H^ STB/day"; QSLTIF

***": LOCATE 18, 56: PRINT "***•': COLOR 7 SIGNIFIES A DEPENDENT CALCULATION. CANNOT CHANGE DIRECTLY";

TYPE SELECTION LETTER TO CHANGE, RETURN TO CONTINUE. OR ESC TO EXIT' 1: PRINT STRING$(80, " - " ) ;

THEN GOTO 1660 ELSE IF 1$ - "A' THEN GOTO 1720 ELSE IF 1$ - "D'

THEN GOTO 1700 ELSE IF 1$ THEN GOTO 1730 ELSE IF 1$

"B" THE CHR$(2"

1670 IF 1$ ELSE IF 1$ 1680 IF 1$ ELSE IF 1$ 1690 IF 1$ - " 1700 INPUT " 1710 INPUT " 1720 INPUT " 1730 INPUT " 1740 LOCATE 24 1750 1$ - INKEY$: IF 1$ -N CLS ; STOP 1760 GOTO 1750 1770 INPUT

THEN GOTO 1770 ELSE IF 1$ THEN GOTO 1790 •F" THEN GOTO 1780 ELSE IF 1$ - "G" THEN GOTO 1800 ELSE IF 1$ - "Q" THEN GOTO 1810 ELSE GOTO 1680 THEN GOTO 1820 ELSE IF 1$ - "J" THEN GOTO 1830 ELSE IF 1$ - "K" THEN GOTO 1840 THEN GOTO 1850 ELSE IF 1$ - CHR$(13) THEN GOTO 1940 ELSE GOTO 1690

M" THEN GOTO 1860 ELSE IF 1$ - "0" THEN GOTO 1920 ELSE GOTO 1660 ENTER NEW INITIAL RESERVOIR PRESSURE"; PI; GOTO 1930 ENTER RESERVOIR PERMEABILITY (md)"; K: GOTO 1930 ENTER RESERVOIR THICKNESS (FT)"; H: GOTO 1930 ENTER WELLBORE RADIUS (FT)"; RWC; GOTO 1930 25: PRINT " ARE YOU SURE -- Y/N";

THEN GOTO 1750 ELSE IF 1$ - "N" THEN GOTO 1930 ELSE IF 1$ - "Y" THE

1780 INPUT 1790 INPUT 1800 INPUT 1810 INPUT 1820 INPUT 1830 INPUT 1840 INPUT 1850 INPUT

ENTER FLUID VISCOSITY (CP)"; VISC: GOTO 1930 ENTER FORMATION VOLUME FACTOR"; BTAW: GOTO 1930 ENTER SYSTEM COMPRESSIBILITY"; CMP: GOTO 1930 ENTER POROSITY (fraction)"; PHI: GOTO 1930 ENTER FLOW RATE. + FOR PROD.. - FOR INJECT. "; QSURF: GOTO 1930 ENTER FIRST TIME STEP (Sec.)"; DT(1): FLAG - 1: GOTO 1930

ENTER NODAL SPACING MULTIPLIER"; DU: GOTO 1930 ENTER ITERATIONS PER TIME STEP"; ITER: FLAG - 1: GOTO 1870 ENTER TIME STEP GEOMETRIC MULTIPLIER"; TSM: FLAG - 1; GOTO 1930

N: FLAG - 1 1860 INPUT " ENTER y/ OF TIMES TO USE TIME-STEP MULTIPLIER 1870 IF ITER * N <- 600 THEN GOTO 1930 1880 CLS : LOCATE 11, 20: PRINT "ITERATIONS * N MUST BE < 500, YOURS IS 1890 LOCATE 13, 20: PRINT "STRIKE ANY KEY TO CONTINUE" 1900 IF INKEYS - "" THEN GOTO 1900 1910 CLS : ITER - 20: N - 25: FLAG - 0: GOTO 1360 1920 INPUT " ENTER NODES TO USE IN COMPUTATIONS 1930 LOCATE 23. 1: PRINT SPC(79); : LOCATE 24. 1: SE GOTO 1270 1940 CLS 1950 LOCATE 1. 25: COLOR 3 I960 PRINT "WELLBORE STORAGE DEFAULT DATA" 1970 LOCATE 2. 1: PRINT STRING$(80. " - " ) ; 1980 COLOR 7 1990 PRINT USING "A) NO WELLBORE STORAGE (\ \ ) " ; rtS 2000 PRINT USING "B) ONE CONSTANT WELLBORE STORAGE (\ \ ) " ; B$ 2010 PRINT USING "C) CHANGING WELLBORE STORAGE (\ \ ) " ; C$

