DECLINE CURVE ANALYSIS FOR NATURALLY FRACTURED RESERVOIRS

CHAPTER ONE

1.0 INTRODUCTION

Fractures are a universal element in sedimentary rock layers, so

much that they are virtually omnipresent in outcrops of

sedimentary rocks. Think of all the outcrops of sedimentary rocks

that you have ever seen and try to recall a layer that was

completely un-fractured, with the possible exception of extremely

ductile rock, such as salt or certain shale, you will not be able

to recall any un-fractured rocks simply because they do not

exist. Further, it has been demonstrated over and over again that

the vast majority of fractures observed in outcrop are not solely

the result of surface conditions. In other words, the fractures

seen in outcrop also exist in the subsurface. Therefore, it

follows that hydrocarbon reservoirs in sedimentary rock all

contain fractures and most of them are fractured enough to be

treated as fractured reservoirs.

Though the geological fractures necessary to conclude that

fractures are common in the subsurface have been known for at

least the last half century, the practice of treating reservoirs1

as fractured rock masses has been extremely slow in becoming a

standard industry practice. Why is this so? Probably the greatest

contributor to the widespread reluctance to face the reality of

fractured reservoirs is because fractured reservoirs are

extremely complex and therefore, much more difficult to deal with

than are un-fractured reservoirs. The complexity comes from the

vast number of both dependent and independent variables that

dictate final reservoir response. Consider for a minute just a

few of the obvious, straightforward reservoir variables, and

their interactions, that must be included in a reservoir

analysis. Calculating reservoir storage depends on knowing both

matrix and fracture porosities. Fracture permeability, matrix

Permeability, and especially their interaction, all contribute to

the behavior of a given reservoir. Fracture geometry, fracture

spacing, fracture surface area, and fracture opening all combine

with fracture morphology and pore space distribution to create

true reservoir permeability and/or permeability anisotropy. Fluid

pressure decline with time changes the value of some variables

2

but not the value of others. Therefore, initial calculations do

not apply throughout the life of the reservoir and some

parameters must be recalculated at several intervals during the

life of the reservoir.

Anyone who has dealt with fractured reservoirs realizes that

these variables are only a few of the numerous variables that

have to be evaluated and properly combined in order to predict

reservoir performance. Is there any doubt, then, that reservoir

complexity is a major contributor to the reluctance even to

attempt systematic treatment of reservoirs as fractured rock

masses?

Another factor that is a deterrent to doing systematic fractured

reservoir analysis is that almost all fractured reservoirs

respond in a manner unique to that specific reservoir. That is,

despite the existence of a good, working fractured reservoir

classification, each fractured reservoir responds in its own

distinctive way. As a consequence, applying general rules of

3

thumb to specific fractured reservoirs can be dangerously

misleading.

Indeed, fractured reservoirs are more complicated than matrix

reservoirs, and they do require more time and money to be

evaluated correctly. The tendency is to ignore the presence and

effect of natural fractures for as long in the field history as

possible. The problems with this denial or avoidance include:

1) Often irreparable loss of recovery factor;

2) Primary recovery patterns that is inappropriate for

secondary recovery;

3) Inefficient capital expenditure during development;

4) Drilling of unnecessary in-fill wells; and

5) Improper assessment of economic opportunities.

It is important to determine the effect of natural fractures in

our reservoirs as early as possible so that our evaluations and

planning can be done correctly from day one. Fracture denial does

nothing positive for our exploration and development activities

and can only lead to poorer technical and economic performance.

4

Fractured reservoirs are very complicated and difficult to

evaluate. Effective evaluation, prediction, and planning in these

reservoirs require an early recognition of the role of the

natural fracture system and a systematic approach to the

gathering and analysis of pertinent data. However, care should

always be taken to make sure that the degree of analysis and

evaluation is commensurate with the particular problem being

addressed. It is easy to get lost in detail and data acquisition,

and lose sight of the economic questions.

Constant producing pressure solutions that define declining

production rates with time, for a naturally fractured reservoir,

with transient inter-porosity flow represented. The solutions for

the dimensionless flow rate are based on a model presented by

Cinco-Ley8, Samaniego and Kucuk22.

The model was extended to include constant producing pressure in

both infinite and finite systems. The results obtained for a

finite no flow outer boundary are new and surprising. Similarly

to Da Prat et al.11, it was found that the flow rate for

5

conditions of fracture skin greater than 10, shows initially

rapid decline, becomes nearly constant for a period, and then a

final decline in rate takes place. The same criterion established

by Da Prat et al11. for the estimation of the outer radius of a

reservoir, reD, requiring that the almost constant flow rate

period be reached by the data is applicable to the present

transient inter-porosity flow model. However, the estimation of

reD from this method is higher for fixed values of the ω and λ

parameters. A field example is presented to illustrate the method

of type curve matching for a naturally fractured reservoir with

transient inter-porosity flow.

The values of ω and λ are determined from the best match and this

is particularly important for the case of production forecasting

by numerical simulation. The results show that the initial

decline could be a key factor in deciding whether to complete or

abandon a well, and for a practical viewpoint, given an initial

value for the flow rate, it is important to know the time

required to deplete the two porosity system.

6

1.1 DECLINE CURVES ANALYSIS

Evaluation of reservoir parameters through decline curve analysis

has become a common current practice (Fetkovich15 1980; Fetkovich

et al15. 1987). The main objectives of the application of decline

analysis are to estimate formation parameters and to forecast

production decline by identifying different flow regimes.

During the period of severe production, curtailment, which is now

behind us, production-decline curves lost most of their

usefulness and popularity in prorated areas because the

production rates of all wells, except those in the stripper

class, were constant or almost constant. While production-decline

curves were thus losing in importance for estimating reserves, an

increasing reservoir consciousness and a better understanding of

reservoir performance developed among petroleum engineers. This

fact, together with intelligent interpretation and use of

electric logs, sore- analysis data, bottom-hole pressure behavior

and physical characteristics of reservoir of reservoir fluids,

7

eliminated a considerable part of the guess work on previous

volumetric methods and put reserve estimates, based on this

method, on a sound scientific basis. At the same time, a number

of ingenious substitutes were developed for the regular

production\decline curve, which made it possible to obtain an

independent check on volumetric estimates in appraisal work, even

though the production rates were constant.

1.2 DEVELOPMENT OF DECLINE CURVE ANALYSIS

The two basic problems in appraisal work are the determination of

a well’s most probable future production. Sometimes one or both

problems can be solved by volumetric calculations, but sufficient

data are not always available to eliminate all guess work. In

those cases, the possibility of extrapolating the trend of some

variable characteristics of such a producing well may be of

considerable help. The simplest and most readily available

variable characteristic of a producing well is its production

rate, and the logical way to find an answer to the two problems

mentioned above, by extrapolation is to plot this variable

8

production rate either against time or against cumulative

production, extending the curves thus obtained to the economic

limit. The point of intersection of the extrapolated curve with

the economic limit then indicates the possible future life or the

future

1.3 RESERVOIR CHARACTERISTICS AND DECLINE CURVES

In order to analyze what influence certain reservoir

characteristics may have on the type of decline curves, it was

first assumed that we are dealing with the idealized case of a

reservoir where water drive is absent and where the pressure is

proportional to the amount of remaining oil. It was further

assumed that the productivity indexes of the wells are constant

throughout their life, so that the production rates are always

proportional to the reservoir pressure.

