Naturally Fractured Reservoirs

CHAPTER

8

ATU RALLY /:RACTURED ESERvOIRS INTRODUCTION Fractures are d i s p l a c e m e n t discontinuities in rocks, w h i c h a p p e a r as local breaks in the natural s e q u e n c e of the r o c k ' s properties. Most geological formations in the u p p e r part of the earth's crust are fractured to s o m e extent. The fractures r e p r e s e n t m e c h a n i c a l failures of the rock strength to natural geological stresses such as t e c t o n i c m o v e m e n t , lithostatic p r e s s u r e changes, thermal stresses, high fluid pressure, drilling activity, and even fluid w i t h d r a w a l , since fluid also partially s u p p o r t s the w e i g h t of the o v e r b u r d e n rock. Although p e t r o l e u m reservoir rocks can be f o u n d at any depth, at the d e e p e r d e p t h s p r e s s u r e of the o v e r b u r d e n is sufficient e n o u g h to cause plastic d e f o r m a t i o n of m o s t of the s e d i m e n t a r y rocks. Such rocks are unable to sustain shear stresses over a long p e r i o d and flow t o w a r d s an equilibrium condition. Fractures may a p p e a r as microfissures w i t h an e x t e n s i o n of only several m i c r o m e t e r s , or as c o n t i n e n t a l fractures w i t h an e x t e n s i o n of several t h o u s a n d kilometers. They may be limited to a single rock f o r m a t i o n or layer, or p r o p a g a t e t h r o u g h m a n y rock formations or layers. In geological terms, a fracture is any planar or curvi-planar discontinuity that has f o r m e d as a result of a p r o c e s s of brittle d e f o r m a t i o n in the earth's crust. Planes of w e a k n e s s in rock r e s p o n d to c h a n g i n g stresses in the earth's crust by fracturing in o n e or m o r e different ways, d e p e n d i n g

488

INTRODUCTION

489

on the direction of the maximum stress and the rock type. A fracture may consist of two rock surfaces of irregular shape, being more or less in contact with each other. The volume b e t w e e n the surfaces is the fracture void. Naturally fractured rocks can be geologically categorized into three main types, based on their porosity systems: (1) Intercrystalline-intergranular, such as the Snyder field in Texas, the Elk Basin in Wyoming, and the Umm Farud field in Libya; (2) Fracture-matrix, such as the Spraberry field in Texas, the Kirkuk field in Iraq, the Dukhan field in Qatar, and Masjidi-Sulaiman and Haft-Gel fields in Iran; and (3) Vugular-solution, such as the Pegasus Ellenburger field and the Canyon Reef field in Texas [1]. The accumulation and migration of reservoir fluids within a naturally fractured formation having the first type of porosity system are similar to those found in sandstone formations. Consequently, the techniques developed to determine the physical properties of sandstone porous media in Chapter 3 could be directly applied to formations having intercrystalline-intergranular porosity. Unfortunately, this is not the case for reservoirs having the other two types of porosity system. The pores in the matrix of a fracture-matrix formation are poorly interconnected, yielding a pattern of fluid movement that is very different from that of sandstone formations. Rocks with vugular-solution porosity systems exhibit a wide range of permeability distributions varying from relatively uniform to extremely irregular as shown in Figure 8.1.

Figure 8.1. Naturally fractured rock cores taken f r o m wells.

490

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

ORIGIN OF PERMEABILITYIN CARBONATEROCKS A natural fracture is a planar discontinuity in reservoir rock due to deformation or physical diagenesis. Diagenesis--chemical and physical changes after depositionmstrongly modifies the reservoir properties possessed at the time of deposition. The dominant diagenetic process consists of early cementation, selective dissolution of aragonite and reprecipitation as calcite, burial cementation, dolomitization, and compaction-driven microfracturing [2]. Cementation and compaction forces usually completely eradicate any porosity available at the time of deposition. However, chemical changes, usually dissolution, especially in carbonate rocks, modify the initial porosity and recover it partially. Depositional facies, their architecture, systems, and tracts are predominant driving factors in the distribution and quality of current reservoir properties, which are completely different from the properties at the time of deposition. High-permeability vugs, molds, natural fractures, and caverns in carbonate rocks are the result of intense dissolution, which took place before burial as a result of non-reservoir or seal units. Dissolution is also caused by meteoric diagenesis, which is related to subaerial exposure of carbonate rocks and is explained by the general aggressiveness of meteoric water toward sedimentary carbonate minerals. Aragonite is metastable, it dissolves and precipitates into cement, whereas calcite is stable and is less affected by dissolution. Such a type of dissolution causes significant variation in the distribution of porosity and permeability in the reservoirs, thereby defining reservoir quality.

GEOLOGICALCLASSIFICATIONSOF NATURALFRACTURES Natural fracture patterns are frequently interpreted on the basis of laboratory-derived fracture patterns corresponding to models of paleostress fields and strain distribution in the reservoir at the time of fracture [3]. Classification based on stress/strain conditions: Stearns and Friedman proposed classification based on stress/strain conditions in laboratory samples and fractures observed in outcrops and sub-surface settings. On the basis of their work, fractures are generally classified as follows [4]:

(a) Shear fractures exhibit a sense of displacement parallel to the fracture plane. Shear fractures are formed w h e n the stresses in the three principal directions are all compressive. They form at an acute angle

GEOLOGICAL CLASSIFICATIONS OF NATURAL FRACTURES

491

to the m a x i m u m principal stress and at an obtuse angle to the direction of m i n i m u m compressive stress. (b) Extension fractures exhibit a sense of displacement p e r p e n d i c u l a r to and away from the fracture plane. They are formed p e r p e n d i c u l a r to the m i n i m u m stress direction. They too result w h e n the stresses in the three principal directions are compressive, and can o c c u r in conjunction w i t h shear fracture. (c) Tension fractures also exhibit a sense of displacement p e r p e n d i c u l a r to and away from the fracture plane. However, in order to form a tension fracture, at least one of the principal stresses has to be tensile. Since rocks exhibit significantly r e d u c e d strength in tension tests, this results in increased fracture frequency.

Classification based onpaleostress conditions: The geological classification of fracture systems is based on the assumption that natural fractures depict the paleostress conditions at the time of the fracturing. Based on geological conditions, fractures can be classified as in the following paragraphs. Tectonic fractures: The orientation, distribution, and m o r p h o l o g y of these fracture systems are associated with local tectonic events. Tectonic fractures form in n e t w o r k s with specific spatial relationships to faults and folds. Fault-related fracture systems could be shear fractures f o r m e d either parallel to the fault or at an acute angle to it. In the case of the fault-wedge, they can be extension fractures bisecting the acute angle b e t w e e n the two fault shear directions [2, 5]. The intensity of fractures associated with faulting is a function of lithology, distance from the fault plane, magnitude of the fault displacement, total strain in the rock mass, and d e p t h of burial. Fold-related fracture systems exhibit c o m p l e x patterns consistent w i t h the c o m p l e x strain and stress history associated with the initiation and g r o w t h of a fold [6]. Fracture types in fold-related systems are defined in terms of the dip and strike of the beds.

Regiona! fractures: These fracture systems are characterized by long fractures exhibiting little change in orientation over their length. These fractures also s h o w no evidence of offset across the fracture plane and are always p e r p e n d i c u l a r to the bedding surfaces. Regional fracture systems can be distinguished from tectonic fractures in that they generally exhibit simpler and more consistent g e o m e t r y and have relatively larger spacing. Regional fractures are c o m m o n l y developed as orthogonal sets w i t h the t w o orthogonal orientations parallel to the long and short axes of the basin in w h i c h the fractures are formed. Many theories have

492

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

b e e n p r o p o s e d for the origin of the regional fractures, ranging from plate tectonics to cyclic loading/unloading of rocks associated w i t h earth tides. As in the case of tectonic fractures, small-scale variation in regional fracture orientation of up to + 2 0 ~ can result from strength anisotropies in reservoir rocks due to sedimentary features such as across bedding.

Contractional fractures: These types of fracture result from bulk v o l u m e reduction of the rock. Desiccation fractures may result from shrinkage u p o n loss of fluid in subaerial drying. Mud cracks are the most c o m m o n fractures of this type. Syneresis fractures result from bulk volume reduction within the sediments by sub-aqueous or surface dewatering. Dewatering and volume reduction of clays or of a gel or a colloidal suspension can result in syneresis fractures. Desiccation and syneresis fractures can be either tensile or extension fractures and are initiated by internal body forces. The fractures tend to be closely spaced and regular and isotropically distributed in three dimensions. Syneresis fractures have been observed in limestone, dolomites, shales, and sandstones [7]. Thermal contractional fractures may result from contraction of hot rock as it cools. D e p e n d i n g on the d e p t h of burial, they may be either tensile or extension fractures. The generation of thermal fractures is predicted on the existence of a thermal gradient within the reservoir rock material. A classic example of thermally induced fracture is the columnar jointing observed in igneous rocks. Fractures may also result from mineral changes in the rock, especially in carbonates and clay constituents in sedimentary rocks. Phase changes such as the chemical change from calcite to dolomite result in changes in bulk volume, and this leads to c o m p l e x fracture patterns (Figure 8.2). It is clear from the above discussion that the c o m p l e x stress/strain distribution in reservoir rocks results in c o m p l e x fracture patterns. Fracture patterns c o r r e s p o n d i n g to different geological systems have key characteristics that can be used to classify and index natural fracture n e t w o r k s observed in outcrops and subsurface samples (Figures 8.3 and 8.4).

ENGINEERINGCLASSIFICATIONOF NATURALLY FRACTUREDRESERVOIRS Fractures may have either a positive or a negative impact on fluid flow, d e p e n d i n g on w h e t h e r they are o p e n or sealed as a

ENGINEERING CLASSIFICATION

493

Figure 8.2. a, b, c. Different fracture systems in mud, a n d rocks. After Lui et al. [8].

result of mineralization. However, in most fracture modeling studies fractures are considered as open and they have a positive impact on the fluid flow. A sealed small natural fracture may even be undetectable. Nelson identified four types of naturally fractured reservoirs, based on the extent to w h i c h fractures have altered the porosity and permeability of the reservoir matrix [ 1 ]: Type 1: In type 1 reservoirs, fractures provide all the reservoir storage capacity and permeability. The Amal field in Libya, the LaPaz and Mara fields in Venezuela, and Pre-Cambrian basement reservoirs in eastern China are notable type 1 reservoirs (Figure 8.5a).

494

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

Figure 8.3. Modes of fracture formation.

Figure 8.4. Reservoir fluids in shear fractures, Monterey formation, California [9].

Type 1: In type 2 reservoirs, the matrix already has very good permeability. The fractures add to the reservoir permeability and can result in considerably high flow rates, such as in the Kirkuk field of Iraq and Asmari fields in Iran.

Type 3: In type 3 naturally fractured reservoirs, the matrix has negligible permeability but contains most if not all the hydrocarbons. The fractures provide the essential reservoir permeability, such as in the Monterey fields of California and the Spraberry reservoirs of West Texas. Type 4: In type 4 reservoirs, as s h o w n in Figure 8.5b, the fractures are filled with minerals. These types of fractures tend to form barriers to fluid migration and partition formations into relatively small blocks.

INDICATORS OF NATURAL FRACTURES

495

Figure 8.5. Types of naturally fractured reservoirs [10].

These formations are significantly anisotropic and often uneconomic to develop and produce.

