determining individual characteristics of a two- …

DETERMINING INDIVIDUAL CHARACTERISTICS OF A TWO-LAYERED RESERVOIR USING THE TDS TECHNIQUE.

A Thesis presented to the Department of

Petroleum Engineering

African University of Science and Technology

In Partial Fulfilment of the Requirements for the Degree of

Master of Science in the Department of Petroleum Engineering

By

Erhieyovwe Bishop Oghenekaro

Abuja, Nigeria.

December 2017.

1

CERTIFICATION

This is to certify that the thesis titled “Determining Individual Characteristics Of A Two-Layered

Reservoir Using the TDS Technique” submitted to the school of postgraduate studies, African

University of Science and Technology (AUST), Abuja Nigeria for the award of the Master’s

degree is a record of original research carried out by Erhieyovwe Bishop Oghenekaro in the

Department of Petroleum Engineering.

2

DETERMINING INDIVIDUAL CHARACTERISTICS OF A TWO-LAYERED RESERVOIR

USING THE TDS TECHNIQUE.

By

Erhieyovwe Bishop Oghenekaro

A THESIS APPROVED BY THE PETROLEUM ENGINEERING DEPARTMENT

RECOMMENDED: ---------------------------------------------------Supervisor, Prof Djebbar Tiab

--------------------------------------------------Dr. Alpheus IgbokoyiCommittee member

--------------------------------------------------------Head, Department of Petroleum Engineering

APPROVED: ------------------------------------------------Chief Academic Officer

December, 2017-----------------------------------------------

Date

3

ABSTRACT

ERHIEYOVWE BISHOP: DETERMINING INDIVIDUAL CHARACTERISTICS OF A TWO-LAYERED RESERVOIR USING THE TDS TECHNIQUE

“Under the direction of Djebbar Tiab”

The first part of this study discussed in details the two major model that was developed for the

study of a multilayer reservoir system. They are the Dual permeability model proposed by

Bourdet and the semipermeability wall model proposed by Gao. The pressure response of a

crossflow system is observed in three stages; it behaves as a commingled system in the early

stage and then as a single layer homogenous system in the late stage. There is a transition

stage, which connects the early and late stages. The effect of some reservoir parameters such

as the permeability thickness ratio, the storativity ratio and the semipermeability was studied.

The second part of this study focused on a two-layer reservoir system with crossflow when only

one layer is perforated and tested. The pressure response for this test method was observed to

have two infinite acting straight line on which the TDS technique was developed to determine

the individual layer parameters such as permeability thickness for both layers, wellbore storage

and skin effect for the perforated layer.

KEYWORDS: Crossflow, multilayer, TDS

4

ACKNOWLEDGEMENT

I want to first, thank God almighty for His grace and favor that saw me through this

program. It is in Him I live; I move and have my being.

My sincere appreciation goes to my supervisors Prof Tiab Djebbar and Dr. Alpheus

Igbokoyi for their guidance and immense support in completing this work. Appreciation to the

HOD of Petroleum Engineering department, Prof David Ogbe and all other faculties and staffs of

the department for their invaluable support in completing the program.

My profound appreciation also goes to my benefactor, the Pan African Material Institute

(PAMI), AUST, for giving me the opportunity to study at AUST through their prestigious

scholarship. I am very grateful

Special thanks to my parents, Evang. and Mrs. Peter Erhieyovwe and all my siblings for

their love, care, moral financial and support to me. I am so grateful to have you as my

immediate family. I want to appreciate my father in the Lord, Saint Daniel Dikeji MiyeriJesu, the

bishop of the whole world, for his continuous prayers for me.

To all my friends, Judith Takure, Oluwaseyi Osisiogu, John Idoko, Isah Muhammed,

Baduku Sunday and colleagues, thanks for your love, care, understanding, cooperation and

support. I am glad I met you people. I want to specially thank Engr. Musa Muhammed for

helping me with the coding of the equations. I am so grateful.

5

DEDICATION

This work is dedicated to God almighty, the father of my Lord and Savior Jesus Christ. I

owe it all to Him.

6

TABLE OF CONTENTS

ABSTRACT iv

ACKNOWLEDGEMENT v

DEDICATION vi

TABLE OF CONTENTS vii

LIST OF FIGURES ix

LIST OF SYMBOLS xi

CHAPTER ONE 1

Introduction 1

1.1 Introduction 1

1.2 Methods of Testing Two-Layered Reservoir With Crossflow 3

1.2.1 Testing both layers simultaneously 3

1.2.2 Testing each layer separately 4

1.2.3 Testing one zone then both together 5

1.3 Objectives of study 5

CHAPTER TWO 7

Literature Review 7

2.1 Previous Studies 7

2.2 Bourdet model12

2.3 Causes of crossflow 14

7

2.4 Behavior of the crossflow system 15

2.5 Determination of Reservoir Parameters 17

2.5.1 Single layer reservoir 17

2.5.2 Multilayer reservoir system 18

CHAPTER THREE 23

Theory 23

3.1 Concept of the semipermeability wall model 23

3.2 Model Development 24

3.3 Derivation of the short time wellbore pressure equation with wellbore storage and

skin effect for a two-layered reservoir in the infinite acting period 28

3.4 Dual permeability Model 31

CHAPTER FOUR 34

Results and Discussion 34

4.1 Double permeability behavior (Bourdet model) 34

4.2 Semipermeability wall model behavior 38

4.3 Analysis of the infinite acting flow periods 43

4.4 TDS technique for a single layer reservoir system 44

4.5 TDS technique for a two-layer reservoir system 46

4.5.1 Wellbore storage 47

4.5.2 Permeability thickness 48

4.5.2.1 Initial Radial Flow period 48

8

4.5.2.2 Second Radial Flow period 50

4.5.3 Skin Effect 50

4.6 Validation of Results 52

4.6.1 Example one 52

4.6.2 Example two 55

CHAPTER FIVE 58

Conclusion and Recommendation 58

5.1 Conclusion 58

5.2 Recommendation 59

REFERENCES 60

9

LIST OF FIGURES

Figure 1.1: Schematics of commingled reservoir system 2

Figure 1.2: Schematics of a multilayer reservoir system with crossflow 2

Figure 1.3: Pressure response when both layers are tested simultaneously 4

Figure 1.4: Pressure response when only one layer is tested 5

Figure 2.1: Wellbore pressure curves for infinitely large layered system 16

Figure 2.2: Wellbore pressure derivative curves for infinitely large layered system 17

Figure 2.3: Rate Transient for infinitely large layered systems 18

Figure 2.4: The wellbore pressure in drawdown and buildup test when the high diffusivity layer is

perforated 22

Figure 3.1: Schematic of Semipermeability wall model with a partially penetrating

vertical well 24

Figure 3.2: Schematic of dual porosity dual permeability model 31

Figure 4.1: Semi log plot of Double permeability curves with and without crossflow 34

Figure 4.2: Semi log plot of Double permeability curves with and without crossflow 35

Figure 4.3: Log log plot of pressure and pressure derivative for w =0.001 35

Figure 4.4: Log log plot os pressure and pressure derivative for w=0.1 36

Figure 4.5: Log log plot showing the effect of w for k = 0.99 36

Figure 4.6: Log log plot showing the effect of lamda 37

Figure 4.7: Semilog plot of showing the crosspoint between the short and long flow period

39

10

Figure 4.8: Log log plot of Dimensionless pressure and pressure derivative versus time 40



Figure 4.12: Semi log plot of Dimensionless pressure derivative versus time without wellbore

storage and skin effect 44

Figure 4.13: Log log plot of Dimensionless pressure and pressure derivative versus time for a

single layer reservoir 45


Figure 4.15: Semilog plot of Dimensionless pressure versus time showing the initial infinite flow

period 51

Figure 4.16: Plot for Example one - pressure and pressure derivative versus time 53

Figure 4.17: Plot for Example two – pressure and pressure derivative versus time 56

11

LIST OF SYMBOLS

B = Formation volume factor, (bbl/STB)

c t = Total reservoir compressibility, (psi)

C = Wellbore storage constant, (bbl/psi)

h = Thickness of the zone, (ft)

k = Permeability, (md)

~k i = semipermeability between layer i and layer i+1 , (md/ft)

~kDi = Dimensionless semipermeability

(kh)t = Total kh product of the reservoir, (md-ft)

K = constant, Eq.

K 0 = Modified Bessel function of the second kind order zero

K1 = Modified Bessel function of the second kind order one

m = slope of straight line Eq.

m’ = slope of straight line, Eq.

pi = Pressure, (psi)

po = Pressure, (psi)

pD = weighted Pressure, dimensionless

∆ pD = Pressure difference, dimensionless

12

q = Production rate, (STB/D)

r = Radial distance, (ft)

u = Laplace variable

s = Skin

t = Time, (hrs)

α = Interporosity shape factor in Eqn. ()

λ = Interporosity flow parameter defined Eqn. ()

μ = Viscosity, (cp)

∅ = Porosity, (%)

ω = Storativity ratio defined in Eqn. ()

η = diffusivity, ft2/sec

Subscripts:

D = Dimensionless

i = Initial

1,2 = Layer one and two respective

13

CHAPTER ONE

1.1 Introduction

Many reservoirs consist of two or more layers with non-permeable or low-permeability

shales between them. Over the years, well test data has often been interpreted with the

assumption that the reservoir is homogenous and of a single layer (C. Gao, 1987). Pascal, in

1992 pointed out in his study of pressure and rate transient data from wells in multilayered

reservoirs, that there are geological evidences (logs, core analysis) which substantiate the fact

that numerous reservoirs have strong heterogeneous characteristics with respect to the vertical

direction, which indicates that they are multi-layered reservoirs. It has been observed that

pressure transients in multilayered reservoirs often show a significant deviation from the

response that would be obtained in a single layer homogeneous reservoir having equivalent

thickness-averaged properties (Pascal, 1992).