1.4 FRACTAL GEOMETRY

9

Evaluation of reservoir parameters through well-test and decline

curve analysis is a current practice used to estimate formation

parameters and to forecast production decline identifying

different flow régime, respectively. From practical experience,

it has been observed that certain cases exhibit different

wellbore pressure and production behavior. The reason for this

difference is not understood completely, but it can be found in

the distribution of fractures within a naturally fractured

reservoir (NFR). Currently, most of these reservoirs are studied

by means of Euclidean models, which implicitly assume a uniform

distribution of fractures and that all fractures are

interconnected. However, evidence from outcrops, well logging,

production-behavior studies and dynamic behavior observed in

these systems, in general indicates the above assumptions are not

represented in the systems. Fractal theory considers a non-

uniform distribution of fractures and the presence of fractures

at different scales, thus it can contribute to explain the

behavior of many fractured reservoirs. The objective of this

10

project is to investigate the production decline behavior in a

NFR exhibiting single porosity with fractal networks of

fractures. The diffusion equations used in this work are a

fractal-continuity expressions.

In spite of all the work done on decline-curve analysis, the

problem of fully characterizing a NFR exhibiting fractal geometry

by means of production data has not been addressed in the

literature. Thus, the purpose of this work is to present

analytical solutions during both transient and boundary dominated

flow periods and to show that it is possible to characterize a

NFR having a fractal network of fractures with production-decline

data. This would be discussed in detail in chapter three.

1.5 MODELING OF NATURALLY FRACTURED RESERVOIRS with DUAL

POROSITY

Much work has been done on the pressure transient modeling of

naturally fractured reservoirs. However, the rate response and

producing capacity of these reservoirs have not received adequate

attention. This work examines flow-rate decline behavior of11

naturally fractured reservoirs. Naturally fractured reservoirs

are heterogeneous porous media which consist of fractures and

matrix blocks. The matrix blocks store most of the fluid, but

have low permeability. On the other band, the fractures do not

store much, but have extremely high permeability. Most of the

reservoir fluid flows from the matrix blocks into the wellbore

through the permeable fractures. Therefore, the producing

capacity of a naturally fractured reservoir is governed by

matrix-fracture fluid transport capacity, which is called inter-

porosity flow.

Inter-porosity flow was assumed to occur in pseudo-steady state

condition,

12

……………………………1.0

13

Fig 1.1

To describe flow in naturally fractured reservoirs, double

porosity model has been widely used.

Two parameters were defined to characterize the double porosity

behavior:

The inter-porosity flow coefficient:

………………………1.1

Where kf is the fracture permeability, km is the matrix

permeability, rw, is the wellbore radius and α, a geometrical

factor with dimensions of reciprocal area.

The fracture storativity:

…………………………1.2

Where:

Øf is the fracture porosity, 14

Øm the matrix porosity and,

Cf and Cm, the corresponding fluid compressibilities.

The figure above shows the schematic of a naturally fractured

reservoir and its double porosity idealization. This concept was

first proposed by Barenblattet a1.Transient pressure behavior for

this model has been studied by many researchers.

Fig 1.2

15

This work indicates that the double porosity model predicts an

initial high flow-rate followed by sudden rate decline and a

period of constant flow-rate (the inter-porosity flow period).

CHAPTER TWO

2.0 LITERATURE REVIEW

Naturally Fractured Reservoirs (NFR) consists of heterogeneous

porous media where the openings (fissures and fractures) vary

considerably in size. Fractures and openings of large size

generate vugs and interconnected channels, whereas the fine

cracks form block systems which are the main body of the

reservoir.16

2.1 FRACTAL THEORY

Different solutions have been proposed during both transient

(Ehig-Economides13 and Ramey30 1981; Uraiet and Raghavan 1980) and

boundary-dominated (Fetkovich15 1980; Fetkovich et al.15 1987

Ehig-Economides13 and Ramey30 1981; Arps 1945) flow periods. Both

single and double porosity (Da Prat et al. 1981; Sageev et al13.

1985) systems have been addressed. During the boundary-dominated

flow period in homogeneous systems there is a single production

decline but for NFRs in which the matrix participates there are

two decline periods with an intermediate constant-flow period (Da

Prat et al. 1981; Sageev et al13. 1985).

Carbonate reservoirs contain more than 60% of the world’s

remaining oil. Yet, the very nature of the rock makes these

reservoirs unpredictable. Formations are heterogeneous, with

irregular flow paths and circulation traps. In spite of this

complexity at present, all studies on constant-bottom-hole

pressure tests found in petroleum literature assume Euclidean or

standard geometry is applicable to both single-porosity

17

reservoirs and NFRs (Fetkovich15 1980; Fetkovich et al15. 1987

Ehig-Economides13 and Ramey30 1981;Uraiet and Raghavan 1980; Arps

1945; Da Prat et al 1981; Sageev et al. 1985) even though real

reservoirs exhibit a higher level of complexity.

Specifically, natural fractures are heterogeneities that are

present in carbonate reservoirs on a wide range of spatial

scales. It is well known that flow distribution of fractures

(i.e. geometrical complexity). There could be regions in the

reservoir with clusters of fractures and others without the

presence of fractures. The presence of fractures at different

scales represents a relevant element of uncertainty in the

construction of a reservoir model. Thus, highly heterogeneous

media constitutes the basic components of an NFR. So, Euclidean

flow models have appeared powerless in some of these cases.

Alternatively, fractal theory provides a method to describe the

complex network of fractures (Sahimi and Yortsos39 1970)

The power law behavior of fracture-size distributions,

characteristics of fractal systems has been found by Laubach and18

Gale23 (2006). Distributions of attributes such as length, height

or aperture can frequently be expressed as power laws. Scaling

analysis is important because it enables us to infer fractures

attributes such as strike, number of fracture sets and fracture

intensity for larger fractures from the analysis of micro-

fractures found in oriented sidewall cores. This approach offers

a method to overcome fracture-sampling limitations with micro-

fractures as proxies for related macro-fractures in the same rock

volume. (Laubach and Galle23 2006; Ortega et al, 2006)

The first fractal model applied to pressure transient analysis

was presented by Chang and Yortsos6 (1990). Their model describes

an NFR that has, at different scales, poor fracture connectivity

and disorderly spatial distribution in a proper fashion. Acuna et

al1. (1995) applied this model and found the well bore pressure

is power-law function of time. Flamenco-Lopez17 and Camacho-

Velazquez (2003) demonstrated that to characterize a NFR fully

with fractal geometry, It is necessary to analyze both transient

and pseudo steady-state flow well pressure tests or to determine

19

the fractal-model parameters from porosity well logs or another

type of source.

Regarding the generation of fracture networks, Acuna et al1.

(1995) used a mathematical method for this purpose.

2.2 TYPE CURVE TECHNIQUE FOR NATURALLY FRACTURED RESERVOIR –

DUAL POROSITY

In the past, the analysis of short time flow rate data to obtain

reservoirs parameters was not a common technique, mainly due to

the difficulties in obtaining accurate measurements of the flow

rate as compared to high resolution pressure measurements.

However, the advent of new production tools, like the real time

flow meter has made possible the analysis of simultaneously

measured pressures and flow rates in a transient well test. The

advantage of incorporating the measured flow rate, is that the

type curve matching technique is improved, giving more

information regarding the uniqueness as to the type of reservoir

being dealt with, i.e., fractured, multi-layer, composite, etc.

