and well test methods more than those related to core analysis. The Repeat Formation ...... plotting method, suggested by Horner (John lee 1984). If the well has.
(Permeability Evaluation of Sandstone Rocks in Oil Fields, Southern Iraq Using Well Logs). Researcher: AL-AMEEDY, U. Supervisor: Asst. Prof. AVEDISIAN, A., M. Supervisor: Prof. Dr. AL-MUSSAWY, S., University of Baghdad College of Engineering Petroleum Engineering Department September 2003
ABSTRACT This study is focused on the evaluation of formation permeability for a sandstone reservoir in three southern Iraqi fields at depth ranging from 3100 to 3400 m. Zubair formation in three oil fields (A, F & M) located in southern of Iraq including sixteen wells, have been chosen for this study. Methods of determining permeability from well logs are reviewed. A case history is presented showing how an optimum empirical relationship to predict permeability can be developed, with commonly available core and log data for a particular formation and area. Verifiable and accurate permeability prediction from well logs in a well with no core measurement data is presented by improved log derived permeability relationship. Examining a cross plot of permeability to median pore–throat size shows the strong linear correlation between these two variables (R=0.92). Thus, the improved log derived permeability relationship is more accurate than other methods because permeability transform is provided for each pore type (cementation exponent, m) and taken in consideration the pore throat size (Rp). The recent and best techniques for determining the parameters (Rp & m) from well log data were introduced in this study; moreover, a computer program was constructed to calculate the cementation exponent, m. In this study, the resistivity ratio method was presented as an alternative method to eliminate the need to know values of irreducible water saturation or pore radius as required in previously published correlations, which also required expensive core analysis. Sandstone grain size and sorting, and hence permeability, are often related to the amount of interstitial silt and shale, and this in turn may be reflected in the natural radioactivity. Because of this, in some areas permeability of sandstone can be estimated from carefully calibrated gamma ray curves, as the following proposed relationship:
which was applied in the upper shale member with acceptable accuracy. For the M and F fields it is found that the permeability is a function of porosity, irreducible water saturation, and volume of clay : It is also found that the correlation coefficient between predicted and observed permeability for the above function is increased by taking the square root of the permeability ( ). The above equations were developed using multiple regression analysis and nonlinear estimation solutions available in Statistica Software Package. The formation permeabilities of three wells was estimated by different well log methods, and compared with those measured by well test and core analysis. The comparisons showed that in most wells there is a relative good agreement between the results of well logging method and well test methods more than those related to core analysis. The Repeat Formation Tester (RFT) was reviewed, which can make unlimited number of accurate pressure tests in one run in open hole. This technique has applications in the determination of reservoir pressure, fluid density, fluid contacts, deferential depletion, reservoir intercommunication, and effective permeabilities of invaded and uninvaded zones. Theory and principles of these applications as well as a field example, illustrating the use of this type of log for permeability evaluation, are presented. Furthermore, comparisons between RFT and well log derived permeabilities indicate that permeability values by RFT data are larger than or equal to the corresponding log data.
Acknowledgment I am very pleased to express my deep appreciation and gratitude to my supervisors, Mr. Entwan M. Avedisian and Prof. Dr. Sabah AL-Mussawy for their guidance, and encouragement during the time of this work. I am indebted to Prof. Dr. Akram H. AL-Hiti, Head of Petroleum Engineering Department for his efforts in facilitating all the requirements of this work. I am very grateful to Dr. Mohammed S. Al-jawad and Miss. Waffa Al-katan for their kind help. I wish to acknowledge the efforts and valuable contributions of informations especially Mr. Dhiaa Ghanim, and Basheer. I am very grateful to Mrs. Najla Shakir of S.O.C. for her advice and kind help. Thanks are devoted to my colleagues, Dorgham and all my friends. My best thanks to my family especially my mother and my father for their patience and care at all stages of this work.
Nomenclatures Symbols
Description
Unit
a
Coefficient in Archie Equation.
dimensionless
F
Formation Resistivity Factor.
dimensionless
GR k ko, kw & kg kro, krw & krg
Gamma Ray Log.