Most oil and gas reservoirs are layered (stratified) to various degrees because of

sedimentation process over long geological time (Kucuk, 1986). Layered reservoirs are

composed of two or more layers that may have different formation properties and fluid

characteristics. Wells in such reservoirs may produce from more than one layer (Kucuk, 1986).

Pressure behaviour in this kind of vertically heterogeneous system is not necessarily like that of

a single-layered system; it reveals only the average properties of the entire system. To identify

the characteristics of the individual layers is important, especially if water flooding or an

enhanced recovery scheme produces the reservoir fluid. (Horne, 1989)

The deviation of the pressure transient observed in multilayered reservoirs from that of a

vertically homogeneous reservoir, became particularly obvious with the advent of high

resolution, downhole electronic gauges, allowing very accurate pressure measurements.(Kucuk,

1986)

Two different multilayered models have been proposed, depending on the presence or

absence of interlayer crossflow. A multilayered reservoir is called a crossflow system if fluid can

move between layers, and a commingled system if layers communicate only through the

wellbore (Horne, 1989). Hence, a distinction must be made between them. A crossflow system

occurs when a non-zero permeability shale layer exists between two adjacent layers. A wellbore

commingled system occurs when an impermeable shale layer exist between any two or more

layers that are penetrated by the wellbore (Ehlig-economides & Joseph, 1987).

F

igure 1.1 Schematic of commingled reservoir (Chaudhry, 2003)

Figure 1.2 Schematic of a multilayer reservoir with crossflow (Chaudhry, 2003)

1.2 Methods of Testing Two-Layered Reservoir With Crossflow

Bourdarot in 1998 discussed in his book, several methods that can be considered in

testing a two-layered reservoir. They do not all allow the individual characteristics of each layer

to be determined.

The methods include;

i. Testing both layers simultaneouslyii. Testing each layer independentlyiii. Testing one layer then both together.

1.2.1 Testing both layers simultaneously

This method consists in putting the two zones in production and testing them together. The

common wellbore pressure and the individual layer flow rate is measured in this case. This

method generally provides only a little information about the reservoir. The reservoir usually

behaves as if it were homogenous. Three different flow periods occur in this method;

Early-time flow period. At early time, before the crossflow is established, the pressure

response of a crossflow system is identical to that of the commingled system. Late-time flow period. In this period, the crossflow is established and the pressure

response of the system produces a semilog straight-line similar to that of the

homogenous single layer system. This period characterizes the total producing system

(total permeability-thickness, total storativity) Intermediate flow period. This is the period where a transition occurs between the early

time and late time.

Figure 1.3: Pressure response when both layers are tested simultaneously

1.2.2 Testing each layer separately

This method consists in producing only one of the two layers at a time in order to test it. The

procedure is often difficult to apply to each layer in succession. However, this one gives the

most information. The pressure change and especially the derivative, show that three flow

periods come one after the other during testing. Fig. 1.3 shows the three-flow period that exists

in this method. They are:

An initial radial flow period. Only the zone tested is involved in the flow. A stabilization

of the pressure derivative curve corresponds to it. The analysis of this flow is used to

determine the kh and the skin of the tested zone. A transition period; when the derivative decreases. The decrease shows that more

reservoir thickness is becoming involved in the flow. The time it begins depends on the

vertical communication (semipermeability) between the two layers. The better the flow,

that is, the higher ~k i is, the earlier the transition.

A second radial flow period. Both layers are involved in the flow and thus behaves as

a single layer homogeneous reservoir. The stabilization of the pressure derivative curve

at 0.5 corresponds to it. The analysis of this flow is used to determine (kh )t and one

skin. It is difficult to analyze the skin. It is a function of the skin of each layer, the

transmissibility contrast between the layers and the communication between the two

layers.

Figure 1.4: Pressure response when only one layer is tested

1.2.3 Testing one zone then both together

This method consists in putting the first zone in production, testing it then putting the

second zone in production and testing it together with the first one.

1.3 Objectives of study

The purpose of well test analysis is for the quantification of reservoir parameters such as

permeability, skin and wellbore storage. The purpose of this study is to use the Tiab’s Direct

Synthesis method to analyze well test data (wellbore pressure data) in order to determine the

reservoir parameters for each layer of a two-layered reservoir with formation crossflow when

only one layer is perforated at the well.

This work studies the pressure behaviour of a two-layer reservoir with crossflow when

only one layer is tested. The main objective of this study is to:

1. To present and discuss the semipermeability wall model developed by Gao and the

dual permeability model developed by Bourdet for the study of multilayer reservoir

systems.2. To obtain the individual layer parameters such as permeability-thickness, the skin

and wellbore storage using the Tiab’s Direct synthesis technique.

CHAPTER TWO

2.0 LITERATURE REVIEW

In the last few decades, many investigations and research have been carried out on

multilayer models to adequately describe a multilayer reservoir system with or without crossflow.

The two prominent models used today are the double-porosity double-permeability model by

Bourdet in 1985 and the semipermeable model proposed by Gao in 1983.

In this chapter, we will give a summary of some of the previous researches and review in

details the model developed by Bourdet.

2.1 Previous Studies

Lefkovits et al, in 1961 established a rigorous study of the multilayered system without

crossflow. His study, which addressed an arbitrary number of layers with distinct layer

properties, including thickness, porosity, permeability and skin, provides an analytic model and a

host of practical observations that have served as the basis for much of the work that has

followed. He derived the analytical solutions for the wellbore pressure and the layer production

rates in a bounded multilayered reservoir, where each layer had different reservoir parameters.

In 1976, Robert Otuomagie, et al. obtained in their study a set of expressions and

procedure that can be used to determine the reservoir parameters from pressure buildup curves

for an infinite two-layered oil reservoir without crossflow. The study also shows under what

condition an infinite two-layered oil reservoir is similar to a single-layered reservoir. Equations

were derived that describes theoretical buildup for this system when the well is shut in. From the

equation, the permeabilities and skin factors for the individual zones of an infinite two-layered

reservoir can be estimated on the condition that the initial pressure in either zone is known or if

the difference in pressure between the two zones is known. They concluded that in order to

obtain equations to compute the reservoir parameters of an infinite two-layered reservoir, it is

necessary to take into consideration the force of gravity. If the force of gravity is neglected, the

equations obtained can be used only to study the effects of the reservoir parameters, skin and

wellbore storage.

Ramey, in 1978 did an extension the study of the commingled system by considering the

effect of wellbore storage and skin. A major contribution of their research was the use of the

Stehfest algorithm (Stehfest, 1970) to invert the solution in the Laplace space into real space

numerically. Since the method is easier than the method of direct inversion by complex analysis,

this algorithm has been used in many studies concerning well test analysis.

Larsen, in 1982 introduces a general method to determine both the skin factor and the

flow capacity of individual layers in a two-layered reservoir by using only wellbore pressure. The

method is based on infinite-acting approximations derived for systems without crossflow, but

can also be used for crossflow systems to an extent. For the method to work, the wellbore

pressure behaviour must deviate from that of a single-layered reservoir.

Dean and Gao, in 1983a developed a very useful idea to reduce the dimension of the

governing equation by one. The terms for the vertical movement were represented by the

pseudo-steady state approximation in this semi-permeable wall model. They derived

approximate solutions that describes the short and long periods of the wellbore pressure when

each layer of a two-layer reservoir produces independently (Gao, 1987). They also derived an

approximate analytical solution to estimate the formation crossflow velocity. (Gao, 1983b). A

detailed description of the semipermeable model will be given in the next chapter.

Bourdet, in 1985 derived a complete solution for the pressure response in a two-layered

reservoir with crossflow; with the consideration of wellbore storage and skin effect. He also

showed that his solution was the general form of many other reservoir models by showing that

the solution was identical to other solutions of various others when some reservoir parameters

took certain limiting values. A description of the model will be given in the next section.

Cheng Gao, in 1985 studied the behaviour of a two-layer reservoir with crossflow when

only one layer is completed to the single well. He demonstrated the limiting behaviour of the

crossflow during drawdown and build up test, with numerical solutions of the model. He

proposed a well test procedure and interpretation method for determining the parameters of the

individual layers and the semi-permeability between the layers. This procedure has been shown

to give good estimates of the reservoir parameters in more than thirty simulated well tests for

various combinations of parameters.

Prijambodo et al. in 1985 investigated the pressure response of the two-layered

reservoirs with formation crossflow. They showed that the flowing pressure response of a well at

early times can be divided into three flow periods. The first period in which the reservoir

behaves as if it were stratified (no crossflow) system, followed by a transitional period; and then

the third period in which the reservoir behaves as a single system. Unlike most other authors,

they applied the concept of the thick skin. However, the overall effects of the wellbore pressure

were not different from the results of other authors, who had applied the concept of thin skin.