In a fractured formation we may have wells initially producing at a high rate

20

where in some cases, production starts to decline after a few hours without any

clear explanation. Therefore, analyzing the transient flow rate

behavior in a well completed in a fractured formation will add

more information that will result in a more complete evaluation

analysis. From an engineering and economic viewpoint, the initial decline

could be a key factor in deciding whether to complete or abandon a well.

Naturally Fractured Reservoirs have been studied extensively in the

petroleum literature. One of the first such studies was published

by Pirson in 1953. Pollard presented one of the first pressure

models available for interpretation of well test data; however,

the graphical technique proposed is susceptible to error caused

by approximations in the mathematical model.

The first to present a detailed discussion of the transient

radial flow of a slightly compressible fluid through a naturally

fractured reservoir were Barenblatt and Zheltov and Barenblatt et

al.; these authors assume that the flow occurs only in the

21

fracture medium and that the matrix blocks are a source that

delivers flow to the fracture system and that this flow could be

considered under pseudo-steady state flow conditions.

The double porosity concept was introduced in 1960 by

Barenblattet a1. As stated above, it assumed the existence of two

porous regions of distinctly different porosities and

permeabilities within the formation. Also, a continuum was

assumed, where any small volume contained a large proportion of

both media. Hence each point in space had associated with it two

pressure values, Pi in the permeable medium and I?, in the porous,

less permeable medium. Inter-porosity flow was assumed to occur

in pseudo-steady state condition;

…………………2.1

The solution was completed in 1963 by Warren and Roots who

described the reservoir geometry as an orthogonal system of

continuous, uniform fractures, each parallel to the principle

22

axis of permeability. Two parameters were defined to characterize

the double porosity behavior:

The inter-porosity flow coefficient:

………………………..2.2

Where kf is the fracture permeability, km, the matrix

permeability, rw,the wellbore radius and α, a geometrical factor

with dimensions of reciprocal area.

The fracture storativity:

…………………….2.3

Where:

Øf, is the fracture porosity,

Øm is the matrix porosity and,

Cf andCm,are the corresponding fluid compressibilities.

Pseudo-steady state flow was assumed for the matrix as a suitable

approximation for late time data. The results were analyzed on

23

semi-log plots, characterizing the inter-porosity flow region for

different values of X and w.

………………………..2.4

Where

f (s) = ………………………….2.5

Warren and Root obtained analytical solutions useful for well

test analysis found that data in NFR by using the formulation of

Barenblatt et al.; they for a pressure test show two parallel

semi-log straight lines, whose slope is related to the flow

capacity of the formation. These theoretical results were

supported later by two field examples presented by the same

authors9; this model is considered the forerunner of modern

interpretation of two porosity systems.

24

Odeh29 presented a model also assuming pseudo-steady state flow

conditions in the matrix and concluded that a fractured system

behaves like a homogeneous one. Odeh29 suggested in 1965 that

wellbore storage effects dominated pressure response at early

times, and hence the first straight line may not be observed.

Later, Adams et al. presented field examples of pressure test of

a fractured reservoir, and used a radial discontinuity model as

an interpretation tool. The field data exhibited two straight

line portions, such that the first had a slope twice the slope of

the second.

Kazemi20 was first to consider transient matrix flow in a

numerical radial model assuming the NFR made up of horizontal

fractures inter-bedded with matrix strata; his results are

similar to those of Warren and Root, with the exception of a

smooth unsteady state transition behavior in between the two

25

semi-log straight lines, compared to the pseudo-steady state zone

of Warren and Root model.

Gringarten and Witherspoon reviewed the theory on transient

pressure analysis for both hydraulically and NFR.

Type curves for analyzing wells with wellbore storage and skin in

double porosity reservoirs were introduced by Bourdet and

Gringarten. It was claimed that even in the absence of the first

straight line on the semi-log plot, a log-log type curve analysis

could yield all reservoir parameters. Dimensionless parameters

were defined. The idea of computing fissure volume and matrix

block size was presented but was not convincing.

Later de Swaan presented analytical transient solutions for a

well producing at constant rate; his model exclusively involves

flow properties and dimensions of the fracture and the matrix

systems and introduced new diffusivity definitions useful for

reservoir characterization.

26

Crawford et al. presented some of the best field examples of

pressure transient tests on NFR, concluding that a test properly

conducted can be interpreted by the Warren and Root model and

indicated that ω and λ should be obtained from field performance.

Strobel et al. presented another remarkable field behavior

example of a naturally fractured gas reservoir, demonstrating

that both fracture permeability and fracture porosity can be

estimated from type curve analysis of pressure buildup,

interference and pulse tests.

Mavor and Cinco-Ley presented solutions for wellbore storage and

well damaged conditions for a NFR; they used the pseudo-steady

state matrix flow condition, and also considered production, both

at constant rate and at constant pressure. However, little

information is presented concerning the effect of the size of the

system on pressure buildup behavior.

27

Streltsova34 presented a complete review on the work done on the

behavior of NFR. He showed that the transient flow in the matrix

did not cause an inflection point on the pressure profile on the

semi-log plot. A transition straight line was proposed with a

slope equal to one half the slope of the early or late time

straight lines. This facilitated a Horner plot analysis.

Najurieta27 further advanced the de Swaan´s model by presenting

an approximate solution, showing that pressure behavior can be

fully described by five basic parameters.

Kucuk and Sawyer22 described a comprehensive model for gas flow

in a NFR; they considered transient flow in both cylindrical and

spherical matrix blocks.

Gringarten discussed the interpretation of pressure data and

clearly showed the relationship among the parameters used in

different models. Da Prat et al. have discussed the application

28

of the Muskat method to NFR to calculate the permeability-

thickness product. A remarkable work done on the analysis of

pressure data for NFR under practical conditions (influenced by

wellbore storage and skin), has been presented by Bourdet and

Gringarten, Gringarten et al., and Gringarten. They discussed the

use of a new type curve for both identification of the flow

periods and estimation of parameters. Although the use of the

pseudo-steady state matrix flow model was recommended, these

authors where the first to identify the semi-log straight line

during the transition period for the transient state matrix-

fracture flow conditions; however, no application of this feature

was discussed.

Cinco-Ley and Samaniego8 considered transient inter-porosity flow

and included wellbore storage and skin effect; the matrix

fracture transfer was presented in a convolution form which

permits the use of different matrix blocks geometries.

29

Streltsova34 andSerra et al. have presented detailed studies on

the pressure behavior of NFR; both papers thoroughly treat the

transition period commented on the previously published papers,

showing again that during this flow period a semi-log plot of pwD

vs. tD exhibits a straight line of slope 0.5756.

Chen et al7. presented a model with transient state matrix-

fracture flow conditions using linear flow in the matrix for

bounded NFR. Moench and Ogata, discussed the consideration of the

skin in the fractures in NFR.

Cinco-Ley, Samaniego and Kucuk8 presented a model with transient

inter-porosity flow, that considers multiple matrix block size

and matrix-fracture flow restriction (fracture skin in this work),

for a well producing at constant rate, with wellbore storage and

skin in an infinite system; the model that considers only one

matrix block size without fracture skin is the same model of

Cinco-Ley and Samaniego8. With this model it is possible to

30

generate the pseudo-steady inter-porosity flow using a big enough

matrix-fracture flow restriction.