INDICATORSOF NATURALFRACTURES Steams and Friedman reviewed the multiple roles played by fractures in exploration and exploitation of naturally fractured reservoirs [4]. They showed that fractures could alter the matrix porosity or the permeability, or both. If the fractures or connected vugs are filled with secondary minerals, they may restrict the flow. However, even in rocks of low matrix porosity, fractures and solution channels increase the pore volume by both increasing porosity and connecting isolated matrix porosity and therefore help the recovery of petroleum fluids economically. Hence, the ability to estimate a fracture's density and its distribution of porosity is essential for reservoir evaluation. One should keep in mind, however, that fractures alone constitute less than 1% of the porosity [11, 12]. Early recognition of a fractured reservoir and an estimate of its rock

496

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

Pores

Natural Fractures

v

Induced Fractures

v

v

Mud Pit Level

(a)

(b)

(c)

Figure 8.6. M u d loss i n d i c a t i o n a n d p i t level b e h a v i o r in pores, n a t u r a l fractures, a n d i n d u c e d f r a c t u r e s : ( a ) g r a d u a l b u i l d u p in loss ratio w i t h pressure; (b) s u d d e n start a n d e x p o n e n t i a l decline; a n d (c) loss can occur on increase in ECD as p u m p s are t u r n e d o f f a n d on [14].

characteristics, such as porosity and permeability, will influence the location and n u m b e r of subsequent d e v e l o p m e n t wells and, therefore, is of major economic significance. Steams and Friedman [4], Aguilera [13], Saidi [22] and Nelson [1] reviewed many of the approaches used to detect and analyze naturally fractured reservoirs [1, 13]. Some of these methods are as follows (see Figure 8.6): (1) Loss of circulating fluids and an increase in penetration rate during drilling are positive indications that a fractured, cavernous formation has been penetrated (Figure 8.6). (2) Fractures and solution channels in cores provide direct information on the nature of a reservoir. A detailed systematic study of the cores must be made by the geologist in order to distinguish natural fractures from those induced by the core handling process. Careful examination of fracture faces and determination of density, length, width, and orientation of fractures may lead to the ability to distinguish fractures induced during coring from natural fractures. Preferably, a naturally fractured formation should be analyzed with full diameter cores. Plug data, w h i c h do not reflect the permeability of fractures, often indicate a nonproductive formation, w h e r e a s full diameter core data indicate hydrocarbon production. If actual production rates are several-fold higher than those calculated from permeability determined by core analysis, natural fractures not observed in the core are suspected [15]. Low core recovery efficiency--less than 50%msuggests a highly fractured carbonate formation. (3) Logging tools are designed to respond differently to various wellbore characteristics, such as lithology, porosity, and fluid saturations, but not to natural fractures [16, 17]. The presence of a large n u m b e r of open fractures, however, will affect the response of

INDICATORS OF NATURAL FRACTURES

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some logging tools. Well logging m e a s u r e m e n t s based on sonic wave propagation, w h i c h are negligibly affected by the borehole conditions, are used as fracture indicators. Measurements by the caliper log, density log, or resistivity log, u n d e r p r o p e r conditions, can be very effective in locating fractured zones. D i p m e t e r data on the FIL (fracture identification log) provide effective m e t h o d s for fracture detection. (4) The subject of pressure buildup and flow tests in naturally fractured reservoirs has received considerable attention in the p e t r o l e u m literature. Warren and Root assumed that the formation fluid flows from the matrix to fractures u n d e r pseudosteady state and s h o w e d that a semilog pressure buildup curve similar to that s h o w n in Figure 8.7 is typical of a fractured formation [18]. If the existing fractures dominantly trend in a single direction, the reservoir may appear to have anisotropic permeability. If e n o u g h observation wells are used, pressure interference and pulse tests provide the best results. (5) Natural vertical fractures in a non-deviated borehole can be identified as a high amplitude feature w h i c h crosses other bedding planes. (6) D o w n h o l e direct and indirect viewing systems, including d o w n h o l e p h o t o g r a p h i c and television cameras, are also used to detect fractures and solution channels on the borehole face. The borehole televiewer is an excellent tool that provides useful pictures of the reservoir rock, especially with the recent d e v e l o p m e n t s in

498

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

signal processing. Vertical fractures a p p e a r as straight lines w h e r e a s dipping fractures tend to appear as sinusoidal traces because the televiewer shows the wellbore sandface as if it w e r e split vertically and laid flat. Another useful televiewer tool for detecting natural fractures is the formation m i c r o s c a n n e r (FMS) device. This tool can detect fractures that range from f e w millimeters to several centimeters long, distinguishes t w o fractures as close as 1 cm apart (see Figure 8.8), and distinguishes b e t w e e n o p e n and closed fractures. Only fractures that are at least partially o p e n contribute to production. (7) Very high productivity index. A productivity index of 500 STB/D/psi or higher is typical of naturally fractured wells p r o d u c e d u n d e r laminar flow. Some wells in Iranian oilfields r e p o r t e d a productivity index of 10,000 STB/D/psi. In these wells 95% of flow is t h r o u g h fractures [22]. (8) A considerable increase in productivity of the well flowing after an artificial stimulation by acidizing is a strong indication of a naturally fractured formation. Acidizing is done essentially to increase the width of fractures and channels. (9) Because of the high permeability of the fractures, the horizontal pressure gradient is typically small near the wellbore as well t h r o u g h o u t the reservoir [22]. This is primarily true in Type-1 and to a lesser degree in Type-2 fractured reservoirs. O t h e r indicators of the existence of the natural fractures in the reservoir are: (a) (b) (c) (d)

local history of naturally occurring fractures; lack of precision in seismic recordings; extrapolation from observations on outcrops; and pressure test results that are incompatible with porosity and permeability values obtained from core analysis a n d / o r well logging.

As can be d e d u c e d from the p r e c e d i n g discussion, no m e t h o d used alone provides a definite p r o o f of the p r e s e n c e of fractures. FMS logs and borehole televiewers often give a reliable indication of the p r e s e n c e of major features; however, they do not resolve the full complexity of many of the smaller-scale fracture systems. Fracture detection is most certain w h e n several i n d e p e n d e n t m e t h o d s confirm their presence. Different naturally fractured reservoirs require different combinations of m e t h o d s of analysis. A combination of core analysis, pressure transient test analysis, and various fracture-finding logs is strongly r e c o m m e n d e d for detecting and locating fractures. Table 8.1 summarizes the many techniques available for detecting natural fractures.

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VISUAL IDENTIFICATIONOF FRACTURES Nelson [1] defined, for consistency, four useful terminologies to describe cracks in a rock as: (a) F r a c t u r e : any break in the rock; (b) F i s s u r e : an o p e n fracture; (c) J o i n t : one or a group of parallel fractures w h i c h has no detectable displacement along the fracture surface; and (d) Fault: a fracture with detectable displacement. All these features can be visually identified on a core or borehole electrical images. Figure 8.8 shows three types of fracture that may be visually detected: (a) Natural vertical fractures in a non-deviated borehole can be identified as a high-amplitude feature that crosses other bedding planes. They occur in all lithologies. Fractures may be open, mineral filled, or vuggy. Visual inspection of cores and borehole electric images may be used only as a guide for interpretation. Core flow tests and actual production tests are r e c o m m e n d e d for interpreting the morphology of natural fractures. Production and recovery efficiency in reservoirs is influenced by the angle. The angle most often used by oil companies as a criterion is 75 ~ Fractures with dip angles of more than 75 o are treated as vertical fractures, while those less than 75 o are treated as high-angle fractures. Vertical fractures are more c o m m o n in sandstone rocks. (b) Syneresis fractures have a braided appearance and are often referred to as "chicken wire" fractures. They normally occur only in carbonate formations. (c) Mechanically induced fractures are sometimes unintentionally created during the drilling operations, or by hydraulic fracturing to stimulate the formation. Fracture morphology can also be visually detected on cores and/or borehole images. Figure 8.9 shows four detectable fracture morphologies: vuggy, mineral-filled, partially mineral-filled, and open.

PETROPHYSICALPROPERTIESOF NATURALLY FRACTUREDROCKS Although advanced well-logging tools such as nuclear magnetic resonance (NMR) are currently being used to estimate rock permeability

PETROPHYSICAL PROPERTIES

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502

PETROPHYSICS" RESERVOIR ROCK PROPERTIES

d o w n h o l e , the technology is not yet fully developed. The only m e t h o d to estimate permeability reliably is to c o m b i n e core-derived p a r a m e t e r s with c o m p u t e r - p r o c e s s e d log data to establish, statistically, a relationship b e t w e e n the permeability of the fracture-matrix system and o t h e r parameters such as porosity and irreducible w a t e r saturation. Efforts have also b e e n made to incorporate grain diameter and shale fraction in such models to reduce the scatter in the data. With such a relationship established, the formation petrophysical parameters, including permeability distribution, can be d e d u c e d from log data alone in wells or zones w i t h o u t core data. However, in carbonate formations, w h e r e structural heterogeneities and textural changes are c o m m o n and, unfortunately, only a small n u m b e r of wells are cored, the application of statistically derived correlations is e x t r e m e l y limited. Such correlations cannot be used to identify hydraulic flow units or bodies in naturally fractured reservoirs.

FRACTUREPOROSITYDETERMINATION The range of fracture porosity, ~f, is 0.1 to 5 percent, d e p e n d i n g on the degree of solution channeling, as s h o w n in Figure 8.10, and on fracture w i d t h and spacing, as s h o w n in Tables 8.2 and 8.3. In some fields, like the La-Paz and Mara fields in Venezuela, fracture porosity may be as high as 7 percent. Accurate m e a s u r e m e n t of fracture porosity is essential for the efficient d e v e l o p m e n t and economical exploitation of naturally fractured reservoirs. If oil is trapped in both the matrix and fissures, then the total

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Figure 8.10. Development of fracture porosity in carbonate rocks that have low insoluble residue, (a), (b), (c), and high insoluble residue, (d) and (e) [20].

503

PETROPHYSICAL PROPERTIES

TABLE8.2 POROSITYOFVARIOUSNATURALLYFRACTUREDRESERVOIRS[21] Field

Porosity Range (%)

Beaver Gas Field Austin chalk General statement South African Karst Zone CT scan examples Epoxy injection examples Monterey

0.05-5 0.2 1 1-2 1.53-2.57 1.81-9.64 0.01-1.1

TABLE8.3 FRACTUREWIDTHAND SPACINGSOFVARIOUSNATURALLYFRACTURED FORMATIONS[21] Field

Spacing/ Width Range (mm) Average Frequency

Spraberry

0.33 max.

Selected dam sites La Paz-Mara field Small joints Extension fractures Major extension fractures Monterey

0.051 to 0.10 6.53 max. 0.01 to 0.10 0.1 to 1.0 0.2 to 2

0.051

Few inches to a few feet 4 t o 14ft

0.2 0.01

3 to 36 ft

oil in p l a c e in t h e r e s e r v o i r is g iv e n by t h e f o l l o w i n g e q u a t i o n [10]" Not (STB) -- Nom + Nof

(8.1)

w h e r e Nom an d Nof are, r e s p e c t i v e l y , t h e oil v o l u m e s t r a p p e d in t h e m a t r i x a n d fractures. A s s u m i n g a v o l u m e t r i c s y s t e m , t h e s e t w o v o l u m e s , e x p r e s s e d in STB, are c a l c u l a t e d as follows"

Nom = N,ff --

7 , 7 5 8 A h e m (1 - Swm) Bo 7 , 7 5 8 A h ~ f (1 - Swf)

where: A h

Bo -- s u r f a c e area o f t h e reservoir, a c r e s -- a v e r a g e r e s e r v o i r t h i c k n e s s , ft

(8.2) (8.3)

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PETROPHYSICS: RESERVOIR ROCK PROPERTIES

Cf = 0m = Swf = Swm = Bo =

fracture porosity, fraction matrix porosity, fraction w a t e r saturation in fractures, fraction w a t e r saturation in matrix, fraction oil formation volume factor, bbl/STB.

Fracture porosity can be expressed as the ratio of the fracture pore volume (Vpf) over the total bulk volume (Vbt):

(8.4a)

Vpf ~)f = Vbt

The total porosity is: Vpm Vpf ~ Vpm = Vpf ~ ~)t -~ Of + (~m -- Vbt Vbm Vbt (1 -- ~)f)Vbt

(8.4b)

The sonic log only measures the matrix porosity. However, neutron porosity is the combination of both the matrix and fracture porosity. Thus fracture porosity can be estimated from well logs as [22]: 0f -- ~)Neu -- ~)Son

(8.5)

Fracture porosity can also be estimated with the help of well test analysis in such reservoirs, using Equation 8.76. Aguilera developed the following equation that relates the total formation resistivity factor, Ft, for dual porosity systems, to the total porosity based on the Pirson model of fully water-saturated rocks [23]: RwRo Ft =

v0tR~ + (1 - v)Rw Rw

) (8.6)

w h e r e Ro is the resistivity of porous rock 100 p e r c e n t saturated with brine, and Rw is the formation water resistivity, both expressed in Ohm-m. If only the matrix porosity is present in the system, the porosity partitioning coefficient, v, is equal to zero. Thus Equation 8.6 simplifies to Equation 8.7, w h i c h is the same as for a consolidated matrix:

Ro F = Rw

(8.7)

If only fracture porosity is present in the system, such as in Type-1 naturally fractured reservoirs, the porosity the partitioning coefficient

PETROPHYSICAL PROPERTIES

505

is equal to unity. In this case the f o r m a t i o n resistivity factor can be e x p r e s s e d as" 1

F = -mr Cpf-

(88a)

Laboratory tests indicate that the t o r t u o s i t y factor, x, and the fracture p o r o s i t y e x p o n e n t , mf, are a p p r o x i m a t e l y unity in systems w i t h o p e n and well c o n n e c t e d fractures. In Type-2 and Type-3 naturally fractured reservoirs, the f o r m a t i o n resistivity factor can be m o r e generally e x p r e s s e d as:

F =

(1 - ,ff)r

(8.8b)

.~_ o f f

W h e r e mm is the matrix porosity. If only matrix p o r o s i t y is p r e s e n t , i.e. Of = O, Equation 8.8b simplifies to Equation 4.40 w h e r e m = mm and a = I:. On the o t h e r hand, if only fracture p o r o s i t y is p r e s e n t s u c h as in Type-l, Equation 8.8b simplifies to Equation 8.8a. If only the total p o r o s i t y is k n o w n , t h e n F can be e s t i m a t e d from: a

F =

(8.8c)

,p

The fractures should be c o n s i d e r e d as being well c o n n e c t e d if the i n t e r p o r o s i t y coefficient, ~, w h i c h is d e t e r m i n e d from a p r e s s u r e transient test is high, i.e. 10 -4 or 10 -5. If the i n t e r p o r o s i t y factor is low, i.e. ~. is a p p r o x i m a t e l y 10 -8 or 10 -9, the fractures are p o o r l y i n t e r c o n n e c t e d a n d / o r partially mineral-filled. In this case mf and m a y b e as high as 1.75 and 1.5, respectively. For 10 -6 > ~L > 10 -7, 1.75 > mf >_ 1 and 1.5 >_ 1: >_ 1.