API
Formation Permeability.
md
Effective permeability of oil, water and gas respectively.
md
Relative permeability of oil, water and gas respectively.
dimensionless
m
Porosity (Cementation) Exponent.
dimensionless
n
Saturation Exponent.
dimensionless
Pc
Capillary Pressure.
Psia
PHIE
Effective Porosity.
p.u
q
Flow Rate.
cc/sec
Rc
Clay Resistivity.
ohm-m
Rmf
Mud Filtrate Resistivity.
ohm-m
Ro
Resistivity of Clean Formation 100% Saturated.
ohm-m
Rw
Formation Water Resistivity.
ohm-m
Rp
Effective Pore Radius.
m
SP
Spontaneous Potential.
Sw
Water Saturation.
fraction
Swi
Irreducible water Saturation.
fraction
Sxo
Water Saturation in Invaded Zone.
fraction
tp1
Propagation Travel Time in Formation (invadedzone).
Vsh φ φe ρo, ρw, ρma τ
Volume of Shale. Porosity. Effective porosity. Density of oil, water and matrix, respectively. Tortuosity factor.
millivolts
ns/m fraction p.u p.u gm/cm3 dimensionless
Symbol P φ
Greek Symbols Differential Pressure. Porosity. Viscosity. Density.
Subscripts e p 1 2
Effective. Pore. Onset of pretest. Second of pretest.
Abbreviations SOC CPI Psi NMR RFT CSU TDT EPT md
South Oil Company. Computer Processing Interpretation. Pound per square inch. Nuclear Magnetic Resonance Repeat Formation Tester Computer Process System Thermal Decay Time Electrical Propagation Tool Millidarcy.
Units Psia dimensionless cp gm/cc
Contents Title
Page CHAPTER ONE: Introduction
1.1. REGION OF THE STUDY 1.2. AIMS OF THE STUDY CHAPTER TWO :
2 4 Theory
2.1. GENERAL 2.1.1. Absolute Permeability 2.1.2. Effective Permeability 2.1.3. Relative Permeability 2.2. GEOLOGICAL FACTORS AFFECTING SPECIFIC PERMEABILITY 2.2.1. Continuity of Pores and Degree of Interconnection 2.2.2. Size of Pore Spaces 2.2.3. Presence and Nature of Clay 2.2.4. Tortuosity 2.2.5. Cementation Materials 2.2.6. Fractures 2.3. RELATION OF TEXTURE WITH POROSITY AND PERMEABILITY 2.3.1. Grain shape 2.3.2. Grain size 2.3.3 Grain sorting 2.3.4. Sediments fabric 2.4. RELATIONSHIP BETWEEN PERMEABILITY AND PETROPHYSICAL PROPERTIES 2.4.1. Porosity, Specific Surface Area, Tortuosity, and Permeability Relationship 2.4.2. Connate Water Saturation and Permeability Relation 2.4.3. Capillary Pressure and Permeability Relation 2.4.4. Relationship between Formation Resistivity Factor and Permeability 2.4.5. Acoustic Wave Velocity and Permeability Relation 2.4.6. Relation of the Nuclear Magnetic Resonance (NMR) to Permeability 2.5. MAJOR METHODS OF PERMEABILITY DETERMINATION 2.5.1. Core Analyses Method
5 5 6 6 6 7 7 7 10 10 11 11 12 12 13 13 15 15 17 17 19 22 23 24 23
2.5.2. Well Test Analyses 2.5.3. Well Logging Methods
24 26
CHAPTER THREE :Techniques of Permeability Evaluation 3.1. POROSITY CORRELATION METHOD 3.2. IRREDUCIBLE SATURATION METHOD 3.3.CEMENTATION EXPONENT, m, AND EFFECTIVE PORE RADIUS, RP, MODEL AS AN IMPROVE LOG-DERIVE PERMEABILITY 3.4. RESISTIVITY RATIO MODEL 3.5. TRANSFORM'S APPROACH 3.6. PERMEABILITY OF WATER BEARING MODEL 3.7. REPEAT FORMATION TESTER (RFT) LOG