Although decline curve analysis is widely used, specific methods

for NFR with transient inter-porosity flow are not available. It

is the objective of this project to develop a model with the

above characteristics to study decline curve analysis for a NFR.

The Cinco-Ley, Samaniego and Kucuk8 model would be chosen as the

basis for this work due to its consideration of transient inter-

porosity flow, with multiple sizes, matrix block and matrix-

fracture flow restriction. In this study the model would be

extended using constant producing pressure in both infinite and

finite systems, with only one matrix block size and new

approximate analytical solutions would be presented for small and

long times.

31

CHAPTER THREE

3.0 METHODOLOGY

32

After a fractured reservoir has been discovered with its initial

well, early evaluations determine if the prospect will be

economic to develop. This requires proper reservoir management

based on (1) the knowledge of how the reservoir compares to

already-developed reservoirs having similar properties, and (2)

the knowledge of how the geological, engineering, and petro-

physical data integrate into a coherent reservoir/depletion

model. The following sections will address these aspects in

detail.

3.1 CLASSIFICATION OF FRACTURED RESERVOIRS

Reservoir Types

Once the origin, continuity, and reservoir properties of the

fracture system are determined, and the flow interaction between

the fractures and the matrix has been investigated, the reservoir

must be classified on the basis of what positive effects the

fracture system provides to overall reservoir quality. The

following classification has proven useful in this regard:

33

Type 1: Fractures provide the essential reservoir porosity

and permeability.

Type 2: Fractures provide the essential reservoir

permeability.

Type 3: Fractures assist permeability in an already

producible reservoir.

Type 4: Fractures provide no additional porosity or

permeability but create significant reservoir anisotropy

(barriers).

This classification is an expansion of that proposed in Hubbert

and Willis (1955). The first three types describe positive

reservoir attributes of the fracture system. The fourth, while

somewhat non-parallel to the others, de-scribes those reservoirs

in which fractures are important not only for the reservoir

quality they impart, but for the inherent flow anisotropy and

reservoir partitioning they create. A depiction of this

classification in graph form is given in Figure 3.1.

34

The advantages of this classification are that it delineates the

parameters of the fracture system, which are most important in

quantifying a particular reservoir, and it allows for prediction

of the types of production problems that are likely to occur.

In the first type of fractured reservoir, where the fracture

system provides the essential porosity and permeability, an early

calculation of fracture porosity or fracture volume attainable

per well is of paramount importance. An accurate knowledge of

this volume must be gained as early as possible to evaluate total

reserves obtainable per well and to predict if initially high

flow rates will be maintained or drop rapidly with time. In these

estimations, fracture width and fracture spacing values are

critical. Accurate fracture porosity calculations in fractured

reservoir Types 2 through 3 are much less important because the

fracture system provides only permeability—the matrix supplies

any significant porosity or storage volume. In those types, the

matrix pore volume (usually several orders of magnitude greater

than the fracture volume) overshadows the fracture volume so much

35

that to make an accurate, early calculation of fracture porosity

is unimportant.

Figure 3.1: A schematic cross plot of percent reservoir porosity

versus percent reservoir permeability (percent due to matrix

versus percent due to fractures) for the fractured reservoir

classification used by this author. These reservoirs, however, an

early knowledge of fracture/matrix interaction is extremely

important to determine whether the matrix porosity can be drained

by the fracture system.

36

Figure 3.1

3.2 FRACTURED RESERVOIRS WITH FRACTAL GEOMETRY

RATE BEHAVIOR FOR SINGLE-POROSITY NFRs

Appendix A presents decline curve for infinite reservoirs and

constant bottom-hole pressure, in which the rate responses

corresponding to the diffusion equations from Chang and Yortsos6

(1990) and O’Shaughnessy and Procaccia28 (1985). Equation A-9 and

Metzler et al25. (1994), equation A-26 are compared with the

traditional Euclidean response. The inversion of these

expressions was attained with the Stehfest33 algorithm

(Stehfest33, 1970). The parameter dmf is fixed and four values of

θ are considered. We can observe that big differences exist at

late times between the Euclidean and fractal responses. As

expected, the flow rate from fractal systems is smaller than that37

from Euclidean systems in which diffusion is faster. Thus, an

analyst cannot use the semi-logarithmic approach or conventional

type-curve analysis to interpret transient- rate data with

fractal behavior. Also, there are differences between the OP and

MGN results, demonstrating that the inclusion of the time

fractional derivative has a definite effect on the rate response.

The long-time behavior for an infinite reservoir is derived in

APPENDIX A and is given as follows:

……………3.01

Where:

……………….3.02

When γ = 1, the OP short-time behavior is attached (see equation

A-11) with θ different from 0. From equation 16, we can observe

that at long times during the transient period, a log-log plot of

rate vs. time will yield a straight line with slope γv. The 38

short-time approximation for the rate in the time domain is given

by:

…………..3.03

When γ = 1, the OP long-time behavior is attained (see equation

A-13). Figure 2 presents some of the rate solutions from figure 1

with their corresponding short and long-time approximations. We

can observe that both approximations reproduce the correct rate

behavior. For practical purposes, however only the long-time

approximation will be considered.

Figure 3 shows the cumulative production during the transient

flow period corresponding to the rate responses presented in

figure 2 and the long-time approximation given by equation A-29

as follows:

39

……………3.04

We can observe that the above approximation reproduces the actual

values for both the OP and the MGN diffusion equations. As

expected, the amount of oil that can be extracted from fractal

systems is smaller than the amount that can be extracted from

Euclidean reservoirs because in fractal systems the oil is

produced from infinite connected clusters only. Thus, it is

important to determine the appropriate place for a new well for a

Euclidean reservoir. This is even more crucial for fractal

systems.

Figure 4 presents a comparison of fractal rate responses with

both OP and MGN expressions and the traditional single porosity

response. Two values of ReD are considered. Again, there are big

differences between fractal and Euclidean responses during both

transient and boundary-dominated flows period and the flow rate

from fractal systems is smaller than that from the Euclidean

40

system because the diffusion is slower in fractal reservoirs than

in traditional ones. Also, we can observe some discrepancies

between rate values obtained with the OP and MGN equations.

The long-time behavior for closed reservoirs and OP diffusion

equation is given as follows:

………….3.05

Where a1, b1 and b2 are given by equations B-5, B-6 and B-7

respectively. The corresponding expression for the MGN equation

is as follows:

………….3.06

Form equations 3.05 and 3.06, we can observe that the long-time

approximations with the OP and MGN equations do not coincide.

Figure 4 also presents the corresponding long-time approximations

from the OP and MGN formulations. We can observe that at long

41

times all approximations collapse into the appropriate fractal

responses.

3.3 MODEL DEVELOPMENT for DUAL POROSITY USING TYPE CURVES

The basic partial differential equations for fluid flow in a two-

porosity system were presented by Warren and Root in 1963. The

model has been extended by Mavor and Cinco Ley8 (1979) to include

wellbore storage and skin effect. Da Prat11 (1981) extended the

model and developed a method to determine the permeability

thickness product, Kh.. Deruyck et al. (1982) applied with

success the warren and Root model to study interference data from

a geothermal field.

The basic partial differential equations are: Da Prat (1981).

= …….3.07

Initial condition:

………………..…3.08

Internal boundary: constant pressure and skin.