Kucuk, in 1986 suggested a new testing method in a two-layered commingled reservoir.

They used the technique of nonlinear parameter estimation by coupling the sandface production

rate of each layer with the wellbore pressure. The coupling of the layer production rate is very

significant for multilayered reservoirs because the rate transient of each layer reveals

information about the layer, while the wellbore pressure determines the average reservoir

parameters.

Cheng Gao, in 1987 presented an approximate expression of the pressure response

and flow rates of individual layers when a well produces with a constant production rate from an

infinite multilayer reservoir with and without interlayer crossflow. The permeability-thickness

product and skin factor of each layer can be determined by one drawdown test if the common

wellbore pressure and the rate of each layer can be measured at the same time. The work also

discusses the behaviour of the crossflow caused by skin factor. A new method for determining

the vertical permeabilities of the shales between layers was developed that uses the crosspoint

of the two straight lines of drawdown or buildup curves. He discovered that if a n -layered

reservoir is divided into two sections and each section produces independently at a constant

rate, both the drawdown and buildup curves have two straight lines which can be used to

determine roughly the semipermeability between the two sections.

Ehlig-Economides and Joseph, in 1987 studied the behaviour of pressure transient for

n homogenous-layered systems with or without crossflow and provided an analytical solution

to the general problem. They also studied the wellbore storage effects and solutions provided

for both infinite acting and bounded systems. The individual layer properties are determined by

the limiting forms of the analytic solution, which are used to identify certain characteristics

patterns in the flow-rate transients. Their work demonstrates that the analysis of layer flow-rate

transients provides a means for determining the layer permeabilities and skins and the effective

vertical interlayer communication by investigating the early time and late time behaviour of the

production rate of each layer. The model they developed and the testing and interpretation

methodology that it implies, offer a theoretically sound means for obtaining a complete layered

reservoir description for an arbitrary number of layers with or without crossflow between any two

adjacent layers.

Horne, in 1989 did a study that investigated the effects of the reservoir parameters such

as permeability, vertical permeability, skin, wellbore storage coefficient and the outer boundary

conditions on the wellbore response (pressure and layer production rate). They made the

following discoveries:

a) The vertical permeability is the only parameter governing the initiation and termination

of the transition from the early time commingled system response to the late time

homogeneous system response.b) The outer boundary conditions have only a small effect on the wellbore response as

long as the transition terminates before the system feels the boundary.

They also applied the nonlinear parameter estimation method to determine the layer

parameters.

Ahmed, et. al. in 1994 examined the effect of unequal initial layer pressures on the

wellbore pressure and sandface rate responses of wells producing from commingled reservoirs.

The main objective was to provide techniques and methodology to determine the individual

layer permeability, porosity and skin factors from the pre-production well test data. The result of

this study provided a useful foundation for the design of a new pre-production well test

procedure that ensures that the individual layer properties are measured adequately and early

in the life of the reservoir when they are most needed.

N. M. Al-Ajmi et al. in 2003, carried out a study to estimate the storativity ration in

layered reservoir with crossflow from pressure transient data. They used an analytically derived

formula for the storativity ratio, in terms of the separation between the two semi-log straight lines

on pressure versus log-time plot. An expression for the vertical separation between the late and

early-time semi-log straight lines was obtained from the early and late-time approximations of

the pressure transient solution of the double permeability systems. They concluded that an

estimation of the individual layer properties can be known if the layer flow rates are available

from production logs. (N. M Al-Ajmi et al. 2003)

Jing Lu et al, in 2015 presented a new solution to pressure transient equation in a two-

layer reservoir with crossflow. They reckoned that a two-layer reservoir with crossflow was

erroneously taken as a dual-porosity-dual-permeability naturally fractured reservoir in literatures;

and thus the model cannot account for the characteristics of two-layer reservoir. They presented

a new mathematical model and an analytical solution to the pressure transient equation of a

uniform-flux. They proposed a parameter to describe the crossflow in a two-layer reservoir

quantitatively and verified that the crossflow coefficient is not constant, but a function of the time

and distance from the wellbore. The solutions which they obtained by using Laplace transform,

double Fourier transform and Green’s functions method with numerical methods, were proven to

give accurate enough results for practical applications. It also allows the effects of formation

properties on pressure behavior both at large distance from the wells and at the wells. The

major distinguishing assumptions they made are: the reservoir is homogenous and infinite

extension in x and y direction and that the reservoir consists of two parallel, contiguous and

overlying layers with no restriction to flow across their common boundary. (Jing Lu et al. 2015)

2.2 Bourdet model

In the literature, dual-permeability and dual-porosity system definitions are mostly

associated with layered systems and fractured systems respectively. The dual-porosity model in

principle is usually considered, as a limiting case of the dual-permeability model, and as such,

resembles those of dual-permeability in many ways.

Warren and Root (Warren, et al. 1984) showed that the transient pressure responses of

wells in a fractured system, modelled with the dual-porosity system are controlled by two

parameters:

Interporosity flow coefficient, λ, is defined by

2 mw

f

krk

--------2.1

The interporosity flow coefficient is a measure of the efficiency of fluid flow from the

matrix to the fissure. km and k f are the permeability of the matrix and fissure system

respectively. Σ is the shape factor; and it is a function of the size and shape of the matrix blocks.

Storativity ratio, ω is defined as

t f

t tf m

c h

c h c h

--------2.2

The storativity is a ratio that relates the storativity of the fissure to the total storativity

(fissure and matrix). The corresponding equations for the interporosity and storativity for a

layered system with crossflow modelled by a dual-permeability system is given in Eqn. 8 and

Eqn. 9 respectively.

Bourdet considered a layered system that is infinite with only two communicating layers.

His model is an extension of the Warren and Root model (Warren, et al. 1984) to the case

where both media are in contact with the producing well. Unlike the dual porosity model, where

the matrix permeability is neglected and assumed to be zero, it is not neglected in this model

and a radial flow rate occurs within the matrix system as well as from the matrix to the well. This

has been referenced in literatures as double permeability model.

The equations describing fluid flow in medium 1 and 2, can be expressed in the

dimensionless form as:

2 11 2 1

DD D D

D

pp p p

t � �

� --------2.3a

2 22 2 11 1 DD D D

D

pp p p

t � �

� --------2.3b

The following dimensionless terms are used:

1 1 2 21 ,2 1,2141.2D D i

k h k hp p p

qB

--------2.4

1 1 2 2

2

1 2

0.000263D

t t w

k h k h tt

c h c h r

� �� ---------2.5

1 1

1 1 2 2

k h

k h k h

-----------2.6

2 1 1

1 1 2 2w

k hrk h k h

-------------2.7

1

1 2

t

t t

c h

c h c h

-----------2.8

2

1 2

0.8698D

t t w

CC

c h c h r � �� ----------2.9

Kappa is the ratio of permeability thickness for layer 1 to the total permeability thickness. This is

usually not considered in the double porosity model hypothesis where κ=1 . After

transforming the above system in the Laplace space, it is solved analytically according to the

method proposed by Kutljarov. The wellbore storage effect and one skin for both layers are

introduced into the solution. The solution is presented in chapter three.

2.3 Causes of crossflow

Three conditions cause crossflow between layers in a single-phase fluid flow. They are:

1. Different boundary pressures for different layers2. Non-proportional change in permeability in different layers. These two crossflow causes

can happen in both steady and unsteady flow.3. Different diffusivities in different layers, which applies only in unsteady state flow.

2.4 Behavior of the crossflow system

It has been stated that the flow in a multilayer reservoir with crossflow behaves as if the

reservoir has only a single layer for most of the time, and as if there were no crossflow between

the remaining layers. These two periods are defined as the short period and the long period.

The transition period between them is defined as the middle period. The response of the well

during this period depends on the contrast in horizontal permeabilities and on the degree of

communication between the layers. In the short time period, semipermeability kDi can be

treated as zero. (Russel & Prats, 1962)

The drawdown or buildup for an infinite multilayer reservoir with crossflow consists of

two parallel straight lines with a transition period between them. The first straight line belongs to

the short time period, and the second belongs to the long period. The transition between them

belongs to the middle period. For an actual reservoir, these three parts may not develop

completely. If kDi is large, the first straight line and even the transition part may disappear. If

kDi is very small and/or the reservoir is not large, the outer boundary may be reached before

the second straight line develops. (Horne, 1989)

The typical wellbore responses in a multilayered reservoir with formation crossflow as

presented by H. park (Horne, 1989) are shown in Fig. 2.1 – 2.3.

Figure 2.1: Wellbore Pressure Curves for infinitely Large Layered System

Shown in Fig. 2.1 are two cases, A and B. In case A, wellbore pressure responds to

production at a constant rate in three progressive stages. At the early time (tD<t D1) , pressure

response of a crossflow system is identical to that of a commingled system with same reservoir

parameters. At the late time (tD>t D2) , wellbore pressure response reaches a semilog

straight line behaviour similar to that of a homogenous single layered system whose reservoir

parameters are the arithmetic average of the parameters of each layer, such that:

1 1 2 2

1 2

n n

n

k h k h k hk

h h h

LL ----------2.10

1 1 2 2 n n

t

q s q s q ss

q

L

--------------2.11

At this stage, the pressure may be expressed as:

10.404535

2D Dp In t s ------------2.12

Fig. 2.2, gives a better picture of the three-stage response.

