Diagnosis and Analysis of Pressure Transient in Fractured ...naturally fractured reservoir and engineering evaluations rely on direct and indirect sources of information. Direct sources

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University of Missan

Engineering College

Petroleum Department

Diagnosis and Analysis of Pressure Transient in

Fractured Reservoir

A project submitted to the college of engineering of the university of

missan in partial fulfilment of requirement for the degree of B.Sc. Of

science in petroleum engineering

BY

Ahmed Taqi Khadim

Azhar Majid

Yasser Hameed

SUPERVISOR

Dr. Ahmed Al-shara

May

2017 1438

بسم اهلل الرحمن الرحيم

ذين وتواوليعلم ال

علم أ

نه ال ربك من الحق أ

خبت به فيؤمنوا لهاد الله وإن قلوبهم له فت

ذين مستقيم صراط إلى آمنواال

العظيمالعلي صدق اهلل

i

DEDICATION

We dedicate this research:

To the fountains of patience and optimism and hope

our dear mothers

To the big hearts Our dear fathers

To those who illuminated our way and supported us

our brothers

To the people who paved our way of science and knowledge

All our teachers

To who sacrifice his soul for our peace

Our Holy crowd

To those who tasted the most beautiful moments with them

our friends

ii

ACKNOWLEDGEMENTS

First, we thank God who has blessed us with the grace of mind. We would

like to express sincere gratitude and appreciation to our supervisor, Dr.

Ahmed Al-shara who has cheerfully answered our queries, provided us

with materials, checked our examples, assisted us in a myriad way with the

writing and helpfully commented on earlier drafts of this project. We

would like to express our appreciation to A.L. Ali Nooruldeen and A.L.

Dhiaa Salman for their assistance and encouragement.

We would also like to thank the faculty and staff at the Department of

Petroleum Engineering for their support during our studies at missan

university.

iii

ABSTRACT

Well pressure transient analysis is widely used in reservoir management to

obtain reservoir information needed for reservoir simulation, damage

identification, well optimization and stimulation evaluation.

In this project, well test interpretation method based on the analysis of the

time with rate of change of pressure in fracture reservoir by two type of

well test draw down and build up.

Interpretation models used in well test analysis for fractured reservoir

identification are based on the fluid flow behavior of reservoir.

In well test interpretation used derivative of pressure versus time because

the derivative is directly represented in one term of the diffusivity

equation, which is the governing equation for all the models of transient

pressure behavior currently in use in well test analysis and the derivative

response is more sensitive to small phenomena.

This work is presented the interpretation of pressure transient tests for dual

porosity model. This research is based on the transient matrix flow model

and the pseudo state matrix flow model. The differences between these

models occur during the transition flow period.

The data in this project were handled theoretically and programmatically

using the Ecrin program is issued by Kappa. It is the industry standard for

the analysis of dynamic data. Ecrin includes modules for Pressure

Transient Analysis (Saphir NL) Production Analysis (Topaze NL) full field

numerical History Matching (Rubis) and Well Performance Analysis

(Amethyste).

iv

LIST OF FIGURES

CHAPTER ONE

Figure (1- 1) Cross plot showing the relative positions of fractured

reservoir ...................................................................................................... 4

CHAPTER TWO

Figure (2- 1)Warren and Root (1963) dual-porosity idealization; cubed

dual-porosity model representation. ........................................................... 9

Figure(2- 2) Kazemi (1969) dual-porosity idealization; layered dual-

porosity model representation. .................................................................. 10

Figure (2- 3) Derivative Type Curve for Double-Porosity Reservoir,

Pseudo-Steady State Flow ........................................................................ 11

Figure (2- 4) fissure system flow .............................................................. 13

Figure (2- 5)Infinite-Acting Radial Flow (IARF) .................................... 14

Figure (2- 6) matrix contribution .............................................................. 15

Figure (2- 7) drawdown and build up test sequence ................................. 18

CHAPTER THREE

Figure (3- 1) Actual and idealized dual porosity reservoir model............ 19

Figure (3- 2) Slab model for naturally fracture reservoir ......................... 21

Figure (3- 4) Build up semi log plot for dual porosity system ................. 23

Figure (3- 5) Extrapolation of second semi log straight line to p* ........... 26

Figure (3- 6) type curve for pseudo steady – state matrix flow ................ 27

Figure (3- 7) derivative type curves for pseudo steady state matrix flow 28

Figure (3-8) characteristic flow regimes in a dual porosity system with

transient matrix flow ................................................................................. 31

Figure (3- 9) estimeting ώ and λ’ with data from flow regimes 1 and 2 .. 36

Figure (3- 10) type curves for transient matrix flow ................................ 38

Figure (3- 11) Example of using type curves for transient matrix flow ... 39

Figure (3- 12) Dual Porosity Transient Interporosity Flow ...................... 40

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CHAPTER FOUR

Figure (4- 1) semi log plot of pwf vs t for problem (4.1) ......................... 44

Figure (4- 2) log-log plot of pwf vs t for problem (4.1) ............................. 46

Figure (4- 3) type curve match for problem (4.1) ..................................... 48

Figure (4- 4) solution problem (4.1) by Ecrin. ......................................... 49

Figure (4- 5) semi log plot of pwf vs t for problem (4.2) .......................... 52

Figure (4- 6) log-log plot of Δp & Δp` vs t for problem (4.2) .................. 54

Figure (4- 7) type curve match for problem (4.2) ..................................... 55

Figure (4- 8) solution problem (4.2) by Ecrin .......................................... 56

vi

LIST OF TABLES

Table (4-1) ....................................................................................... 43

Table (4-2) ....................................................................................... 48

Table (4-3) ....................................................................................... 51

vii

NOMENCLATURE

Symbol Description

λ …………………...……………………… interporosity flow coefficient

α…… parameter characteristic of system geometry in dual-porosity system

ω …….…………...……………. storativity ratio in dual porosity reservoir

ct ……………………...….…………………… total compressibility, psi–1

C……………………...……………. wellbore storage coefficient, bbl/psi

CD …………………...….…… dimensionless wellbore storage coefficient

γ ………………………………...Euler’s constant, = 1.781, dimensionless

(ϕVct)f ………………....... fracture "storativity" for dual porosity reservoir

(ϕVct)f+m ………………..…... total "storativity" for dual porosity reservoir

h……………………………………..……………. formation thickness, ft

hm ………………………………….………………. thickness of matrix, ft

Km …………………………….….………………matrix permeability, md

𝐾− ………………………………..…………......average permeability, md

Kf ……………………………….………....…. fracture permeability, md

Kh ……………………………….…………... horizontal permeability, md

Kv ……………………………….…………….... vertical permeability, md

L………………...…………… distance from well to no-flow boundary, ft

q……………………………………………... flow rate at surface, STB/D

rw ………………………………………………………. wellbore radius, ft

t ……………………………………………………… elapsed time, hours

tp …………………………………………………… producing time, hours

NFRs……………………………………………..natural fracture reservoir

viii

Symbols Description

Pi………...………………………………… initial reservoir pressure, psia

Δp1hr ….... pressure change from start of test to one hour elapsed time, psi

Δt…………………………………… time elapsed since start of test, hours

Δt* …………………… time at point of intersection of two semi log lines

Θ…………………………………………………….…. intersecting angle

μ …………………………………...…………………………viscosity, cp

γ ………………………………...Euler’s constant, = 1.781, dimensionless

(ϕVct)f ………………..... fracture "storativity" for dual porosity reservoir

(ϕVct)f+m …………………... total "storativity" for dual porosity reservoir

M…………………………………………………………… Mobility ratio

D…………………………………………………………. Storativity ratio

ix

TABLE OF CONTENTS

DEDICATION ............................................................................................ i

ACKNOWLEDGEMENTS ....................................................................... ii

ABSTRACT .............................................................................................. iii

LIST OF FIGURES .................................................................................. iv

LIST OF TABLES .................................................................................... vi

NOMENCLATURE ................................................................................ vii

TABLE OF CONTENTS .......................................................................... ix

CHAPTER ONE

INTRODUCTION ...................................................................................... 1

1.1 Background of the study .................................................................... 1

1.1.1 Compressibility .......................................................................... 3

1.1.2 Naturally fractured reservoir classification system ..................... 3

1.1.3 Identification of Natural Fractures in tools: ................................ 5

1.2 Objectives of the study .................................................................... 5

1.3 Problem Description: ......................................................................... 6

1.4 Scope of the Work ............................................................................. 7

CHAPTER TWO

LITERATUE REVIEW .............................................................................. 8

2.1 General Review ................................................................................. 8

2.2 Dual Porosity Response: .................................................................. 13

2.2.1 Infinite-Acting Radial Flow (IARF): ........................................ 14

2.2.2 Basic assumptions of the double porosity model: ..................... 15

2.2.3 Behavior of the double porosity model: .................................... 16

2.3 Well testing: ..................................................................................... 16

2.3.1 Description of a well test ........................................................... 16

2.3.2 objectives of well testing: .......................................................... 16

2.3.3 Information obtained from well testing ..................................... 16

x

2.3.4 Types of Tests ............................................................................ 17

CHAPTER THREE

MATHEMATICAL MODEL OF NATURALLY FRACTURE

RESERVOIR ............................................................................................ 19

3.1 naturally fractured reservoir models: ............................................... 19

3.2 Pseudo steady-state matrix flow model: .......................................... 22

3.2.1Semilog analysis technique ........................................................ 22

3.2.2Type curve analysis technique ................................................... 26

3.3 Transient matrix flow model ........................................................... 30

3.3.1Semilog analysis techniques ....................................................... 32

3.3.2 Type-curve analysis technique .................................................. 38

CHAPTER FOUR

RESULT AND DISCUSSION ................................................................. 43

CHAPTER FIVE

CONCLUSIONS AND RECOMMENDATIONS ................................... 58

5.1 conclusions ...................................................................................... 58

5.2 recommendations ............................................................................. 59

REFERENCES ......................................................................................... 60

CHAPTER ONE INTRODUCTION

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CHAPTER ONE

INTRODUCTION

1.1 Background of the study

Naturally fractured reservoirs are heterogeneous porous media wherein the

openings (fissures and fractures) vary in size. Fractures and openings of

large size form vugs and interconnected channels, whereas the fine cracks

form block systems which are the main body of the reservoir. The porous

blocks store most of the fluid in the reservoir. Fracturing occurs when the

rock fails because the differential forces acting upon it exceeds its elastic

limit thereby causing a rupture. Fractures improve reservoir flow and

connectivity [1].

Natural fractures are more common in carbonate rocks and a large fraction

of the world’s natural hydrocarbon reserves can be found in carbonate

reservoirs that have a complex, heterogeneous porosity system.

Permeability within fractures is usually much higher than the permeability

within the matrix and primary flow within the reservoir may be through the

fracture system, the fractures are assumed continuous throughout the

formation and to represent the paths of principal permeability. The high

diffusivity of a fracture results in a rapid permeability. The high diffusivity

of a fracture results in a rapid response along the fracture to any pressure

change such as that caused by well production. Pressure transient tests can

be used to interpret the characteristics of flow within a reservoir [2]. These

tests are based on the solution of the diffusivity equation response to

pressure changes that occur in the surrounding fractures. Such no

concurrent responses cause pressure depletion of the fracture relative to the

matrix, which in turn induces matrix-to-fracture crossflow. This period of

transient crossflow takes place immediately after the fracture pressure


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response and before the matrix and the fracture pressures equilibrate, after

which the formation acts as a uniform medium with composite properties.

The effect of assumptions made on the nature of matrix and properties. The

effect of assumptions made on the nature of matrix and fracture interaction

is manifested during this transitional period of matrix-to-fracture fluid

transfer. The flux of fluid released by the matrix depends on the matrix

size, porosity, permeability, and the matrix/fracture pressure difference. At

the matrix/fracture interface, the matrix flux contribution to fracture flow

may be assumed proportional to either the pressure difference between

matrix and fracture or to the averaged pressure gradient throughout the

matrix block [3].