42

. …………………..3.09

External boundary: infinite reservoir,

………………..3.10

For closed reservoir;

…………………..3.11

The matrix to fracture flow functions are given as:

For strata:

………………..3.12

For spheres:

………………..3.13

The dimensionless flow rate into the wellbore is given by:

………………..3.14

The cumulative production is related to the flow rate by:

………………….3.15

3.4 Method of Solution

43

A common method for solving a flow equation under the conditions

given is to use the Laplace transformation. The advantages of

this method have been described by Van Everdingen and Hurst38.

The equations are transformed into a system of ordinary

differential equations which can be solved analytically. The

resulting solution in the transformed space is a function of the

Laplace parameter and the radius. To obtain the solution in real

time, the inverse Laplace transform is used. In our work, the

inverse was found using the Stehfest algorithm this approach was

introduced in the reservoir flow studies by Ramey, and used has

been successfully by many authors. Included in this work is a

short and longtime analysis, which provides simple expressions in

real time; these expressions can be used to verify results

obtained from the numerical algorithm, as well as to select the

adequate N parameter (equal to 10 for the present study) to be

used to perform the numerical inversion, in addition to being

useful in interpretation of results. The analytical solutions in

44

Laplace space for both the infinite and the closed outer

boundaries are given in the following sections.

3.5 INFINITE OUTER BOUNDARY

The transient solution obtained in this work for the

dimensionless flow rate is given by:

…………………..3.16

Where the transfer function is:

…………………..3.17

For strata:

…………………..3.18

45

And for spheres:

…………………..3.19

For small times, a solution to Eq. 3.19 can be obtained

substituting the modified Bessel functions by their asymptotic

expansions. The dimensionless flow rate can be expressed:

…………………..3.20

For Sw = 0:

…………………..3.21

In terms of cumulative production:

…………………..3.22

For Sw = 0;

46

…………………..3.23

For conditions of ω =1 and small times, Eq. 3.22 is identical to

that presented by van Everdingen and Hurst38. As previously

stated the expression obtained for the flow rate can be

associated with a homogeneous reservoir through an effective

time, . Thus, initial production from a NFR in an

infinite medium does not detect the presence of the matrix

porosity; it behaves like a homogeneous reservoir. For long times

the solution depends the matrix-fracture surface exposed to flow;

it can be derived by making a long time approximation for the

general solution expressed by Eq. 3.20; the solution obtained in

this work is given by:

…………………..3.24

If ω = 1 and Sw = 0, the solution reduces to that previously

reported b Jacob and Lohman

47

…………………..3.25

The results obtained through transient inter-porosity flow model,

without fracture skin, two values of ω and four values of λ,

does not show constant flow rate period and give more production

that the case with pseudo-steady state transfer. Also, at long

times, the solution approaches that for the homogeneous case, as

shown in Fig. 4.01.

To generate the pseudo-steady-state flow solution the transient

inter-porosity flow model is used with a value of fracture skin =

6, two values of ω and five values of λ. It can be observed that

the bigger λ, the sooner starts the transition flow, as shown in

Fig. 4.02.

The same analysis to that of Da Prat et. al., for a non-

communicating matrix, λ = 0, was done. Fig. 4.03 presents several

curves as function of ω, at all times the solution depends on tD’

= tD/ω.

48

To generate the pseudo-steady-state flow solution the model with

transient inter-porosity flow is used with a fracture skin = 6

and a non-communicating matrix, λ = 0, as shown in Fig. 4.04. The

ranges used are: and .

3.6 CLOSED OUTER BOUNDARY

Fetkovitch15 discussed the findings of Tsarevich and Kuranov36,

regarding that the exponential decline is a long time solution of

the constant pressure case. The solution obtained in this work

for the dimensionless flow rate, in the Laplace space is given

by:

…………………..3.26

Where:…………………..3.27

…………………..3.28

49

For short times, as for the homogeneous system, there is no

dependence on drainage radius, which means that the system

behaves as an infinite medium. For intermediate times, the value

of the flow rate (during the almost constant rate period) depends

strongly on Cf,Sf, and ω.

For long times, the flow rate given by Eq. 20 can be expressed in

terms of time as:

…………………..3.29

And for the cumulative production:

…………………..3.30

At long times, for a homogeneous reservoir an exponential decline

can be observed for the constant producing pressure case. Thus,

results for homogeneous systems can be extended to fractured

reservoirs. It can be concluded that, as previously stated in a

fractured reservoir, the final decline takes place later in time

50

as compared to the homogeneous case (ω=1). This implies that, it

takes longer time to deplete a fractured system. Eq. 3.29 should

represent the homogeneous solution when either ω = 1 or

tends to infinite.

Taking limits in Eq. 3.29 yields (using L´hopital rule´s):

…………………..3.31

At long times, for a homogeneous system, the flow rate becomes

zero.

At long times (tD ∞), from equation 3.30, the cumulative

production for a naturally fractured reservoir is given by:

…………………..3.32

The long time solution can be used to explain the observed period

of constant flow rate, qD.

51

As time increases, the exponential term in Eq. 3.30 begins to

dominate until the flow rate becomes zero. The series expansion

for the exponential is given as follows:

…………………..3.33

For small argument:

…………………..3.34

Then,

…………………..3.35

Requiring that:

…………………..3.36

From both practical and economic point of view, given an initial

value for the flow rate, it is important to know how long it

takes to completely deplete the fractured reservoir, the flow

rate starts to decline when it reaches the approximate value of

the flow rate given by Eq. 3.30 , which can be expressed as:

52

…………………..3.37

Results obtained with the model with transient inter-porosity

flow, without fracture skin, for two values of ω, five values of

λ and reD=50 are presented in Fig.4.6. It is shown that for the

flow rate first shows a rapid decline, and then it presents a

linear behavior for a long period (the almost constant flow rate

period is not shown) and is longer for smaller λ , after which a

final rate decline takes place, as well as at long times the

solution is dominated by boundary effects.

It can be noticed from a comparison of Fig. 4.6 that the

cumulative production for the case of transient matrix fracture

flow is higher than for the pseudo-steady state case.

However, when using transient inter-porosity flow with fracture skin= 6

(given two values of ω and five values of λ), the results are

surprising: the flow rate at first shows a rapid decline and then

the behavior becomes almost constant for a long period (equivalent

to pseudo-stationary flow), after which a final rate decline takes53

place and shows that the bigger λ, the sooner the transition flow

starts, as shown in Fig. 4.06.Thus, compared to the homogeneous

case (ω =1), a longer time is required to deplete a two porosity

system. The analysis similar to Da Prat et. al., for a non-

communicating matrix, λ =0, where several curves are shown as a

function of ω , in all times the solution depends on tD´ = tD/ω,

is presented in Fig. 4.07.

As previously stated, to generate the pseudo-steady flow was the

transient inter-porosity model with fracture skin =6 conditions

for the specific case of a non-communicating matrix, λ =0, the

solutions, are shown in Fig.4.08

3.7 PRODUCTION FORECAST ANALYSIS

For the observed decline in flow rate from an engineering and

economic point of view; the initial decline could be a key factor

in the completion or abandonment of a well.

Decisions concerning production forecast and estimates of the

size of fractured reservoirs should not be based only on the

observed initial decline. Ignoring the presence of a fractured

54

system can lead to a great error on the estimation of the

cumulative production.