ITER •*• N

(MAX 250)"; M: GOTO 1930 PRINT SPC(79); ; IF FLAG - 0 THEN GOTO 1360 EL

100

2080 PRINT USING 2090 PRINT USING

•TRUE/OPP. - (\ \ ) " ; DS

2013 COLOR 5

3600 ^ ^ " *^^^^ " ^ ^^^^ ° ^ ' ^^" SHUT-IN PERIOD - , » .M*'r'/ .4RS, rD - >i...»»»«-

2030 PRINT USING "E) LENGTH OF RECALIBRATION FLOW PERIOD - ^Hh^^.^ i^ HRS. tD - .»...»*--*>. P / 3600. LTIFP * TDX / 3600 2035 COLOR 7 2040 AWB - PIE * RWC * RWC 2050 PRINT USING "F) X-SECTIONAL AREA OF TUBING (isolated annul us) - ihMU sqf t, 2060 VTBG - ATBG * DEPTH / 5.6146 2070 PRINT USING "G) VOLUME OF TUBING - M.iHHhM bbl"; VTBG

'H) COMPRESSIBILITY OF WELLBORE FLUID - ihM 1/psi"; CF •I) DENSITY OF WELLBORE FLUID - < #.#M Ib./cuft."; DENS

2100 PRINT USING "J) Csl - COMPRESSION. Cs2 - CHG. LIQ LEVEL--2110 IF DS - "TRUE" THEN GOTO 2150 2120 CSl - ATBG * 25.64741923# / DENS 2130 CS2 - VTBG * CF 2140 GOTO 2170 2150 CSl - VTBG * CF 2160 CS2 - ATBG * 25.64741923# / DENS 2170 PRINT USING "K) Csl - M-iHUH^ bbl/psi"; CSl 2180 CSDl - .8937966101# * CSl / (PHI * CMP * H * RWC * RWC) 2190 PRINT USING "L) CsDl - U. im, iHHh iH^" ; CSDl 2200 PRINT USING "M) Cs2 - H-HM bbl/psi"; CS2 2210 CSD2 - CSDl * CS2 / CSl

"N) CsD2 - M-MihM^hW; CSD2 FORCE A SPECIFIED CSD" RETURN TO MAIN PROGRAM MENU"

.: i *

ATBG

2220 PRINT USING 2230 PRINT "0) 2240 PRINT "P) 2250 COLOR 4 2260 PRINT 2270 LOCATE 8, 67: 2280 LOCATE 14, 27: 2290 COLOR 7 2300 LOCATE 20, 2310 LOCATE 21, 2320 LOCATE 21,

PRINT "***-: PRINT

LOCATE 9, 39: PRINT LOCATE 15, 28: PRINT

LOCATE 13, 28: PRINT "*** •: LOCATE 16. 27: PRINT

10 10 15

CC

'C" THEN GOTO 2400 ELSE IF 1$ - 'T

THEN GOTO 2500 ELSE IF 1$ - "I" THEN GOTO 2550 CHR$(27) THEN GOTO

PRINT "CsDl WILL BE USED IF ONLY ONE WELLBORE STORAGE IS CHOSEN" COLOR 4: PRINT "***-: COLOR 7 PRINT "SIGNIFIES A DEPENDENT VARIABLE, CANNOT CHANGE DIRECTLY"