Figure 2.2: Wellbore Pressure Derivative Curves for Infinitely Large Layered Systems

It is also observed that in the late period, the production layer rates becomes constant,

as shown in Fig. 2.3.

2.5 Determination of Reservoir Parameters

2.5.1 Single layer reservoir

Conventional well test analysis for a single layer homogeneous reservoir is performed by

matching the measured data, commonly the wellbore pressure, graphically with the

mathematically or numerically computed data either on a semilog or on a log-log scale.

Figure 2.3: Rate Transient for Infinitely Large Layered Systems

The pressure response of some reservoir models at late time can be approximated as a

straight line on a semilog plot, from which we can directly calculate several reservoir

parameters. When the test is too short for a semilog straight line to develop or when the

reservoir model does not produce a semilog straight line inherently, a log-log type curve

matching technique is applied to analyze the measured data. Tiab, in 1993 also developed a

technique for analyzing pressure data. This technique known as the TDS (Tiab’s direct

synthesis) was developed by the analysis of a log-log plot of pressure and pressure derivatives

versus time.

2.5.2 Multilayer Reservoir

Various methods for determining the individual layer parameters in a multilayer reservoir

have been developed by several authors over the years. Kucuk, in 1986 suggested the

sandface-flowrate convolution method, which is, in essence, a multi-rate test applied to a

multilayered reservoir without formation crossflow (commingled system). They assumed that the

coupling of wellbore pressure and production rate would yield a semilog straight line by noticing

that the wellbore pressure is a function of the reservoir parameters while the layer production

rate is a function of the layer parameters only.

Ehlig-Economides and Joseph, in 1987 analyzed early time data to determine layer

properties, k i and s i . The problem is that the early time interval should be long enough

that the equation develop a semilog straight line. Thus, this method for the crossflow system will

not be adequate when the vertical permeabilities of the layers are big. The behavior of a

crossflow system is such that for most part, it act as if they were a single-layer system.

The slope of the straight line reflects the arithmetic average of the thickness-weighted

permeability, kh (assuming that the fluid properties are identical). That is, if m represents

the slope of the semilog straight line for a pressure test in a well producing from a reservoir

consisting of n layers. (Russel & Prats, 1962)

1

162.6n

i ii

qBkh k h

m

� ;----------2.13

The skin factor, s , corresponding to this value of m may be estimated by the

conventional method as given in equation 2.5:

,

2

( )1.1513 log log 3.2275ws wf s

t w

p t p t t khts

m t t c h r � � � �� --------2.14

Here, ∅C t h , is the thickness-weighted porosity-compressibility product:

1

n

t i ti ii

c h c h

�---------------2.15

Although Larsen, in 1982 does discuss the determination of individual layer properties

and the skin factor in each layer for a two-layered system, this limitation is best overcome if both

wellbore pressures and layer flow rates are measured. (Kucuk et al, 1986a; Economides and

Joseph, Raghavan 1989, Larsen 1994)

Prijambodo et al. showed that the skin factor, s estimated from a pressure test

usually reflects a flow-rate-weighted average of the skin factors in each layer.

1

ni

ii

qs s q

� �� ;

-------------2.16

qi = production rate of layer i at instance of shut-in

For a commingled system, Ramey, (1978) and Prijambodo. et. al., (1985) showed that

s reflects a flow-capacity-weighted average of the skin factor in each layer.

1

ni i

ii

k hs skh

� �� ;

----------------2.17

Horne, (1989) suggested two new methods to obtain the initial estimate of the reservoir

parameters. The first method is applicable only to isotropic two-layered reservoirs. The second

method is generally applicable for n -layered reservoirs and works better when the reservoir

is a commingled system or when the vertical permeabilities are small.

For the first method, (kh)t and s are first measured for the entire system by the

conventional method. The pressure and time are dimensionalized with the following

expressions.

141.6( )t i wf

wDt

kh p pp

q

-----------------2.18

2

0.0002637( )tD

t wt

kh tt

h c r

------------2.19

A semilog plots of the dimensionless pressure and it’s derivative is prepared and the

transition period is identified by comparing with the pD curve of a homogenous system

whose reservoir parameters are (kh)t and s . The transition is entirely determined by the

crossflow, which is governed by the parameter λ . Since this method assumes isotropic

layers, λ is defined as:

2

2 21 21 1 2 1 2 2

2 1

2 wr

k kh h h h h h

k k

� � � �

� � � �� --------2.20

The type curve method is used to determine ( k1

k2) . With this ratio and (kh)t , we can

calculate k1 and k2 . The skin factor can be calculated by measuring the late time

production rate of the bottom layer.

Cheng Gao, in 1985 developed another method to determine the reservoir properties of

the individual layer. He suggested that n layers be separated into two sections by a packer

and produce only from the upper section, until two different straight lines appear. The first

straight line determines the average permeability of the producing section and the second line

determines the permeability and the skin of the entire system. This would be followed by

(n –1) drawdown tests, lowering the parker by one formation every time. With these n

average values, the individual layer permeability can be calculated. Fig 2.4 shows the typical

flow behavior for drawdown and buildup well tests when the higher diffusivity layer is perforated

for a two-layer reservoir. It can be seen that at the beginning of production, p1 increases and

has a slope 1/w1 on the semilog plot. After this period, crossflow begins to influence the flow

significantly.

The lower diffusivity layer feeds the higher diffusivity layer, so that p1 increases more

slowly. This transient period is not short. It takes more than two cycles on a plot of p vs.

log t in this case. After the transient period, the system reaches the second straight-line

region on the semi-log plot. The second straight line has a slope of 1/ (w1+w2 ) , so from this

second straight line, we can get the total transmissibility of the system.

Figure 2.4: The wellbore pressure in drawdown and buildup test when the high diffusivity

layer is perforated.

The methods suggested by Economides and Horne requires that the two layers are

produced together and the common wellbore pressure are measured alongside the individual

layer flow rates. On the other hand, the method suggested by Cheng Gao for a two-layer

reservoir system with crossflow requires that only the top layer is perforated. In this case, we

measure only the wellbore pressure to determine the individual layer parameters. This method

seems to be more convenient, less procedures and can give a quick estimate of the layer

properties.

CHAPTER THREE

3.0 Theory

This work aims to study the behaviour of a two-layered reservoir with crossflow,

perforated in one layer only and to determine the layer parameters. The study of the signature

produced by the pressure derivative curve helps us to understand how this particular system

behaves.

The fluid in the reservoir flows into the wellbore at the point where it is perforated. The

fluids entering into the wellbore will be from the perforated layer for a period, if there is an

semipermeable shale bed between the layers, until the crossflow begins. Thereafter, the fluids

entering into the wellbore will be from both layers.

The fluids in the reservoir normally move from the region of high pressure to a region of

lower pressure. The perforation in the first layer causes a reduction in pressure in the layer and

because there exists a communication between the two layers, fluids from the second layer

begin to flow into the first layer. The rate of flow depends on the permeability and thickness of

the shale between the layers and the vertical permeability and thickness of the two reservoir

layers. The higher the semipermeability between the layers, the higher the rate of crossflow

between the two layers. Equation 3.4 gives the expression for the semipermeability.

3.1 Concept of the Semipermeability Wall Model

The horizontal dimensions of reservoirs are generally much greater than their thickness

between the top and bottom boundaries. The pressure change in the vertical direction is

generally small if there is no low-permeable shale within a layer. Thus, the pressure at the

midway point in the vertical direction of a layer is a good representation of the average pressure

in the layer. This gave rise to the concept of vertical equilibrium, which implies that the vertical

permeability in each layer is infinite; i.e. perfect vertical communication. (Coats, et al. 1971).

Because of the small pressure change in the vertical direction within a layer, we can

concentrate the vertical resistance to flow at an imaginary wall between any two layers and let

the vertical resistance within the layer be zero. The resistances of the imaginary walls between

the layers is taken to be equivalent to the actual resistance of the reservoir. These imaginary

walls, called “semipermeable walls,” are a remedy to the assumption that the vertical

permeability within layers is infinite. (Gao, 1984).

The five assumptions used in the semipermeable model presented by Gao are:

1. The reservoir is homogenous in vertical direction in each layer.2. The reservoir pore space is filled with a slightly compressible single-phase fluid.3. The thickness of each layer is constant.4. The horizontal permeability of each layer is finite, but the vertical permeability is

infinite.5. Gravity force is negligible.

3.2 Model Development

Figure 3.1: Schematic of semipermeability wall model with a partially perforated verticalwell

The figure above shows the schematic of a two-layer reservoir model. This schematic

shows that crossflow exists between the two layers and only layer 1 is perforated to the well.

The following assumptions are used in developing a mathematical model for the system(Gao

1984):

Crossflow,

, Layer 2

Layer 1 Crossflow

a) The reservoir is of infinite lateral extent, consisting of two homogeneous layers with

homogenous low-permeability shale between them. The top and bottom boundaries are

impermeable. The storage of the shale can be neglected compared to the storage of the

producing layers.b) The fluid is slightly compressible and flow is single-phase, obeying Darcy’s law.c) Gravitational forces are negligible and pressure gradients are smalld) The semipermeability wall model can be used in place of an actual two-dimensional

reservoir model.