45 46 48 49 51 53 57
CHAPTER FOUR: Results and Discussion 4.1. POROSITY CORRELATION 4.2. IRREDUCIBLE SATURATION METHOD 4.3. IMPROVED LOG-DERIVED PERMEABILITY BY, RP AND m, PARAMETERS 4.4. RESISTIVITY RATIO MODEL 4.5. TRANSFORM APPROACHS 4.6. COMPARISON OF PERMEABILITIES MEASUREMED BY DIFFERENT APPROACHES 4.7. VERTICAL PERMEABILITY (kv) BY CAPILLARY PRESSSURE METHOD 4.8. PERMEABILITY OF WATER BEARING ZONES 4.9. REPEAT FORMATION TESTER (RFT)
69 72 78 80 82 91 92 101 103
CHAPTER FIVE: Conclusions and Recommendations 5.1. CONCLUSIONS
115
5.2. RECOMMENDATIONS
118
REFERENCES APENDIX
A
APENDIX
B
APENDIX
C
119
CHAPTER TWO
THEORY
CHAPTER TWO THEORY 2.1. GENERAL 2.1.1. Absolute Permeability (k) This property permits the passage of a fluid through a net work of the interconnected pores of a rock (its effective porosity) without damage of rock. In other words, permeability is a measure of the fluid conductivity of the rock(Calhoun 1953, Pirson 1950). Darcy’s experiment was made with water flowing through sand filter beds and resulted in the formulation that the rate of flow through a sand bed was proportional to the pressure head above the filter and a cross sectional area of the filter, but inversely proportional to the thickness of the filter. This law has been extending to apply to any fluid flowing in any direction through porous material. The general formulation of Darcy’s law in differential form is: v
k dp μ dl
(2.1)
where; v=microscopic velocity of flow, =viscosity of the flowing fluid, dp k=permeability constant, = pressure gradient in direction of flow. dl A permeability of 1 Darcy may be define as the permeability, which will allow the flow of 1 cc per second of fluid of 1 centipoises cp through an area of one cm² under pressure gradient 1 atm per cm Pirson (1950).
5
CHAPTER TWO
THEORY
2.1.2. Effective Permeability (ke): It describes the passage of a fluid through a rock in the presence of other pore fluids. When an oil flows through a rock saturated with other fluids (water and /or gas), the effective permeability of oil will be (ko), while for water (kw) and for gas (kg). The effective permeability depends on rock characteristics and the percentages saturation of the fluid present in the pores (Selley 1998). 2.1.3.Relative Permeability: It is the ratio of the effective permeability for a given fluid at definite saturation to the permeability at 100% saturation (Calhoun 1953). The relative oil permea-bility is kro=(ko/k), for water is krw=(kw/k) and for gas is krg=(kg/k) (Fig.2-1). 100
80
60
Percent gas saturation 4 20
0.8 kg/k
Relative permeability
0.6
ko/k
0.4 Equilibrium of gas saturation
Equilibrium of Oil saturation 0.2
0 0
2
40
6
80
100
Percent oil saturation
Fig. (2-1): Typical curves for the relative permeabilities of oil and gas for different saturation (after Selley 1998).