EXAMPLE The following characteristics of a Type-2 naturally fractured f o r m a t i o n w e r e o b t a i n e d from core analysis: Cf -- 0.037

~)m - -

Ro -- 1.77 o h m - m

O. 15

mf -- 1.5

mm - 2

Rw -- 0.035 o h m - m

Estimate the tortuosity factor for this formation.

506

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

SOLUTION Using Equation 4.5 the f o r m a t i o n resistivity factor is"

F --

Ro Rw

=

1.77 0.035

= 50.57

The tortuosity is calculated from Eq. 8.8b"

I:-- F((1--,r &mf~&mm J't'm +(~7 f) -- (50.57) ((1 -- 0.03715)(0.152) + 0.0371"5) -- 1.5

POROSITY PARTITIONINGCOEFFICIENT Reservoirs w i t h a fracture-matrix p o r o s i t y s y s t e m - - s u c h as f o u n d in m a n y c a r b o n a t e rocks due to the e x i s t e n c e of vugs, fractures, fissures, and j o i n t s m d i f f e r considerably from reservoirs having only o n e p o r o s i t y type. The s e c o n d a r y porosity strongly influences the m o v e m e n t of fluids, w h e r e a s the primary p o r e s of the matrix, w h e r e m o s t of the reservoir fluid is c o m m o n l y stored ( m o r e than 96% in Type-3 naturally fractured reservoirs), are poorly i n t e r c o n n e c t e d . The Spraberry field of West Texas is an e x a m p l e of a naturally fractured s a n d s t o n e oil reservoir, w h i c h is c o m p o s e d of alternate layers of sands, shales, and limestones. The Altamont trend oilfield in Utah is a n o t h e r naturally fractured s a n d s t o n e reservoir w i t h a porosity of 3% to 7% and an average matrix p e r m e a b i l i t y less than 0.01 mD [ 13]. Laboratory-measured values of permeability for naturally fractured cores can be significantly different from the in-situ values d e t e r m i n e d by well p r e s s u r e analysis. The difference is attributed to the p r e s e n c e of fractures, fissures, joints and vugs, w h i c h are not adequately s a m p l e d in the core analysis. One of the earliest m e t h o d s used to analyze full-sized naturally fractured cores was d e v e l o p e d by Locke and Bliss [30]. The m e t h o d consists of injecting w a t e r into a core sample and m e a s u r i n g the p r e s s u r e values as a function of the cumulative injected v o l u m e of w a t e r (Figure 8.11). The s e c o n d a r y p o r e space, Vt, b e c a u s e of its high permeability, will be the first to fill up w i t h water. A sharp increase in p r e s s u r e is r e c o r d e d later, indicating that the matrix p o r o u s space, Vm, has to fill up. The total p o r e volume, Vt = Vf + ~)mVm, is c o n s i d e r e d to be filled up w h e n a pressure of 1,000 psi is r e a c h e d in the test. If the fraction of total p o r e v o l u m e in the s e c o n d a r y porosity is v, then:

Vf Vf -Vt V + ~)mVm

v -- ~

(8.9)

PETROPHYSICAL PROPERTIES

1200

507

...........................................................................................................................................................................................................................

1000

9~

8oo

::::I

600

13. 400

---i q

200

0 ~ 0

1 O0

200

300

Volume of water injected, cc Figure 8.11. Locke and Bliss Method for Estimating the Pore ,Space of Fractures.

The term v is c o m m o n l y referred to as the " p o r o s i t y p a r t i t i o n i n g coefficient." This coefficient represents the apportioning of total porosity (t~t) b e t w e e n the matrix (intergranular) porosity, 0m, and secondary pores (vugs, fractures, joints, and fissures), ~t. The value of v ranges b e t w e e n zero and unity for dual porosity systems. For total porosity equal to matrix porosity (absence of fracture porosity), v = 0. For total porosity equal to fracture porosity, v = 1. This coefficient can be estimated from core analysis using the Locke and Bliss method, pressure analysis, and well logging data. By assuming that the fractures and matrix are c o n n e c t e d in parallel, as s h o w n in Figure 8.20, Pirson suggested the following equations for short and long normal or induction tools [25]. If the drilling fluid used is non-conductive, the following correlations can be used to estimate the porosity partitioning coefficient and fracture intensity index: (a) Short Normal 1 Rxo

=

Rxo =

v(DtSxo (1 -- V) ~t $2 + Rw Rmf

(8. lOa)

FRmf $2o

(8.10b)

$08

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

(b) Long Normal 1

Vt~tSw

R-~---- Rw

(1 - V) t~tS2w ~

(8.11)

Rmf

w h e r e : Rxo = Borehole c o r r e c t e d invaded zone, short normal, resistivity, o h m - m Rmf = Mud filtrate resistivity, o h m - m Rt = Borehole c o r r e c t e d true, long normal, resistivity, ohm-m Rw = W a t e r resistivity, o h m - m ~)t = total p o r o s i t y of the formation, fraction Sw = w a t e r saturation, faction Sxo = Saturation of m u d filtrate in the flushed zone, fraction Most of these p a r a m e t e r s can be m e a s u r e d on e i t h e r cores or well logs. The w a t e r saturation of the flushed zone can be e s t i m a t e d only from correlations. Each formation s e e m s to require a slightly different correlation. O n e of these correlations is: Sxo = S,Cx w

(8.12)

T h e e x p o n e n t Cx (typically 0.20 to 0.25) is a s s u m e d arbitrarily, d e p e n d i n g on the e x p e r i e n c e of the mud-log analyst and results o b t a i n e d in n e a r b y wells. In a high-porosity and high-permeability f o r m a t i o n Cx ~ Sw. Because m u d filtrate and formation w a t e r are miscible, in a w a t e r - b e a r i n g zone Sxo ---- 1. In an oil-bearing zone Sxo = 1 - Soxo, w h e r e Soxo is the residual oil saturation in the flushed zone, typically in the range of 0.20 to 0.30. It is generally a s s u m e d that the a m o u n t of residual oil or gas is the same in b o t h the flushed zone and the invaded zone. T h e flushed zone is that w h i c h i m m e d i a t e l y s u r r o u n d s the w e l l b o r e (3 to 6 in. radius). The invaded zone is that b e y o n d the flushed zone (several feet thick). The p r e s e n c e of fissures n e a r the b o r e h o l e may increase the radius of b o t h zones. In low-porosity (Ot < 10%) and low-permeability formations (k < 5 mD), any m u d invasion w o u l d be very limited; but if k is high t h e n m u d filtration could be high and d e e p into the formation. In this case the range of the residual oil saturation is 10 to 20 p e r c e n t . In high-porosity (Ot > 15%) and high-permeability (k > 100 m D ) formations, a low m u d invasion results, w i t h residual oil saturation of a p p r o x i m a t e l y 30%. In the case of high porosity and l o w permeability, Soxo is in the o r d e r of 20%. T h e s e ranges of Soxo are applicable primarily in w a t e r - w e t s a n d s t o n e

PETROPHYSICAL PROPERTIES

509

formations. The presence of fractures near the wellbore and their density are factors that must be taken into account w h e n estimating Soxo. Combining Equations 8.10a and 8.11 and solving explicitly for the porosity partitioning coefficient, v, yields: Rw

11

V -- (~t (Sw -- Sxo)

Rt

1 ) (8.13)

Rxo

If the total porosity ~, is k n o w n from logs or cores, the matrix porosity and fracture porosity may be estimated from: ~ m - - ~)t (1 -- v)

(8.14)

~f = ~t -- ~m

(8.15)

The porosity partitioning coefficient v, c o m m o n l y used by the petrophysicist, is physically equivalent to the storage capacity ratio, co, w h i c h is more commonly used in well test analysis. But, because of the difference in scale, it is unlikely that the two values would ever be equal for the same formation. Note that even equations 8.13 and 8.9 will yield slightly different values of v, because one is obtained from well logs (Equation 8.9) while the other is measured in cores. Logs seem to yield slightly lower values of v, because the measurements are done under in-situ conditions.

EXAMPLE A newly drilled well in a naturally fractured reservoir was logged. The average total porosity of the system was estimated from cores as 14%. Other k n o w n characteristics are: A = 3,000 acres, Bo = 1.25 bbl/STB Rmf = O. 17 ohm-m,

h = 52ft, Rw = O. 19 ohm-m,

Sw = 0.22, Rt = 95 ohm-m,

m = 1.75.

(1) Estimate the porosity partitioning coefficient. (2) Estimate the matrix porosity and fracture porosity. (3) Calculate the total oil in place, STB.

SOLUTION (1) In order to calculate the porosity partitioning coefficient v from Equation 8.13, we need to determine first the resistivity in the

510

PETROPHYSICS: RESERVOIR R O C K PROPERTIES

f l u s h e d zone. Using Eq. 8.10b:

(31.2)(0.17)

FRmf Rx

O

--

$2o

.~

0.7382

= 9.70hm-m

T h e f o r m a t i o n resistivity f a c t o r F a n d t h e w a t e r s a t u r a t i o n in t h e i n v a d e d z o n e Sxo are e s t i m a t e d f r o m e q u a t i o n s 8.8c a n d 8.12, r e s p e c t i v e l y , a s s u m i n g 1; -~ 1 a n d Cx -- 0.2"

F=

0.141.75

=31.2

Sxo -- 0.220.20 -- 0 . 7 3 8 Using E q u a t i o n 8.13 t h e p o r o s i t y p a r t i t i o n i n g c o e f f i c i e n t is:

V --

0.19

( 1

0.14(0.22 --0.738)

95

1 ~ _ 0.24 9.72

/

This value i n d i c a t e s that f r a c t u r e s c o n t r i b u t e 24 p e r c e n t of t h e total p o r e space. (2) W e n o w c a n e s t i m a t e t h e m a t r i x p o r o s i t y a n d f r a c t u r e p o r o s i t y f r o m E q u a t i o n s 8.14 and 8.15" ~)m -- ~)t(1 - v) -- O. 14(1 - 0.24) -- O. 106 ~f -- ~t - qbm -- O. 14 - O. 106 -- 0 . 0 3 4 (3) A s s u m i n g t h e w a t e r s a t u r a t i o n in t h e f r a c t u r e s is e q u a l to t h e w a t e r s a t u r a t i o n in t h e m a t r i x , t h e initial oil in p l a c e in t h e m a t r i x a n d f r a c t u r e s are c a l c u l a t e d f r o m E q u a t i o n s 8.2 a n d 8.3, respectively"

N--

( 7 , 7 5 8 ) ( 3 , 0 0 0 ) ( 5 2 ) ( 0 . 1 1 ) ( 1 -- 0.22)

1.25

= 8 3 , 0 7 1 , 4 2 3 STB N-

8 3 , 0 7 1 , 4 2 3 + 2 5 , 5 4 0 , 5 6 5 -- 1 0 8 , 6 1 1 , 9 8 8

T h e total oil in p l a c e in this naturally f r a c t u r e d r e s e r v o i r is" Not -- 8 0 , 1 8 6 , 7 0 0 + 2 5 , 5 4 0 , 5 6 5 -- 105.7x106 STB This total oil v o l u m e is c o r r e c t , a s s u m i n g t h e p o r o s i t y p a r t i t i o n i n g coefficient is t h e s a m e in t h e e n t i r e reservoir. This is h i g h l y unlikely

5| 1

PETROPHYSICAL PROPERTIES

Figure 8.12. Frequency of occurrence o f natural fractures near major faults [25].

in naturally fractured formations, w h e r e p o r o s i t y varies o v e r short distances.