Suppose a homogenous two-layered reservoir is penetrated by a vertical well and both

layers produce independently with constant dimensionless rates qD1 and qD2

respectively. Using the semipermeable wall model, the equations for a two-layer system in the

dimensionless form is given as:

1 1 11 2

1 1

1 10D D D

D D DD D D D D

p p kpr p p

t r r r w� ��

%

-------------3.1a

2 2 21 2

2 2

1 10D D D

D D DD D D D D

p p kpr p p

t r r r w� ��

%

-------------3.1b

The dimensionless expressions are given by:

Dw

rr

r

,2

0.0002637 eD

w

tt

r

,

141.2

o itDi

kh p pp

q B

2

( )i w

Dit

k rk

kh

%%

, ( )i i i tw k h kh

, Di iq q q

And

Di i e -----------3.2

Where

1

( )n

t i ii

kh k h

�,

ii

i

k

c

,

1

1 11

n n

e i i t i ii i

ni k h c h

i i

w

� � � ��

� ��

and

1

n

ii

q q

�-------------3.3

~k i is the semipermeability between layer 1 and layer 2 and is defined by

1

1

1

12

i i

i

i i

v v v

kh hh

k k k

� �

� ��

%

---------------3.4

Where i=1,2

Gao in 1987 presented the approximate solution for both the short and long time period

without wellbore storage effect.

The approximate solution for the short period gotten by neglecting the last term in Eq. 3.1, is

given by:

2

, 1, 22 4Di D

Di ii Di D

q rp E i

w t� �

� �� -------------3.5

The wellbore pressure, pwfD1∧pwfD2 are:

1 11 1

1

41

2D D D

wfD

q tp In s

w

� � � �� --------------3.6a

2 22 2

2

41

2D D D

wfD

q tp In s

w

� � � �� ---------------3.6b

If only layer one is perforated and produced, i.e. qD2=0 then pwfD2=0

The approximate solution for the long period is given by:

11 2 1

12

41 2

2D D

wfD

t qp In w K In s

w

� �� --------3.7a

22 1 2

22

41 2

2D D

wfD

t qp In w K In s

w

� �� ---------3.7b

For layer 1 and layer 2 respectively.

Where K=( qD1

w1

−qD2

w2) , λ2=

kD1

w1w2

and Euler constant γ=1.781

pD 1=−12

Ei(−rD2

4 tD )+wD 2[K K 0 (√⋋rD )√⋋K1 (√⋋rD )

−ξ1wD1wD2

2~kD1t D

(1−e−β1 t D)e−rD

2

4 tD ] --------3.8a

pD 2=−12

E i(−rD2

4 tD )−wD1[K K 0 (√⋋ rD )√⋋K 1 (√⋋ rD )

−ξ1wD1wD2

2~kD1 tD

(1−e−β 1t D) e−r D

2

4 t D ] --------3.8b

pD=pD1wD1+ pD2wD2 ----------3.8c

Equation 3.8a to 3.8c gives the dimensionless pressure for layer 1, layer 2 and the total

pressure respectively.

In the next section, we derived the dimensionless pressure equation in the infinite acting

period by considering both the wellbore storage and the skin.

3.3 Derivation of the short time wellbore pressure equation with wellbore storage and

skin effect for a two-layered reservoir in the infinite acting period.

Suppose a homogeneous two-layer reservoir is penetrated by a vertical well. Layer 1

and Layer 2 produce independently with constant dimensionless rates qD1 and qD2 ,

respectively. Assuming also that the pressure regime is in the short time period i.e the pressure

response is only from the penetrated layer. The problem is expressed as follows:

1 10Di Di

DDi D D D D

p pr

t r r r� �� --------- 3.9

This is same as Eq. 3.1 but without the semipermeability term. We neglected the

semipermeability term because we assume that the effect of the crossflow is not felt during this

period.

Initial Boundary conditions

, 0 0Di D Dp r t ---------- 3.10

Outer boundary conditions

, 0Di D Dp r t ��------------3.11

Inner boundary conditions

Wellbore storage effect

1D

wDi DiD i D Di

D D r

dp pC w r q

dt r

� �� ------3.12

Skin effect

1D

DiwDi Di i D

D r

pp p s r

r

� �� -----------------3.13

With the given boundary equations, we now provide a solution to the model above in the

Laplace domain.

Laplace Transform of Eq. 3.10 gives

Di o D o DDi Di

u up AI r BK r

� � � �

� � � �� ---------3.14

Where u is the Laplace transform variable with respect to tD

Taking the Laplace transform of Eq. 3.11 gives

, 0Di D Dp r t ��------------ 3.15

Putting Eq. 3.15 into Eq. 3.14 and solving, we have

Di o DDi

up BK r

� �

� �� ------------ 3.16

Taking Laplace of Eq. 3.12 gives

,

1

0D

Di DiD wDi D D i D

D r

p qC up p r t w r

r u

� �� --------- 3.17

The Laplace transform of Eq. 3.13 gives

1D

DiwDi Di i D

D r

pp p s r

r

� �� ------------3.18

Differentiating Eq. 3.16, we have

1Di

DD Di Di

p u uB K r

r � �� -------------- 3.19

Substituting Eq. 3.16 and Eq. 3.19 into Eq. 3.13, we have

1wDi o D i D DDi Di Di

u u up BK r s r B K r

� ��

� �� ------------ 3.20

Making B the subject of the formula, Eq. 3.20 becomes

1

wDi

o D i D DDi Di Di

pB

u u uK r s r K r

� ��

� �� --------3.21

Substituting Eq. 3.19 into Eq. 3.17, we have

1Di

D wDi i D DDi Di

qu uC up w r B K r

u � ��

� �� ------------- 3.22

Substituting Eq. 3.21 into 13.22 and putting rD=1 , we have

1

1

i wDiDi Di Di

D wDi

o iDi Di Di

u uw p K

qC up

uu u uK s K

� ��

� ��

� �� ---------- 3.23

Rearranging Eq. 3.23 gives,

1

1 1

1Di o i

Di Di Di

wDi

D o i iDi Di Di Di Di

u u uq K s K

pu u u u u uuC K s K w K

� ��

� �� ------3.24

Equation 3.24 is the solution of the transient pressure distribution with skin and wellbore

storage in the Laplace transform domain; where K0 and K1 are modified Bessel function

of the second kind of order zero and order one respectively.

To complete the solution of the problem, Eq. 3.24 must be inverted into the real-time

space. The real inversion of Eq. 3.24, however, is not available in terms of standard functions.

Hence, it is inverted numerically using the Stehfest algorithm.

3.4 Dual Permeability Model Solution

Figure 3.2: Schematic of Dual-permeability modelThe Dual permeability model is an extension of the dual porosity model proposed by

Warren and Root. Contrary to the dual porosity model, the matrix system permeability is not

neglected in this case. A radial flow rate occurs within the matrix system as well as from the

matrix to the well. In other words, Both the matrix and the fissure, contributes to the flow of fluid

into the wellbore.

Barenblatt (Barenblatt et al, 1960) describes the equation describing fluid flow in layer 1

and 2 as:

2 11 2 1

DD D D

D

pp p p

t � �

� ---------3.25a

2 22 2 11 1 DD D D

D

pp p p

t � �

� ------------3.25b

The solution to the dual permeability model proposed by Bourdet for a layered system as

given in Eqn 2.25 is given below:

0 01 1 1 2

( ) ( )1

( ) ( )D D

D u E up u uC

s K s K � �

� � � � ---------3.26

where

0 01

1

( )

( )

KK

K

----------3.27

2 1

2 1

1 1a aD u

a a

----------3.28

1 2

1 2

1 1a aE u

a a

-------------3.29

21 11 1 1 1a u � � � � -----------3.30

22 21 1 1 1a u � � � � ------------3.31

21

11

2 1

u u

� � � � � �� -----------3.32

22

11

2 1

u u

� � � � � �� ----------3.33

2 21 4

1 1

u u

� � � � � � ----------3.34

A limiting form of Eqn. 3.26 is when there is no crossflow between the two layers. In this

case, λ becomes zero. The solution for this case is given below:

0 01 1

11

11

D Dp u uC

s K u s K u

� ��

� � � �� ---------3.35

Where

1 1

1 1 2 2

k h

k h k h

----------3.36

2 2 2

1 1 2 2w

k hrk h k h

------------3.37

1

1 2

t

t t

c h

c h c h

--------------3.38

In the following chapter, semi-log and log-log plots of dimensionless pressure versus

dimensionless time are generated to show the behavior of a two-layered reservoir system with

the dual permeability model and that of a semipermeability wall model.

CHAPTER FOUR

4.0 Presentation and Discussion of Results

4.1 Double permeability behavior (Bourdet model)

The curves generated with the double permeability model are plotted with the

dimensionless pressure pD versus the dimensionless time group tD/CD . And they are

defined by four parameters that constitutes Eqn. 3.24 to Eqn. 3.32.

- CDe(2 s) defines the condition of the well, i.e the wellbore storage effect and skin

- κ and ω defines the contrast in properties between the two layers- λ Defines the interaction (crossflow of fluid from one layer to the other) between

the two layers.

Figures 4.1 to 4.5 shows the log-log plot of the double permeability curves. The values of

CDe(2 s) has been fixed to 1 and the crossflow parameter λ to 4 x 10−4 .