2.2.GEOLOGICAL FACTORS AFFECTING SPECIFIC
PERMEABILITY: In addition to the textural features, which have a great influence on the porosity and permeability of the sediments, as it will be discussed later, the following factors affecting specific permeability. 6
CHAPTER TWO
THEORY
2.2.1.Continuity of Pores and Degree of Interconnection: Permeability increase with increasing the continuity of pores. Most sandstone is known to be porous and permeable. Sandstone may be porous but the interconnecting channels may be not continuous hence it is of low permeability (Muskat 1949, Pirson 1959). Claystone for example is known to be porous, but it's pores are not interconnected, therefore, this type of rocks is known by its low permeability. 2.2.2.Size of Pore Spaces: Permeability is also a function of pore size. The conduction rate of a single pore is proportional to the fourth power of its diameter; therefore, if the pore diameter is double then the rate flow under the same pressure 4
gradient will be (2 ) or 16 times than the similar pore (Al-Rekabi 1982, Hager 1949). 2.2.3. Presence and Nature of Clay It is a factor of great importance in controlling the value of specific permeability to a certain fluid. The amount and the kind of clay as well as its distribution through the reservoir rock, has an important bearing on liquid permeability, particularly on whether or not the flowing fluid reacts with clays (Muskat 1949, Pirson 1959). 2.2.3.1. Clay mineral types: Clay minerals are the most abundant in sedimentary rocks, comprising perhaps as much as 40% of the minerals in these rocks. Half or more of the clay minerals in the earth’s crust are illites, followed in the order of relative abundance by montmorillonite, mixed layer illite-montmorillonite, chlorite, mixed layer chlorite–montmorillonite, and kaolinite. It is expected that the montmorillonite has more affect on permeability than chlorite and illite and the leatests have greater influence on permeability than kaolinite (schlumberger Nov. 1961). 7
CHAPTER TWO
THEORY
2.2.3.2. Shale distribution and its effect on permeability and porosity : Clays are often, found in sands, siltstone, and conglomerates. Core analysis, petrographic thin-section examination, and most recently, scanning electron microscopy has revealed that clay material, which is often referrers to as “shale,” may be distribute in sand formations in three different forms: laminated, structural, and dispersed (Fig. 2-2). Within a sand body, thin lamina of clay and other fine-grained material may occur. They are of detrital origin; i.e. they are form outside the sandstone framework. These detrital clays called laminated clay and shale. The laminae themselves do not affect the porosity and permeability of the sand streaks. These laminae, however, are more or less continuous and act as vertical permeability barriers (Avedisian 1988, Bassiouni 1994) . Dispersed Laminated
φe
φe φe
Structural Fig. (2-2): Schematic of clay mineral distribution through clastic reservoirs rocks (after Avedisian 1988).
Clay can be diagenetic origin (i.e., formed within the sand framework). A source of diagenetic clay is the in-situ alteration of nonquartz particles owing to the reaction with formation water. The most common alterations are those of feldspar to kaolinite and of hornblende to chlorite. This alteration leads to structural clays. Also considered structural clays are those originating when pellets or clasts of clays are deposited as an integral part of matrix predominated by sand. Diagenetic sands, however, usually occur as dispersed clays. Dispersed clays developed when clay crystals precipitate from pore fluids. Precipitation occurs in response to pore-water chemistry 8
CHAPTER TWO
THEORY
changes brought about by filtration through shale or by changing temperature and pressure during the burial and compaction of the sediments. Structural Laminated
k (md)
Dispersed
φ Fig. (2-3): Effect of dispersed clays through pores on porosity and permeability (after Avedisian 1988).
Dispersed clays can occur in pores as discrete particles, intergrown crystal lining the pore walls that form a relatively thin and continuous coating, and crystals extending far into or completely across a pore or pore throat. Dispersed clay can markedly reduce the permeability of formation (Fig. 2-3). Fig. (2-4) relates the position of a point on a Neutron-Density cross plot to the three types of shale distribution previously mentioned.
Fig. (2-4): Idealized shale type distribution (after Schlumberger Nov. 1981). 9
CHAPTER TWO
THEORY
The presence of a small amount of structural shale has little effect on the effective porosity of the formation. Since the shale grains are equivalent to quartile grains, as the shale content increases, however, the effective porosity decreases rapidly since the shale phase becomes continuous. Laminated shale plot on a line connecting the 100% clean sand point ( φ max) with pure shale point. Individual sandy layers contain their initial permeability but the overall permeability thickness product (kh) varies in inverse proportion to the percentage of shale lamination. Since dispersed shale occupies original pore space (Fig. 2-2), the effective porosity (effective porosity sharing in fluid flow) decreases rapidly with increasing shale content. The maximum quantity of dispersed shale is however, limited to the original porosity of clean sand at the one depth burial. This somewhat idealized distribution will rarely be observed in practice since the relative quantities of each type of shale are not independent. For example, as the laminated shale percentage increases, both structural and dispersed clay content increase within the sandy layers such that the effective porosity reaches zero (the point of Fig.(2-4)) before the laminated shale percentage reaches 100%. 2.2.4. Tortuosity: Permeability varies inversely with length of flow, therefore inversely with tortuosity (see section 2.4.1 & 2.4.4); so whatever shortness the path will increase the permeability. 2.2.5. Cementation Materials: The rock may be full of joints one or another being closed either by cementation, i.e. by clay or, mud, a rock of this type is to be low in permeability (Pirson 1959).