FRACTUREINTENSITYINDEX Tension stress causes rock failure along major faults, giving rise to fracture porosity (~)f), and fractures of decreasing w i d t h (wf) and length (hf) and f r e q u e n c y of o c c u r r e n c e (FII) away from the fault plane, as s h o w n in Figure 8.12 [25]. Thus p e r m e a b i l i t y is m u c h m o r e affected by fracture d i m e n s i o n s than the matrix or total porosity. The curve fit e q u a t i o n s for b o t h u p t h r o w n and d o w n t h r o w n blocks are as follows: (a) U p t h r o w n block: 1

dLu -- 4.2 x lO-2FII 2 -- 2.43 x 10 -5

(8.16)

(b) D o w n t h r o w n block: 1

dLn-

9.44 X 10 -3 exp ( F I I ) - 9.3 • 10 -3

(8.17)

w h e r e : dLu = Lateral distance to the fault for u p t h r o w n block, ft dLn -- Lateral distance to the fault for d o w n t h r o w n block, ft FII = fracture intensity index, fraction. The in-situ value of the fracture intensity i n d e x is estimated from: FII =

1/Rxo - 1/Rt 1/Rn~ - 1/Rw

(8.18)

512

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

Equation 8.16 (R 2 = 0.989) is applicable for a distance range of 250 to 5,000 ft and an FII range of 7 to 25%, and Equation 8.17 (R 2 = 0.998) is applicable for a distance range of 250 to 1,250 ft and an FII range of 7 to 25%. These correlations were developed from well data obtained by Pirson near the Luling-Mexia fault in the Austin chalk. The primary application of these two correlations is in the exploration stage and w h e n the presence of a nearby fault is k n o w n a priori from seismic data, as they provide only an order of magnitude of the distance to the fault. It is important to emphasize that (a) FFI is influenced by several factors, including the n u m b e r of fractures and fracture geometry, and (b) not all natural fractures are the result of faulting. The following equations can be used to estimate fracture width and fracture permeability in a type 1 naturally fractured reservoir:

wf kf-

0.064

[(1 - Siw) FII] 1"315

1.5 • 107r

[ ( 1 - Swi) FII] 2"63

(8.19a) (8.19b)

w h e r e porosity, FII, and irreducible water saturation are expressed as fractions, and fracture width and fracture permeability in cm and mD, respectively. The fracture porosity can be directly estimated using the following empirical correlation [22]:

~)f-

E (1

1)]c

Rmf RLLS RLLD

(8.20)

w h e r e the range of the coefficient CT is b e t w e e n 2/3 (typical for Type-1 fractured reservoir) and 3/4. Rmf, RLLS and RLLD are, respectively, the mudfiltrate, laterolog shallow, and laterolog deep resistivities in ohm-m. RLLS and RLLD are equivalent to Rxo and Rt, respectively.

EXAMPLE Seismic surveys and geological studies have indicated that the well in the previous example is located in a naturally fractured zone and in an u p t h r o w n layer. Using the given data, calculate the FII and estimate the distance to the nearest fault, if the resistivity of the invaded zone is 7.5 ohm-m.

PETROPHYSICAL PROPERTIES

513

SOLUTION Using Equation 8.18, the fracture intensity i n d e x is"

FII --

1/Rxo -

1/R t

1 / R m f - m/Rw

1/9.7 -

=

1/0.17-

1/95

1/0.19

=0.15

The distance to the nearest fault is e s t i m a t e d f r o m the c o r r e l a t i o n c o r r e s p o n d i n g to the u p t h r o w n b l o c k and can n o w be e s t i m a t e d using Equation 8.16" dLu -dLu --

4.2 • IO-2FII 2 -

2.43 x 10 -5 1

4.2 x 1 0 - 2 ( 0 . 1 5 ) 2 -

2.43 x 10 -5

~ 1,100 ft

The distance to the fault can be directly estimated using Figure 8.13. For the FII value of 15%, the distance is a p p r o x i m a t e l y 1100 ft.

EXAMPLE Resistivity survey in a well yielded the following data: w e l l b o r e c o r r e c t e d m u d filtrate resistivity -- 0.165 ohm-m, w a t e r resistivity -O. 18 ohm-m, invaded zone resistivity -- 12 ohm-m, and d e e p f o r m a t i o n

Figure 8.13. Fault p r o x i m i t y index (FPI) as a f u n c t i o n o f fracture intensity index (FII) in Austin chalk [25].

514

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

resistivity = 85 ohm-m. The reservoir average p o r o s i t y (17%) w a s d e t e r m i n e d from a N e u t r o n log. Substantial m u d loss w a s o b s e r v e d during drilling of this well as well as in n e i g h b o r i n g wells. Pressure test analysis as well as cores c o n f i r m e d the p r e s e n c e of extensive natural fractures in the well. (a) Estimate the fracture intensity i n d e x and the p o r o s i t y partitioning coefficient. (b) If the average irreducible w a t e r saturation e s t i m a t e d f r o m log analysis is 24%, d e t e r m i n e the fracture width. Note that the coefficient Cx in e q u a t i o n 8.12 is typically 0.25. (c) Estimate the fracture width, fracture p e r m e a b i l i t y and fracture p o r o s i t y of the formation.

SOLUTION (a) K n o w i n g RLLS = Rxo = 12 and RLLD = Rt = 85, the fracture intensity index is estimated from Eq. 8.18:

1 FII --

1

Rxo 1

Rt = 1 / 1 2 - 1/85 = 0.1417 1 1 / 0 . 1 6 5 - 1/0.18

Rmf

Rw

The saturation of m u d filtrate in the flushed zone is: Sxo -- SCwx -- 0.240.25 -- 0.70 The porosity partitioning coefficient is calculated using e q u a t i o n 8.13:

aw (1

V --

Ot(Sw - Sxo)

Rt

1) Rxo

( 1

0.18 0.17(0.24 -- 0.70)

85

1 ~ = 0.165 12 /

(b) Using Equation 8.19a:

Wf

--

O.O64

[(1

-- Siw)

FII] 1"315

Ot 0.064 0.17

[(1 - 0.24)(0.1417)] 1"315 -- 0.02 c m

PETROPHYSICAL PROPERTIES

5| 5

(c) Using Equation 8.19b: kf-

1.5 • 107~)t [ ( 1 - Swi) FII] 2"63

= 1.5 • 107 ( 0 . 1 7 ) [ ( 1 - 0 . 2 4 ) 0 . 1 4 1 7 ] 2.63 = 7,265 mD Using equation 8.20, where CT = 3/4, the fracture porosity is:

C~f-- [ R m f (

1 RLLS

1 )]CT--[0"165(12

1

3/4 1 85)] = 0.0358

RLLD

For C T - - 2/3, the fracture porosity is 0.052; thus the value of Cg is between 0.036 and 0.052. The matrix porosity is: t~m = t~t(1 -- ~) = 0.17(1 -- 0.165) = 0.142 Note that the sum of ~f (for CT = 3/4) and ~)m is O. 177, which is approximately equal to the total porosity obtained from well logs. Therefore the fracture porosity of this reservoir is 3.6%.

PERMEABILITY--POROSITYRELATIONSHIPSIN DOUBLEPOROSITYSYSTEMS Petroleum reservoirs can be divided into three broad classes based on their porosity systems: (1) intergranular; (2) intercrystalline-intergranular; (3) solution channels and/or natural fractures. Reservoirs with vugular solution channels and/or fractures differ from those having intercrystalline-intergranular porosity in that the double porosity system strongly influences the movement of fluids. The double porosity can be the result of fractures, joints, and/or solution channels within the reservoirs. Carbonate reservoirs with a vugular-solution porosity system, such as the Pegasus Ellenburger Field and Canyon Reef Field in Texas, exhibit a wide range of permeability. The permeability distribution may be relatively uniform or quite irregular. The double porosity reservoir with a uniform permeability distribution can be analyzed as follows. Consider a rock sample with two dominant pore radii, as shown in Figure 8.14. The total flow through such systems is the sum of

5| 6

PETROPHYSICS" RESERVOIR ROCK PROPERTIES

Figure 8.14. Unit model with two d o m i n a n t p o r e radii. The systems possess different petrophysical properties such as porosity a n d permeability. individual flow rates through each system, the systems having different petrophysical properties such as porosity and permeability. qt -- ql + q2

(8.21)

Using Darcy's law (for qt) and Poiseuille's (for ql and q2), we have: I nl Xrcl 4 + n2xr42 ] AP ~

AP

kAt ~-~ --- ~

8

~L

(8.22)

The total area for the system is: nlxrcl2

At =

~)1

+

n2xr22

~)2

(8.23)

Also, we know from Chapter 3 that:

rc =

2

(8.24)

Svp

Substituting Equation 8.23 in 8.22 and Equation 8.24 in the resulting equation gives:

1

k -- -

[mjSvpl § ljSv 2]

2 [1/(~)lS2pl ) + 1/(~2S4p2)]

(8.25a)

PETROPHYSICAL PROPERTIES

$17

The general form of this e q u a t i o n is"

n 1

i=~l (1/S4pi)

k~ ~ 2 ~ (1/(~)iS2pi) ) i=1

(8.25b)

For a single porosity system, this equation r e d u c e s to" 1 (1/Sv4p) k -- 2 (1/((~S2p))

2S2p

(8.26)

The constant 2 in Equation 8.26 is related to the shape of the capillaries and their tortuosity and can be r e p l a c e d by KT"

n(

)

Z 1/Sv4pi i--1

k -

i=l

(8.27)

(KTi / (t~iS2pi))

where KT -- 6fsp't:

(8.28)

Methods for estimating the p o r e shape factor fsp and the tortuosity of the capillaries 1: are discussed in C h a p t e r 3. In the case of formations containing a very small n u m b e r of channels p e r unit p o r e volume, such as in reservoirs w i t h high storage capacity in a rock matrix, and very low storage capacity in channels, nl >> n2, Equation 8.25a can be w r i t t e n as: k =

~)1 2Spv 1

=

~)lr21

(8.29a)

8

w h e r e the subscript 1 stands for primary p o r e space, w h i c h stores most of the fluid. In the case of n2 >> nl, i.e. rocks in w h i c h the fluid is stored mainly in secondary p o r e spaces such as fissures and vugs, Equation 8.29a becomes: k =

r

22 2Spv

2

= t~2rc2

8

,,,. ~-,M fK'~ x.,~.

t-~a"~l[ "X ,,.= / ,~.j

w h e r e the subscript 2 stands for s e c o n d a r y p o r e space. Thus, in cases w h e r e nl >> n2 and n2 >> nl, double porosity systems may be

518

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

approximated by a single pore space system, and consequently the methods developed in Chapter 3 for clastic rocks can be used in carbonate formations. In the case where n i is approximately equal to n2, and since it is impossible to determine n l and n2, an alternative to the above approach is to take the geometric mean of the two capillary systems, i.e." k - - ~ ( 0 1 r 2r1c)l r c 2((~2r22) 4 ( ~8 l ( ~ 2 -8

8

(8.30)

Using an average value of rcl and rc2, and an average value of ~)1 and ~2, Equation 8.30 becomes similar to the Kozeny equation.

POROSITYAND PERMEABILITYRELATIONSHIPSIN TYPE ] NATURALLY FRACTUREDRESERVOIRS As mentioned earlier in regard to type 1 reservoirs, fractures provide all the storage capacity and permeability and the fluid flow behavior is controlled by the fracture properties. The equation for volumetric flow rate, combined with Darcy's law, provides the basic approach for estimating fracture permeability. Consider a block of naturally fractured rock with n fractures, as shown in Figure 8.15. Assuming the fractures are rectangular, smooth, and do not contain any mineral, the Hagen-Poiseiulle equation gives: q =

nhfw 3 AP 12

BL

(8.31a)

Figure 8.15. Unit model used in calculation of fracture permeability in type 1 naturally fractured reservoirs.

PETROPHYSICAL PROPERTIES

$19

and Darcy's law is. AP q = kA~ gtL

(8.31b)

Equating these two equations and solving for permeability results in:

k --

nhfw 3 12A

(8.32)

The physical difficulty in using equation 8.32 is that the n u m b e r of fractures, fracture height, and fracture width have to be known. Since, by definition, Vp -- Vb

nhfwfL AL

(8.33)

and

A =

nhfwf

(8.34)

substituting Equation 8.34 in 8.32 yields:

k --

(8.35)

12

Equation 8.35 is similar to Equation 3.14, k -- ~r2/8, w h e r e the capillary radius r and constant 8 have been replaced by the fracture width wf and 12 respectively. Equation 8.35 is c o m m o n l y used to calculate fracture permeability. Equation 8.35 can be used to calculate wf if the porosity and permeability are k n o w n from well logs or well testing:

wf --

~/

k 12~

(8.36)

520

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

FRACTURES POROSITY AND APERTURE FROM CORES Oil-beating fractured granite is a major productive formation in some parts of the world, such as in the Bach Ho field offshore Vietnam. Fractured granite consists of three main elements: macrofractures, low permeability matrix with microfractures, and tight non-permeable matrix [33]. Tan et al. selected ten of the most representative w h o l e cores (D = 6.7 cm) from this oilfield, with total porosity from 3.03 to 9.93 percent, and permeability from 226 to 19,250 mD, as s h o w n in Table 8.4 [33]. After trimming and cleaning, the cores w e r e saturated with a brine, then the total porosity ((~f) w a s determined. The saturated samples w e r e then loaded into a capillary cell (porous plate technique) and Pc was increased in steps from 0.05 to 5 bars. The water saturation Sw was recorded at each step. They observed that the larger the fracture width, the lower the capillary forces. From this observation, they demonstrated that the sudden change in the slope of Pc versus Sw, as s h o w n on Figure 8.16, corresponds to the volume of fractures. The fracture porosity and porosity partitioning coefficients w e r e then calculated from:

~f = ~t (1 -- Sws)

(8.37a) (8.37b)

V -- ~)f/~)t -- 1 -- SWS

w h e r e Sws is the water saturation corresponding to the sudden change in slope of the Pc curve. Values of c~f and v are s h o w n in Table 8.4. This table shows very high values of v, w h i c h indicates a very high density of microfractures in the matrix.