Let us start the analysis of the initial decline by considering

the simplest case of a non-communicating matrix, λ = 0. In this

case, the behavior is the same as that for a homogeneous system,

but with tD´ = tD/ω. Figs. 4.07 and 4.08 show the dimensionless

flow rate behavior, in terms of qDvs. tD for different values of

ω, for values of the fracture skin 0 and 6, respectively. All

curves show a defined decline as the final depletion state

approaches. An expression for the flow rate can be derived from

the dimensionless wellbore pressure function for constant rate

production after the onset of pseudo steady state.

Van Everdingen and Hurst showed that knowing the pressure in the

well, it is possible to find the flow rate by applying the

inverse Laplace transformation to the following relationship:

…………………..3.38

55

Mavor and Cinco-Ley showed that for a closed, bounded two

porosity system with pseudo steady state matrix to fracture flow

system, pwD at constant rate is given by:

…………………..3.39

Applying the Laplace transform to the well bore pressure:

…………………..3.40

Substituting Equation 32 into equation 10

…………………..3.41

Inverting to real time:

…………………..3.42

In the case of a non-communicating matrix, the initial decline is

exponential in nature, and can be described by Eq. 3.42. Also

included in the work of Mavor and Cinco-Ley are the evaluation of

parameters λ and ω from decline curves and a study of the

observed initial decline in production rate. A procedure for

56

using log-log type curve matching to analyze rate-time data is

presented.

From the match point, the fracture permeability, ω, ηmD, reD and Sw

can be calculated. This methodology is easy to apply for well

data.

3.8 DECLINE CURVE ANALYSIS USING TYPE CURVES

Fetkovich15 described a procedure for using log-log type curve

matching to analyze rate-time data for a homogeneous system. The

same method can be applied to naturally fractured reservoirs as

pointed out by Da Prat et. al11.and Sageev et al30. However, the

relationship between qD vs. tD is controlled by ω and ηmD, as well

as by CfbD, Sw, Sf; we present a method where ω and ηmD can be obtained

using only flow rate transient data; the production decline

procedure presented for log-log type-curve matching should be

simple if ω and ηmD can be obtained independently from pressure

build up analysis, if this information is not available, it may

be necessary to use several type curves to obtain the best match.

In this case, for a given reD, it would be necessary to consider

57

several 3 or 4 values of ω and 4 or 5 values of λ for each ω. As

a result, many pairs of ω and λ might be found for a known reD.

If ω and λ can be obtained from pressure buildup analysis, the

particular type-curve to be used in production calculations or

matching for estimation of reservoir size can be properly

defined.

The solution for homogeneous system may be obtained by setting ω

= 1 and it is shown in the figures for comparison with results

for a fractured system. The homogeneous system case is the same

solution as that presented by Fetkovich15. The type curves

corresponding to ω = 0.001 and λ =1E-6 are shown in Fig. 4.9b for

a range of reD from 100 to 150,000.In this case, the constant

flow rate period is shown for large values of the reD.

Once ω and ηmD are known, a type curve can be used to compute

production rates for a particular reservoir. A type curve match

should provide information about the fracture permeability, kfb and

total storativity, (φ ct)t.

58

The production rate as a function of time may be graphed on

tracing paper, and then placed over the desired type curve. From

a match point, the bulk fracture permeability may be obtained

from the dimensionless-real flow rate match:

…………………..3.43

Similarly, from the dimensionless-real time match point, the

total storativity may be obtained:

…………………..3.44

In a similar manner, using the definition of dimensionless

fracture storage, the fracture storactivity can be obtained:

…………………..3.45

59

In addition, because ω and λ were determined by selection of the

type-curve, information about the matrix block geometry and

dimensions can be obtained, as indicated by the shape factor α,

if km can be obtained from core analysis. Using the following

equation (equaling lamda of definitions):

…………………..3.46

Where the characteristic dimension is:

…………………..3.47

3.9 THE USE OF TYPE CURVES

The basic steps used in type curve matching of declining rate-

time data for the naturally fractured reservoirs are as follows:

1. The actual rate vs. time data is plotted in any convenient

unit on log-log tracing paper of the same size cycle as the

curve to be used. (For convenience all type curves would be

plotted on the same log-log scale so that various solutions

could be tried.)

60

2. The tracing paper data curve would be placed over a type

curve; the coordinate axes of the two curves being kept

parallel and shifted to a position that represents the best

fit of the data to a type curve. More than one of the type

curves presented in this project may have to be tried to

obtain a best fit of all the data.

3. A line would be drawn through and extended beyond the rate-

time data (of the naturally fractured reservoir) overlain

along the uniquely matched type curves. Future rates then

would be simply read from the real-time scale on which rate

data is plotted.

4. To evaluate decline curve constants or reservoir variables,

a match point would be selected anywhere on the overlapping

portion of the curves, and the coordinates of this common

point on both sheets would be recorded.

5. If none of the type curves would fit all the data

reasonably, the departure curve method would be tried. This

method assumes that the data is a composite of two or more

61

different decline curves. After a match of the late time

data has been made, the matched curve is extrapolated

backward in time, and the departure, or difference, between

the actual rate and rates determined from the extrapolated

curve at corresponding times would be re-plotted on the same

log-log scale. An attempt would then be made to match the

departure curve with one of the type curves. Future

predictions then would be made as the sum of the rate

determined from the two (or more if needed) extrapolated

curves.

An example would be presented to illustrate the method of using

type curve matching to analyze typical declining rate-time data

for a naturally fractured reservoir (NFR). Type curve approach

provides solutions on which engineers can agree or shows when a

unique solution is not possible with a type curve only.

62

CHAPTER FOUR

RESULTS AND DISCUSSION

4.1 ANALYSIS OF A FIELD CASE FOR A NATURALLY FRACTURED RESERVOIR

WITH FRACTAL GEOMETRY

A combined analysis of well testing and production data for a

well producing in a field in the southern region of Mexico is

presented to explain the methodology proposed in this work.

This field includes a deep, complex NFR with an underlying

aquifer. The formation was deposited during the Upper Jurassic,

and it consists of calcareous dolomite. Detailed geological,

petrographic and geophysical work has identified normal and

reverse faults, a microcrystalline matrix containing mostly open

micro-fractures and macro-fractures and vugs of approximately 3%

porosity. Furthermore, recent well-testing and well-logging63

studies have estimated average permeability values as low as

0.01md.

Many of the wells have produced oil and associated gas at high

production rates for their productive lives to date and

experience has indicated that some zones of this formation are

more productive in certain wells. Because the matrix in this

field is compact and does not participate in the production, a

single-porosity approximation will be used.

To analyze the formation, we will use Barker’s diffusion equation

mainly because it is the simplest fractal expression – it

considers porosity and permeability to be independent of the

scale, so the corresponding analysis is simplified. In case

additional information is available, other fractal expressions

can be used.

The transient long-time behavior of Barker’s expression is given

as follows:

64

………….4.01

Where:

………….4.02

And

………….4.03

Figure 4.13 shows a pressure-buildup test showing well defined,

parallel straight lines for both shut-in pressure and pressure

derivative. This characteristic power-law behavior is a proof of

the presence of a fractal fractures system, in which a broad

distribution of fracture sizes exists besides a poor connectivity

among them. Then, the fractal dimension can be estimated from the

slope, v, obtaining dmf = 0.3256

As Flamenco Lopez and Camacho Velazquez17 (2003) mentioned the

determination of the four parameters of a fractal model is not

65

possible from a single-well transient test. Therefore, the use of

boundary-dominated flow information is an option to overcome this

problem. Thus, figure 101 shows the historic behavior of oil

production and bottom-hole pressure during the boundary-dominated

floe period for the same well in which the build-up test was run.