10
CHAPTER TWO
THEORY
2.2.6. Fractures: Permeability is strongly increased by fractures even those with small width(Chilingar 1972, Muskat 1949). Fig. (2-5) shows types of fracture and their affects on permeability, where fracturing, shattering, joint planes and bedding planes especially, increase permeability greatly by the large crosssectional area of the tabular openings(Leverson 1967). Fig. (2-5): Idealized section showing how the wall material and widen the fractures, which connect otherwise isolated cavities and pore. Other factors through permeability digenetic Fractures aid in come increasing solutions passing through dissolved (after Leverson 1967)
Cast
Beddin g Fracture s
Other factors coming through digenetic processes, could also affect permeability such as: re-crystallization of some minerals, replacement of one mineral by another, solution ...etc, which would increase or decrease porosity as well as permeability of rocks.
2.3. RELATION OF TEXTURE WITH POROSITY AND
PERMEABILITY The texture of the sediments is closely correlated to its porosity and permeability. The texture of a reservoir rock is related to the original depositional fabric (The fabric of rock is the porosity concerned with orientation in space of component particles of the rock) of the sediments, which is modified by subsequent diagnosis. The diagnosis may be negligible in some type of rocks like sandstone, where in carbonates it may be sufficient to obliterate all traces of original depositional features (Selley 1998). The textural parameters of unconsolidated sand that may affect porosity and permeability are shown as follow:
11
CHAPTER TWO
THEORY
2.3.1. Grain shape: The two aspects of grain shape that have to be considered, are roundness and sphericity. As in Fig. (2-6), these two properties are quite distinct. Roundness describes the degree of angularity of the particle, whereas sphericity describes the degree to which the particle approaches spherical shape. The permeability of sand composed of angular grains is greater than that of sand composed chiefly, of spherical grains of similar size. This may be due to the fact that angular grains are packed more loosely and also develop bridging (Leverson 1967). Rocks composed mainly of flat, mica-shape particles needle-like crystals pack loosely, have a high porosity, and in general, probably have a high permeability.
Fig. (2-6): Sand grains showing the difference between shape and sphericity (after Selley 1998).
2.3.2. Grain size: Theoretically, porosity is independent of grain size for uniformly packed and graded sands. In practice, however coarser sands sometimes have higher porosities than of finer sand or vice versa. This disparity may be due to separate, but correlative factors, such as sorting and /or cementation. Permeability declines with decreasing grain size because diameter of pore decreases and hence capillary pressure increases.
12
CHAPTER TWO
THEORY
Thus, a sand and shale may both have porosities of 10 percent; the former may be a permeable reservoir where the latter may be an impermeable cap rock. 2.3.3 Grain sorting Porosity increases with improved sorting. As sorting decreases, the pores between the larger, frameworks forming are filled by the smaller particles. Permeability decreases with sorting decrease for the same reason. As mentioned earlier, sorting sometime varies with grain size of particular reservoir sand, thus indicating a possible correlation between porosity and grain size. Fig. (2-7) summarized the effect of sorting and grain size on porosity and permeability in unconsolidated sands. 100000 Well SORTING
Moderate
10000
Coarse
Poor Medium Fine
1000
GRAIN SIZE
Very poor
V.well
Permeability
Very fine 100
10
1
0.1 0
10
20
30
40
50
Porosity
Fig. (2-7): Graph of porosity against permeability showing the relationship to grain size and sorting for un-cemented sands (after Selley 1998).