TABLE8.4 RESULTSOF COREANALYSISOF FRACTUREDRESERVOIR

kf, mD

~)t, %

V

~f, %

(])m, %

1800 19250 15220 1704 8520 386 824 514 226 302

8.33 8.63 4.25 4.67 3.03 4.34 9.93 5.54 4.06 7.69

0.328 0.302 0.435 0.4 0.396 0.26 0.266 0.462 0.37 0.223

2.73 2.61 1.85 1.87 1.20 1.13 2.64 2.56 1.50 1.71

5.60 6.02 2.40 2.80 1.83 3.21 7.29 2.98 2.56 5.98

PETROPHYSICAL PROPERTIES

521

Figure 8.16. A i r / w a t e r capillary pressure curves [33].

Tan et al. also performed simultaneous m e a s u r e m e n t of permeability (to water) and resistivity on naturally fractured core samples, using a Hassler coreholder e q u i p p e d with two silver-coated electrodes. For each core sample, resistivity and permeability w e r e determined at various overburden pressure from 15 to 400 bars. Samples of two saturation states w e r e investigated: Full brine saturation, and partial brine saturation. Resistivity measurements w e r e performed on 31 fully brine saturated cores with k < 3 mD then the calculated formation resitivity factor (F = Ro/Rw), was plotted against fracture porosity (Figure 8.17). A curve-fit of the data points shows that the cementation factor is significantly low, w h i c h is typical of systems w i t h high porosity partitioning coefficient. The log-log plot of resistivity versus permeability (Figure 8.18) allowed t h e m to investigate the relationship b e t w e e n fracture permeability and fracture width. They concluded that wf calculated from resistivity (Equation 8.40b) represents the real value of the fracture aperture. Expressing fracture porosity in percent and fracture width or aperture in micrometer (~tm) Equation 8.35 becomes: kf -- 8.44 x

10-4w~)f

(8.38)

w h e r e kf is expressed in Darcy units. The fracture porosity is calculated from: ~f --

O.04nfwf ~D

(8.39)

522

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

Figure 8.17. Formation resistivity factor vs. porosity [33].

Figure 8.18. Cross-plot of resistivity vs. permeability [33].

The fracture width or aperture can be estimated from the following equations:

Wf--

xDkf ) 33.32 • lO-6nf

(8.40a)

523

PETROPHYSICAL PROPERTIES

Or, a s s u m i n g t h e fracture l e n g t h is equal to t h e l e n g t h of t h e c o r e sample: 106RwL wf =

(8.40b)

nfrofD

Where: nf = n u m b e r of fractures in f r a c t u r e d c o r e s a m p l e Rw = Brine or w a t e r resistivity, O h m - m rof = electrical r e s i s t a n c e of t h e f r a c t u r e d c o r e sample, O h m D = D i a m e t e r of f r a c t u r e d c o r e s a m p l e as s h o w n in Figure 8.28A, c m L = Length of f r a c t u r e d c o r e s a m p l e as s h o w n in Figure 8.28A, c m kf = Fracture p e r m e a b i l i t y , Darcy ~f = Fracture porosity, p e r c e n t wf = Fracture w i d t h or a p e r t u r e , m i c r o m e t e r (gtm)

EXAMPLE Resistivity m e a s u r e m e n t s w e r e p e r f o r m e d on a 100% w a t e r s a t u r a t e d c o r e s a m p l e c o n t a i n i n g 5 fractures. T h e following results w e r e obtained" D -- 7.62 c m

Rw -- O. 15 O h m - m

rof = 675 O h m

~)t- 10.3%

L -- 30.48 c m Calculate: (a) (b) (c) (d)

Fracture Fracture Fracture Porosity

width porosity Permeability p a r t i t i o n i n g coefficient and m a t r i x porosity.

SOLUTION (a) T h e fracture w i d t h or a p e r t u r e is o b t a i n e d f r o m Eq. 8.40b:

wf --

106RwL nfrofD

=

106 • O. 15 • 30.48 5 • 6751 x 7.62

= 178 gtm

(b) T h e fracture p o r o s i t y is e s t i m a t e d f r o m Eq. 8.39:

~f-

O.04nfwf xD

=

0.04 • 5 x 178 7.62x

= 1.5%

524

PETROPHYSICS" RESERVOIR ROCK PROPERTIES

(c) The fracture permeability is estimated from Eq. 8.38: kf -- 8.44 x 10 -4 x 1782 x 1.5 -- 39.6 Darcy (d) The porosity partitioning coefficient and matrix porosity are determined from Equations 8.37b and 8.15: V

Of Ct

~

- -

~ -

1.5 =0.14 10.3

~ m - - ~t - - t ~ f - - 8 . 8 %

SPECIFIC AREAOF FRACTURES Let Spv be the internal surface area per unit of pore volume, where the surface area for n fractures is n(2wfL + 2hfL) = 2n(wf + hf)L, and the pore volume is n(wfhfL), assuming the fracture provides all of the storage and permeability. The specific surface area per unit pore volume is: Svp =

2n (wf+hf)L n2wfhfL

= 2

(1 hff

+~

1)

(8.41)

Wf

Using the same assumptions, the specific surface area per unit grain volume is" Sgv --

2n (wf + hf) L AL(1 - r

(8.42)

Multiplying and dividing by wfhf, and simplifying, yields:

Sgv

--

2nwfhf ( 1 A(1 - 0 ) ~ +

1)

(8.43)

wf

Substituting for A from Equation 8.37 and simplifying results in:

Sgv=2( 1 -~~ )(~f

1)

+

wf

(8.44)

The term l/hf is very small in comparison to l/wf because hf >> wf. Thus equation 8.44 reduces to"

2(0)

Sgv -- wf

1- ~

(8.45)

PETROPHYSICAL PROPERTIES

525

Combining Equations 8.41 and 8.44 gives:

S g v - - ( ~ 1) - ~ ) Spy

(8.46)

Since l / w f >> 1/hf, Equation 8.41 reduces to" 2 2 Svp = - - or wf -wf Svp

(8.47)

Substituting for wf, Equation 8.35 becomes, k --

(8.48) 3S~v

Combining Equations 8.46 and 8.36 gives:

1(0, )

k -- 3Sg2v

(1 - (~)2

(8.49)

The derivation of Equations 8.33 through 8.49 assumes that the fractures are rectangular, smooth, uniform, and that fracture length is equal to the length of the rock sample. The constant 3 is specific to the shape of the fracture. Equations 8.48 and 8.49 can be generalized for all fracture shapes as follows:

k --

(8.50) KTfS~v

'

k -- KTfS2v

(1 -- ~))2

t

(8.51)

w h e r e KTf = Ksf'C, Ksf being the fracture shape factor and I; the tortuosity. This equation is similar to the generalized Kozeny equation. Unlike sandstone formations, identification and characterization of flow units in carbonate formations is not possible because of extreme variations of fissures, both in terms of geometry and intensity. However, in reservoirs w h e r e the geometry and distribution of fissures are uniform t h r o u g h o u t the reservoir, one could use the same concepts of reservoir quality index (RQI), flow zone index (FZI), and Tiab's hydraulic unit characterization factor (HT) as were presented in Chapter 3.

526

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

Figure 8.19. Effect of fracture shape on the permeability-porosity relationship.

EFFECTOF FRACTURESHAPE Consider a fracture with an elliptical cross-section as shown in Figure 8.19. Assuming a type 1 naturally fractured reservoir, the specific surface area per unit pore volume, Spv, is: Spv=

Ase Vp

(8.52)

where Ase is the surface area of the elliptical fracture and is given by: Ase =TZ

Lo.75(wf + h f ) -

0.5v/wehfJ L

(8.53)

and Vp-- ~wfhfL

(8.54)

Combining the above three equations, and simplifying: S v p = 3 ( 1~ + Wf

~f) -

2 ~/wfhf

(8.55)

Since 1/hf << l/wf, the above equation reduces to:

Svp__( 3 )

2

WZ -- ~/Wfhf

(8.56)

PETROPHYSICAL PROPERTIES

527

A s s u m i n g w f h f >> Wf, Equation 8.56 further reduces to (with less than 5% error)"

3 Svp = ~ wf

(8.57)

It is clear from the above equations that the value of the facture shape factor Ksf changes with fracture shape.

HYDRAULIC RADIUSOF FRACTURES The effective or hydraulic radius of a fracture (rhf) can be obtained by representing the fracture as a capillary tube. Equating Equations 3.1 O, w h i c h is valid for a capillary tube system, and 8.3 l a, w h i c h accounts for fracture geometry, yields:

xr4f

8

=

hfwf

3

12

(8.58a)

Solving for the radius results [32]"

rhf --

2 hfw ') /4

~-~

(8.58b)

Equation 8.58b is very important equation in a sense that it interprets the fracture geometry in terms of equivalent hydraulic radius and thus can be incorporated in any tube model (Figure 8.20).

Figure 8.20. Fracture h y d r a u l i c r a d i u s in f r a c t u r e d (left) a n d u n f r a c t u r e d rocks (right). A rock s a m p l e with o n e f r a c t u r e (left) o f h y d r a u l i c r a d i u s e q u a l to 1 c m is e q u i v a l e n t to a rock s a m p l e w i t h one solution c h a n n e l (right) o f r a d i u s rc = 1 cm.

528

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

Substituting Equation 8.58b in 3.10 ( w h e r e r = rhf) yields:

C f . / - -2h f w f 3 -- 0.05758 Cf C h f w 3 kf = -8-V3x

(8.59)

The fracture permeability in this equation is in cm 2, porosity is a fraction, and fracture height and w i d t h are in cm. If fracture width, we, and fracture porosity, ~f, are determined from core analysis and permeability is determined from well testing, fracture height can be determined from Equation 8.59. Another application of Equation 8.59 is to decide w h a t radius a horizontal well has to have in order to get the same benefit of a hydraulic fracture of w i d t h wf and height hf.

EXAMPLE Core analysis, well logs and pressure data yielded the following data:

~f -- 0.027

wf -- 0.015 cm

kf = 51.3 Darcies

Estimate: (a) Fracture height and (b) Hydraulic radius of the fractures.

SOLUTION (a) Equation 8.59 can be rearranged for fracture height as follows"

kf hf--

0.05758r

)2

1

(8.60)

w3

Since 1 D a r c y 9.87 x 10 -7 cm 2, therefore 51.3 Darcies -(51.3)(9.87 • 10 -7) = 5.06 • 10 -5 cm 2.

5 . 0 6 x 10 -5 hf --

0.05-7--58(0.027)

)2 (0.015) 3

= 314

cm-

10.3 ft

PETROPHYSICAL PROPERTIES

$29

(b) Hydraulic radius can be calculated using Equation 8.58b. 1/4 rhr-

/--hfwl;! \3x ~/

rrff =

(2 )3) 1/4 = 0.12 c m ~-~ (314)(0.015

This value of rhf implies that a fracture w h i c h has a height of 314 c m and a width of 0.015 cm, is equivalent to a cylindrical channel w h i c h has a hydraulic radius of O. 12 cm.

TYPE 2 NATURALLYFRACTUREDRESERVOIRS In this type of reservoir the matrix has a good porosity and permeability. Oil is trapped in both the matrix and fractures. Consider a representative block containing two parallel layers, as s h o w n in Figure 8.21. The average permeability in the matrix can be m o d e l e d using the capillary tube model and equations developed in Chapter 3. The average permeability in the fracture system can be expressed by the equations developed in previous sections in this chapter. For nc capillaries and nf fractures, the following a p p r o a c h can be followed to estimate permeability in type 2 naturally fractured reservoirs. Total flow rate from both matrix and fractures can be expressed as: qt

--

qf + qm

(8.61)

Figure 8.21. Representative elementary rock volume containing two parallel systems of matrix and fracture. The fluid is stored in both matrix and fractures (type 2 reservoirs).