2330 COLOR 3: LOCATE 25, 1: PRINT STRING$(80. " - " ) ; 2340 LOCATE 22. 8: PRINT "TYPE SELECTION LETTER TO CHANGE OR TYPE RETURN TO EXECUTE FORMAT. LOR 7 2350 IS - INKEYS: IF IS - "" THEN GOTO 2350 ELSE IF 1$ - CHR$(13) THEN GOTO 2620 ELSE IF 1$ " THEN GOTO 2380 ELSE IF IS - "B" THEN GOTO 2390 ELSE IF 1$ -" THEN GOTO 2410 ELSE GOTO 2360 2360 IF IS - "E" THEN GOTO 2490 ELSE IF IS - "H ELSE IF IS - "J" THEN GOTO 2560 ELSE IF IS - "P" THEN GOTO 2370 ELSE IF 1$ 2370 ELSE IF 1$ - "0" THEN GOTO 2510 ELSE GOTO 2350 2370 CLS : GOTO 1360 2380 AS -2390 BS -2400 CS -2410 INPUT "ENTER APPROXIMATE TIME TO END FIRST SHUT-IN PERIOD. HRS' 2420 IF TWBS2 < LENGTH THEN TWBS2XX - TWBS2: GOTO 2570 2430 CLS : COLOR 2: LOCATE 11, 6: PRINT "TIME TO BEGIN 2ND WELLBORE STORAGE .MUST BE < SIMULATION RUN TIME" 2440 LOCATE 13. 21: PRINT USING "SIMULATION RUN TIME - MM-H hrs.": TL

PRINT USING "YOU HAVE CHOSEN iHHHhhM hrs."; rWBS2 / 3600 PRINT "STRIKE ANY KEY TO CONTINUE" LOCATE 18. 2: IF INKEYS - "" THEN GOTO 2470

2480 TWBS2 - TWBS2XX: GOTO 1940 2490 INPUT "ENTER NEW LENGTH FOR RECALIBRATIQNFLOW PERIOD (HRS) GOTO 2570 2500 INPUT "ENTER WELLBORE FLUID COMPRESSIBILITY 1/psi"; CF: GOTO 2570

YES": YES": YES":

BS - "NO": AS - "NO": AS - "NO";

CS - "NO": CS - "NO": BS - "NO";

GOTO GOTO GOTO

1950 1950 1950

LFSI: LFSI LFSI 3600

2450 LOCATE 15, 23: 2460 LOCATE 17, 25: 2470 PRINT CHRS(7):

LTIFP. LTIFP - LTIFP * 3600

101

ICSD ZolO INPUT "ENTER DESIRED CSDl, NOTE THAT THE WELLBORE DENSITY WILL BE AFFECTED 2^20 DENS - 22.92357637# * ATBG / (FICSD * PHI * CMP * H * RWC " RWC 2530 LOCATE 23. 1: PRINT SPC(77); : LOCATE 23, 1: INPUT "ENTER DESIRED CSD2. NOTE THAT COMPR. WILL BE AFFECTED"; F2CSD 2540 CF - F2CSD * PHI * CMP * H * RWC * RWC / ( . 893796610iy/ * VTBG): GOTO 2570 2550 INPUT "ENTER WELLBORE FLUID DENSITY, Ib./cuft."; DENS; GOTO 2570 2560 IF D$ - "TRUE" THEN D$ - "OPP." ELSE IF D$ - "OPP." THEN D$ - "TRUE"- GOTO 2570 2570 LOCATE 23, 1: PRINT SPC(77); : GOTO 1950

2590 REM * ALL DATA INITIALIZATION AND SIMULAION FORMAT 2600 REM * IS OVER AT THIS POINT 2610 REM *AAjtAi>A*A***^

2620 CLS : LOCATE 1, 1 2630 COLOR 3 2640 PRINT STRING$(80, " - " ) ; 2650 IF TS - "PDD" THEN PRINT TAfl(29):

WB

* *

THEN PRINT TAB(29) 2660 IF TS - "PBB' 2670 COLOR 7 2680 PRINT USING ' SURF 2690 IF AS - "YES" THEN PRINT TAB(15) 2700 REM 2710 IF CS - "YES" THEN PRINT USING "