Based on this information, figure 3.15 shows the semi-log graph

of the rate normalized by the pressure drop vs. time. We observe

that a straight line is defined, implying that the long-time

behavior during boundary-dominated flow period. For the case of

Barker’s equation, we just need to consider θ = 0 in equation

4.03: thus, in real variables, this expression can be written as

follows:

………….4.04

From the intercept defined in the transient pressure graph shown

in figure 4.13, combined with the slope of Equation 4.14

66

(determined in figure 4.15) the drainage-radius value is

estimated with the real porosity value. Then, the permeability is

estimated from the intercept of equation 30 and the computed

value of drainage radius. Thus, re = 4.050 ft. and k = 0.011md.

If we ignore the fractal behavior and use the equations proposed

by Fetkovich et al. (1987) for Euclidean systems together with

the estimated porosity value and the slope of drainage radius and

permeability are unrealistic.

Note that if the OP equation is used instead of Barker’s

equation, we will also have four equations, slopes and ordinates

to the origin of figures 4.13 and 4.15, but the four unknowns

would be: aVs, m, dmf and θ, which would simply imply that

porosity and an estimate of re are given. The parameters ω and λ

do not play a role because the matrix does not participate in the

solution.

Observing figures 4.05, 4.10 and 4.11, it is obvious that the

existing non-unique problem of matching rate data in the

67

Euclidean case will be present also for the fractal cases,

especially when only early-time transient rate data are

available, also, considering the similarity of different type

curves, it is expected to have several equally plausible matches.

For this reason, it is worth stressing that flow-rate measurement

is often not enough to evaluate the effects of a multiplicity of

scales and a transient test is necessary.

68

69

Fig. 4.01: Decline Curves for theOP and MGN models. Infinite

Fig 4.04:Decline Curves for the OP and MGN models. Closed reservoirs

70

Fig 4.02: STA and LTA for the OP and MGN models. Infinite

Fig 4.05: Cumulative production for the OP and MGN models. Closedreservoirs

Fig 4.06: The influence of ω on the OP model with dual porosity. Infinite reservoir

Fig 4.03: Cumulative production for the OP and MGN models. Infinite reservoirs

Fig 4.07: The influence of dmf

on the OP model with dual porosity, STA and LTA.

Fig 4.10: The influence of dmf

on the OP model with dual porosity, STA and LTA. Closed

71

Fig 4.09: Cumulative production for the OP model with dual porosity. Infinite reservoirs

Fig 4.11: The influence of reD

on the OP model with dual porosity, STA and LTA.

Fig 4.08: The influence of θ on the OP model with dual porosity, STA and LTA.

Fig 4.12: Cumulative production for the OP model with dual porosity and LTA. Closed

Fig 4.13: Pressure-buildup test in field case Fig 4.15: Rate normalized by the

pressure drop vs. time, field case

4.2 TYPE CURVE MATCHING EXAMPLE

The production decline field data for an oil reservoir is shown

in in Table 1. The production rate as a function of time was

graphed on tracing paper and placed over the type curve

corresponding to ω = 0.001, ηmD =10−6, Sf= 6and = 5000 reD generated

with Cinco-Ley et al8. model. The fracture permeability, kfb, can

72

Fig 4.14: Pressure and productionhistories in field case

be calculated from the dimensionless-real flow rate match point,

using data in Table 1 and Eq. 35:

………….4.06

TABLE 1: SHOWING THE PRODUCTION RATE DATA FOR A NATURALLYFRACTURED RESERVOIR

73

Q(bbl/day) T (days)10150 1009690 3008850 4008175 6007425 8006486 12005475 17004378 21003872 27003517 34003314 40002767 50002405 70002270 90002270 120002198 20000

TABLE 4.2: SHOWING THE INPUT PRODUCTION-RATE DATA

Pi Pwf Sw μ rw B ω λ re reDI h J

rw

'

11500

5000

-4.09 1

0.25 1

0.001

1.00E-06

1500 100

480

1.561538

15

FIGURE 4.1: TYPE-CURVE MATCHING EXAMPLE FOR A NATURALLY FRACTUREDRESERVOIR

74

4.3 CALCULATIONS

From Eqn.4.06

Similarly, from the real-dimensionless time match and using Eq.

36, the total storativity is obtained:

The product for the fracture:

For cubic blocks:

The inter-porosity flow shape factor is:

The size of matrix blocks is:

75

4.4 DIMENSIONLESS FLOW RATE BEHAVIOR FOR CONSTANT PRESSURE PRODUCTION CONDITIONS IN A NATURALLY FRACTURED RESERVOIR

76

Fig. 4.16. Log-log dimensionless flow ratebehavior for constant pressure production conditions, infinite Naturally fractured reservoir, Sf=0, Sw=0.

Fig. 4.17. Log-log dimensionless flow rate behavior forconstant pressure production conditions,infinite

77

Fig. 4.18. Log-log dimensionless flow rate behavior for constant pressure production conditions, infinitenaturally fractured reservoir, km = 0,

Fig. 4.19. Log-log dimensionless flowrate behavior for constant pressure production conditions, infinitenaturally fractured reservoir, km = 0,

Fig. 4.20. Log-log dimensionless flow rate behavior forconstant pressure production conditions

Fig.4.21. Log-log dimensionless flow rate behavior forconstant pressure production conditions

Fig. 4.22. Log-log dimensionless flow rate behavior for constant pressure production conditions in a bounded naturally fractured reservoir, the reserves are located only in the

Fig. 4.23. Log-log dimensionless flow rate behavior for constant pressure production conditions in a bounded naturally fractured reservoir, the reserves are located only in the

78

Fig. 4.24. Type curves used for decline curve analysis inNFR for reD: 100, 200, 500, 1,000, 5000,10,000,

CHAPTER FIVE

5.0 CONCLUSIONS AND RECOMMENDATION

5.1 CONCLUSIONS

The main purpose of this work is to present a more general

decline curve analysis with type curves for NFR and decline curve

analysis of NFR with fractal geometry based on single porosity

model for transient and boundary-dominated flow and also the

transient inter-porosity flow, including the fracture skin

effect.

From the results of this study using type curves, the following

conclusions can be established:

1. The model permits an easy change of the matrix block geometry.

79

2. Approximate analytical solutions for short and long times are

presented; others previously presented solutions are particular

cases (Chen et. al.).

3. For decline curve analysis, the use of the Warren and Root

Model for the decline analysis of double porosity systems can be

justified by using a matrix-fracture flow restriction.

4. The fracture skin can be confirmed by other sources, such as

that from thin section of cores.

5. The bulk fracture parameters of permeability and the

storativity and the outer radius can be estimated through the

methodology of this study.

6. The estimated outer radius considering transient matrix to

fractures transfer obtained in this work is higher than the value

of pseudo-steady state given by Da Prat et. al.

On the basis of the results presented with the consideration of

fractal geometry, the following conclusions can be made:

1. It has been shown that oil production from disordered

fractured media exhibits anomalous behavior that cannot be

80

explained by the conventional Euclidean model based on the

classical diffusion equation.