Naturally fractured reservoirs are elusive systems to characterize and

difficult to engineer and predict. It is important to establish some basic

criteria for recognizing when fractures are an important element in

reservoir performance and to recognize the nature and performance

characteristics of a naturally fractured reservoir. Characterization of the

naturally fractured reservoir and engineering evaluations rely on direct and

indirect sources of information. Direct sources of information include

cores, drill cuttings, down hole photographs and videos of the borehole.

Indirect sources include outcrops, drilling history, mud log, conventional

and specialized well logs, seismic information (preferentially 3D), well

testing, inflatable packers, and production history. For reservoirs where the

fractures have a positive or negative effect on fluid flow, it is of paramount

importance to have knowledge of magnitude and direction of in-situ

principal stresses; azimuth, dip, spacing, and aperture of fractures; matrix

and fracture porosity, matrix and fracture permeability, and matrix and

fracture water saturation. These data help in calculations of how the in-

place hydrocarbons are distributed between matrix and fractures, and the


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flow capacity of the wells. Naturally fractured reservoirs can be open,

permeable pathways, or they can be permeability baffles resulting from the

presences if secondary mineralization or other fine-grained material filling

the gaps. Most natural fractures are more or less vertical. Horizontal

fracture may exist for a short distance, most horizontal fractures, however,

are sealed by overburden pressure. Both horizontal and semi-vertical

fractures can be detected by various logging tools [2-3]. The best method to

analyze the Naturally fracture reservoir is well test because it is detecting

the parameter of Naturally fracture reservoir like (K , 𝜔. 𝜆 ) .

1.1.1 Compressibility

If the reservoir is composed by matrix and fractures, the compressibility of

the fractures is bigger than the compressibility of the matrix. The relative

difference between the two compressibility’s depends on various factors

including the amount of secondary mineralization within the fractures, the

orientation of the fractures and in-situ stresses, and if the reservoir is over-

pressured, normally pressured, or under-pressured [4].

1.1.2 Naturally fractured reservoir classification system

Most if not all reservoirs contain fractures. It is the degree to which

fractures influence fluid flow through a reservoir that should dictate the

level of resources needed to identify characterize and model fractures. The

effects of fractures can change throughout the productive life of the

reservoir as pressures and fluid types change during primary and secondary

stages moreover fractures don’t always conduct fluid they are often barriers

to flow. Fractures reservoirs are classified based on the interaction between

the relative porosity and permeability contribution from both the fracture

and matrix system [2]:

Type1: reservoir with fractures providing both primary porosity and

primary permeability typically have large drainage areas per well and


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required fewer wells for development these reservoir, show high initial

production rates.

Type2: reservoirs can have surprisingly good initial production rates for a

low permeability matrix but can have difficulties during secondary

recovery if the communication between the fracture and matrix is poor.

Type3: reservoirs are typically more continuous and have good sustained

production rates but can have complex directional permeability

relationships leading to difficulties during the secondary recovery phase.

Type M: reservoir have impressive matrix qualities but are sometimes

compartmentalized causing them to secondary recovery effectiveness

variable within the same field.

Type4: reservoir would plot near the origin because the fracture

contribution to permeability in type4 reservoir is negative3.

Figure (1- 1) Cross plot showing the relative positions of fractured

reservoir


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1.1.3 Identification of Natural Fractures in tools:

It is essential to identify fractures during exploration and drilling. Well logs

are useful tools in identifying these natural fractures. A program for

fracture detection based on logging techniques consists of using different

logs.

• Borehole televiewer logs can be used to identify induced and natural

fractures.

• Acoustic, receptivity, and temperature logs can be used to obtain a

realistic picture of fractures and their width and orientation in the vicinity

of wellbore.

• Variable-density logs can define fracture intervals. Both core analysis and

logging are valuable techniques for explored wells in detecting fracture

porosity Øf, permeability kf, and nature of the matrix or intergranular

porosity. However, there are many wells drilled where no core was taken

and the logs do not show any evidence of fractures. Therefore, well test

analysis is the only technique used for getting information about the

fractured nature of formation, and can provide information on fractured

reservoir parameters such as km, Øm, km, Øf, size and shape of the matrix

blocks, and the nature and orientation of the fracture pattern, in addition to

determining p or pi and skin factor (s).

1.2 Objectives of the study

The main objectives of this project are to analyze, interpret and categorize

the pressure transient responses observed in naturally fractured to

understand the heterogeneities of the porosity system. The specific

objectives are:

Estimate:

1. Find out if there was a fracture in the reservoir.

2. The parameter of Naturally fracture reservoir (K, ω, λ).


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3. Hydraulic connectivity of the reservoir.

4. Knowledge of boundaries that surrounds the well.

5. Formation damage (S).

6. Formation flow capacity (kh)

1.3 Problem Description:

Well pressure transient analysis is a widely used in reservoir management

to obtain reservoir information needed for damage identification, well

optimization and stimulation evaluation. Well test analysis helps to

understand the type of fluid flow within a reservoir as well as providing

information necessary for other reservoir management processes such as

numerical simulation. Some of the information obtained from pressure

transient analysis include effective permeability, average reservoir

pressure, distance to drainage area boundaries and skin. Well test data

obtained from multiple wells in naturally fractured field is analyzed using

commercial well testing software. The different well test profiles are then

analyzed using the pressure transient analysis models such as

homogeneous, dual porosity. Well test interpretations from the test field

are then categorized according to reservoir and boundary model

interpretations. Pressure transient interpretations can be non-unique due to

the complex porosity system found in naturally fractured reservoirs. It is

therefore possible to obtain a good match for the pressure responses with

various reservoir models. The multiple

interpretations of the well test data are made unique by comparing with

additional information obtained from the geological map, production data,

drilling logs and well logs.


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1.4 Scope of the Work

The main objective of this study was the identification of fracture through

the analysis and interpretation of data coming from well tests (buildup,

draw down). The research is focused on pressure transient analysis using

dual porosity model with two assumptions transient and pseudo steady

state matrix flow. In this work, two methods were used, semi-log and type

curve. In addition, these results were compared with the results of the

ECRIN program.

CHAPTER TWO LITERATURE REVIEW

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CHAPTER TWO

LITERATUE REVIEW

2.1 General Review

one of the rock heterogeneities that has deserved the attention of many

investigators is the presence of fractures crossing the producing formations.

Many of the important producing fields around the world are found in

fractured formations. In spite of this is during the last decade's most of the

reservoir engineering studies have been oriented towards homogeneous

systems. It appears that the first study on the performance of a fractured

reservoir was published in 1953[2].

In 1959, Pollard presented one of the first pressure transient models

available for interpretation of well test data from two-porosity systems.

The most complete analysis of transient flow in two-porosity systems was

presented in 1960 by Barrenblattet and Zheltov .

Cinco Ley [5,6] described well testing as an ideal tool to find reservoir-flow

parameters and to detect and evaluate heterogeneities that affect the flow

process in carbonate formations. He discussed five different reservoir flow

models that can be used to portray the behavior of naturally fractured

reservoirs. The models are homogeneous reservoir, composite reservoir,

anisotropic medium, single fracture system and double porosity medium.

He discussed the application and limitations of these models in well-test

analysis.

For naturally fractured reservoirs, two dual-porosity models, by Warren

and Root (1963) and Kazemi (1969), are widely used in the petroleum

industry. These models are the idealized versions of naturally fractured

reservoir systems that comprise matrices and fissures. Warren and Root

introduced the model in which the matrices and fractures are uniformly


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distributed and overlapping (Figure 2-1). The fluid transfer from the matrix

to the fracture network is in a pseudo steady state and follows Darcy’s Law.

There is no direct communication between matrix elements and between

matrix system and wellbore; flow is only between matrix and fractures and

between fractures and wellbore.

Figure (2- 1)Warren and Root (1963) dual-porosity idealization; cubed

dual-porosity model representation.

Kazemi[7](1969) introduced a different idealization for naturally fractured

reservoirs. This model consists of horizontal matrices and fractures

extending throughout the reservoir (Figure 2-2). It considers the flow

within the matrix and transient fluid transfer from the matrix to the fracture.

Flow in the matrix system is one dimensional (1-D) and in a normal

direction to the fracture system. This is essentially a layered reservoir

model in which matrix and fracture have distinct properties. Similar to the

Warren and Root (1963) model, communication only exists between

matrix and fractures and between fractures and wellbore.


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Figure(2- 2) Kazemi (1969) dual-porosity idealization; layered dual-

porosity model representation.

De-Swaan[8] (1976) developed an analytical double-porosity model

assuming transient interporosity flow in the matrix for different geometries

than those used by Kazemi[7]. He presented early and late time region

solutions for spherical and slabs idealized models but the solutions for the

transient period was not presented.

Najurieta[9] (1976)extended the solutions presented by De Swaan[8] in order

to properly describe the transitional period taking into account the transient

behavior in the matrix. He presented a simplified model for slabs and cubes

idealized models and he also proposed a systematic approach for analyzing

well tests in naturally fractured reservoirs.

Bourdet and Gringarten[10] (1980) proposed a new set of type curves for

analyzing wells with wellbore storage and skin effects in dual porosity

systems. They developed the type curves by rearranging the parameter

combinations in the solutions presented by Mavor and CincoL[5].

Gringarten [11] illustrated the application of these type curves to evaluate

matrix block size and fissure volume in fissured reservoirs from actual well

test data.


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Bourdet et al. [12,13] (1981) introduced the use of pressure-derivative type

curves in well-test in naturally fractured reservoirs interpretation and

discussed the application of the new type curves to interpret well test data.

An example of the derivative type curve is shown in Figure

(2-3). They showed the use of the derivative pressure curve as diagnostic

plots in performing well test analysis. For NFR, they considered both

pseudo steady-state and transient flow and the effects of wellbore storage

and skin was included.

Figure (2- 3) Derivative Type Curve for Double-Porosity Reservoir,

Pseudo-Steady State Flow

Authors others worked on Dual-Porosity

Giovanni Da Prat (1981) [14] presented pressure transient analysis for

constant rate production and transient rate analysis for constant pressure

production are presented for a naturally fractured reservoir. Constant

producing pressure solutions, which define declining production rates with

time, were presented. The solutions for the dimensionless flowrate and

pressure are based on a model presented by Warren and Root (1963). It

was found that the flow rate shows a rapid decline initially, becomes nearly


12 | P a g e

constant for a period, and then a final decline in rate takes place. A. striking

result of the present study is that ignoring this of a constant flowrate period

in a type-curve match can lead to erroneous estimates of the dimensionless

outer radius of a reservoir.

A dual-porosity model based on the modified Barenblatt et al.'s (1993)[15]

formulation has been presented for the simulation of naturally fractured

reservoirs. The behavioral transition between a homogeneous and a

naturally fractured reservoir can be expressed as a function of

compressibilities of fluid, fractures and solid grains, together with the

porosities of matrix blocks and fracture network. The linear behavior of

some special naturally fractured reservoirs, as identified by Odeh, can be

simulated by the present model when the fluid transfer between fracture

and matrix block is assumed to be negligible. The equations of single phase

flow through 'double-porosity' fractured reservoirs were formulated by

Barenblatt et al. using the thermodynamic analogy of mixtures. In the

formulation, the actual composite medium of fractures and blocks is

replaced by two overlapping continua, and pair of average properties are

defined over the common bulk volume at each spatial point. Separate

constitutive equations are written for matrix and fractures. Two media

interact through a quasi-steady state flow representing, fluid transfer via a

source/sink term.

Khalid Aziz (1995) [16] derived Matrix-fracture transfer shape factors for

dual-porosity simulation of naturally-fractured reservoirs were derived by

combining analytical solutions of pressure diffusion for various flow

geometries. The resulting equations describing the matrix-fracture flow

were cast in a form similar to those proposed by Barenblatt et al[21] . and

Warren and Root [2], but without making the pseudo-steady state

assumption. Shape factors representing one, two and three parallel sets of

https://www.researchgate.net/profile/Khalid_Aziz4


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fractures were obtained. A generalized matrix-fracture transfer function for

a non-isotropic, rectangular matrix block was also derived. Fine-grid

single-porosity and one-block dual-porosity models were used to verify the

results. The results of fine-grid and one-block dual-porosity models were

in good agreement. The shape factors consistent with observations made in

several publications on dual-porosity simulation studies.