'CONSTANT RATE DRAWDOWN' 'PRESSURE BUILD UP TEST'

INIT. PRESS. - iHHHhM psia QSURF - ####.## STB/day": PI

NO WELLBORE STORAGE, Qsurf - Qsand AT -ALL TIMES":

' bbl/psi. CS2 CSl, CS2;

•FIRST SHUTIN PERIOD ENDS APPROXIMATELY AT TIME STEP JUST PRIOR TO 44.,hh>.4 HRS

LTIFP / 3600

EXPECTED SEMILOG SLOPE IS iHHHHhU^ psi/cycle"; -162.6 '^ QSURF

CHANGING WELLBORE STORAGE, CSl - 'Hh^ U.iHr^^^ bbl/psi" 2715 COLOR 5 2720 PRINT USING ."; LFSI / 3600 2730 PRINT USING "FULL BUILDUP PERIOD BEGINS AT TIME STEP PRIOR TO ##,###.# HRS + LFSI / 3600 2735 COLOR 7 2740 REM 2750 PRINT USING " * BTAW * VISC / (K * H) 2760 COLOR 3 2770 PRINT STRING$(80. " - " ) ; 2780 COLOR 7 2790 PRINT " Time Sandface Pressure Qsand": 2800 PRINT " (hrs) (psia) (STB/day)"; 2810 COLOR 3 2820 PRINT STRINGS(80. " - " ) ; 2830 LOCATE 24. 1: PRINT STRINGS(80. " - " ) ; 2840 LOCATE 25, 20: PRINT "STRIKE SPACEBAR TO PAUSE/PRINT OR ESC TO QUIT 2850 LOCATE 25. 1: PRINT "TS LEFT -"; 2860 COLOR 7 2870 REM 2880 REM ***** 2890 2900 2910 2920 2930 2940 2950 2960 2970 2980

2990 REM 3000 IF AS - "YES" 3010 LOCATE 25, 11 3020 REM *****

CONVERTING UNITS PI - PI / 14.7: REM (TO ATMS) PICHECK - PI - .00001: REM INFLUENCED BOUNDARY CHECKER CMP - CMP * 14.7: REM (TO ATMS) RWC - RWC * 30.48: REM (TO CM) K - K / 1000: REM (TO D) H - H * 30.48: REM (TO CM) QSURF - QSURF * 1.84 CSl - CSl * 2337120.173#: REM (TO CC/ATM) CS2 - CS2 * 2337120.173y/: REM (TO CC/ATM) COUNT - N * ITER - 1: REM TOTAL NUMBER OF TIME STEPS -1

THEN cess - 0 ELSE CCSS - CSl PRINT USING "M^r: COUNT + 1; INITIALIZING PRESSURE ARRAYS *****

lo:

3022 OPEN "PRDIST" FOR INPUT -AS >/l 3030 FOR I - 1 TO M 3032 INPUT </l, I. PNEW(I) 3034 PNEW(I) - PNEW(I) / 14.7 3050 POLD(I) - PNEW(I) 3060 NEXT I 3065 CLOSE #1 3070 REM 3080 REM ***** DEFINING LOGARITHMIC SPACED GRID POINTS ***** 3090 J - 0 3100 FOR I - 0 TO M 3110 R(I) - RWC * EXP(J) 3120 J - J + DU 3130 NEXT I 3140 REM 3150 RW - R(0) 3160 REM 3170 REM ***** DEFINING GRID BOUNDARIES ***** 3180 FOR I - 0 TO M - 1 3190 RPLUS(I) - (R(I -(- 1) - R(I)) / (LOG((R(I + 1) / R(I)))) 3200 NEXT I 3210 RPLUS(M) - RWC * EXP(M * DU) + RWC * EXP(M * DU) - RPLUS(M - 1) 3220 REM 3230 REM *** CALCULATING A AND C COEFFICIENTS *** 3240 ALPHA - 1 / (DU * DU) 3250 FOR I - 1 TO M - 1 3260 A(I -t- 1) - ALPHA 3270 C(I) - ALPHA 3280 NEXT I 3290 REM 3300 REM **** DEFINING TERMS BASED ON Q **** 3310 RIPOINT - 1.014 * (R(l) + RPLUS(O)) / 2 3320 QT - RIPOINT * VISC * BTAW / (2 * PIE * K * H * (RPLUS(l) - RPLUS(O))) 3330 QT(1) - QT * QSURF 3340 CTERMC - PHI * VISC * CMP / (2 * K) 3350 FOR KK - 1 TO N 3360 DT - DT(KK) 3370 CSTERM(l) - QT * CCSS / (DT * BTAW) 3380 FOR NTS - 1 TO ITER 3385 IF FLAG9 - 3 GOTO 3450 ELSE GOTO 3390 3390 IF CUMTIME < LFSI THEN GOTO 3450 ELSE GOTO 3410 3400 REM 3410 IF CUMTIME < LFSI + LTIFP THEN GOTO 3420 ELSE GOTO 5000 3420 IF FLAG7 - 2 THEN GOTO 3450 ELSE GOTO 3425 3425 FLAG3 - 1: GOTO 5500 3427 QSURF - 750 * 1.84: QT(1) - QT * QSURF 3430 TCHG - CUMTIME 3440 DT(1) - DT / 3: FLAG? - 2: GOTO 1270 3445 QTOLD - QT(1) 3450 FOR I - 1 TO M 3460 IS - INKEYS: IF IS - CHRS(27) THEN GOTO 3980 ELSE IF 1$ - CHR$(32) THEN GOTO 4010 3470 CTERM - CTERMC * R(I) * (RPLUS(I) + RPLUS(I - 1)) / DT 3480 B(I) - -C(I) - A(I) - CTERM - CSTERM(I) 3490 D(I) - QT(I) - (CTERM + CSTERM(I)) * POLD(I) 3500 NEXT I: REM FINISHED CALCULATING COEFICIENT MATRIX 3510 BETA(l) - B(l) 3520 GAMMA(l) - D(l) / BETA(l) 3530 FOR I - 2 TO M 3540 BETA(I) - B(I) - A(I) * C(I - 1) / BETA(I - 1) 3550 GAMMA(I) - (D(I) - A(I) * GAMMA(I - I)) / BETA(I)

103

3 560 NEXT I 3570 PNEW(M) - GAMMA(M) 3580 GOTO 3640 3590 CLS : COLOR 2: LOCATE 11, 16 3600 PRINT "A BOUNDARY HAS BEEN REACHED, PROGRAM TERMINATED" 3610 LOCATE 13, 15: PRINT "RETURN TO PRINT 'GOOD' DATA OR ESC TO BEGIN AGAIN" 3620 LOCATE 14, 30: 1$ - INKEYS: IF IS - "" THEN GOTO 3620 ELSE IF 1$ - CHR$(13) THEN GOTO 4050 ELSE IF IS - CHR$(27) THEN GOTO 3930 ELSE GOTO 3620 3630 REM 3640 REM *** 3650 FOR I -3660 L - M -3670 PNEW(L) 3680 NEXT I 3690 CUMTIME 3700 PWB(JJ) 3710 TWB(JJ)

THOMAS ALGORITHM FOR SOLVING THE SYMMETRIC TRIDIAGONAL MATRIX 2 TO M I + 1 GAMMA(L) C(L) * PNEW(L + 1) / BETA(L)

DT 14.