2.3.4. Sediments fabric: a. Grain packing: The two important characteristic features of the fabric are how the grains are packed and how they are oriented. The classic studies showed that spheres of uniform size have six theoretical packing geometries.
13
CHAPTER TWO
THEORY
These geometries range form the loosest cubic style with a porosity of 48% to the lightest rhombohedra style with 26% porosity (Fig. (2-8)).
Cubic Packing (48% Porosity)
Rhombohedra Packing (26% Porosity)
Fig. (2-8): the loosest and tightest theoretical packing for spheres of uniform diameter (after Selley 1998).
b. Grain orientation: The preceding analysis of packing was based on the assumption that, grains are spherical, which is generally untrue of all except oolites. Most quartz grains are actually prolate spheroids, slightly elongated with respect to their crystallographic axis. Sand also contains flaky grains of mica, clay and other constituents. Thus the second element of fabric, namely orientation, is perhaps more significant to porosity and permeability than packing itself. How grains are oriented has a little effect on porosity, but a major effect on permeability. The sediments are mostly, stratified. The layering being caused by flaky grains, such as mica, shells and plant fragments, as well as by clay laminae. Because of this stratification, the vertical permeability, in general, is considerably lower than the horizontal permeability, the ratio of vertical permeability to horizontal permeability in a reservoir is important because of its effect on coning as oil and gas are produced. Variation in permeability also occurs parallel to bedding. In most sands, the grains generally show a preferential alignment within horizontal plain. Fig. (2-9) shows that permeability will be greatest parallel to grain orientation, since in this orientation the fabric alignment is with least resistance to fluid movement.
14
CHAPTER TWO
THEORY
Z
Current
X Y
Fig. (2-9): Block diagram of sand showing layered fabric with grains oriented parallel to current. Generally, kx>ky>kz (after Selley 1998).
2.4. RELATIONSHIP BETWEEN PERMEABILITY AND
PETROPHYSICAL PROPERTIES The physical properties of reservoir rocks that are of importance in formation evaluation and well log interpretation are their electrical properties when fully or partially saturated in water (fresh or saline); their radioactive properties, either natural or induced by neutron bombardment; their elastic and acoustic wave propagation properties; their density and their thermal conductivity (Pirson 1963). 2.4.1. Porosity, Specific Surface Area, Tortuosity, and Permeability Relationship: The quantitative relationship between porosity and permeability is obscured and variable. Beyond the fact that permeable rock must be porous; this seems to be only a general connection. As shown in Fig. (2-10) there is no more than a rough relationship between the porosity and permeability of two reservoir rocks (Donald 1983). The interconnected net work of pores in a reservoir rock may be viewed as a system of tortuous capillaries of mean radius ( r p ) the physical length of the average path of fluid particle, when traveling through the tortuous net work, being ( ) times larger than the short distance (Pirson1963). A more reliable correlation that combines most petrophysical properties ( φ, Sv,,rp ) is given by Kozeny equation (see Appendix A for derivation) k 10
8
φ3 2 2 Sv 2(1 φ)2
15
(2.2)
CHAPTER TWO
THEORY
The main factor, which influences grain specific surface area (Sv), is grain size. This explains why fine-grain rocks like claystone have low permeability even though their porosity may be high (Pirson 1963).
Fig. (2-10): Relation between porosity and permeability in two reservoir rocks (after Donald 1983)
*
Fig. (2-11): Relation between the percentage of interstitial water and the permeabilities of various oil reservoirs (after Leverson 1967).