2.2 Dual Porosity Response:

The double-porosity models assume that the reservoir is not homogeneous,

but made up of rock matrix blocks, with high storativity and low

permeability, connecting to the well by natural fissures of low storativity

and high permeability. The matrix blocks cannot flow to the well directly,

so even though most of the hydrocarbon is stored in the matrix blocks it

has to enter the fissure system in order to be produced.

Figure (2- 4) fissure system flow

When the well is first put on production, the first flow regime will be

fissure system radial flow – i.e. the fissure system is producing, and there

is no change in pressure inside the matrix blocks. This first flow regime is

typically over very quickly, and is frequently masked by wellbore storage.

If not, it will be manifested by an IARF (Infinite-Acting Radial Flow)

response on the pressure derivative.


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2.2.1 Infinite-Acting Radial Flow (IARF):

Fluid flows towards the wellbore equally from all directions – the pressure

drop expands radially. The upper and lower bed boundaries are parallel and

clearly defined, the reservoir rock between them is homogeneous, and the

wellbore is perpendicular to the bed boundaries:

Figure (2- 5)Infinite-Acting Radial Flow (IARF)

The initial radial flow (IARF) regime is called infinite-acting because until

the first boundary is reached, the flow pattern and corresponding pressure

drop at the wellbore are exactly as would be obtained if the reservoir were

truly infinite.

Once the fissure system has started to produce, a pressure differential is

established between the matrix blocks, still at initial pressure pi, and the

fissure system, which at the wellbore has a pressure Pwf. The matrix blocks

then start to produce into the fissure system, effectively providing pressure

support, and the drawdown briefly slows down, creating a transitional ‘dip’

in the derivative.


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Figure (2- 6) matrix contribution

2.2.2 Basic assumptions of the double porosity model:

1. Each point in the reservoir is associated with two pressures, namely pf

the pressure of the fluid in the fractures, and pm, the pressure of the fluid

in the matrix pore volume.

2. The fluid flows to the well through the fractures system only; the matrix

blocks are not connected.

3. Most of the reservoir fluid is stored in the matrix blocks porosity, the

Storage of the fractures network is only a small fraction of the reservoir

Storage.

4. Three matrix block geometries are usually considered, depending upon

the number n of fissure plane directions:

• n = 3, the matrix blocks are cubes

• n = 2, the matrix blocks are cylinder

• n = 1 matrix blocks like slab.

5. Two different types of matrix to fissure flow have been considered:

• Flow under pseudo-steady state conditions.

• Flow under transient flow conditions.

6. In the double porosity models, all matrix blocks are homogeneous, and

they have the same size.


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2.2.3 Behavior of the double porosity model:

When a well is opened in a fractures reservoir, a rapid pressure response

occurs in the fractures network due to its high diffusivity. A pressure

difference is created between matrix and fissure, and the matrix blocks start

to produce into the fractures. The pressure of the matrix blocks (pm)

decreases as flow progresses and, finally, tends to equalize with the

pressure of the surrounding fissures (pf) [2].

2.3 Well testing:

Well testing is a tool for reservoir evaluation and characterization.

Information obtained from flow and pressure transient tests about in situ

reservoir conditions are important to determining the productive capacity

of a reservoir.

2.3.1 Description of a well test

During a well test, a transient pressure response is created by a temporary

change in production rate. The well response is usually monitored during

a relatively short period of time compared to the life of the reservoir,

depending upon the test objectives. well evaluation, tests are frequently

achieved in less than two days. the case of reservoir limit testing, several

months of pressure data may be needed.

2.3.2 objectives of well testing:

• Characterizing formation.

• Evaluating and predicting well performance.

2.3.3 Information obtained from well testing

Reservoir description.

• Formation flow capacity (kh).

• Reservoir heterogeneities (natural fractures, layering, change of

characteristics


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• boundaries (distance, site and shape)

• Pressures (initial p and average p)

well description:

• Production potential (productivity index Pl and skin factor)

• Well geometry [17].

2.3.4 Types of Tests

In some cases, the type of test performed is governed by the test

objectives. In other cases, the choice is governed by practical limitations

or expediencies.

A- Drawdown Test

B- Buildup Test

C- Injection Test

D- Falloff Test

E- Interference Test

F- Drill Stem Test (DST).

in this research concern on two type A and B.

2.3.4A- Pressure Drawdown Tests

In a drawdown, a well that is static, stable and shut-in is opened to flow,

for the purposes of transition, the flow rate is supposing to be constant, but

in practice, this is difficult to achieve as shown in figure (2-7).

A pressure drawdown test is conducted by producing a well, starting

ideally with uniform pressure in the reservoir. Rate and pressure are

recorded as functions of time. The objectives of a drawdown test usually

include estimates of permeability, skin factor, and, on occasion, reservoir

volume. These tests are particularly applicable to:


18 | P a g e

1. new wells.

2. wells that have been shut in sufficiently long to allow the pressure

to stabilize.

3. wells in which loss of revenue incurred in a buildup test would be

difficult to accept. Exploratory wells are frequent candidates for

lengthy drawdown tests, with a common objective of determining

minimum or total volume being drained by the well. [18]

2.3.4 B- Pressure Buildup test

the increase of bottom hole pressure after shut in is used for analysis.

Before the build-up test, the well must have been flowing long enough to

reach stabilized rate. During shut-in periods, the flow rate is accurately

controlled(zero) as shown in figure (2-7). It is for this reason testing is the

most familiar transient well-testing technique, which has been used

extensively in the petroleum industry [19].

Figure (2- 7) drawdown and build up test sequence

CHAPTER THREE MATHEMATICAL MODEL OF NFRs

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CHAPTER THREE

MATHEMATICAL MODEL OF NATURALLY

FRACTURE RESERVOIR

3.1 naturally fractured reservoir models:

A characteristic of naturally fracture reservoirs is the presence of two

distinct types of porous media called the matrix and fracture, because of

the different fluid storage and conductivity characteristic of the matrix and

fracture, these reservoirs called dual-porosity reservoirs figure (3-1).

illustrates a naturally fractured reservoir composed of a rock matrix

surrounded by an irregular system of vugs and natural fracture.

Fortunately, it has been observed that a real, heterogeneous naturally

fractured reservoir has a characteristic behavior that can be interpreted with

an equivalent homogeneous dual- porosity model like that shown in the

idealized sketch.

Figure (3- 1) Actual and idealized dual porosity reservoir model.

Several models have been proposed to represent the pressure behavior in a

naturally facture reservoir. These models differ conceptually only in the

assumption made to describe fluid flow in the matrix. Most dual porosity


20 | P a g e

models assume that production from the naturally fracture system goes

from matrix to the fracture and then to the wellbore (the matrix dose not

produced directly into the wellbore). Furthermore, the models assume that

the matrix has low permeability but a large storage capacity relative to the

natural fracture system, while the fracture have high permeability but low

storage capacity. Warren and root [2] introduced two dual-porosity

parameters in addition to the usual single-porosity parameters that can be

used to describe dual-porosity reservoir.

Interporosity flow is the fluid exchange between the two media (the matrix

and fracture) constituting a dual porosity system, warren and root defined

the interporosity flow coefficient) λ ( In a dual porosity reservoir, the

interporosity flow coefficient is the ratio of the permeability of the matrix

(km) to that of the fractures (kf) defined as:

𝜆 = 𝑎 𝑟𝑤2 𝑘𝑚

𝑘𝑓 …………………… 3-1

Where km = permeability of the matrix, kf = permeability of the natural

fractures and α = parameter characteristic of the system geometry.

Note that the geometric coefficient () accounts for the shape of the matrix

blocks. The interporosity flow coefficient is usually in the range of

10-4 to 10-9.

The interporosity flow coefficient is a measure of how easily fluid flows

from the matrix to the fracture, the parameter α is defined by [20]

𝛼 =4𝑗(𝑗+2)

𝐿2 ………………….….3-2

Where L = a characteristic dimension of a matrix block and j = the number

of normal sets of planes limiting the less –permeable medium (j=1,2,3) for

example j = 3 in the idealized reservoir block model in fig (3-1) on the


21 | P a g e

other hand for the multilayered of “slab” model figure (3-2), j=1 for the

slab model by letting L=hm (the thickness of an individual matrix block) λ

becomes,

𝜆 = 12 𝑟𝑤2 𝑘𝑚

𝑘𝑓ℎ𝑚2 …………………………3-3

The storativity ratio ω is defined by

𝜔 =(∅𝑣𝑐𝑡)𝑓

(∅𝑣𝑐𝑡)𝑓+𝑚=

(∅𝑣𝑐𝑡)𝑓

(∅𝑣𝑐𝑡)𝑓+(∅𝑣𝑐𝑡)𝑚 ………………3-4

Where v= ratio of the total volume of on medium to the bulk volume of the

total system and Ø= ratio of pore volume (PV) of one medium to the total

volume of that medium. The range of typical value for 𝜔 is from 0.001 to

0.5. The subject f and f+m refer to the fracture and to the total system

(fractures plus matrix). respectively. Consequently, the storativity ratio is

a measure of the relative fracture – storage capacity in the reservoir.

Figure (3- 2) Slab model for naturally fracture reservoir

Although many models have been developed for naturally fracture

reservoir, we present two common models – pseudo steady – state and

transient flow – describing flow in the less – permeable medium, the

matrix. Bare blatt et al [21]. and subsequent authors [2,22-23] assumed pseudo

steady – state flow, while others, [24-25] notably swan [24], assumed transient

http://petrowiki.org/File:Vol5_Page_0798_Image_0001.png


22 | P a g e

flow in the matrix. intuition suggests that. In a low – permeability matrix,

very long times should be required to reach pseudo steady – state and

transient matrix flow should dominate; however, test analysis indicates that

pseudo steady – state flow is quite common. A possible explanation of this

seeming inconsistency is that matrix flow is almost always transient but

can exhibit a pseudo steady – state – like behavior if there is a significant

impediment to flow from the less- permeable medium to the more –

permeable.

3.2 Pseudo steady-state matrix flow model:

The Pseudo steady-state matrix flow model assumes that, at a given time,

the pressure in the matrix is decreasing at the same rate at all points and

thus flow from the matrix to fracture is proportional to the difference

between matrix pressure and pressure in the adjacent fracture.

Because it assumes a pressure distribution in the matrix that would be

reached only after what could be a considerable flow period the pseudo

state flow model obviously is oversimplified again however this model

seems to match a surprising number of field tests. One possible reason is

that damage to the face of the matrix could cause the flow from matrix to

fracture Pseudo steady-state flow.

3.2.1Semilog analysis technique

The pseudo steady state matrix flow solution developed by warren and

root[2] predicts that two parallel straight line will develop on a semilog

graph of test data.


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Figure (3- 3) Build up semi log plot for dual porosity system

Description :

1- The initial straight line reflects flow in the fracture system only at

this time the formation is behavior like a homogeneous formation with

fluid flow originating only from the fracture system with no

contribution from the matrix. Consequently, the slop of the initial semi

log straight line is proportional to the permeability thickness product

of the natural fracture system just as it is for any homogeneous system.

2- The fluid in the matrix begins to flow into the fracture and a rather

flat transition region appears.

3- Finally, the matrix and fracture reach an equilibrium condition and

a second straight line appears. At this time, the reservoir again behaves

like a homogeneous system, but now the system consists of both the

matrix and the fracture. The slope of the second semi log straight line

is proportional to the total permeability thickness product of the matrix

/ fracture system.

Because the permeability of the fracture is much greater than that of matrix

the slope of the second line is almost identical to that of the initial line.


24 | P a g e

The shape of the semi log plot of test data from naturally fracture reservoir

is almost never the same as predicated by warren and root model. Wellbore

storage almost always obscures the initial straight line and often obscures

part of the transition region between the straight line.