$30

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

Using Darcy's law (for qt) and Poiseuille's law (for qm and qf) gives:

kAt ~

--

Encrcnfhf l P 8

+

pL

12

(8.62)

The total area of matrix and fracture can be expressed as: At =

ncXrc2

*c

r

nfhfwf

(8.63)

Of

Assuming equal storage capacity of both systems (matrix and fracture), i.e., the porosity partitioning coefficient v is approximately 0.50 and therefore nc = nf and ~f = r Equation 8.63 simplifies as: (8.64)

At -- n (Xrc2 + hfwf)

Thus the average permeability can be extracted first by substituting Equation 8.64 in 8.62 and then solving for k:

k - - xr 2 + h f w f

~

+

hfw:3 12

(8.65)

For a unit block area, hf = 1. While hf and wf can be relatively easily measured, this is not always the case with rc. A rather simplistic a p p r o a c h to determine average permeability in type 2 reservoirs is to calculate the geometric mean of the two systems:

(8.66)

Assuming the average porosity 0 -- V/~)f~)c Equation 8.66 becomes: k

--

(rcwf \ 9.8 ) r

(8.67)

It is obvious from this discussion that in naturally fractured carbonate formations, w h e r e structural heterogeneities and textural changes are c o m m o n and only a small n u m b e r of wells are cored, the practice of using statistical core permeability-porosity relations to characterize flow units is not r e c o m m e n d e d . The main parameters that influence the flow units in naturally fractured reservoirs include: secondary porosity (fractures,

FLUID FLOW MODELING IN FRACTURES

531

fissures, and vugs), matrix porosity, fracture intensity index, fracture dimensions (shape, width, and height), tortuosity, porosity, partitioning coefficient, specific surface area, and irreducible w a t e r saturation. These parameters must be i n c o r p o r a t e d in the definition of flow units in o r d e r to effectively characterize them.

FLUID FLOW MODELINGIN FRACTURES Fractures are m o d e l e d as flow channels or cracks. Their t w o main properties from the fluid flow point of view are the storage capacity and the fluid transmission or transfer capacity, also k n o w n as fracture conductivity. These t w o properties are d e p e n d e n t on the dimensions of length, width, and height.

FRACTUREAREA Fracture area is d e t e r m i n e d by the shape and relative dimension of the fracture, and influences the mechanical behavior of the rock mass. Fractures are usually assumed to be circularly shaped, w i t h constant radius, or parallelogram shaped, using a rectangle or square shape as a simplifying assumption. Fracture area is influenced by the e x t e n t of the fracture. There are three cases: (1) fractures are infinitely laterally extensive, (2) fractures terminate on o t h e r fractures, and (3) fractures terminate in intact rock. However, from fluid transfer point of view they are m o d e l e d as rectangular planes of a certain w i d t h w, height h, and length L or x, as s h o w n in Figure 8.22. Three-dimensional fracture g e o m e t r y systems can be r e p r e s e n t e d in: (1) three principal planes: defining matrix blocks, Figure 8.23(a);

Figure 8.22. Fracture dimension for flow modeling point of view.

532

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

Figure 8.23. An idealized schematic o f elementary blocks [19].

(2) two principal planes: defining matches, Figure 8.23(b), (b'); and (3) one series of parallel planes: defining sheet, Figure 8.23(c).

FRACTURESTORAGECAPACITY In contrast to the matrix porosity, fracture porosity contributes only a few percent to the total porosity. Fracture aperture is typically up to a few millimeters in width, and typical fracture spacing is in the centimeter to meter range. Because fracture apertures are generally significantly greater than typical matrix pore-throat sizes, they contribute the major portion of the total transmissivity of the petroleum rocks, and consequently are an important factor in the m o v e m e n t of fluids. Fracture porosity initially is very high, but, over time, fractures may b e c o m e partially filled with fines. This filling process considerably reduces the fracture porosity to less than five percent. Since only fracture conductivity is necessary in flow calculations, not m u c h attention has been given to fracture porosity or storage capacity. The overall fracture storage capacity, w h i c h indicates h o w m u c h fluid is held within the fracture n e t w o r k of a particular reservoir, is best estimated from pressure buildup tests.

FLUID FLOW MODELING IN FRACTURES

533

FRACTURECONDUCTIVITY In reservoir engineering, fractures have b e e n typically categorized on the basis of their fluid transmission capacity or conductivity as follows: (1) F i n i t e c o n d u c t i v i t y : Finite conductivity fractures allow a limited a m o u n t of the fluid to flow. If the fracture has dimensionless conductivity FCD = ( k f w f ) / ( k r x f ) less than 300, it is t e r m e d a finite conductivity fracture. (2) I n f i n i t e c o n d u c t i v i t y : Infinite conductivity fractures are highly conductive and their fluid transferring capacity is greater than that of the finite conductivity fractures. If FCD = (kfwf)/(krxf) > 500 then the fracture is infinitely conductive. This n u m b e r is a c c e p t e d by many researchers; however, some w o r k s assume FCD > 300 for infinite conductivity. (3) U n i f o r m flux: Uniform flux fractures allow the fluid to flow t h r o u g h t h e m such that there occurs a certain pressure drop but the a m o u n t of the fluid entering and leaving the fracture is constant. These three categories of fractures w e r e d e v e l o p e d for hydraulic fractures since physical dimensions of hydraulic fractures can be controlled by increasing the injection pressure and the a m o u n t of fluid and p r o p a n t that control the fracture opening. Natural fractures, on the o t h e r hand, rarely s h o w infinite conductivity behavior. This is because no p r o p a n t is present in natural fractures and the fracture surface with time develops a skin due to the chemical and physical changes that take place with time and to the p r e s e n c e of reservoir fluids. Total reservoir conductivity is controlled by the fracture frequency, w i d t h or aperture, and length. Fracture frequency is the n u m b e r of fractures p e r unit length (depth). Fracture frequency determines the fracture volume in a rock and is n e e d e d in order to d e t e r m i n e the porosity due to fractures. Fracture aperture or w i d t h is the fracture o p e n i n g and is a critical p a r a m e t e r in controlling fracture porosity and permeability. Fracture length determines the distance the fracture is penetrating the reservoir rock from the weUbore. Fractures are rarely straight, as s h o w n in Figure 8.24. They are curvilinear and create a tortuous path as c o m p a r e d with straight tubes. The term fracture tortuosity is frequently used to define the irregular shape of the fractures and flow paths in reservoir rocks. Tortuosity is the ratio of the actual fracture length c o n n e c t i n g t w o points and m i n i m u m fracture length, therefore the m o r e the fractures are interconnected, the less the value of x.

534

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

Figure 8.24. Examples of fracture frequency and fracture tortuosity.

Figure 8.25. Realistic and idealized naturally fractured rocks, regenerated after Warren and Root [18].

CHARACTERIZINGNATURALFRACTURESFROM WELL TEST DATA Warren and Root first m o d e l e d the transient flow of fluids in naturally fractured rocks, assuming that the rock consists of a fracture n e t w o r k as s h o w n in Figure 8.25 [18]. This widely p o p u l a r model for flow analysis in naturally fractured reservoirs is referred to as the "sugar cube model" Warren and Root assumed that the entire fluid flows from the matrix to the fractures and only fractures feed the wellbore. Since not all naturally fractured reservoirs behave similarly, the degree of fluid flow is controlled by the matrix and the fracture properties. Thus Warren and Root introduced t w o key parameters to characterize naturally fractured reservoirs: (1) the storage capacity ratio, co, w h i c h is a measure of the fluid stored in fractures as c o m p a r e d with the total fluid p r e s e n t in the reservoir; and

CHARACTERIZING NATURAL FRACTURES

535

(2) the inter-porosity flow parameter, ~,, w h i c h is a measure of the heterogeneity scale of the system and quantifies the fluid transfer capacity from matrix to the fracture and vice versa. A value of unity for ~, indicates the absence of fractures or, ideally, that fractures behave like the matrix such that there is physically no difference in petrophysical properties; in other words, the formation is homogeneous. Low values of ~,, on the other hand, indicate slow fluid transfer b e t w e e n the matrix and the fractures. The actual range of ~ is, however, 10 -3, w h i c h indicates a very high fluid transfer, to 10 -9, w h i c h indicates p o o r fluid transfer b e t w e e n the fractures and the matrix. The storage factor co has a value b e t w e e n zero and unity. A value of 1 indicates that the all fluid is stored in the fractures, whereas a value of zero indicates that no fluid is stored in the fractures. A value of 0.5 indicates that the fluid is stored equally in matrix and fractures. Mathematically, the storage capacity ratio and the inter-porosity flow parameter are defined as follows:

co --

(~)Ct)f = ((~Ct)f ((~Ct)t (OCt)f+ (~Ct)m

(8.68)

and

- ot _2_r_______~wkm kf

(8.69)

w h e r e cz is the geometry parameter, given by:

0~ =

4n (n + 2)X 2

(8.70)

w h e r e n is 1, 2, and 3 for sheet, matches, and cube models respectively, as s h o w n in Figure 8.23. For cubical and spherical geometries [26]:

cz =

60

(8.71a)

w h e r e Xm represents the side length of the cube or the diameter of the sphere block. For long cylinders

cz =

32

(8.71b)

$]6

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

w h e r e Xm is the diameter of the cylinder. formations 12 cz = ~

For layered or slab

(8.71c)

w h e r e hf is the fracture height, usually taken as the formation thickness of the fractured zone stacked in b e t w e e n the other layers. Knowing the inter-porosity parameter X from well test analysis, the fracture height hf can be calculated from:

hf -- rw

~12km ~kf

(8.72)

For the sugar-cube model, the side length of each matrix block is obtained from:

Xm = rw

~6

0km ~kf

(8.73)

Figure 8.26. (a) Spherical, Co) cubical, (c) cylindrical, a n d (d) layered or stacked matrix blocks with natural fractures.

CHARACTERIZING NATURAL FRACTURES

537

EXAMPLE A well is c o m p l e t e d in a naturally fractured reservoir. K n o w i n g the following data, w h i c h w e r e obtained from core analysis and a single pressure d r a w d o w n test: -- 2.5 • 10 -6

rw -- O. 3 ft

kf -- 39,000 mD

km -- O. 185 mD

(a) Calculate the side length of the matrix blocks. (b) What w o u l d be the height of the fracture zone if the system w e r e layered?

SOLUTION (a) The side length Xm is calculated from Equation 8.73" /60km Xm-rwV~ f =0.3

6 0 x 0.185

2.5 x 10 -6 x 3 9 , 0 0 0

= 15.3 ft

(b) The fracture height is calculated from Equation 8.72:

hf - rw

12 x 0.185 ~12km _ 0.3~ = 1.4 ft Xkf 2.5 x 10 -6 x 39,000

Both of the Warren and Root parameters, ~. and co, are preferably obtained from well test data by using either the conventional semi-log analysis or the type curve matching techniques. Using the Tiab's Direct Synthesis t e c h n i q u e both parameters can be d e t e r m i n e d from the log-log plot of the pressure derivative versus time w i t h o u t using the type-curve matching t e c h n i q u e [29]. Figure 8.27 is the semi-log pressure test, with a typical t w o parallel lines indicating p r e s e n c e of natural fractures. The storage capacity ratio can be estimated from this figure, using the following equation: co -- exp

-2.303~ m

(8.74)

w h e r e 8P is the pressure difference b e t w e e n the t w o parallel lines in Figure 8.24 and m is the slope of either line. The degree of fracturing in each segment of the reservoir can influence the estimated value of co; consequently testing different wells can yield different values of co.

538

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

6,500

...........................................................................................................................................................................................................................

!

S

','00 6,300

/ /

6,200 6,100

~-

inflectionpoi~ /f ~

f

]'

5,800st -

5,700 !......

" / , ......

100,000

10,000

~ AtH4nf~ 1,000 100 AtH= (tp + At)lAt

10

1

Figure 8.27. A

typical pressure test curve showing two parallel lines, a strong indication o f the presence o f natural fractures in petroleum reservoir rock.

The slope is used to estimate the f o r m a t i o n permeability, k, from:

k --

162.6q~tBo mh

(8.75)

The units of p r e s s u r e and slope are psi and psi/log cycle respectively. O n c e co is estimated, the fracture p o r o s i t y can be estimated if matrix p o r o s i t y (~m, total matrix compressibility Cm, and total fracture compressibility cf are k n o w n , as follows:

r --

0•mCm cf(1 - co)

(8.76)

Fracture compressibility m a y be different from matrix compressibility by an o r d e r of magnitude. Naturally fractured reservoirs in Kirkuk field (Iraq) and Asmari field (Iran) have fracture compressibility ranging from 4 x 10 -4 to 4 x 10 -5 p s i - 1. In Grozni field (Russia) cf ranges from 7 x 10 -4 to 7 x 10 -5. In all these reservoirs cf is 10 to 100 folds h i g h e r than Cm. Therefore the practice of assuming cf -- Cm is n o t acceptable.