• PWB(JJ - D ) / ((LOG(TWB(JJ) / TWB(JJ - 1)))) PWB(JJ)) / (PI * 14.7 - PWB(JJ - 1))) / (LOG(rWB(JJ) / rWB(JJ -

1: PRINT SPACES(80); : NEXT Y: SCROLL - 11

'im': COUNT; ###./;(#y,l#-; TWB(JJ), PWB(JJ), QSFPRT(JJ);

600 THEN GOTO 4050 + 1

1

CUMTIME PNEW(l) * CUMTIME / 3600

3720 QSFPRT(JJ) - (QSURF + CCSS * (PNEW(l) - POLD(l)) / (BTAW * DT)) / 1.84 3730 IF JJ - 1 THEN GOTO 3780 3740 SL(JJ) - CLOG * (PWB(JJ) 3750 ZL(JJ) - LOG((PI * 14.7 • ))) 3760 IF SCROLL < 24 THEN GOTO 3780 3770 FOR Y - 11 TO 23: LOCATE Y, 3780 LOCATE SCROLL, 1 3790 PRINT USING "M-M' 3800 LOCATE 25, 11: PRINT USING 3810 SCROLL - SCROLL + 1 3820 FOR J - 1 TO M 3830 POLD(J) - PNEW(J) 3840 NEXT J 3845 IF JJ -3850 JJ - JJ 3860 COUNT - COUNT 3870 NEXT NTS 3880 NEXT KK 3890 GOTO 4050 3900 GOTO 1060: REM END OF SIMULATION RUN, RETURN TO BEGINNING 3910 REM 3920 REM **** 3930 PI - PI * TCHG - 0 3940 CMP - CMP / 14.7: QSURF - QSURF / 1.84: FLAG5 - 0: CUMTIME - 0: SCROLL - 11 3945 IF FLAG3 - 0 THEN CLS : GOTO 1350 3950 FLAG3 - 0: N - 30: ITER - 20: DT(1) - 2: CLS : GOTO 1270 3960 REM 3970 REM **** ARE YOU SURE SUBROUTINE **** 3980 COLOR 3: LOCATE 25, 20: PRINT " ARE YOU SURE Y/N 3990 IS - INKEYS: IF 1$ - "" THEN GOTO 3990 ELSE IF 1$ - "Y" THEN GOTO 3930 ELSE IF 1$ - "N" THE N LOCATE 25, 20: PRINT "STRIKE SPACEBAR TO PAUSE OR ESC TO'QUIT"; : COLOR 7: GOTO 3470 4000 REM **** PAUSE SUBROUTINE **** 4010 COLOR 3: LOCATE 25, 20: PRINT " STRIKE P TO PRINT OR SPACEBAR TO RESUME "; 4020 IS - INKEYS: IF IS - "" THEN GOTO 4020 ELSE IF IS - "P" THEN GOTO 4050 ELSE LOCATE 25, 20: PRINT "STRIKE SPACEBAR TO PAUSE/PRINT OR ESC TO QUIT "; : COLOR 7 4030 GOTO 3470: REM RESUME WHERE IT LEFT BEFORE PAUSE 4040 REM A .A AAA A A A******** PRINT SUBROUTINE ******************* 4050 GOTO 4090 4051 COLOR 7: LPRINT CKR$(15): WIDTH "Iptl:", 137: LPRINT CHR$(27); "G" 4060 CLS : LOCATE 11, 20: PRINT "PRINTING. PRESS ESC KEY TO CANCEL" 4070 LPRINT CHR$(27); "1"; CHR$(0): REM SETS LEFT .MARGIN ON PRINTER

SUBROUTINE TO PARTIALLY RE-INITIALIZE FOR A NEW RUN 14.7: K - K * 1000: H - H / 30.48: RWC - RWC / 30.48 JJ FLAG3 - 0: LL