The reservoir permeability thickness product, kh (actually the kh of the

fracture or khf because the khm is usually negligible) can be obtained from

the slope m of the two semilog straight line. Storativity ω can be

determined from their vertical displacement ơp. The interporosity flow

coefficient ʎ can be obtained from the time of intersection of a horizontal

line drawn through the middle of the transition curve with either the first

or second semilog straight line [20].

When semilog analysis is possible (when the correct semilog straight line

can be identified) the following procedure is recommended for semilog

analysis of buildup or drawdown test data from well completed in naturally

fracture reservoir.

1- From the slope of the initial straight line (if present) or final straight line

(more likely to be present) determine the permeability thickness product

kh. In either case the slope m is related to the total kh of the system,

total system kh is essentially all in the fractures. The permeability

thickness produce is given by

khf =k- h=162.6 qµB

m …………………….3-5

Where k−= khf/h strictly speaking, the slope of the second straight line is

related to (khf +khm), but khm ordinarily is much less than khf

2- If both initial and final straight line can be identified (or the position of

the initial line can at least be approximated) and the pressure difference

δp established, then the storativity ratio ω is calculated from


25 | P a g e

ω = 10δp

m⁄ …………………………….3-6

if the times of intersection of a horizontal line drawn through the midpoint

of the transition data with the first and second semi log straight line are

denoted by t1 and t2, respectively, the storativity ratio also may be

calculated from

ω =t1

t2 …………………………3-7

for a buildup test, where the time of intersection of a horizontal line drawn

through the midpoint of the transition data with the first and second

semilog straight lines are denoted by [(tp+∆t)/∆t]1 and [(tp+∆t)/∆t]2,

respectively, the storativity ratio may be calculated from

ω = [(tp+∆t)/∆t]2

[(tp+∆t)/∆t]1 ……….............3-8

3- The interporosity flow coefficient λ, is calculated3 for a drawdown test

by

λ = (Ø v ct )f µ rw2

γ k−t1=

(Ø v ct )f+m µ rw2

γ k−t2 …………3-9

Or for a buildup test by:

λ = (Ø v ct )f µ rw2

γ k−t1(

tp+∆t1

∆t) =

(Ø v ct )f µ rw2

γ k−t2(

tp+∆t2

∆t) ……………3-10

Where ɣ is the exponential of Euler’s constant (ɣ= 1.781). because t1 and

t2 often are approximated, the value of λ obtained by this method may not

be accurate but usually is of the same order of magnitude as the correct

value.

The term (Ø v)m and (ct)m in Eq.(3-10) are obtained by conventional

method. A porosity log usual reads only the matrix porosity not fracture

porosity) and thus gives Øm while (ct)m is the sum of co so, cg sg, cow sw and

cf. vm usually can be assumed to be essentially 1.0. from the definition of ω

in eq:


26 | P a g e

(Ø v ct )f = (Ø v ct )m (ω

1−ω)………………….3-11

4- The second semi log straight line should be extrapolated to P1hr and the

skin actor is

s=1.151[ ∆p1hr

m- log (

k

∅µctrw2) +3.23] ………….3-12

Where ∆ P1hr = (pi – p1hr) for a drawdown test or [p1hr – pwf (∆t =0)] for a

buildup test.

5- The second semilog straight line should be extrapolated to p* from p*,

p can be found with conventional method the Matthew – Browns

Hazebroek[4] .

Figure (3- 4) Extrapolation of second semi log straight line to p*

3.2.2Type curve analysis technique

Particularly because of the wellbore storage distortion, type curve is quite

useful for identifying and analyzing dual-porosity system. Figure (3-5)

shows an example of the Bourdet et al [26] Type curve developed for pseudo

steady – state matrix flow.

Description :

1. Initially test data follow a curve for some value of CDe2s where CD is the

dimensionless wellbore storage coefficient. In the example in

figure (3-5), the earliest data for well a follow the curve for CDe2s =1.



27 | P a g e

2. The data then deviate from the early fit and follow a transition curve

characterized by the parameter λe-2s. in figure (3-5), the data follow

the curve for λe-2s = 3*10-4.

3. When equilibrium is reached between the matrix and fracture system,

the data then follow another CDe2s = 0.1 curve.

Figure (3- 5) type curve for pseudo steady – state matrix flow

Figure (3-6) illustrates derivative type curve for a formation with pseudo

steady state matrix flow [26].

Description :

The most notable feature, characteristic of naturally fractured reservoir

curve is the dip below the homogeneous reservoir curve.

• The curve dipping downward are characterized by a parameter λ

CD/ω(1-ω).

• the curve returning to the homogeneous reservoir curve are

characterized by the parameter λ CD / (1- ω).

Test data that follow this pattern on the derivative type curve can be

interpreted reasonably as identifying a dual porosity reservoir with pseudo


28 | P a g e

steady sate matrix flow (a theory that needs to be confirmed with

geological information and reservoir performance). Pressure and pressure

derivative type curve can be used together for analysis of a dual porosity

reservoir. The pressure derivative data are especially useful for identifying

dual porosity behavior. Manual type-curve analysis for well in naturally

fractured reservoirs is tedious, and the interpretation involved is difficult.

Most current analysis uses commercial software.

Figure (3- 6) derivative type curves for pseudo steady state matrix flow

Calculation of fracture and reservoir parameter from type curve:

Perform a quantitative type curve analysis the objective of this analysis is

either to confirm the result from the semi log analysis or to estimate

reservoir properties when no semi log analysis is possible.


29 | P a g e

The following procedure is recommended for type curve analysis of build

up or draw down test data for well completed in naturally fracture reservoir

following the pseudo steady sate model.

1. Plot pressure change and pressure derivative data on log –log graph

paper or on tracing paper. Plot (pi-Pwf) vs.t for a drawdown test or (pws

– Pwf(∆t=0)) vs. [(tp +Δt)/ Δt] for a buildup test.

2. If an estimate of permeability is available from the semi log analysis,

calculate a pressure match point with Eq (3.13)

(∆p)=141.2 q B µ

kh ( 𝑃𝐷) ……………… 3.13

Where pD is an arbitrarily chosen value. Record a time match point (∆t,

tD/CD).

3. With the type curve in matched position, determine the value of CDe2s

characterizing the fit of the earliest data on the pressure type curve. The

fit on the derivative type curve should confirm this parameter value.

The earliest data are those that appear before the transition region. This

values of CDe2s, which we call (CDe2s )f , characterizes the fracture

system.

4. With the type curve remaining in the same position, determine the value

of CDe2s, which we call (CDe2s )f+m , that characterizes the test data after

the transition to total system flow has been completed.

5. Read the value of λ e-2s that characterizes the horizontal transition curve

crossed by the test data on the pressure type curve at an intermediate

time in the transition region. There is a significant uniqueness problem

in determining the best fitting transition curve.

For test B in figure (3-5) the transition curve chosen was for λ e-2s = 10-7 .

λ = (λ e-2s) e2s ………………. 3.14


30 | P a g e

6. Calculate the storativity ratio, ω, from the ratio of the two CDe2s values

𝜔 =(𝑐

𝐷 𝑒2𝑠) 𝑓+𝑚

(𝑐𝐷 𝑒2𝑠) 𝑓

………………. 3.15

7. If no semilog analysis is possible, calculate permeability from the

pressure match point (∆p, pD)

k= 141.2 q Bµ

∆p h pD ………………. 3.16

Recorded a time match point (∆t, tD ,/CD).

8. Compute the wellbore – storage coefficient from the time match point,

(∆t, tD ,/CD).

CD= 0.0002637 k

∅µct rw2 (

∆ttD CD

) ……………….3.17

9. From the value of CD determined in step 8 and the value of (CDe2s )f+m

determined in step 4 ,compute the skin factor , s.

s=0.5 ln((c

D e2s) f+m

CD ) …………… 3.18

Using the estimate of (S) from step 8 and the parameter λ e-2s from step (4)

calculate λ. This value should be consistent with the estimate of λ from

semilog analysis. If it is not, then the type curve match should be refined.

3.3 Transient matrix flow model

The more probable flow regime in the matrix is unsteady-state or transient

flow (i.e., flow in which an increasing pressure drawdown starts at the

matrix/fracture interface and moves farther into the matrix with increasing

time). Only at late times should pseudo steady-state be achieved, although

a matrix with a thin, low-permeability damaged zone at the fracture face

may behave as predicted by the pseudo steady-state matrix flow model

even through the flow in the matrix is actually unsteady-state.


31 | P a g e

A semilog graph of the test data for a formation with transient matrix flow

has a characteristic shape different from that for pseudo steady-state flow

in the matrix. Three distinct flow regimes have been identified that are

characteristic of dual-porosity reservoir behavior with transient matrix

flow. Figure (3-7) illustrates these flow regimes on a semilog graph as

regimes 1, 2, and 3.

Figure (3-7) characteristic flow regimes in a dual porosity system with

transient matrix flow

Description :

Flow regime 1 occurs at early times when all production comes from the

fractures. Flow regime 2 occurs when production from the matrix into the

fracture begins and continues until the matrix to fracture transfer reaches

equilibrium. This equilibrium point marks the beginning of flow regime 3,

during which total system flow, from matrix to fracture to wellbore, is

dominant. The same three flow regimes appear when there is pseudo

steady-state matrix flow. The duration and shape of the transition flow

regimes, however, is considerably different for the two matrix flow

models.


32 | P a g e

Serra 𝑒𝑡 𝑎𝑙. 𝑒[25] observed that pressures from each of these flow regimes

will plot as straight lines on conventional semilog graphs. Flow regimes 1

and 3, which correspond to the classic early –and late-time semilog

straight-line periods, respectively have the same slope [2-23]. Flow regime 2

is an intermediate transitional period between the first and the third flow

regimes. The semilog straight line of flow regime 2 has a slope of

approximately one-half that of flow regimes 1 and 3. If all or any two of

these regimes can be identified, then a complete analysis is possible with

semilog methods alone. Certain noni deal conditions, however, may make

this analysis difficult to apply.

Flow regime 1 often is distorted or even totally obscured by well bore

storage, which often makes this flow regime difficult to identify. Flow

regime 2, the transition region, also may be obscured by well bore storage.

Flow regime 3 sometimes requires along flow period followed by along

shut-in time to be observed, especially in low-permeability formations.

Furthermore, boundary effects may appear before regime 3 is fully

developed. Sec. 3.3.1 and 3.3.2 present semilog and log-log plotting

analysis techniques for well tests from dual-porosity reservoir exhibiting

transient matrix flow.

3.3.1Semilog analysis techniques

Serra et al.3 presented a semilog method for analyzing well-test data in

dual-porosity reservoirs exhibiting transient matrix flow. They found that

the existence of the transition region, flow regime 2, and either flow regime

1 or flow regime 3 is sufficient to obtain a complete analysis of drawdown

or buildup-test data.

Assumptions :

• unsteady-state flow in the matrix.

• no wellbore storage.


33 | P a g e

• rectangular matrix-block geometry(slab). on the basis of our

experience, the rectangular matrix block geometry is adequate,

although different assumed geometries (spheres) lead to slightly

different interpretation results.

The major weakness of the Serra et al. method is that it assumes no well

bore storage. In many cases, flow regimes 1 and 2 are partially or even

totally obscured by well bore storage, making analysis by their method

impossible or difficult. Despite this limitation, the Serra et al. method has

great practical value when used in conjunction with type-curve method.

The following discussion presents calculation procedures for application

of the Serra et al. method these calculations apply to both buildup- and

drawdown-test data.

Semilog analysis based on flow regimes1 and 2.

This case is rarely observed because of well bore storage. If these flow

regimes are present however the following procedure can be used to

analyze the test data.

1. Estimate kf hft=kh from the slope of the semilog straight line (m or

m*, where m*=m/2),

kh≈kfhft =162.6qBµ

m=

81.29qBµ

m* ……………….3.19

Where hft = net fracture thickness.