CHARACTERIZING NATURAL FRACTURES

$]9

The fracture compressibility can be e s t i m a t e d from the following e x p r e s s i o n [22]" 1 -- ( k f / k f i ) 1/3

Cf --

(8.77)

AP

k~ -- Fracture permeability at the initial reservoir p r e s s u r e Pi kf -- Fracture permeability at the c u r r e n t average reservoir p r e s s u r e 1~ In d e e p naturally fractured reservoirs, fractures and the stress axis o n the formation generally are vertically oriented. Thus w h e n the p r e s s u r e d r o p s due to reservoir depletion, the fracture p e r m e a b i l i t y r e d u c e s at a l o w e r rate than o n e w o u l d expect, as indicated by Equation 8.77. In Type-2 naturally fractured reservoirs, w h e r e matrix porosity is m u c h g r e a t e r than fracture porosity, as the reservoir p r e s s u r e d r o p s the matrix p o r o s i t y decreases in favor of fracture porosity [22]. This is not the case in Type-1 naturally fractured reservoirs, particularly if the matrix p o r o s i t y is very low. A representative average value of the effective p e r m e a b i l i t y of a naturally fractured reservoir may be o b t a i n e d from: k-

v/kmaxkmin

(8.78)

where kmax = m a x i m u m permeability m e a s u r e d in the direction parallel to the fracture plane (Figure 8.28), thus kmax ~ kf kmin -- m i n i m u m permeability m e a s u r e d in the direction p e r p e n d i c u l a r to the fracture plane (Figure 8.28), thus kmin ~ km

4"

-.

-

W

--~

L

D

llliiX

h=hf

! kmin

Wf

(A)

Wf

Figure 8.28. M a x i m u m a n d m i n i m u m permeability.

(B)

540

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

Equation 8.78 b e c o m e s : k-

v/kfkm

T h e fracture p e r m e a b i l i t y (plus c o n n e c t e d e s t i m a t e d from:

(8.79) vugs) can t h e r e f o r e

be

k2

kf =

(8.80) km

W h e r e km is the matrix permeability, w h i c h is m e a s u r e d f r o m r e p r e s e n t a t i v e cores, and k is the m e a n p e r m e a b i l i t y o b t a i n e d f r o m p r e s s u r e transient tests. C o m b i n i n g e q u a t i o n s 8.77 and 8.80 yields: 1 - (k/ki) 2/3 Cf =

AP

(8.81)

Where ki = average p e r m e a b i l i t y o b t a i n e d from a transient test run w h e n the reservoir p r e s s u r e was at or n e a r initial conditions Pi and k = average p e r m e a b i l i t y o b t a i n e d from a transient test at the c u r r e n t average reservoir pressure, 15 Ap = P i - 1 ~ Matrix p e r m e a b i l i t y is a s s u m e d to r e m a i n c o n s t a n t b e t w e e n the t w o tests. Note that e q u a t i o n s 8.77 and 8.81 are valid for any t w o c o n s e c u t i v e p r e s s u r e transient tests, and t h e r e f o r e Ap = I~1 - 1~2 . The time b e t w e e n the t w o tests must be long e n o u g h for the fractures to d e f o r m significantly in o r d e r to d e t e r m i n e an a c c u r a t e value of cf. T h e fracture p e r m e a b i l i t y can also be e s t i m a t e d from the following correlation [31 ], if the fracture w i d t h (or a p e r t u r e ) wf is k n o w n from logs or c o r e m e a s u r e m e n t s : kf -- 3 3 o 0 t w f 2

(8.82)

w h e r e wf is in m i c r o n s (1 m i c r o n = 10 -6 m), the storativity ratio co and total pososity ~)t a r e e x p r e s s e d as a fraction and kf in mD. T h e effective p e r m e a b i l i t y of a naturally fractured reservoir may also be a p p r o x i m a t e d from the following equation: k = km + Ofkf

(8.83)

This e q u a t i o n should preferably be used for verification p u r p o s e s , i.e. o n c e kf is calculated from e q u a t i o n s 8.80 or 8.82 and Of from Eq. 8.76,

CHARACTERIZING NATURAL FRACTURES

$4 |

they should be substituted into Eq. 8.83. If kf and ~f are correct, t h e n the effective permeability k from Eq. 8.83 should be approximately equal to that obtained from well test analysis. If the fracture width cannot be measured from logs or core analysis, and kf can be calculated from Eq. 8.80, then equation 8.82 may be used to estimate wf"

Wf --

kf 3 30X~t

(8.84)

The inter-porosity fluid transfer coefficient is then estimated as:

( 1 ) ) ~, - 3792(~ct)f+mgrew ( c01n kfAtinf

(8.85)

The reservoir permeability, k, is in mD, fluid viscosity, It, in cP, wellbore radius, rw in ft, inflection time, Atinf, in hrs, porosity in fraction, and total compressibility, ct, in psi-1. The test time corresponding to the inflection point, Atinf, is obtained from the semilog plot of the pressure drop AP versus shut-in time At. Sometimes, however, the inflection point is not obvious on a semilog plot due to the presence of a nearby boundary or nearowellbore effects such as wellbore storage and skin. It is thus r e c o m m e n d e d that a pressure derivative plot be used as a guide for locating this inflection point. If a H o m e r plot is used, i.e., a plot of the shut-in pressure versus H o m e r time, Atu = (tp + At)/At, then the point of inflection is obtained from:

Atinf --

tp (AtH)inf- 1

(8.86)

w h e r e (AtH)inf is simply ((tp + At)/At)inf, as s h o w n on Figure 8.24, tp is the production time, and At is the test time during a pressure buildup test. On the log-log plot of the pressure derivative (t • AP) versus test time At, the inflection point Atinf is easily recognized on the pressure derivative plot. It corresponds to the time at w h i c h the m i n i m u m value of the trough is reached. Applying the Tiab direct synthesis (TDS) technique, k, 03, and X can be obtained from the log-log plot w i t h o u t

542

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

using type-curve matching [29]. The permeability is obtained from: k =

70.6qBBo h (t • Ap') R

(8.87)

where (t • Apt)R is obtained from the horizontal line of the pressure derivative, which corresponds to the infinite acting radial flow regime. The inter-porosity fluid transfer coefficient is given by: 42.5h~ctrw2 ) (t • Ap') m i n qBo Atmin

)~ __

(8.88)

w h e r e (t • AP')min and Atmin are the coordinates of the minimum point, as shown in Figure 8.32. The fracture storage ratio is given by: CO0~ =

(8.89)

e -XtDmin

where the dimensionless time corresponding to the minimum time, tnmin, is calculated from:

tDmin --

0.0002637k ) Atmin ~tr2w(~)Ct)m+f

(8.90)

Atmin is the time coordinate of the minimum point of the trough on the pressure derivative curve. Atmin on the log-log plot of the derivative curve is equivalent to &tint on the semilog plot of pressure versus time. Equation 8.89 is plotted in Figure 8.29. Curve fitting the points and solving explicitly for co yields: co -

(2.9114 - 3 . 5 6 8 8 ln(Ns)

-

6"5452) -1 Ns

(8.91)

where Ns -- e -~'tDmin

(8.92)

The fracture storage ratio co can be directly determined from Figure 8.30 or Equation 8.91.

EXAMPLE Pressure tests in the first few wells located in a naturally fractured reservoir yielded a similar average permeability of the system of 85 mD.

$43

CHARACTERIZING NATURAL FRACTURES

Pressure curve

<~ ,.!

Pressure derivative curve

e~

m. _o

Flat portion-indicating flow from Matrix

Unit slope portion indicating wellbore storage and skin effect

Trough, a typical indication of Presence of natural fractures

~og ( At ) Figure 8.29. Effect o f natural fractures on pressure derivative on a log-log plot o f pressure a n d pressure derivative against time.

0.35 ..... q---i ......

0.30

~---i--~---,:- . . . .

-I

~---,:---i--~-

.... i---i--|]---i--i--i---it--!---i--i--i-I ..... i..i ......

0.25 (-

._o 0.20

o .m a:::

r

i_.i..~___,~ .... ~_..~..i..i.

t :t

=

114 9

3.5688 -

~

-

f t

6.54521 ~

--!

I::: __.

t

O o (D

r 0.15 ct:l t,_

0.10

0.05

0.00 0.65

0.70

0.75

0.80

N,

0.85

0.90

0.95

1.00

Figure 8.30. Storage Coefficient f r o m the time coordinate o f the m i n i m u m p o i n t o f the pressure derivative curve.

$44

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

An interference test yielded the same average reservoir permeability, w h i c h implies that fractures are uniformly distributed. The total storativity, ( 0 C t ) m + f = 1.4 x 10 -6 psi -1, was obtained from this interference test. Only the porosity, permeability and compressibility of the matrix could be determined from the recovered cores. Figures 8.31 and 8.32 s h o w the behavior of pressure and pressure derivative of a recent pressure buildup test c o n d u c t e d in a well. The pressure data can be found in reference 28 (Well R-6). The pressure drop from the initial reservoir pressure to the current average reservoir pressure is 974.5 psia. The characteristics of the rock, fluid and well are given below:

--

4.15 x

10 -6

km = 0.15 mD

q = 17000 STB/D

Bo = 1.74 RB/STB Ctm

~t = 0.47 cP

rw = 0.292 ft

h = 1150 ft

psi-1

~)m --"

14%

(1) Using: (a) conventional semilog analysis, and (b) the TDS technique, calculate the current: 1. formation permeability, 2. Storage capacity ratio, and 3. fluid transfer coefficient (2) Estimate the four fracture properties: permeability, porosity, width and matrix block dimensions.

i-~;,-r

..... -t

......

.t,tt

+

~

~- +-~+-

t - d

5240 _ [ / ; . ~ . 7 . } : ~ c , r ~ + _

E

~/-~---~

[- ; - - t a ~ d : t - T T ~

. . . . . . .

4---~

!T'~

. . . . .

'. . . . ~ - ~ - 4 -

' '

.

.

.

.

.

.

.

.

.

t

....

?

--

1 ---+-~

~__,___+._._,_~ . . . . . . . . . . . . ,_ _~_....,-..~,-..i-q-g..-~

-4 . . . . . .

---~.-

+-~ .

4--::~ -~ ~ -

~-~

.

.

.

.-

.

?-

:

i

- , .- ~

!

i

'

-

I

+--~4-4-.+

In 5220

........

0.01

~

..............................

0.1

I

Time, hr Figure

8.31.

Pressure buildup test data plotted against shut-in time.

10

545

CHARACTERIZING NATURAL FRACTURES

100

i m

w D. m,

e m

t= i I

i m

L_

~D !._

.o m_

1 0.01

0.1

1

10

Time, hr

Figure 8.32. Pressure derivative group plotted against shut-in time.

SOLUTION l(a) Conventional m e t h o d From Figure 8.31" 0P -- 15 psi, m = 25 psi/log cycle and Atinf -0.22 hrs 1. The average permeability of the formation is estimated from the slope of the semilog straight line. Using Equation 8.75 yields:

k =

162.6ql.tBo mh

=

162.6 (17000) (0.47) (1.74) (25) (1150)

= 78.6 mD

2. Fluid storage coefficient is estimated using Equation 8.74:

o-

exp ( - 2 . 3 0 3 ~

exp (-2

--0.25

The storage coefficient of 0.25 indicates that the fractures o c c u p y 25% of the total reservoir pore volume.

546

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

3. The inter-porosity fluid transfer coefficient is given by Equation 8.85: 3792(1"4• (78.6) (0.22)

0.251n

0.25

= 4.27 x 10 - 6 The high value of fluid transfer coefficient indicates that the fluid transfer from matrix to fractures is very efficient. l(b) TDS t e c h n i q u e From Figure 8.32, the following characteristic points are read: Atmin -- 0.22 hrs

(t*APt)R -- 10.9 psia

(t*APt)min = 4.76 psia

1. Using the TDS technique, the value o f k is obtained from Equation 8.87: 70.6q~tBo k

m

h(t*Ap')R

70.6 (17000) (0.47) (1.74)

=

(1150) (10.9)

= 78.3 mD

2. The inter-porosity fluid transfer coefficient is given by Equation 8.88: 2 ) (t*Ap' )min 42.5h (~)Ct)m+f rw --

qBo

--

Atmin

(17000) (1.74)

0.22

= 4.267 x 10 - 6

3. The storage coefficient co is calculated from Equations 8.91"

tDmin --

0.0002637k ) Atmin ~tr2(~)ct)m+f

_ (0.0002673• 0.47 x 0.2922 x 0.0000014

x 0.22

80,968.8

Ns -- e -~'tnmin = e -4"267.80'968"8 = 0.70783 co --

1

2.9114 - 3.5668/1n(0.70783) -- 6.5452/0.70783

= 0.25

The conventional semilog analysis yields the same values of k, co and as the TDS technique. The main reason for this m a t c h is that b o t h parallel straight lines are well defined.