0 0 0.1 10 1000 100000 10000000 10000000000

0

0.01

0.1

1

10

100

homogenous

k=0.6

k=0.9

k=0.99

k=0.999

k=0.6

k=0.9

k=0.99

k=0.999

tD/CD

PD

an

d P

D'

Fig 4.1:Semi-log plot of Double permeability curves with and without crossflow for

w=0.001

1 10 100 1000 10000 1000000.1

1

10

k=0.6

k=0.9

k=0.99

k=0.999

tD/CD

PD

Fig 4.2:Semi-log plot of Double permeability curves with and without crossflow for w=0.1

0 0 0.01 1 100 10000 1000000 1000000000

0

0.01

0.1

1

10

100

k=0.6k=0.6k=0.9k=0.9k=0.99k=0.99k=0.999k=0.999homogenous

tD/CD

PD

an

d P

D'

Fig 4.3: Log-log plot of pressure and pressure derivative for w = 0.001

0 0 0.01 1 100 10000 1000000 1000000000

0

0.01

0.1

1

10

100

k=0.6k=0.6k=0.9k=0.9k=0.99

tD/CD

PD

an

d P

D'

Fig 4.4: Log-log plot of pressure and pressure derivative for w = 0.1

0 0 0.01 1 100 10000 1000000 1000000000

0

0.01

0.1

1

10

100

w=0

w=0

w=0.001

w=0.001

w=0.01

w=0.01

w=0.1

w=0.1

tD/CD

PD a

nd P

D'

Fig 4.5: Log-log plot showing the effect of w for k = 0.99

0 0.1 10 1000 100000 100000000.01

0.1

1

10

100

l=4E-3l=4E-4l=4E-5l=4E-6l=4E-3l=4E-4l=4E-5l=4E-6

tD/CD

PD'

Fig 4.6: Log-log plot showing the effect of Lamda

On figure 4.1 and 4.2, the light dashed curves represent the early and late time behavior

for a system without crossflow ( λ=0¿ , and the thick dashed curves represent the total

homogenous response. The solid lines represent the global double permeability curves for a

system with crossflow.

It is seen that three different regimes occurs in a double permeability response with

crossflow:

at early time, before the crossflow is established, the response is the same as that

without crossflow at late time, the system response is same as the homogenous response which

characterizes the total producing system. at intermediate time, a transition behavior is established.

Each parameters has significant effect on the shape of the pressure response. The kappa (

κ ) affects the shape of the transition. The ‘S’ shape in the pressure curve is more

pronounced when kappa value is close to 1, therefore producing a very characteristic

heterogeneous response. For low κ (0.6), the ‘S’ shape is less pronounced. On the

derivative response, the depth of the transition increases with an increase in the value of κ .

Figure 4.1 to 4.4

Omega ( ω ) affects the nature of the global response and the length of the transition. For

small ω , the early time curve is very different from the late time curve. The transition is long

and therefore, deep in derivative curves. For large values of ω , the global response tends to

the homogenous response curve. Figure 4.5 shows the curve for various ω .

For a given value of λ , all the curves merge at the end of the transition. It defines the

pressure at the start of the total system behavior. Figure 4.6 shows that the smaller the value of

lamda, the longer and deeper the transition and the longer time it takes the total system

behavior to start. In conclusion, the heterogeneous characteristics of the responses are much

more pronounced for large permeability-thickness ratios κ , small storativity ratios ω

(high diffusivity contrast) and small values of λ .

4.2 Semipermeability wall model behavior

The following figures below shows the behaviour of the pressure and pressure derivative

curve for a wellbore storage of 100, skin values of 0, 2, 5 and w1 values of 0.1, 0.3, 0.5, 0.7,

0.9. The type curves are based on a K v value of 10md and shale thickness of 5ft

100 1000 10000 100000 1000000 100000000

5

10

15

20

25

30

35

40

Skin = 2, CD = 100

w1=0.3w1=0.3w1=0.5w1=0.5w1=0.7w1=0.7

Dimensionless time, tD

Dim

esio

nles

s Pre

ssur

e, P

D

F

ig 4.7: Semilog plot of Dimensionless Pressure versus Dimensionless Time showing the

crosspoint between the short and long flow period

Figure 4.7 shows the crosspoint between the pressure curve for the short time and long time

flow. The crosspoint shows the point where the second layer becomes involved in the flow. This

point varies for different values of w1 . It is worthy to note that the slope of the second infinite

acting flow period is the same for all values of w1 , but the slope of the initial infinite acting

flow line is dependent on the value of w1 .

As stated in the previous chapter, w1 represents the permeability-thickness ratio of layer

one to the total permeability-thickness ratio. w1 of 0.1 means that the transmissibility of layer

1 is lower than layer 2. It also depicts that the diffusivity of layer 1 is lower than the diffusivity of

layer two.

0.01 0.1 1 10 100 1000 10000 1000000.01

0.1

1

10

100

S = 0, CD = 100

w1=0.1w1=0.1w1=0.3w1=0.3w1=0.5w1=0.5w1=0.7w1=0.7w1=0.9w1=0.9

tD/CD

PD a

nd (t

D/CD

)PD'

Figure 4.8: Log-log plot of Dimensionless Pressure and pressure Derivative versus

Dimensionless Time

The pressure derivative curve for w1 = 0.1, do not have the initial infinite-acting radial

flow period. The figure shows that the wellbore storage effect is still in effect before the transition

period occurred. Hence, we see only the second initial flow line that represents the total system.

This phenomenon can be explained as follows: when the semipermeability is high and/or the

permeability-thickness of the perforated layer is low, the fluids flows into layer 1 (the perforated

layer) even before the wellbore storage effect is over. The system thus behaves as a single

layer reservoir at this point.

For w1 = 0.5, figure 4.8 shows that the system stabilizes at the initial infinite flow period at

1/w1 and the transition to the second infinite flow period began at tD/CD of about 1300.

It can be seen that the point of transition increases as w1 increases from 0.1 to 0.5 and

starts to decrease as w1 increases.

For w1 = 0.9, diffusivity and permeability-thickness of the perforated layer is much higher

than the second layer, the transition between the initial infinite flow period and the second

infinite flow period is very small because the pressure in layer 1 must have reduced drastically

before the crossflow from layer 2 is initiated. Because of the of low diffusivity layer 2, the flow of

fluid to layer 1 is slow and gradual hence the small change in the pressure derivative.

The figures 4.9 to 4.10 shows the pressure and pressure derivative curves when skin is 2

and 5 respectively.

0.01 0.1 1 10 100 1000 10000 1000000.01

0.1

1

10

100

S = 2, CD = 100

w1=0.1w1=0.1w1=0.3w1=0.3w1=0.5w1=0.5w1=0.7w1=0.7w1=0.9w1=0.9

tD/CD

PD a

nd (t

D/C

D)P

D'


Dimensionless Time

The higher the value of the skin, the greater the higher the higher the values of the

dimensionless pressure. The skin, which indicates the degree of damage of the wellbore at the

perforated region, reduces the pressure at which the fluid enters into the wellbore. The greater

the skin value, the lower the pressure; hence, a higher dimensionless pressure.

0.01 0.1 1 10 100 1000 10000 1000000.01

0.1

1

10

100

1000

S = 5, CD = 100

w1=0.1w1=0.1w1=0.3w1=0.3w1=0.5w1=0.5w1=0.7w1=0.7w1=0.9w1=0.9

tD/CD

PD a

nd (t

D/C

D)P

D'


Dimensionless Time.

1 10 100 1000 10000 100000 1000000 100000001000000000.1

1

10

100

PD1PDpD2

tD

PwD


Dimensionless Time.

Figure 4.11 shows the pressure response for layer one and layer two which are

described by equation 3.8. PD represents the total pressure of the system as shown by

equation

4.3 Analysis of the Infinite-acting flow periods

If we produce only one layer, e.g. layer 1, the pressure and pressure derivative behaviour,

shows that three flow periods exists during the test. They are:

i. An initial radial flow period. Only the zone tested is involved in the flow. A stabilization of

the derivative at 0.5* (k1h1+k2h2 )/k 1h1 corresponds to it. The analysis of this flow is

used to determine the kh wellbore storage and the skin of the tested zone.ii. A transition period; when the derivative decreases. The decrease shows that more

reservoir thickness is becoming involved in the flow. The time it begins depends on the

vertical communication (semipermeability) between the two layers. The better the flow,

that is, the higher ~k i is, the earlier the transition.

iii. A second radial flow period. Both layers are involved in the flow and thus behaves as a

single layer homogeneous reservoir. The stabilization of the derivative at 0.5

corresponds to it. The analysis of this flow is used to determine (kh )t . It is difficult to

analyze the skin. It is a function of the skin of each layer, the transmissibility contrast

between the layers and the communication between the two layers.

It can be seen from the figures above, that the initial infinite-acting radial flow period forms a

straight line with a value corresponding to 0.5* (k1h1+k2h2 )/k 1h1 or 0.5*1w1

which is used

to calculate k1h1 . The second infinite-acting radial flow period forms another straight line with

a value corresponding to 0.5. This observation is more obvious from Fig. 4.12.

10100

100010000

100000

1000000

10000000

100000000

1000000000

Skin = 0, CD = 0

w1 = 0.8w1 = 0.6w1 = 0.5w1 = 0.4w1 = 0.2

Dimensionless time, tD

Dim

ensi

onle

ss p

ress

ure

Der

ivati

ve, P

D'

Figure 4.12: Semilog plot of Dimensionless pressure derivative versus dimensionless

time without wellbore storage skin effect.