2. Estimate the skin factor, s, in one of two ways.

A. first, assuming a value of n2 km Øm cmt (which will be improved in step

7 as part of an iterative procedure),

s=0.5756 [∆p1hr

*

m*- log [

(kfhft)2

n2km∅mcmtrw4 µ

] +3.729] ……….…3.20

Where n = number of fractures (equal to net matrix, thickness hmt, divided

by individual matrix-block thickness, hm, or net fracture thickness hft,


34 | P a g e

divided by individual fracture thickness, hf); km= matrix permeability, md;

Øm=matrix porosity, fraction; cmt = total matrix compressibility, psi-1; and

kf = fracture permeability, md.

B. alternatively, assuming a value of Øfhftcft (which will be checked in step

6 as part of an iterative process),

s=1.151 {∆p1hr

m- log [

(kfhft)2

hf∅fcfrw2 µ

] +3.23} …….……. 3.21

Where Øf = fracture porosity, fraction, and cft = fracture system total

compressibility, psi-1

3. calculate pwD with

pwD=kfhf∆p

141.2qµB ………….…… 3.22

4. plot pwD vs. Δt on tracing paper or on semilog graph paper with the same

scale as in figure (3-8).

5. align the vertical axis of the data plot with the vertical axis of

figure (3-8). move the data plot in the horizontal direction only to obtain

a time match point, record values of the parameters 𝜆′,ώ ,t*1,t1D,

and (tD, Δt) at the match point, where t*1 is the time (hours) at the point

of intersection of flow regimes 1 and 2 and t1D is the dimensionless time

at this point, defined by

t1D=0.0002637kt1

*

∅µctrw2 …………… 3.23

6. Estimate Øf cft hft from the time match point,

∅fcfthft=2.637×10-4kfhft

µrw2 (

∆t

tD)MP …………….3.24

If the value does not agree with that in step 2B, use this updated value,

return to step 2B, and repeat until there is no further change in the value of

Øf cft hft .


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7. Estimate n2kmØmcmt,

n2km∅mcmt= 532.3µ(∅fcfthft)

2

t1* ………….… 3.25

If the value does not agree with that in step 2A, use this updated value,

return to step 2A, and repeat until there is no further change in the value of

n2kmØmcmt

8. Assume a value of Øm cmt and estimate km/h2m

km

hm2 =

(n2km∅mcmt)

h2∅mcmt ………… 3.26

9. Assume that hmt = h and estimate 𝜆′ and ώ

ω, =∅mcmth

∅fcfthft ……………….3.27

𝜆′ = 12𝑘𝑚ℎ 𝑟𝑤

2

ℎ𝑚2 𝑘𝑓ℎ𝑓𝑡

……………….. 3.28

10. estimate λ and ω

𝜔 =1

1+𝜔′ ………… 3.29

λ ≈ λ′ ………….. 3.30


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Figure (3- 8) estimeting ώ and λ’ with data from flow regimes 1 and 2

Semilog analysis based on flow regimes 2 and 3.

Because of well bore-storage effects, flow regimes 2 and 3 are more

commonly observed in practice. The following procedure can be used to

analyze test data under these conditions.

1. Estimate kf hft = kh from the slope of the semilog straight line (m or m*,

where m*= m/2) with Eq(3-19).

2. Assume a value of Øm cm μ and calculate km/h2m

km

hm2 =

532.3∅mCmtµ

∆t* ………… 3.31

Where Δt* = time at which flow regimes 2 and 3 intersect.

3. Estimate 𝜆′

λ′ = 12kmhmtrw

2

kfhfthm2 ≈ 12

kmhrw2

hm2 kfhft

…………. 3.32

Where h ≈ hmt

4. Estimate the time, tb2 (or Δtb2 for buildup), at which flow regime 2

begins and calculate


37 | P a g e

∅fcfthft=8.33×10-4(kfhft∅mcmthλ'tb2

µrw2 )0.5 ……….. 3.33

5. Estimate ώ, ω and λ

ω′ =∅mcmthmt

∅fcfthft ……………. 3. 34

𝜔 =1

1+𝜔′ ………. 3.35

𝜆 ≈ 𝜆′ …………………3.36

Assuming h ≈hmt

6. Estimate the skin factor, s.

s=1.151[∆p1hr

m- log (

kfhft

∅mcmthµrw2 ) +3.23] …………. 3.37

For a flow test, Δp1hr= pi - p1hr for a buildup test, Δp1hr = p1hr- Pwf (Δt=0)

Both of these semilog methods require some knowledge of or assumptions

about matrix and fracture properties. The advantage of semilog analysis is

the unique interpretation possible when the correct semilog straight lines

are identified. The disadvantages are that

• wellbore storage can distort data to the extent that application of

semilog analysis is impossible

• the test duration is sometimes insufficient to reach flow regime 3

(especially in low-permeability formation), or boundary effects

may occur before flow regime 3 develops.


38 | P a g e

3.3.2 Type-curve analysis technique

Bourdet et al. [12] presented type curves for analyzing well test in dual-

porosity reservoirs that include:

• the effects of wellbore storage

• unsteady-state flow in the matrix.

The type curves are useful supplements to the serra et al. semilog analysis.

Figure (3-9) gives an example of the pressure and pressure-derivative type

curves for transient matrix flow.

Description :

early (fracture-dominated) data are fit on the pressure type curve by a cDe2s

curve. Data in the transition region are fit by curves characterized by a

parameter β′. Finally, data in the homogeneous-acting, fracture-plus-matrix

flow regime are fit by another CDe2s curve.

Figure (3- 9) type curves for transient matrix flow

On the derivative type curve, early data also are fit by a derivative curve

reflecting homogeneous behavior. Figure (3-10) shows an actual example.

If wellbore-storage distortion ceases before transition region begins


39 | P a g e

(unlikely but possible), the derivative data will be horizontal and should be

aligned with the (tD/cD)pD = 0.5 curve. However, if the transition region is

present (recall that is semilog slope is half that of the middle-time straight

line) the derivative curve will flatten and should be aligned with the

(tD/cD)pD = 0.25 curve after wellbore distortion has ceased and before

boundary effects have appeared the homogenous (fracture-plus –matrix)

data should be horizontal on the derivative type curve and should be

aligned with the (tD/cD)pD = 0.5 .

Figure (3- 10) Example of using type curves for transient matrix flow

Figure (3-11) Represent response the two types matrix geometry, namely

slabs and spheres. Though the pressure curves look identical, the

derivatives are different:

• The sphere model response hardly reaches the 0.25 straight line but

remains above it.

• The curve generated for slab matrix is tangential to the 0.25 line, and at

late transition time, the change from 0.25 to the 0.5 level is steeper

Figure (3-11) illustrates that the matrix geometry has only a limited

influence on double porosity responses. ln practice. when the analysis is

made by hand, it is not possible to differentiate between the two solutions.



40 | P a g e

and the same type curve is used When a computer is used. and provided

the quality of the data is good one of the two solutions is sometimes

preferred because the derivative match appears slightly better The choice

of the matrix geometry does not influence the numerical results of analysis.

Figure (3- 11) Dual Porosity Transient Interporosity Flow

We recommended the following procedure for using the Bourdet et al.

transient or unsteady-state matrix-flow type curves.

1. Plot data as Δp vs. Δt on tracing paper or on log-log graph paper with

the same scale as the type curves.

2. Next, perform qualitative type-curve analysis. The purpose of this step

is to identify various flow regimes to determine whether semilog

analysis is possible. Overlay the log-log graph on the type curves and

move the graph horizontally and vertically until a good match is

obtained on the basis of the following criteria: A unit-slope line on the

test data should overlay that line on the derivative type curve similarly,

horizontal lines should overlay the (tD/cD)pD = 0.25 line

(transition)region, or the (tD/cD)pD = 0.5 line (homogeneous-acting

region) on the derivative type curve.


41 | P a g e

3. If the two straight lines are present and can be identified, perform a

semilog analysis with the procedure discussed in the previous sections.

4. Perform a quantitative type-curve analysis. the objective of this analysis

is to conform the results from the semilog analysis is possible

If an estimate of permeability is available from the semilog analysis,

calculate a pressure match with eq. 3.13

(∆p)MP=141.2qBµ

kh (pD)MP ……………….. 3.13

Where pD = an arbitrarily chosen value.

If no estimate of permeability is available, use the match from the

qualitative type – curve analysis. Obtain a time match point (∆t, tD / cD).

If no semilog analysis is possible, match the data with type curves and

obtain pressure and time match points, (∆p, pD) and (∆t,tD / cD ),

respectively. Compute the average system permeability with the pressure

match point.

k=141.2qBµ

h(

pD

∆p)MP ………… 3.38

Calculate (CD)f+m from the time match point (∆t,tD / cD ),

(cD)f+m=0.0002637k

∅mcmtµrw2 (

∆tetDCD

)MP ……………3.39

Calculate the skin factor

S=0.5 ln (cDe2s

cD)f+m ……………….…3.40

Calculate the interporosity coefficient λ

λ=1.8914(cDe2s)f+m

(β')MPe2s …………….. 3.41


42 | P a g e

Calculate the storativity ratio

ω =(cDe2s)f+m

(cDe2s)f …………. 3.42

If (CDe2s )f cannot be determined uniquely then

ω ≦ (cDe2s)f+m

(cDe2s)f ……….. 3.43

Although this procedure can be used to analyze tests that have wellbore –

storage distortion, it does require the existence of flow regimes 3 for a good

interpretation. For best result, the type curve and semilog analysis methods

should be applied simultaneously until a consistence interpretation can be

found with both approaches. Some tests may exhibit behavior that can be

interpreted equally well as single – or dual- porosity. In these cases,

additional well test is needed to determine which interpretation is more

likely. Manual type-curve matching is tedious and difficult, especially with

the interpolation involved. Analysis ordinarily uses commercially

available software to analyze these kinds of tests after the reservoir model

has been identified.

CHAPTER FOURE RESULT AND DISCUSSION

43 | P a g e

CHAPTER FOUR

RESULT AND DISCUSSION

Problem (4.1):

Analysis of pressure-buildup test in naturally fractured reservoir a pressure

buildup test was simulated for an oil well in naturally fractured reservoir

with transient matrix flow behavior. Well and reservoir data are

summarized next, and table below gives the buildup test data. Determine

permeability, skin factor, storativity ratio, and interporosity flow

coefficient with the semi log and type-curve analysis techniques.

qg=333 STB/D; tp=2000 hours; so=0.67; rw=0.33 ft; h=33 ft; Ø=0.063;

Bo=1.3 RB/STB; Ct=2.25*10-5; µo=1.3 cp; sw=0.33; cmt=4.0*10-6;

pi=p=2876 psi; Pwf (Δt=0) =2692.05 psi.

Table (4-1)

Δt pwf Δt pwf Δt pwf

0.001 2697.7 0.09 2803.8 8 2832.1

0.002 2703 0.1 2805.3 9 2833.1

0.003 2708 0.2 2812 10 2833.6

0.004 2712.6 0.3 2814.9 20 2837

0.005 2717 0.4 2816.8 30 2839.4

0.006 2721 0.5 2818.2 40 2841.3

0.007 2725.1 0.6 2819.2 90 2847.6

0.008 2728.8 0.7 2820.1 100 2848.5

0.009 2732.3 0.8 2820.9 200 2854.3

0.01 2735.6 0.9 2821.5 300 2857.6

0.02 2760.5 1 2822 400 2859.8

0.03 2775.8 2 2825.7 900 2865.4

0.04 2785.5 3 2827.8 1000 2866.1

0.05 2792 4 2829.2

0.06 2796.4 5 2830.3

0.07 2799.6 6 2831.2

0.08 2802 7 2831.9


44 | P a g e

Solution:

Step 1: semi log method

Using the data in table (4-2), make semi log plot of Δp vs.t as shown in

figure (4-1).