CHARACTERIZING NATURAL FRACTURES

547

(2) Current p r o p e r t i e s of the fracture (a) The fracture p e r m e a b i l i t y is calculated from Eq. 8.80" k2 kf --

km

78.62 =

0.15

= 4 1 , 2 1 6 mD

The fracture p e r m e a b i l i t y at initial reservoir p r e s s u r e is: k2 kfi -

km

852 =

0.15

= 4 8 , 1 6 6 mD

(b) The fracture porosity is obtained from Eq. 8.76. In fractured reservoirs with d e f o r m a b l e fractures, the fracture compressibility c h a n g e s with declining pressure. The fracture compressibility can be estimated from the following e x p r e s s i o n [22]:

1 - ( k f / k f i ) 2/3 Ctf

AP

1 - ( 4 1 , 2 1 6 / 4 8 , 1 6 6 ) 2/3 Ctf

--

974.5

= 10.12 • 10 - s psi -1

The compressibility ratio is: Ctf Ctm

=

10.12 X 10 - s 4.15 X 10 -6

= 24.4

Thus, the fracture compressibility is m o r e than 24 folds h i g h e r than the matrix compressibility, or ctf -- 24.4Ctm. The fracture porosity is: (O) ) Ctm~)m-- ( 0"25 ) 0"14 = 0.0019 -- 0.19% ~)f -- ~ - - - ~ Ctf 1 -- 0.25 24~4 The total porosity of this naturally fractured reservoir is"

0t -- ~)m + 0f -- 0.14 + 0.0019 -- 0.1419

548

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

Substituting the values of km, kf, and q)f into Equation 8.83 yields approximately the same value of the effective permeability obtained from well testing, i.e. k ~ km + t~fkf -- 0.15 + 0.0019 x 41,216 = 79.4 mD (C) The fracture width or aperture may be estimated from Eq. 8.84

41,216 Wf --

3300t

The side length 8.73b:

60km

Xm --

33 x 0.25 x 0.1419 Xm

= 187 microns -- 0.187 m m

of the cubic block is calculated from Equation

rwv ~,kf = 0.298

60 • 0.15 4.267 x 10 -6 x 41,216

= 2.1 ft

This interpretation assumes that (a) the change in matrix compressibility and porosity of this naturally fractured reservoir is negligible and (b) the Warren and Root sugar-cube model is applicable.

PROBLEMS 1. What are the major factors in the creation of natural fractures in the reservoir rock?

2. Discuss (a) the geological classification and (b) the engineering classification of natural fractures. What are these classifications based on? 3. What are the major indicators of natural fractures? 4. Name the most prolific naturally fractured oilfields of the world. 5. What are the major petrophysical characteristics of the natural fractures. How do these characteristics affect the flow of fluids through the fractures? 6. Name the most c o m m o n techniques used to characterize natural fractures in petroleum bearing rocks. 7. What are the two main parameters involved in the Warren and Root sugar cube model? Discuss their significance and physical meaning.

PROBLEMS

$49

8. Differentiate b e t w e e n fault, joint, and fracture. H o w do they affect the fluid flow in p e t r o l e u m reservoirs? 9. A 5-inch long, 2-inch thick rock sample has only one fracture. The fracture w i d t h was m e a s u r e d as 0.03 cm and it fully p e n e t r a t e s the rock sample over its entire thickness. (a) Calculate the surface area of the space created by the fracture assuming rectangular and elliptical fracture shapes. (b) Calculate the hydraulic radius of the fracture. 10. A resistivity survey in a well s h o w e d a wellbore-corrected m u d filtrate resistivity of 0.1 ohm-m, w a t e r resistivity 0.19 ohm-m, invaded zone resistivity 2 ohm-m, and d e e p formation resistivity 115 ohm-m. The average porosity of 22%, estimated from log data, well matches the porosity estimated from cores. Pressure test analysis as well as cores indicated the p r e s e n c e of natural fractures in the well. Substantial m u d loss was also observed during drilling of this well, and in neighboring wells. Several o u t c r o p s also indicate the p r e s e n c e of natural fractures in the area. Using log data, estimate the fracture intensity index and the porosity-partitioning coefficient. 11. A newly drilled well in an oil reservoir was logged. Seismic surveys and geological studies indicated that the well is located in a faulted naturally fractured zone, w h i c h is a d o w n t h r o w n layer. The average total porosity (15 %) of the system was estimated from cores. O t h e r k n o w n characteristics are: A = 4,500 acres

h = 70ft

Sw = 0.25

Bo = 1.1 bbl/STB

Rw = O. 11 Ohm-m

Rt = 80 Ohm-m

Rnff = 0.15 Ohm-m

m=

1.30

(1) Estimate the porosity partitioning coefficient (2) Estimate the matrix porosity and fracture porosity (3) Calculate the total oil in place, STB (4) Calculate the FII and (5) Estimate the distance to the nearest fault, if the resistivity of the invaded zone is 6.5 ohm-m. (6) Does the p r e s e n c e of a nearby fault change the estimate of total oil in place?

550

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

NOMENCLATURE A Bo C

d FII Ft

hf h HT k L m mf mm No P q R r rw S S V W

Area, cm 2 Formation volume factor, RB/STB Compressibility, p s i - 1 Distance, ft Fracture Intensity Index, unit less Total Fracture Intensity Index, unit less Fracture height, cm or ft Formation Thickness, ft Tiab's hydraulic unit characterization factor, unit less Permeability, mD, Darcy, or cm 2 Length, cm Cementation factor or slope of semilog straight line Fracture or double porosity cementation factor, unit less Matrix cementation or porosity factor, unit less Oil in place, STB Pressure, psi or dynes/cm 2 Flow rate, STB/D or cm3/S Resistivity, ohm-m Radius, cm Wellbore radius, ft Specific surface area, cm 2 Saturation, fraction Volume, cm 3 Width, cm or ktm

SUBSCRIPTS b C C

e f gr h m o

mf f+m P pv S

Bulk Capillary Characterization Ellipsoidal Fracture Grain Hydraulic Matrix; oil Mud filtrate Fracture and matrix Pore; producing Pore volume Surface

REFERENCES

sh

55 |

Shape True Total Wellbore or w a t e r Wellbore flowing Flushed zone

t t

w Wf Xo

GREEKSYMBOLS r ~t 1; V

~P

Porosity, fraction Viscosity, cP Tortuosity, unit less Porosity partitioning coefficient, unit less Vertical separation on the t w o p r e s s u r e curves

REFERENCES 1. Nelson, R. A. "Fractured Reservoirs: Turning Knowledge into Practice." Soc. Petrol. Eng. J., Apr. 1987, pp. 407-414. 2. Massonnat, G. and Pernarcic, E. "Assessment and Modeling of High Permeability Areas in Carbonate Reservoirs," Paper SPE 77591. SPE/DOE IOR Symposium, Tulsa, OK, 13-17 April 2002. 3. Handin, J. and Hager, R. V. "Experimental Determination of Sedimentary Rocks under Confining Pressure: Tests at Room Temperature in Dry Samples." AAPG Bull., Vol. 41, 1957, pp. 1-50. 4. Steams, D. W. and Friedman, M. "Reservoirs in Fractured Rock." Am. Assoc. Petrol. Geol. (AAPG) Memoir 16 and Soc. Expl. Geophys., Special Publ. No. 1O, 1972, pp. 82-100. 5. Yamaguchi, T. "Tectonic Study of Rock Fractures."J. Geol. Soc. Jpn., Vol. 71, No. 837, pp. 257-275, 1965. 6. Charlesworth, K. A. K. "Some Observations on the Age of Jointing in Macroscopically Folded Rocks," in Kink Bands a n d Brittle Deformation, A.J. Baer and D. K. Norris (Eds.), Geological Survey of Canada, 1968, paper 68-52, pp. 125-135. 7. Picard, M. D. "Oriented Linear Shrinkage Cracks in Green River Formation (Eocene), Raven Ridge Area, Uinta Basin, Utah."J. Sedimentary Petrology, Vol. 36, No. 4, pp. 1050-1057. 8. Lui, X., Srinivasan, S. and Wong, D. "Geological Characterization of Naturally Fractured Reservoirs Using Multiple Point Geostatistics." Paper SPE 75246, presented at SPE/DOE Improved Oil Recovery Symp., Tulsa, OK, 13-17 April 2002.

$$2

PETROPHYSICS: RESERVOIR ROCK PROPERTIES

9. Dholakia, S. K., Aydin, A., Pollard, D. and Zoback, M. D. "Development of Fault Controlled Hydrocarbon Migration Pathways in Monterey Formation, California." AAPG Bull., Vol. 82, 1998, pp. 1551-1574. 10. Chilingarian, G. V., Mazzullo, S. J. and Rieke, H. H. Carbonate Reservoir Characterization: a Geologic-Engineering Analysis, Part L Elsevier Sci. Publ., NY, 1992. 11. Choquette, P. W. and Pray, L. C. "Geologic Nomenclature and Classification of Porosity in Sedimentary Carbonates." Am. Assoc. Petrol Geol. (AAPG) Bull., Vol. 54, 1970, pp. 207-250. 12. Chilingarian, G. V., Chang, J. and Bagrintseva, K. I. "Empirical Expression of Permeability in Terms of Porosity, Specific Surface Area, and Residual Water Saturation."J. Petrol Sci. Eng., Vol. 4, 1990, pp. 317-322. 13. Aguilera, R. Naturally Fractured Reservoirs. Petroleum Publ. Co., Tulsa, 1980, 703 pp. 14. Dyke, C. G., Wu, B. and Tayler, M. D. "Advances in Characterizing Natural Fracture Permeability from Mud Log Data." Paper SPE 25022, Euro. Petr. Conf., Cannes, France, 16-18 November 1992. 15. Keelan, D. K. "Core Analysis for Aid in Reservoir Description." Soc. Petrol Eng. J., Nov. 1982, pp. 2483-2491. 16. Schlumberger, Inc. Log Interpretation--Principles/Applications. Schlumberger Educational Services, Houston, 1987. 17. Pirson, S. J . Oil Reservoir Engineering, 2nd ed. McGraw-Hill, New York, 1978. 18. Warren, J. E. and Root, P.J. "The Behavior of Naturally Fractured Reservoirs." Soc. Petrol Eng. J., Sept. 1963. 19. Elkewidy, T. "Characterization of Hydraulic Flow Units in Heterogeneous Clastic and Carbonate Reservoirs." Ph.D. Dissertation, School of Petroleum and Geological Engineering, University of Oklahoma, Norman, OK, 1996. 20. Tkhostov, B. A., Vezirova, A. D., Vendel'shtyen, B. Y. and Dobrynin, V. M. Oil in Fractured Reservoirs. Izd. "Nedra", Leningrad, 1979, 219 pp. 21. Hensel, W. M., Jr. "A Perspective Look at Fracture Porosity." SPE Formation Evaluation, Dec. 1989. 22. Saidi, A. M. Reservoir Engineering o f Fractured Reservoirs. Total Edition Press, Paris, 1987. 23. Aguilera, R. "Analysis of Naturally Fractured Reservoirs from Conventional Well Logs."J. Petrol Tech., July 1976, pp. 764-772. 24. Lake, W. and Carroll, H. B., Jr. Reservoir Characterization. Academic Press, New York, 1968, p xi. 25. Pirson, S.J. Geologic Well Log Analysis, Gulf Publishing Co., Houston, TX, 1970. 26. Home, R. N. M o d e m Well Test Analysis, 2nd ed. Petroway Publ. Co., 1995.

REFERENCES

55]

27. Tiab, D. Advances in Pressure Transient Analysis. Lecture Notes Manual, Norman, OK, 2003. 28. Stewart, G. and Ascharsobbi, F. "Well Test Analysis for Naturally Fractured

Reservoirs", SPE 18173, presented at 63rd SPE ATCE, Houston, Texas, 2-5 Oct. 1988. 29. Engler, T. and Tiab, D.: "Analysis of pressure and pressure Derivative without type curve matching, 4. Naturally fractured reservoirs," Journal of Petroleum Science & Engineering, 1996, pp. 127-138. 30. Locke, L. C. and Bliss, J. E. "Core Analysis Technique for Limestone and Dolomite," World Oil, Sept. 1950. 31. Bona, N., Radaelli, F., Ortenzi, A., De Poli, A., Peduzzi, C. and Giorgioni, M. "Integrated Core Analysis for Fractured Reservoirs: Quantification of the Storage and Flow Capacity of Matrix, Vugs, and Fractures," SPERE, Aug. 2003, Vol. 6. pp. 226-233. 32. Tiab, D. M o d e m Core Analysis, Vol. I Houston, Texas, May 1993, 200 pp.

Theory, Core Laboratories,

33. Tuan, P. A., Martyntsiv, O. F., Dong, T. L., "Evaluation of Fracture Aperture and Wettability, Capillary Properties of Oil-Bearing Fractured Granite," Paper SCA-9410, Proceedings, 1994 Intern. Symp. of SCA, Stanvenger, Norway, Sept. 12-14, 1994.