SL - 0

104

4080 LPRINT CHR$(27); "Q"; CHR$(137): REM SETS RIGHT .MARGIN ON PRINTER ^090 LPRINT CHR$(27); "&LOH": I PRINT TAB(IO); "RESERVOIR AND WELLBORE SIMULATION INPUT DATA" 4100 LPRINT TAB(IO); : LPRINT STRING$(80, "-") ' llO IF AS - "YES" THEN LPRINT TAB(54); : LPRINT "NO WELLBORE STORAGE" 4120 IF BS - "YES- THEN LPRINT TAB(IO); : LPRINT USING "SINGLE WELLBORE STORAGE - iHHh'hHHM^^"^" bl/psi"; CSl / 2337120.173# 4130 IF BS - "YES" THEN LPRINT TAB(IO): / (PHI * CMP * H * RWC * RWC) 4140 IF CS - "YES" THEN LPRINT TAB(50); : LPRINT USING "CHANGING WELLBORE STORAGE, CSl - -hht .-.--MiHHH^ b b l / p s i , CS2 - im.iHmimi^ b b l / p s i " : CSl / 2337120.173#, CS2 / 2337120.173<< 4150 IF CS - "YES" THEN LPRINT TAB(50); : LPRINT USING "CSDl - M-iHHm CSD2 - */y . .f«««« ' * * " : .1592277277// * CSl / (PHI * CMP * H * RWC * RWC). . 1592277277y|( * CS2 / (PHI * CMP * H -^ R, C * RWC) 4160 LPRINT TAB(IO):

LPRINT USING "CSD - M-iHHHHK ll^llllim^ * CSl

4165 LPRINT TAB(IO) 3600 4170 LPRINT TAB(IO) 4180 LPRINT TAB(IO) 4190 LPRINT TAB(IO) 4200 LPRINT TAB(IO) 4210 LPRINT TAB(IO) 4220 LPRINT TAB(IO) 4230 LPRINT TAB(IO) 4240 LPRINT TAB(IO) 4250 LPRINT TAB(IO) 4260 LPRINT TAB(IO) 4270 LPRINT TAB(IO) 4280 REM 4290 LPRINT TAB(IO);

LPRINT USING "FIRST SHUT-IN PERIOD ENDED AT ^HHm.HHhH^ HRS."; LFSI / 36C

LPRINT USING "FULL TEST STARTED AT iHhiHHhl^ HRS."; LTIFP / 3600 + LFSI /

LPRINT STRINGS(80, "-") LPRINT USING "PERMEABILITY - ^HHHhiHH^ md"; K * 1000 LPRINT USING "POROSITY - M-M percent"; PHI * 100 LPRINT USING "INITIAL PRESSURE - iHHHhM psia"; 14.7 * LPRINT USING "H - iHHHhM ft."; H / 30.48 LPRINT USING "Ct - #y/.#yr*'" 1/psi"; CMP / 14.7

•VISCOSITY - im.^m cp."; VISC •FORMATION VOLUME FACTOR - #.### RB/STB";

LPRINT USING "WELLBORE RADIUS - #.##//## ft."; RW / 30.48 LPRINT USING "LENGTH OF FLOW PERIOD - ^^HHhMiHHH^ hrs."; LPRINT USING "PRODUCTION RATE - HHhM STB/DAY"; QSURF /

LPRINT USING LPRINT USING

PI

BTAW

CUMTIME / 3600 1.84

SQR(K * (CUMTIME

iHm. Mim ##. #y/yw/ Sandface Pressure tD

(ps ia )

M.iHHHH^ pD

'HHHi.^^r Qsand": (s tb /dav

TCHG / 3600; HRS": PRINT " SECOND CHANGE OCCURR

LPRINT USING "RADIUS OF INVESTIGATION - UMM-im ft."; - (LFSI -t- LTIFP)) / (2302.585094y/ * PHI * VISC * CMP))

4300 LPRINT TAB(IO); : LPRINT USING "EXPECTED SEMILOG SLOPE - MiHhiHHHH^ psi/cyc"; -2.69350435« * QSURF * BTAW * VISC / (K * H) 4310 LPRINT STRINGS(80, "-")