Figure (4- 1) semi log plot of pwf vs t for problem (4.1)

From figure (4-1) Slop for fracture flow =20 psi/cycle

kh≈kfhf =162.6qBµ

m

kh≈kfhf =162.6*333*1.3*1.3

20=4575.32 md-ft

For h=33; k =4575.32/33 =138.64 md

(Δt*) the point of intersection of two semi log lines at a Horner time ratio

is found from figure (4-1).

(Δt*+tp)/Δt*=60; Δt*=2000/59=33.9 hours

calculate km /h2m from available data we have Øm=0.0633, cmt=4*10-6 and

µo=1.3 cp .

2680

2700

2720

2740

2760

2780

2800

2820

2840

2860

2880

1 10 100 1000 10000 100000 1000000 10000000

pw

f

log(t+Δt/Δt)

pwf vs log( t)pwf vs log( t)


45 | P a g e

km

hm2 =

532.3∅mCmtµ

∆t*=

532.3*0.063*4*1.3*10-6

33.9= 5.16849*10-6 md/ft2

Estimate λ'

λ'=12kmhmtrw

2

kfhfthm2 =

12*5.16849*33*0.332*10-6

4575.3=4.87155*10-8

Δtb the beginning of flow regime 2 is found from figure (4-1)

∆tb+tp

∆tb=2200

Δtb= 2000/ (2200-1); then Δtb= 0.91 hour

∅fcfthft is calculated from Δtb

∅fcfthft=8.33×10-4(kfhft∅mcmthλ'tb2

µrw2

)0.5

∅fcfthft=8.33×10-4 [4575.3*0.063*4* 10-6*33*4.87155* *0.91

1.3*0.332]

0.5

∅fcfthft = 1.00254*10-7 ft/psi

Estimate ώ, ω and λ where h ≈ h m

ω'=∅mcmthmt

∅fcfthft=

0.063*4*33*10-6

1.00254*10-7=83.344

The parameter ω is

ω =1/ (1+ ώ) =1/ (1+83.344) = 0.011

Then λ ≈ λ' =4.87155*10-8

From figure (4-1) calculate the skin factor we need P1hr on the semilog

straight line of flow regime 3 at Δt=1

)Δt+2000)/Δt=(2000+1)/1= 2001


46 | P a g e

At this value P1hr = 2807.4 psi

The skin factor is

s=1.151[∆p1hr

m- log (

kfhft

∅mcmthµrw2

) +3.23]

𝑠 = 1.151 ∗ [2807.4 − 2692.05

20− 𝑙𝑜𝑔 (

4575.32

0.0633 ∗ 2.52 ∗ 10−5 ∗ 33 ∗ 0.332) + 3.23]

s= 0.01≈0

From the semi log plot shown as figure (4-1), it is not obvious fracture flow

period because is distorted by well bore storage, which makes this flow

regime difficult to identify. This is followed by a transition period to matrix

flow of duration of half one to thirty hours approximately. The transient

flow from both matrix and fracture started approximately after Forty hours

of flow.

Step 2: type curve matching.

Prepare a log- log plot of pwf vs t from the data table (4-2). the log-log plot

shown as figure (4-2). In this example, because of unavailability of

pressure derived we could not plot)∆P′ Vs. ( tp +Δt)/ Δt).

Figure (4- 2) log-log plot of pwf vs t for problem (4.1)

1

10

100

1000

110100100010000100000100000010000000

Δp

( tp +Δt)/ Δt

Δp vs ( tp +Δt)/ ΔtΔp vs ( tp +Δt)/ Δt


47 | P a g e

From figure (4-3) Pressure match point (PMP)

(pD/ Δp) MP = (1/18.1) = 0.055

k=141.2qBµ

h(

pD

∆p)MP=

141.2*333*1.3*1.3*0.055

33=132.4 md

kh=4369.2 md-ft

From figure (4-3) Time match point (TMP)

[Δt/(tD/cD)] MP = (0.32/100) =3.2*10-3

(cD)f+m=0.0002637k

∅mcmtµrw2

(∆te

tD

CD

)MP

(cD)f+m=0.0002637*132.4*3.2*10-3

0.063*2.25* 10-5*1.3*0.332=555.263

𝑐𝐷𝑒2𝑠 = 570

From figure (4-3), type curve match parameters were estimated as

follows:

(𝑐𝐷 𝑒2𝑠 )𝑓+𝑚 = 570 ;(𝑐𝐷 𝑒2𝑠 )𝑓= 104 and β′ = 1010

Then estimate S

S=0.5 ln (cDe2s

cD)f+m = 0.5 *ln (

570

555.263) =0.013

And estimate λ

λ=1.8914(cDe2s)f+m

(β')MPe2s=1.8914*

570

1010e2*0.013=1.05*10-7

Finally, calculate ω

ω =(cDe2s)f+m

(cDe2s)f=

570

104=0.057


48 | P a g e

Table (4-2) ( tp +Δt)/ Δt Δp ( tp +Δt)/ Δt Δp (tp +Δt)/ Δt Δp

2000001 5.65

25001 109.95

334.3333 139.15

1000001 10.95

22223.22 111.75

286.7143 139.85

666667.7 15.95

20001 113.25

251 140.05

500001 20.55

10001 119.95

223.2222 141.05

400001 24.95

6667.667 122.85

201 141.55

333334.3 28.95

5001 124.75

101 144.95

285715.3 33.05

4001 126.15

67.66667 147.35

250001 36.75

3334.333 127.15

51 149.25

222223.2 40.25

2858.143 128.05

23.22222 155.55

200001 43.55

2501 128.85

21 156.45

100001 68.45

2223.222 129.45

11 162.25

66667.67 83.75

2001 129.95

7.666667 165.55

50001 93.45

1001 133.65

6 167.75

40001 99.95

667.6667 135.75

3.222222 173.35

33334.33 104.35

501 137.15

3 174.05

28572.43 107.55

401 138.25

Figure (4- 3) type curve match for problem (4.1)


49 | P a g e

The derivative curve shows evidence of double porosity behavior which is

indicated by the characteristic dip below the homogeneous behavior level.

Also, the derivative curve in figure (4-3) shows transient flow from the

combined system of fracture and matrix is represented by the stabilized

leveling of the derivative curve after the dip.

Step 3: solution by ecrin

Figure (4- 4) solution problem (4.1) by Ecrin.

In this example, the skin factors, storativity ratio and the flow capacities

estimated from type curve matching, semi log methods and by Ecrin are

reasonably in close agreement. The value of the skin is close to zero and

this means that there is no damage in the reservoir so the matrix flow

behavior is transient behavior. The value of the storativity ratio within the

specified range (0.01-0.05) and This value indicates that the storage in

fracture is weak relative to the total system. The flow capacities in this

example indicate that the flow capacities are high. there is poor agreement


50 | P a g e

in the interporosity flow coefficient calculated from type curve matching,

semi log methods and by Ecrin. The interporosity flow coefficient

calculated from semi log method can be considered as gross estimates

because of the difficulties associated with estimating the correct value for

semi log parameters (Δtb , Δt*).

Under these circumstance, the value of the interporosity flow coefficient

calculated from type curve matching close to the value of the interporosity

flow coefficient calculated by Ecrin and can be considered to be more

representative of the actual values. The most accurate value can be taken

from the Ecrin program. Note the results in which the program usually

ranges between the calculated results in semi log and a curved type method.

The interporosity flow coefficient is usually in the range of 10-4 to 10-9 The

interporosity flow coefficient calculated from all methods within this

range. The interporosity flow coefficient calculated by Ecrin is (3*10-7)

This means that the permeability of the fracture is much greater than the

permeability of the total system because the interporosity flow coefficient

is the ratio of the permeability of the total system to that of the fractures.


51 | P a g e

Problem (4.2) [27]:

A drawdown test was conducted on a well in a naturally fractured reservoir.

the test data are shown in table calculate fracture and reservoir parameters

from the test data. well and other reservoir properties are given as follows:

h=30 ft; Øm=0.12; rw=0.5 ft; Bo=1.3 RB/STB; Ct=14*10-6; µo=1.2 cp;

p=4480 psi; qo=2950 STB/D.

Table (4-3)

t pwf Δp tΔp` t Pwf Δp tΔp`

0 4480 0 5.93 3923 557 95.16

0.05 4462 18 128.87 6.93 3907.69 572.31 115.62

0.06 4440 40 136.28 7.93 3890.08 589.92 134.18

0.08 4403 77 147.34 8.5 3880.64 599.36 137.5

0.09 4379.2 100.8 141.03 10.5 3850.62 629.38 138.42

0.12 4338.5 141.5 135.82 11.6 3837 643 144.72

0.17 4296.5 183.5 137.56 13.4 3814.45 665.55 140.49

0.2 4270 210 138.68 15 3800 680 144.4

0.25 4240 240 145.16 16.3 3787 693 146

0.33 4196 284 144.7 17.4 3778 702 144.8

0.4 4170 310 141.94 18.35 3770 710 146.74

0.51 4135 345 149.17 19.2 3763.5 716.5 144.14

0.57 4117 363 139.44 20.5 3754 726 143.71

0.65 4100 380 146.64 21.5 3747.2 732.8 144

0.74 4080 400 120.92 22.5 3740.6 739.4 143.85

0.92 4070 410 99.51 23.5 3734.4 745.6 143.76

1 4060 420 101.18 25 3725.4 754.6 148.33

1.1 4052.4 427.6 76.42

1.4 4036 444 67.38

1.65 4025 455 67

1.93 4014.49 465.51 69.02

2.3 4002 478 72.41

2.93 3984.07 495.93 78.05

3.93 3959.73 520.27 85.48

4.93 3939.9 540.1 89.7


52 | P a g e

Solution:

Step 1: semi log method

Prepare a semi log plot of pwf vs t from the data table (4-3). the semi log

plot shown as figure (4-5).

Figure (4- 5) semi log plot of pwf vs t for problem (4.2)

The slope of semi log straight line in figure (4-5) m = 311 psi/ cycle.

kh≈kfhf =162.6qBµ

m=

162.6*2950*1.3*1.2

311=2406.06 md-ft

Kf =(kh)total = 2406.06/30 = 80.2 md

From figure (4-5), the vertical pressure displacement δp=155.56 psia

ω = 10δp/m =10-(155.56/311) = 0.32

a straight line through the middle of the transition curve intersects the two

parallel straight line as shown in figure (4-5), t1=1.7, t2 =3.7, respectively

λ= (Ø v ct )f+m µ rw2

γ k-t2=

0.12*14*10-6 *1.2*(0.5)2

1.78*80.2*3.7=9.54*10-10

(Ø v ct )f=(Ø v ct )m (ω

1-ω) = (

0.32

1-0.32) *(0.12*14*10-6 )

3600

3700

3800

3900

4000

4100

4200

4300

4400

4500

0.01 0.1 1 10 100

pw

f

log(t)

pwf vs tpwf vs t


53 | P a g e

(Ø v ct )f =7.9059*10-7 psi -1

Now we can use the question at (t1) to Estimate (λ)

λ= (Ø v ct )f µ rw2

γ k-t1=

7.9059*10-7 *1.2*(0.5)2

1.78*80.2*1.7=9.77*10-10

Then Estimate S

s=1.151[ ∆p1hr

m- log (

k

∅µctrw2) +3.23]

s=1.151*[ 4480 – 4190

311- log (

80.2

0.12*14*10-6 *1.2*(0.5)2) +3.23]

s = -4.65

From the semi log plot shown as figure (4-5), it is obvious that fracture

flow lasted for about one hour. This is followed by a transition period to

matrix flow of duration of one to nine hours. Pseudo steady-state flow from

both matrix and fracture started approximately after ten hours of flow.

These flow periods conform to the flow periods is identifiable on the

derivative curve of figure (4-6).

Step 2: type curve matching.

Using the data in table (4-3), make a log-log plot of Δp vs. t as shown in

figure (4-6). Also, plot the derivative curve of Δp` vs. t on the same graph

using the data in table (4-3).