4.4 TDS technique for a single layer reservoir system

In 1993, Tiab introduced a technique for interpreting pressure tests using log-log plots of

the pressure and pressure derivative versus time to calculate reservoir and well parameters

without type curve matching. This technique known as Tiab’s Direct Synthesis (TDS) consists of

obtaining characteristic points of intersection of various straight-line portions of the pressure and

pressure derivative curve, slopes and starting times of these straight lines. These points, slopes

and time are used with appropriate equations to solve directly for permeability, wellbore storage

and skin for a single-layered homogeneous reservoir.

The dimensionless pressure PwD , dimensionless time, tD , and dimensionless

wellbore storage coefficient are expressed as follows:

141.2D

khp p

qB� �

� �� ------4.1

2

0.0002637D

t w

kt t

c r� �

� �� -------4.2

2

0.8935D

t w

c cc hr

� �� -----4.3

The figure below shows the pressure and pressure derivative curve of a well producing

from a single layer homogeneous reservoir.

Figure 4.13: Log-log plot of dimensionless pressure and pressure derivative versus

dimensionless time of a single layer reservoirTiab developed an expression for calculating the permeability, skin and wellbore storage

of a single layer homogeneous system by analyzing the characteristics points observed in the

pressure and pressure derivative plot.

70.6

r

q Bk

h t P

�

------4.4

24

qB tC

t P� �� ----------4.5

20.5 7.43r r

t wr

p kts In

t p c r� �� -------4.6

Where Eq. 4.4, 4.5 and 4.6 are the equations for permeability, wellbore storage and skin

respectively. The subscript r stands for radial flow line.

4.5 TDS technique for a two-layer reservoir system

The dimensionless pressure PwD , dimensionless time, tD and dimensionless wellbore

storage coefficient are expressed as follows:

141.2i

tD o i

khp p p

qB

-------4.7

2

0.0002637t

Dt wt

kh tt

c h r

-------4.8

2

0.8935D

t wt

c cc h r

� �� ------4.9

Where 1 2t t ttc h c h c h

and 1 2tkh kh kh

It can be observed from figure 4.7 to 4.10 that the dimensionless pressure curve for a

well producing from a two-layered homogenous reservoir has a unit slope line during the early

time of the initial flow period for all w1 . These lines corresponds to pure wellbore storage flow.

The equation of this straight line is

pD=tDcD --------4.10

Combining Eq. 4.8 and 4.9 gives

42.95 10 tD

D

kht t

C C� �

�� -----4.11

The derivative of Eq. 4.10 with respect to (tD /CD ) is given as

D DD

D D

t tp

c c

� ��

� � ----4.12

Where the dimensionless pressure derivative is given as

226.856 w t tD

r c hp p

qB

� ��

� � -----4.13

Combining Eq. 4.11 and 4.13 gives

141.2

tDD

D

khtp t p

C q B� ��

� � � �� -----4.14

4.5.1 Wellbore Storage

The wellbore storage effect of a two-layer reservoir that is producing independently can be

derived as follows:

By combining Eq. 4.11, Eq. 4.12 and Eq. 4.14, we have

4

12.95 10

141.2t t

r

kh khtt p

C q B � � � �

� �� ------- 4.15

Rearranging Eq. 4.15 to solve for C gives

124r

qB tC

t p� �� ---------- 4.16

Where t is any point on the wellbore storage line and (t∗∆ p’) is the corresponding

pressure derivative value.

4.5.2 Permeability-Thickness

4.5.2.1 Initial Radial Flow Period

As stated earlier, for a two-layered reservoir with crossflow that is perforated and

produced in the top zone only, the initial infinite acting radial flow period observed is a straight

line. It can be clearly seen from Fig 4.12, that the equation of the straight line, during the initial

infinite acting radial flow period for various values of w1 , can be gotten as follows:

From Eq. 4.12,

DD

D

tp

c

� ��

� � has a constant value that corresponds to 1

10.5

w

� �� for every

value of

D

D

t

c within the initial infinite acting radial period. Therefore,

11

10.5D

DD r

tp

C w

� ��

� � � �� ------- 4.17

Note that the subscript r1 and r2 stands for infinite acting radial flow region in the short

and long flow period respectively.

0.01 0.1 1 10 100 1000 10000 1000000.01

0.1

1

10

100

S = 5, CD = 100

w1=0.1w1=0.3w1=0.5w1=0.7w1=0.9

tD/CD

PD a

nd (t

D/C

D)P

D'

Figure 4.14: Log-log plot of Dimensionless pressure derivative versus dimensionless

time

At this period, only the zone tested (the top zone in this case) is involved in the flow. The

permeability-thickness in this period will be that of the top zone (layer 1) only.

Substituting Eq 4.17 into 4.14

1

1 1

0.5

141.2t t

r

kh kht p

k h q B� �

� � �� ------- 4.18

By rearranging Eq. 4.18, k1h1 becomes

1

1 1

70.6

r

q Bk h

t p

�

-------------4.19

4.5.2.2 Second Radial Flow Period

In this period, both layers are involved and a second infinite acting radial flow period is

observed. The equation of this line corresponds to that of a single layer homogeneous reservoir.

From Fig. 4.8 it is obvious that the equation of the straight line, during the second infinite acting

radial flow period for all values of w1 , can be gotten as derived by Tiab (Tiab, 1993):

From Eq. 4.12,

DD

D

tp

c

� ��

� � has a constant value that corresponds to 0.5 for every value of

D

D

t

c within the second infinite acting radial period. Therefore,

2

0.5DD

D r

tp

C

� ��

� �� ------- 4.20

Substituting Eq. 4.20 into 4.14

2

0.5141.2

tr

kht p

q B� �

� � �� ------- 4.21

By rearranging Eq. 4.18, k1h1 becomes

2

70.6t

r

q Bkh

t p

�

--------4.22

4.5.3 Skin Effect

The skin effect of the wellbore in the perforated layer can be derived as in the case of a single

layer reservoir. Tiab in 1933 also derived an expression for this. The derivation is as follows

For test data on the semilog straight line for the initial flow period, the dimensionless

pressure can be modelled with the logarithmic approximation to the line source equation.

1

10.5 0.80907 2rD Dp Int s

--------4.23

1 10 100 1000 10000 100000 1000000 100000000.01

0.1

1

10

100

1000

Skin = 5, CD = 100

w1=0.1w1=0.3w1=0.5w1=0.7w1=0.9

tD/CD

PD

Figure 4.15: Semilog plot of Dimensionless pressure curve versus Dimensionless time

showing the initial infinite flow period

By adding and subtracting ¿CD inside the brackets in 4.23 yields

11

20.5 0.80907r

sDD D

D

tp In In C e

C

� �� -----------4.24

From Fig. 4.15, we observe that the reservoir system can be treated as a single layer

homogeneous reservoir during the initial flow period because only one layer (layer 1 in this

case) is active. In this first straight-line period, the well test theory of a single layered reservoir is

effective for a multilayered system perforated in one layer. Hence, we can derive the equation

for the skin effect by following the conventional procedure used for a single layer homogeneous

reservoir as derived by Tiab (Tiab, 1993).

For a single layer homogeneous reservoir, the infinite-acting radial flow portion of the

pressure derivative is a horizontal straight line. The equation of this line is:

0.5DD

D r

tp

C

� ��

� �� ------------4.25

In this case, r represent r1 ; the initial radial flow period.

An expression relating the initial infinite-acting radial flow line portions of the pressure

and pressure derivative curves can be derived by dividing Eq. 4.24 with Eq. 4.25 which gives:

1

1

10.80907 2rD

D

DD

D r

pIn t s

tp

C

� ��

��

-----------4.26

Substituting for the dimensionless term and solving for skin gives

1 1

1

1 20.5 7.43r rt

t wr t

p kh ts In

t p c h r

� �� ---------4.27

Where t r1 is any convenient time during the initial infinite acting radial flow line and

∆ pr1 is the corresponding value.

4.6 Validation of Result

4.6.1 Example One

The table below shows the reservoir properties of a two-layer reservoir that is produced

from the top zone only.

Table 1

Property

Pay thickness, ft

Porosity, %

Permeability, md

Total compressibility, psi-1

Wellbore storage, bbl/psi

Viscosity, u, cp

Skin

Layer 1

100

0.35

300

0.00006

0.0693

0.5

5

Layer 2

100

0.2

300

0.00006

-

0.5

-Flow rate, q = 800 bbl/stb; Shale permeability, Kv = 10 md; shale thickness, h= 5 ft; B = 1

The plot of pressure and pressure derivative versus time from the pressure data given in the

appendix is shown below.

0 0 0 0.01 0.1 1 10 100 10001

10

100

time, hrs

∆P a

nd (t

*∆P

') ps

i

Figure 4.16: Plot for Example one – Pressure and pressure derivative versus time

1. Wellbore storage, C bbl/psiThe equation for the wellbore storage is given in equation 4.16 as

24

qB tC

t p� �� -------- 4.16

Where t = 0.001791 hrs , (t∗∆ p’) = 0.803576 psi are values gotten from line

ab (wellbore storage line).