2. For the first time, expressions that represent the

asymptotic short and long-time behavior are presented both

for cases in which the matrix blocks participate,

considering pseudo-steady-state matrix-to-fractal fracture

transfer function and when the matrix does not play a role

in the rate response. Transient and boundary-dominated flow

conditions are considered. Two fractal diffusion equations

are used, including a fractional diffusion equation.

3. The rate expressions during the boundary-dominated flow

shows that the typical semi-log straight-line behavior for

the rate response of the Euclidean case may also be present

for the more general case. When the fractional diffusion

equation applies, however, this behavior is different.

4. The necessity of analyzing both transient and boundary-

dominated flow information to fully characterize a NFR with

fractal geometry is demonstrated.

81

NOMENCLATURE

A = Drainage area

A = drainage area, ft2

aVs= Fractal parameter related to the porosity of the fracture

network

Bo = Oil formation volume factor, RB/STB

c = Compressibility at initial conditions

CA= dimensionless pseudo steady state shape factor.

CFB= fracture area; is the ratio between matrix surface and rock

volume, ft-1.

82

Ct= compressibility, psi-1.

d = Euclidean dimension

dmf = Mass fractal dimension

h = formation thickness, ft.

H = matrix block size, ft.

In = modified Bessel function, first kind, nth order.

K = Modified Bessel function, second kind

k = permeability, mD.

Kn = modified Bessel function, second kind, nth order.

m= Fracture-network parameter

n = number of normal set of fractures.

Np= cumulative production, bbl.

p = pressure, psi.

pI= Laplace transform of p .

PwD = Dimensionless wellbore pressure

Pwf = Wellbore flowing pressure

pwf= wellbore flowing pressure, psi.83

q = Oil flow rate

q(t) = volumetric rate, bbl/day.

qD = Dimensionless rate

r = Radial distance

rD= dimensionless radius.

re= outer boundary radius, ft.

reDI= effective dimensionless well outer boundary radius.

rw = wellbore radius, ft.

rwI= effective wellbore radius, ft.

s = Laplace space parameter.

s = Mechanical skin factor

Sf= fracture skin.

Sw= Van Everdingen and Hurst skin factor.

t = Time, t hours

tD = Dimensionless time

tDA= dimensionless time based on drainage area A.

u = Laplace-transform variable84

V = ratio of total volume of medium to bulk volume.

v = Variable defined

x = thickness, ft.

α = inter-porosity flow shape factor, ft-2.

β = Fractional- derivative order

γ = Gamma function

η = diffusivity.

θ = Conductivity index

λ = dimensionless matrix-fracture permeability ratio, reflects

the intensity of the fluid transfer matrix-fractures.

μ = oil viscosity, cp.

ς = characteristic dimension of the heterogeneous medium, ft.

σ = Matrix/fracture-interactive index

Φ = Porosity, fraction

ψ = Variable defined

85

ω = dimensionless fracture storativity, is the ratio of the

storage capacity of the fracture to the total capacity of the

medium.

Subscripts

b = bulk (matrix and fractures).

d = damaged zone.

D = dimensionless.

e = external

f = fracture

m = matrix

surf = matrix-fracture surface

t = total

w = wellbore

86

5.3 REFERENCES

1. Acuna J.A., Ershaghi, I., and Yortsos. Y.C. 1995. Practical

Application of fractal Pressure-Transient analysis in

Naturally Fractured Reservoirs. SPEFE 10 (3): 173-179;

Trans., AIME, 299. SPE-24705-PA. DOI: 10.2118/24705-PA

2. Adams, A. R., Ramey, H. J., Jr. and Burguess, R. J., 1968:

Gas Well Testing in a Fractured Carbonate Reservoir, JPT (Oct.) 1187-

1194; Trans., AIME 243.

3. Arps. J.J Analysis of Decline Curves. 1945. Trans. AIME,

160: 228-247

4. Ashok Kumar Belani (August 1988): Department of Petroleum

Engineering of Stanford University

87

5. Barker. J.A. 1988. A Generalized Radial Flow Model for

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APPENDIX

Appendix A – Analytical Solutions for infinite Reservoir without

matrix participation

Chang-Yortsos Diffusion Equation

The equation proposed in Chang and Yortsos (1990) is given by:

……………………………………………………..A-1

The inner boundary condition is given as

……………………………………………………..A-2

The initial condition is given as

……………………………………………………..A-3

And the outer boundary condition is given as

……………………………………………………..A-4

Considering the above conditions from Watson (1944), we obtain in

the Laplace space

95

……………………………………………………..A-5

Where

……………………………………………………..A-6

And

……………………………………………………..A-7

Thus, the dimensionless oil-production rate is given by

……………………………………………………..A-8

The rate solution with mechanical skin is given by

……………………………………………………..A-9

When θ = 0, which implies v=0. Eq. A-9 is the corresponding

Euclidean expression. Also, if we consider the solution for

constant oil rate, pwD, we can check that Duhamel’s principle is

satisfied:

96

.…………………………………………………..A-10

The long time approximation for the rate expression in the time

domain is given by

……………………………………………………..A-11

And the corresponding equation for the cumulative production by

……………………………………………………..A-12

The short time approximation for the rate in the time domain is

given by

……………………………………………………..A-13

For the case of variable rate and pressure, we could use the

following definition for dimensionless pressure

……………………………………………………..A-14

With an inner boundary given by

97

……………………………………..A-15

Thus, the rate solution In Laplace space is given by

………………………………………..A-16

Where F(u) is the Laplace transform of f(t). The inverse of this

expression for the case of mechanical skin different from zero is

given as:

………………………………………..A-17

In terms of real variables, in field units, we have

…………………………………..A-18

Where u is the Laplace transform with respect to time, given in

hours and

……………………………………………………..A-19

……………………………………………………..A-20

98

……………………………………………………..A-21

And

……………………………………………………..A-22

Metzler-Glockie-Nonnenmacher Diffusion equation.

By use of the Laplace transform of a fractal derivative given by

Metzler et al (1944), the differential equation in the Laplace

space is given as

……………………………………………………..A-23

Where λ = 2/(θ+2), which represents the anomalous diffusion

co-efficient. If = 1, this equation is the same as that

proposed by Chang and Yortsos (1990), by use of the same inner,

outer and initial conditioned above, we obtain in Laplace space

……………………………………………………..A-24

99

Where v is given by Eq. A-6 and

……………………………………………………..A-25

Thus, the dimensionless oil-production rate is given by

……………………………………………………..A-26

When θ = 0, which implies v=0 and = 1. This equation is the

corresponding Euclidean expression. The corresponding solution

for constant oil rate, PwD is given as follows

……………………………………………………..A-27

Which collapses to the Chang and Yortsos91990) solution when =

1. Considering both solutions at constant rate, we can check that

Duhamel’s principle is satisfied.

100

The long time approximation for the rate expression in the time

domain is given by

……………………………………………………..A-28

And the corresponding equation for the cumulative production is

given by:

……………………………………………………..A-29

The short-time approximation for the rate in the time domain is

given by

……………………………………………………..A-30

For the case of variable rate and pressure, we use the definition

of dimensionless pressure given by Eq. A-14 with the same inner

boundary as Eq. A-15, obtaining the following rate solution in

Laplace space.

101

……………………………………………………..A-31

Where F(u) is the Laplace transform of f(t)

In terms of real variables, in field units, for the case of

mechanical skin different from zero, we have;

…………………………………..A-32

Where u is the Laplace transform with respect to time, given in

hours, and

…………………………………………………..A-33

102