54 | P a g e

Figure (4- 6) log-log plot of Δp & Δp` vs t for problem (4.2)

From figure (4-7) Pressure match point (PMP)

(pD/ Δp) PMP = (0.35/100)

From figure (4-7) Time match point (TMP)

[Δt/(tD/cD)]TMP = (1/10)

(∆𝑝) =141.2 𝑞 𝐵 µ

𝑘ℎ ( 𝑝𝐷 )

100=(141.2*2950*1.3*1.2*0.35)/(kh)total

(kh)total = 2274.31`md-ft

K=2274.31/30=75.81`md

CD= 0.0002637 k

∅µct rw2 (

∆ttD CD

) = 0.0002637*75.81

0.12*1.2*14*10-6 *(0.5)2 (

1

10 )

CD = 3966.49

From figure (4-7), Type curve match parameters were estimated as

follows:

(𝑐𝐷 𝑒2𝑠 )𝑓+𝑚 = 0.4 ;(𝑐𝐷 𝑒2𝑠) 𝑓= 2 and λ e-2s = 0.06

1

10

100

1000

0.01 0.1 1 10 100

Δp

log(t)Δp vs t Δp` vs t


55 | P a g e

𝜔 =(𝑐

𝐷 𝑒2𝑠) 𝑓+𝑚

(𝑐𝐷 𝑒2𝑠) 𝑓

= 0.4

2 = 0.2

s=0.5 ln((c

D e2s) f+m

CD ) = 0.5 ln(

0.4

3966.49) = -4.601

λ = (λ e-2s) e2s = (0.06) e2*(-4.601)

λ =6.05*10-6

Figure (4- 7) type curve match for problem (4.2)

The derivative curve clearly shows evidence of double porosity behavior

which is indicated by the characteristic dip below the homogeneous

behavior level. Also, the derivative curve in figure (4-6) shows pseudo

steady-state flow from the combined system of fracture and matrix which

is represented by the stabilized leveling of the derivative curve after the

dip.


56 | P a g e

Step 3: solution by ecrin

Figure (4- 8) solution problem (4.2) by Ecrin

The skin factors, storativity ratio and flow capacities calculated from type

curve matching, semi log method sand by Ecrin are reasonably in close

agreement. the values of the skin are negative these mean that improved

reservoir permeability.the values of the storativity ratio within the range

(0.001-0.5) and theses value indicate that the storage in fracture is strong

relative to the total system. The flow capacities values in this example

indicate that the flow capacities are medium. The value of the interporosity

flow coefficient calculated from semi log out of the range (10-4-10-9) and

this indicates that the value is unreasonable. The values of the interporosity

flow coefficient calculated by the Ecrin program is approximated to the

value calculated by the type curve matching and within the required range

and can be considered the correct value. This difference in results is due to

the difficulty of finding the correct values from the plots.


57 | P a g e

In both Problems, we observe that the semi log method has a percentage of

errors, especially in the calculation of the interporosity flow coefficient.

Therefore, we use the type curve method to confirm the results. These

errors are due to the difficulty in determining the flow periods in the plot

because of the phenomenon of well bore storage, which distort the period

of flow in the fracture and the difficulty of finding the correct values from

semi log plot.

CHAPTER FIVE CONCLUSIONS

58 | P a g e

CHAPTER FIVE

CONCLUSIONS AND RECOMMENDATIONS

5.1 conclusions

Based upon work performed for this thesis the following conclusions were

drawn:

1- Pressure transient testing is a reliable method for detecting and

evaluating reservoir heterogeneities that affect reservoir fluid flow and

reservoir performance in naturally fracture reservoirs.

2- Field test transient pressure behavior validates the use of the transient

matrix flow model for test interpretation in naturally fractured

reservoirs.

3- There are essential differences between the pseudo steady state matrix

flow and the transient matrix flow models.

4- The transient flow behavior of a well in a naturally fractured reservoir

may exhibit three flow periods:

a) fracture storage dominated flow period.

b) the transient matrix linear flow dominated period.

c) the pseudo steady State matrix low dominated period.

5- The use of the pressure derivative curve is essential for determining

a suitable model for the transient behavior.

6- The derivative approach improves the definition of the analysis plots

and therefore the quality of the interpretation because the derivative

response is more sensitive to small phenomena.

7- The transient matrix linear flow is characterized by a semi log

straight line whose slope is equal to half the slope of the two classical

parallel straight lines (m). This flow period will exist whenever 𝜔 less is

or equal to 5 x 10-2 for strata and 10-2 for spheres for values higher than

CHAPTER FIVE CONCLUSIONS

59 | P a g e

these there exists an apparent straight line during the transition period

whose slopes ranges from 0.5 m to m.

8- It is unlikely to identify the fracture storage dominated straight line for

small values of𝜔 because wellbore storage usually masks the

corresponding semi log straight line.

9- A proper analysis of data includes the application of type curves to

identify when the influence of wellbore storage ends. Frequently, data

dominated by wellbore storage are misinterpreted as falling on the first

semi log straight line.

10- If the first semi log straight line cannot be identified in a test,

approximate values for 𝜔 can be estimated from the slope ratio of the

second and third semilog straight lines.

11- The design of a transient pressure test for a well in a naturally

fractured reservoir should aim at the detection of at least the second and

third straight lines through minimizing wellbore storage effects;

furthermore, for pressure build up tests it is necessary that the flowing

time prior to the shut in be long enough to reach the third semilog

straight line.

12- Information obtained from well tests should be combined with other

sources such as geological information, well logs, core sample data,

production data and drilling information in order to properly

characterize the well and reservoir properties.

5.2Recommendations

1. Work on more than one model like homogeneous and dual porosity

and triple porosity reservoir models.

2. Work on a field study.

3. Calculate the area of fracture in the reservoir.

60 | P a g e

REFERENCES

1. John lee “pressure transient testing” SPE, New York (1982) Dallas.

2. Warren, J.E. and Root, P.J “the behavior of naturally fractured

reservoirs” SPEJ (September 1963) 245; Trans AIME, 228.

3. Nelson RA: “evaluating fractured reservoirs: introduction,” geologic

analysis of naturally fractured reservoirs, 2nd ed. Woburn,

Massachusetts.

4. Aguilera, Roberto: Recovery Factors and Reserves on Naturally

Fractured Reservoirs, Journal of, Canadian Petroleum Technology,

Distinguished Authors Series (July 1999), p. 15-18.

5. Cinco-Ley, H.: “Well-Test Analysis for Naturally Fractured

Reservoirs,” JPT (Jan. 1996) 6.

6. Mavor, M.J. and Cinco-Ley, H.: “Transient Pressure Behavior of

Naturally Fractures Reservoirs,” paper SPE 7977 presented at the 1979

SPE California Regional Meeting, Ventura, California, 18-20 April.

7. Kazemi, H.: “Pressure Transient Analysis of Naturally Fractured

Reservoirs with Uniform Fracture Distribution,” SPEJ (December

1969) 451.

8. De Swaan O. A.: “Analytic Solutions for Determining Naturally

Fractured Reservoir Properties by Well Testing,” SPEJ (June 1976)

117.

9. Najurieta, H.L.: "A Theory for the Pressure Transient Analysis in

Naturally Fractured Reservoirs," paper SPE 6017 presented at the

1976 SPE Annual Technical Conference, New Orleans, 3-6 October.

10. Bourdet, D. and Gringarten, A.C.: “Determination of Fissure Volume

and Block Size in Fractured Reservoirs by Type-Curve Analysis,”

paper SPE 9293 presented at the 1980 Annual Fall Technical

Conference and Exhibition SPE-AIME, Dallas, 21-24 September.

61 | P a g e

11. Gringarten, A.C., Burgess, T.M., Viturat, D., Pelissier, J., and Aubry,

M.: “Evaluating Fissured Formation Geometry from Well Test Data:

A Field Example,” paper SPE 10182 presented at the 1981 Annual Fall

Technical Conference and Exhibition SPE-AIME, San Antonio,

Texas, 5-7 October.

12. Bourdet, D., Alagoa, A., Ayoub, J.A., Pirard, Y.M., and Melun, J.:

“New Type Curves Aid Analysis of Fissured Zone Well Tests,” World

Oil (April 1984) 111.

13. Bourdet, D., Ayoub J., Whittle T., Pirard Y., and Kniazeff

V.:“Interpreting Well Tests in Fractured Reservoirs,” World Oil

(October 1983) 77.

14. Well test analysis for naturally fractureld reservoir, SGP- TR- 49

(JULY 1981), Stanford Geothermal Program, Interdisciplinary

Research in,Engineering and Earth Sciences,STANFORD

UNIVERSITY (Stanford, California).

15. SPE-26270-MS M. Bai (The U. of Oklahoma) | J-C. Roegiers (The

U. of Oklahoma)

16. Journal of Petroleum Science and Engineering 13(3-4):169-

178 · November 1995

17. - Roland N. Horne, "Modern Well Test Analysis", Petroway Inc., 1995

18. John Lee," Well Testing”, SPE, AIME 1982.

19. Amanat U. Chaudhri, "Oil Well Testing Handbook", GPP, 2004.

20. Gringarten, A.C:” Interpretation of test in fissured and multilayered

reservoir with double porosity behavior: Theory practice “JPT

(December 1983) 549; Trans AIME, 277.

21. Barenblatt G. E. Zheltov, I.P. and Koshina, I.N. “Basic concepts in

the theory of homogeneous liquids in fissured rocks” J.APPL.math

22. Odeh a. s. “unsteady state bchavior of naturally fractured reservoirs

“SPEJ “march 1965’ 60; trans. AIME, 234

https://www.onepetro.org/search?q=dc_creator%3A%28%22Bai%2C+M.%22%29

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62 | P a g e

23. Mavor, M.J. and Cinco, H, “transient pressure behavior of naturally

fractured reservoirs” paper SPE 7977 presented at the 1979 SPE

California regional meeting, ventura, California 18-20 April

24. Desuaan – A.” analytical solutions for determining naturally fractured

reservoirs properties by well testing” SPEJ (June 1976) 117, trans,

AIME 261.

25. Streltsova T.D, “well pressure behavior naturally fractured

reservoirs” SPEJ, (October 1983) 769, trans, AIME, 275

26. Bourdet, d, et al. “interpreting well test in fractured reservoirs” world

oil (October 1983.) 77-87.

27. Nnaemeka Ezekwe ' Petroleum Reservoire Engineering Practice '

(September, 2010) Printed in the United States of America

APPENDIX

1A

APPENDIX

LEARNING OF ECRIN PROGRAM

1. If Al-Saphir program has been installed as a standalone application

click on new project Or click on (ctrl+n). This opens in

succession two dialogs:

Figure ( 1A) Initialization dialog

A. This dialog allows you to choose the test type; reference fluid type (It

means the type of fluid, which were analyzed by), the available fluid

rates (Means the fluid associated with the type of oil hydrocarbons), net

drained thickness, well radius and average porosity keep all other

parameters as the suggested default. Click .

APPENDIX

2A

Figure ( 2A) Initialization dialog 1 of 2

1- Inter the well pore radius(rw)

2- Inter the pay zone thickness (h)

3- Porosity (Ø)

4- Reference phase (oil, gas, water)

5- Available rate (oil, gas, water)

6- Press Next to open new page of the PVT parameters (do not forget

the units).

Figure ( 3A) Initialization dialog 2 of 2

B. input the PVT parameters; formation volumes factor, the fluid viscosity

and the system compressibility. Keep all values at their suggested

APPENDIX

3A

defaults. Click to create the new project. Al-Saphir main

screen is displayed.

1- Inter the formation volume factor β (STB/DAY) or other unites.

2- Viscosity of the fluid µ (cp) or other unites.

3- Inter the total compressibility ct (psi-1) or other unites.

If we have the total compressibility are not available can be entered

saturation and compressibility are available for fluids. It will automatically

calculate the total compressibility as shown in the figure above.

If we have not been available all the information listed above directly,

are calculated by click We will show a dialog box as shown

Figure(4A).

Figure ( 4A) Document PVT defintion

1- Water properties

2- Gas properties

3- Oil properties

4- Condensate properties

When click and inter require information , The data will be needed

by the program will automatically calculate As shown in Figure(5A).