By substituting the values into equation 4.16, C = 0.0713 bbl/psi

2. Permeability- thickness of layer 1yThe equation is given in equation 4.19 as

1

1 1

70.6

r

q Bk h

t p

�

------------ 4.19

From line cd, (t∗∆ p’)r1 = 0.978215

By substitution, k1h1 = 28869 md-ft

3. Permeability- thickness of layer 2The equation is given in equation 4.22 as

2

70.6t

r

q Bkh

t p

�

------------ 4.22

From line ef (t∗∆ p’)r2 = 0.4707 psi

By substitution, (kh )t = 59995.75 md-ft

Therefore, k2h2 = 31126.75 md-ft4. Skin, S1

The equation for the skin is given in equation 4.27 as

1 1

1

1 20.5 7.43r rt

t wr t

p kh ts In

t p c h r

� �� ------4.27

From line cd, t r1 = 2.23779 hrs and the corresponding ∆ pr1 = 20.87529

psi

By substitution, S1 = 4.44.

The TDS technique for yields values that are similar to the exact values for k1h1 , k2h2 ,

S1 and C

4.6.2 Example Two

The table below shows the reservoir properties of a two-layer reservoir that is produced

from the top zone only.

Table 2

Property

Pay thickness, ft

Porosity, %

Permeability, md

Total compressibility, psi-1

Wellbore storage, bbl/psi

Viscosity, u, cp

Skin

Layer 1

100

0.35

1000

0.00006

0.027

0.5

2

Layer 2

200

0.15

600

0.00006

-

0.5

-Flow rate, q = 500 bbl/stb; Shale permeability, Kv = 10 md; shale thickness, h= 5 ft; B = 1

The plot of pressure and pressure derivative versus time from the pressure data given in the

appendix is shown below.

0 0 0 0.01 0.1 1 10 1001

10

time, hrs

∆P a

nd

(t*

∆P

') p

si

Figure 4.17: Plot for Example Two – Pressure and pressure derivative versus time

1. Wellbore storage, C bbl/psiThe equation for the wellbore storage is given in equation 4.16 as

24

qB tC

t p� �� -------- 4.16

Where t = 0.0002718 hrs , (t∗∆ p’) = 0.81358 psi are values gotten from line

ab (wellbore storage line).

By substituting the values into equation 4.16, C = 0.031 bbl/psi


1

1 1

70.6

r

q Bk h

t p

�

------------ 4.19

From line cd, (t∗∆ p’)r1 = 0.183368 psi

By substitution, k1h1 = 96036.99 md-ft


2

70.6t

r

q Bkh

t p

�

------------ 4.22

From line ef (t∗∆ p’)r2 = 0.080227 psi

By substitution, (kh )t = 220000 md-ft

Therefore, k2h2 = 123963.01 md-ft4. Skin, S1

The equation for the skin is given in equation 4.27 as

1 1

1

1 20.5 7.43r rt

t wr t

p kh ts In

t p c h r

� �� ------4.27

From line cd, t r1 = 0.18907 hrs and the corresponding ∆ pr1 = 2.8252 psi

By substitution, S1 = 1.59.

The TDS technique for yields values that are similar to the exact values for k1h1 , k2h2 ,

S1 and C

CHAPTER FIVE

5.0 Conclusion and Recommendation

5.1 Conclusion

In this study, we showed that the Tiab’s Direct Synthesis technique can be used to

determine the individual layer properties such as permeability, skin and wellbore storage of

producing layer. Compared to other methods that uses both the wellbore pressure and sandface

flowrate of each of the layers to determine the layer parameters, this method uses only the

wellbore pressure to obtain the individual layer parameters for a two-layer reservoir system with

crossflow system.

When both layers of a crossflow system are tested together, the pressure response

behaves as though there is no communication between the layers in the early time region and

as though the reservoir is a single layer in the late time region.

Two straight infinite acting straight lines are observed when only one layer is perforated

and tested for a crossflow system. The initial infinite acting straight line is used to obtain the

parameters of the perforated layer and the second infinite acting straight line is used to obtain

the total reservoir parameters.

If the permeability thickness of the perforated layer is very much smaller than the second

layer, and the semipermeability is high, the first infinite acting straight line will not appear.

We saw from the two examples in chapter four that the reservoir parameters gotten with

this technique are close to the actual reservoir parameters. This proves that this technique can

be applied to any two-layer reservoir system with crossflow that is partially perforated by a

vertical well.

5.2 Recommendations

The validation of the results obtained in this work was done with simulated data. This

technique should be tested with actual field data before application. Further studies can be done

for a system with more than two layers using the TDS technique.

REFERENCES

1. Ahmed Aly, et al. "A New Technique for Analysis of wellbore Pressure From Multi-

layered Reservoirs With Unequal Initial Pressures To determine Individual Layer".

Society of Petroleum Engineers Journal, (1994).2. Barenblatt, G.I, Zeltov, Ju.P. and Kocina, I.N,: "Basic Concepts in the theory of Seepage

of Homogenous Liwuids in Fissured Rocks (Strata)", Soviet J. App. Math. and Mech.,

1960.3. Bourdet, D. "Pressure Behavior of Layered Reservoirs With Crossflow". Society of

Petroleum Engineers Journal, (1985).4. Coats, K.H., Dempsey, J.R., and Henderson, J.H.: "The use of vertical Equilibrium in

two-Dimensional simulation of Three-Dimensional Reservoir Performance," Society of

Petroleum Engneers journal (March 1971) 63-71; Trans., AIME, 2515. Ehlig-Economides, C. A., & Joseph, J. "A New Test for Determination of Individual Layer

Properties in a Multilayered Reservoir". Society of Petroleum Engineers Journal, (1985).6. Gao and Dean, "Single-Phase Flow in a Two-Layer Reservoir with Significant Crossflow

- Pressure Drawdown & Buildup Behaviour When Both Layers are Completed at a

Single Well'. Society of Petroleum Engineers Journal, (1984).7. Gao, Chengtai, "The crossflow Behaviour and the Determination of Reservoir

Parameters by Drawdown Tests in Multilayer Reservoirs". Society of Petroleum

Engineers journal, (Sept. 1983)8. Gao Chengtai, "Determination of Individual Layer Properties by Layer-by-Layer Well

Tests in Multilayer Reservoirs with Crossflow". Society of Petroleum Engineers Journal,

(1985).9. Gao Chengtai, "Crossflow Behaviour in a Partially Perforated Two Layer Reservoir; the

Evaluation of Reservoir Parameters by Transient Well Tests", Society of Petroleum

Engineers Journal (1987)10. Gao Chengtai. "Determination of Parameters for Individual Layers in Multilayer

Reservoirs by Transient Well Tests". Society of Petroleum Engineers Journal, March

(1987).11. Gilles Bourdarot "Well testing: interpretation methods" Centre for petroleum engineering

and project development, TECHNIP edition, (1998).12. Horne, P. and R. N. "Well Test Analysis of a Multilayered Reservoir With Formation

Crossflow". Society of Petroleum Engineers Journal, (1989).13. Jing Lu, Botao Zhuo, and M. M. Rahman "New Solution to Pressure Transient Equation

in a Two-Layer Reservoir With Crossflow" Society of Petroleun Engineers (Sept 2015).14. Kucuk, F. "Well Testing and Analysis Techniques for Layered Reservoirs". Society of

Petroleum Engineers Journal (1986).15. Larsen, "Determination or Skin Factors and Flow Capacities of Individual Layers in Two-

Layered Reservoirs". Society of Petroleum Engineers Journal, (1982)16. Lefkovits et al. "A study of the behaviour of bounded reservoirs composed of stratified

layers”. Society of Petroleum Engineers Journal, (1961)17. N. M. Al-Ajmi, H. Kazemi and E. Ozkan "Estimation of Storativity Ration in a layered

Reservoir with crossflow" Spciety of Petroleum Engineers Journal (October, 2003).18. Otuomagie, R. "Analysis of Pressure Buildup in an Infinite Two-layered Oil Reservoir

Without Crossflow". Society of Petroleum Engineers Journal, (1976).19. Pascal, B. "Analysis of Pressure and Rate Transient Data From Wells in Multilayered

Reservoirs: Theory and Application". SPE Annual Technical Conference and Exhibition,

(1992).20. Prijambodo et al. "Well Test Analysis for Wells Producing Layered Reservoirs With

Crossflow". Society of Petroleum Engineers Journal, 25(3), (1985).21. Ramey and Tamey, "Drawdown Behaviour of a well with storage and Skin effect

communicating with layers of Different radii and other characteristics". Society of

Petroleum Engineers Journal, (1978).22. Russel, D. and Prats, M. "Performance of layered reservoirs with crossflow, the single-

compressible-fluid case". Society of Petroleum Engineers Journal, (1962).23. Stehfest, H.: "Algorithm 368, Numerical Inversion of Laplace Transforms," D-5

Communications of the ACM, vol. 13, No.1, Jan. (1970)24. Tiab, D. "Analysis of Pressure and Pressure Derivatives Without Type-Curve Matching:

I-Skin and Wellbore Storage". Society of Petroleum Engineers Journal, (1993).25. Warren, J.E. and Root, P.J.: “Behavior of Naturally Fractured Reservoirs”, Society of

Petroleum Engineering Journal (Sept. 1963)

determining individual characteristics of a two- …

Documents