4320 Gs - "u.imn 4330 LPRINT " Time 4340 LPRINT " (hrs.) )": 4350 LPRINT STRINGS(80, "-") 4351 CLS : PRINT "FIRST CHANGE OCCURRED AT ED AT "; TCHG2 / 3600; " HRS" 4352 INPUT "ENTER THE NAME TO SAVE THIS DATA UNDER"; NM$ 4355 OPEN NMS FOR OUTPUT AS //I 4356 FOR I - 1 TO JJ - 1 4357 WRITE #1, TWB(I). PWB(I). 3601.284152// * K * TWB(I) / (PHI * VISC * CMP * RW * RW) , 6.28280 315# * K * H * (PI - PWB(I) / 14.7) / (750 * BTAW * VISC), QSFPRT(I) 4358 NEXT I 4359 CLOSE 1 4360 FOR I - 1 TO JJ - 1 4370 LPRINT USING G$; TWB(I), PWB(I). 3601.284152// * K * TWB(I) / (PHI * VISC * CMP * RW * RW) , 6.28280315// * K * H * (PI - PWB(I) / 14.7) / (750 * BTAW * VISC), QSFPRT(I) 4380 NEXT I 4390 LPRINT STRING$(80, "-") 4400 REM 4410 LPRINT CHRS(27); "&L0H" 4420 LPRINT USING "RADIAL PRESSURE DISTRIBUTION .AFTER ///M.////// hrs 4425 LPRINT " END OF SECOND SHUTIN" 4430 LPRINT " NODE RADIUS, feet PRESSURE, psia a^^O FFS - " ////// ////y/y;iy^y^y //y . y////////^////// ^.f,^^, ,.,t,f,hut,r 4450 LPRINT STRINGS(78, "-")

CUMTIME / 3600

105

4460 4470 4480 4490 4500 4510 5000 5010 5015 5020 5030 5040 5500 5510 5515 5520 5530 5540 5550 5560 5570 5580 5590 5600 6000 6010 6020 6030 6040 6050 6060 6070 6080 6090 6100 6110 6120

I, R(I) / 30.48, PNEW(I) * 14.7

LPRINT

3 THEN GOTO 3450 ELSE GOTO 5010 GOTO 6000 QSURF - 0

FOR I - 1 TO M LPRINT USING FF$; NEXT I LPRINT STRINGS(78 LPRINT : LPRINT : GOTO 1060 IF FLAG9 -FLAG9 - 3: QT(1) - 0: DT(1) - DT / 3 TCHG2 - CUMTIME GOTO 1280 LPRINT CHRS(27); "&LOH": LPRINT STRINGS(80, "-") LPRINT USING "RADIAL PRESSURE DISTRIBUTION AFTER ////////.////// hrs."; CUMTIME / 3600 LPRINT " END OF FIRST SHUTIN" LPRINT " NODE RADIUS, feet PRESSURE, psia" FFS - " #//// //////////////////.//////////////// UM.iHHHHHHr LPRINT STRINGS(80, "-") FOR I - 1 TO M LPRINT USING FF$; I, R(I) / 30.48. PNEW(I) * 14.7 NEXT I LPRINT STRING$(78. "-") LPRINT : LPRINT : LPRINT GOTO 3427 LPRINT CHR$(27): "&10H" LPRINT STRINGS(80. "-") LPRINT USING "RADIAL PRESSURE DISTRIBUTION AFTER ////////.////// hrs. LPRINT " END OF FLOW PERIOD"

CUMTIME / 3600

LPRINT " NODE FFS - " M^^ LPRINT STRINGS(80. "-") FOR I - 1 TO M LPRINT USING FFS; I. R d ) / 30.48, NEXT I LPRINT STRINGS(78, "-") LPRINT : LPRINT : LPRINT GOTO 5015

RADIUS, feet //////////y/y/M. ////////////////

PRESSURE, p s i a ' ////////.//////////////"

PNEW(I) * 14.7

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that the Library and my major department shall make it freely avail­

able for research purposes. Permission to copy this thesis for

scholarly purposes may be granted by the Director of the Library or

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Disagree (Permission not granted) Agree (Permission granted)

Student's signature Student's signature :k:,^£^2vv,^^

Date

4-^ /^3 /&9 Date ^ /