APPENDIX

4A

click To identify the the components of the gas and then Account

specific gravity As shown in Figure(5A).

Figure ( 5A) Document PVT definition (gas composition)

Click and inter (Tc, Pc) Will be calculated specific

gravity as shown in Figure(6A).

Figure ( 6A) Document PVT definition (critical properties)

C. input the PVT parameters; formation volumes factor, the fluid viscosity

and the system compressibility. Keep all values at their suggested

APPENDIX

5A

defaults. Click to create the new project. Al-Saphir main

screen is displayed.

4- Inter the formation volume factor β (STB/DAY) or other unites.

5- Viscosity of the fluid µ (cp) or other unites.

6- Inter the total compressibility ct (psi-1) or other unites.

2. The main screen is opened with the “Interpretation” page active. This

page (or panel) contains six icons and by clicking consecutively on the

icons from top to bottom, executing the dialogs and instructions, you

will follow exactly the default path of the basic workflow used in

pressure transient analysis.

Figure ( 7A) Al-Saphir main screen

3. Load Q.

A. Click on the icon. This will initialize the load sequence which is

normally a sequence of two dialogs. Specify an ASCII file in the

“Define data source” dialog. rat in the Example directory. A preview of

the file will be shown in the dialog as illustrate.

APPENDIX

6A

Figure ( 8A) Al-Saphir main screen

Click on the file and press the icon file and chose the file that contains

the data and click .

Figure ( 9A) load the flow rate Q

Or use the keyboard – spreadsheet to load the flow rate (q) with the time.

1- Inter the time production (tp)

2- Inter the flow rate at (tp)

3- Inter the final time of the test (∆tf )

APPENDIX

7A

4- Inter the flow rate at the test (q=0) when we shut the well

Figure ( 10A) Load step 1 – Define data source

Figure ( 11A) load the flow rate Q

B. Click to go to the Data Format dialog. Saphir has recognized

the file as valid and has automatically assigned the first column as

“Decimal Time” and the second as “Oil Rate”. This is known as “free

format”. The units are the correct units so there is no need to change the

formatting proposed by Saphir. However, the file contains steps and

values, thus you will choose the radio button.

APPENDIX

8A

Figure ( 12A) Load step 2 – Data Format

1- Lines format

Choose (field) because the formation is data field.

2- Time format

Choose steps: duration.

C. Click on to load the flow rate file. A history plot with the

loaded flow rate file in steps is displayed.

Figure ( 13A) loaded flow rate (history plot)

APPENDIX

9A

4. Loading pressure

A. Click on the icon

Figure ( 14A) load pressure p

Figure ( 15A) load pressure

Select type of data source choose keyboard-spreadsheet

1- Inter the time of test.

2- Inter the pressure.

APPENDIX

10A

Figure ( 16A) Load pressure Step 1 – Define data source

Click to go to the Data Format dialog.

Figure ( 17A) Load pressure Step 2 – Data format

Check the time and pressure unites

Lines format choose field

Time format choose points

APPENDIX

11A

B. Click on to load the pressure file. The history plot contains

now both the flow rate and pressure file.

Figure ( 18A) Load pressure Step 2 – Data format

5. Extracting delta P:

The next natural step is to extract the buildup pressure to the log log plot,

compute the Bourdet derivative to be displayed together with the delta P to

aid the interpreter to choose the model that may finally match the data and

yield to the results.

Figure ( 19A) History plot

APPENDIX

12A

A- Click on the icon. The first dialog is to choose the period to be

extracted, in this case of “build-up”. The second dialog is to set the

derivative smoothing, the filtrations (points/log cycle) and the starting

pressure (P at dt=0, often also called Pwf), the default value of the

pressure has been chosen by al-Saphir as the nearest pressure point to

the change in flow rate (when the flow rate goes to zero for the buildup).

Stay with the default values and click.

Figure ( 20A) Extract dialogs

B- Now al-Saphir screen contains new plots; the log log and the semilog

plot and history plot.

Figure ( 21A) after extract

APPENDIX

13A

6. Modeling:

Once the match is satisfactory the process changes to diagnostic mode. The

diagnostic phase includes the search for all flow regimes that may be

present in the response of the extracted period. This allows the interpreter

to choose the most appropriate model that includes all the flow regimes

identified. Then the next natural process is to run the model and obtain the

match which again will yield the results.

Figure ( 22A) log-log & semi-log plots

Click on the model icon. This will open the model dialog and you

set the combination of wellbore, well, reservoir and boundary models. The

default parameters of the model are displayed in the dialog and you can set

them manually. The default values of C (wellbore storage coefficient) and

kh (permeability thickness product) is deduced from the original automatic

match made by Saphir. Keep the default values as seen in figure (23A).

APPENDIX

14A

Figure ( 23A) Model dialog

Click generate the model is generated and immediately it can

be seen that there is an important mismatch between the model and the data

6.1 Wellbore model

Changing wellbore storage:

In case of multiphase flow at the well bore it is possible to have changing

well bore storage. The WBS starts at one value and then changes to a

second value where it remains constant.

Ci/Cf ratio of (initial C/final C)

Alpha is the dimensionless time of change of storage.

Decreasing WBS: derivative is sometimes to the left of the dp response

and often humps above it

Increasing WBS: initial pressure response in unit slope followed by a

flattening of the slope when the WBS increaser to its final value. The

derivative shows a valley corresponding to the flattening of the slope.

APPENDIX

15A

Figure ( 24A) changing well bore storage

6.2 Well model

A. fracture infinite conductivity

The well intersect a single fracture in the vertical plane

1- The flow is linear.

2- The well is at the center of the fracture.

3- No ∆p losses along the fracture length.

Log-log response

The pressure and derivative curve are parallel and both follow a straight

line with slope n=0.5.

Figure ( 25A) Infinite-Conductivity fracture

APPENDIX

16A

B. finite conductivity vertical fracture:

1- The wells intersect a single fracture in the vertical plane.

2- The flow is bilinear.

3- The well is at the center of the fracture.

4- There is ∆p losses along the fracture length.

Log-log response

The pressure and derivative curve are parallel and both follow a straight

line with slop to n = 0.25 after that the response starts to linear with slope

= 0.5.

Figure ( 26A) Infinite-Conductivity fracture

C. vertical limited entry:

The well produces from a perforated interval smaller than the total

producing interval.

Log-log response

At early times, radial flow may establish and this is masked by WBS. Then

spherical flow develops the derivative curve have slope =-0.5, later radial

flow in the full formation is achieved.

APPENDIX

17A

Figure ( 27A) limited entry response

D. horizontal

The well is horizontal and the vertical or slanted section is not perforated,

both the top and bottom boundary are sealing.

Log-log response

At first radial flow is present, and when the top/bottom boundaries are

reached, linear flow with n=0.5 slope later horizontal radial flow develops

in the formation.


APPENDIX

18A

6.3 Reservoir model

A. Homogeneous

The reservoir is homogenous, isotropic and has constant thickness.

Log-log response:

At early time the pressure response is effects by WBS with unit slope n=1,

when infinite acting radial flow (I.A.R.F) is established the pressure

derivative stabilized and follows a horizontal line


B. Two layers (double permeability):

Log-log response

1- At early times the layers are producing separately without cross flow.

2- At intermediate times, cross flow established (transition flow regime).

3- Later the pressure equalizes in the two layers (homogenous behavior of

the total system).

APPENDIX

19A

Figure ( 30A) Double Permeability Type-Curve

C. radial composite

The radial composite model is characterized by change in mobility and

storativity in the radial direction

Log-log response

1- The pressure behavior describes the homogeneous regime in the inner

region

2- After a transition a second homogeneous regime is achieved in the outer

region

Figure ( 31A) radial composite response

APPENDIX

20A

D. Linear composite

The linear composite model is characterized by change in mobility and

storativity in the linear direction

Log-log response

1- the pressure behavior describes the homogeneous regime in the inner

region.

2- after a transition a second homogeneous regime is achieved in the

outer region.

Figure ( 32A) linear composite response

6.4 Boundary models

A. Single boundary

One linear fault at some distance from the producing well limits the res.

Extension in one direction either sealing or water drive constant pressure

Log-log response

Before boundary effect the res. Response shows (I.A.R.F) after that two-

possible case may exist.

APPENDIX

21A

1- sealing fault: after boundary felt the permeability half of the initial on

Horner plot, boundary shown doubled straight line slop m2=2m1.

2- constant pressure: the water drive support produces, the derivative

drops with unit slope n= -1.

Figure ( 33A) linear boundary response

B. intersecting boundaries

Two intersecting boundaries sealing or constant pressure located at some

distance from the producing well the intersecting angle θ is always less

than 180

Log-log response

Before boundary effect the reservoir response shows (I.A.R.F) with

permeability k1 after that two-possible case may exist

1- Two sealing fault: there will be a single jump to the final stabilization,

at a value 360/q times the initial radial flow stabilization.

2- Constant pressure: if one or both the boundary is water drive the

pressure stabilizes and the derivative drops

APPENDIX

22A

Figure ( 34A) intersecting boundaries

C. parallel boundary

Two parallel boundaries sealing or constant pressure located at some

distance from the well.

Log-log response:

Before boundary effect the res. Response shows (I.A.R.F) with

permeability after that two-possible case may exist.

1- Two sealing fault: when the boundary is reached a liner flow with

n=0.5 slop established

2- Constant pressure: if one or both the boundary is water drive the

pressure stabilizes and the derivative drops.

Figure ( 35A) parallel faults

APPENDIX

23A

D. closed boundary

Each side can be either a sealing or constant pressure.

Log-log response

Before boundary effect the res. Response shows (I.A.R.F) with

permeability after that two-possible case may exist

1- Sealing boundary

Response on drawdown derivative: When all the sealing boundaries Are

reached both the pressure and the derivative curve follow a unit slop n=1

straight line.

Response on build up derivative: When all the sealing boundaries are

reached the res., Pressure tends to stabilize at average res. Pressure and the

derivative curve drops

2- Constant pressure

1- If any of the side acts as a constant pressure boundary the log-log

pressure curve tends to stabilize and the derivative drops

2- Because no depletion is present in this case the pressure derivative trend

is the same for both the buildup and drawdown periods.

Figure ( 36A) closed boundary

APPENDIX

24A

7. Improving:

A-Click on the improve icon. This will allow you to run a nonlinear

regression to improve the match between the model and data. Uncheck the

permeability k and run the regression on the wellbore storage constant C

and the skin. Click .

Figure ( 37A) improve

Figure ( 38A) improve process

Omega: storativity ratio; Lambda: interporosity coefficient

Al-Saphir screen now shows the model match on all the plots.

APPENDIX

25A

Figure ( 39A) al-Saphir screen

8. Sensitivity:

The final step in the workflow is the sensitivity. Click on the icon.

The dialog that follows. Will allow you to specify the parameters to run the

sensitivity on. In this case the choice is the skin value. Move the Skin from

the parameter selection area to the parameter values. And enter the

following sensitivity skin values: 1, 2, 4, 6, 8, 10, 14, and 20.

Figure ( 40A) sensitivity

APPENDIX

26A

Figure ( 41A) sensitivity

Click and a log-log plot with all the skin models will be created.

Figure ( 42A) sensitivity to skin

When you do not know the type of the reservoir and its boundary and the

type of well you can use the option (KIWI) When click The

suggestions window appears and the suggestions are not necessarily

correct after the selection click on as shown figure(43A). The

proposed plot will appear as shown figure (44A).

APPENDIX

27A

Figure ( 43A) kiwi solution

Figure ( 44A) kiwi plot

To display the results, we have obtained in the form of a paper we click

on A dialog box appears that identifies the plot to be extracted and

click as shown figure(45A).

APPENDIX

28A

Figure ( 45A) output window

Figure ( 46A) outputs

Documents

Diagnosis and Analysis of Pressure Transient in Fractured ...naturally fractured reservoir and engineering evaluations rely on direct and indirect sources of information. Direct sources