16
CHAPTER TWO
THEORY
2.4.2. Connate Water Saturation and Permeability Relation: There is no significant and universal relationship between connate water and permeability, the gross trend can be at least regionalized. The differences shown in Fig. (2-11) between the curves for different fields should serve to emphasize the general decline of water content, with increasing permeability. It is evidently an expression of the fact that the capillary forces maintaining the water saturation against the fluid head above the water table increase with decrease average pore-radius this is to be expected from basic physical principle. 2.4.3. Capillary Pressure and Permeability Relation The capillary pressure of a reservoir increases with decreasing pore size or, more specifically, pore-throat diameter ( k f ( Rp) . Capillary pressure is also related to the surface tension generated by the two adjacent fluids, it increases with increasing surface tension. Fig. (2-12) shows capillary pressure curves for various reservoirs. Mercury injection, %
Capillary Pressure Pc
100
0
3 2
1
Displacement Pressure 0
Irreducible water saturation Water saturation, %
100
Fig. (2-12): Capillary pressure curves for various reservoirs. (1) A clean well sorted with uniform pore diameters (2) An intermediate quality reservoir, (3) a poor quality reservoir with a wide range of pore diameter.
17
CHAPTER TWO
THEORY
A mathematical expression has been developed (Leveret 1942) for a general relation between capillary pressure (Pc) and general properties of a porous media: 1/2
φ (2.3) Pc j(Sw) δ Cosθ k where ( δ ) is the interfacial tension dyne/cm2, ( θ ) is the contact angle, solid
to liquid of the wetting phase and j(Sw) is the Leveret’s capillary pressure function. When water saturation (Sw) is unity, a suitable relation is proposed (Rose & Broce 1949) : 1 Pd k
1/2
φ δ t
1/2
(2.4)
Where, the contact angle assumed to be 0, Pd =displacement pressure, and other parameters have been previously defined. Nevertheless, it has been shown, in the theoretical section, that F t1/2 /ts1/2 φ . Hence,
Pd δ/(k ts φ)1/2 F
(2.5)
The displacement pressure, Pd, dose not seem to be obtainable from electric log, but Pc may be derived in principle by using the same technique as that employed by Tixier (1949). The general relationship between Pc and Pd for all porous media is undoubtedly complex, but to a first approximation, the relationship derived by Rose and Bruce (1949) may be used. This relationship states that Pd Pc Sw1/2 . It then follows that, 1/2
1 Pc k
δ F ts
1/2
φ1/2Sw 1/2
(2.6)
or that, k
2 Pc 2 F 2ts Sw
18
(2.7)
CHAPTER TWO
THEORY
In Eq. (2.7) may be assumed constant to a first approximation, and since ts varies from about 2.0 to 2.5, it is also essentially a constant. The porosity, φ , is calculable from log data directly, by assuming a reasonable value for m in the expression F φ m , F may be expressed in terms of φ with an accuracy compatible with other practically essential approximations. Thus, in terms of parameters that is obtainable from appropriate logs (Wyllie & Rose 1950). k cons.
φ 2m1
(2.8)
Pc 2 Sw
Eq. (2.8) can be modify from the relation of Brown and Husseini (1977), φB
kA
SwC Pc D where A, B, C, and D are empirical constants.
Relationship between Factor and Permeability: 2.4.4.
Formation
(2.9)
Resistivity
Formation resistivity factor (F) is a useful measurement in analyzing reservoir fluids and is defined as the electrical resistance of a rock that is saturated with a conducting electrolyte (Ro) such as a brine, dividing by the resistivity of the electrolyte (Rw)(Archie 1942). Ro (2.10) Rw The resistivity factor of sandstone reservoir is seemed to increase as the F
porosity decrease Fig. (2.13) F
1 φm
(2.11)
or in other form 2
2 Le 1 τ F L φ φ
19
(2.12)
CHAPTER TWO
THEORY
Fig. (2-13): Relation of porosity to the formation resistivity factor (after Archie).
The actual fluid flow velocity, ve, within the pores of a medium size is greater than the microscopic velocity, v, implied by q/A, where q is volumetric flow rate and A is the cross-sectional area of the porous medium. The increase velocity is a result of the increased length of the actual flow path, Le, compared with the length, L, across the porous medium, and the decreased area of actual flow. The area is decreased by a factor φ . Thus, the actual fluid velocity given as (Bassiouni 1994). ve
v Le v τ φ L φ
(2.13)
Combining Eq. (2.13) with Darcy‘s law (Eq. 2.14), v
k dp μ dL
(2.14)
result in k
ve μ φ L ve μ φ Le (dp/dL) τ (dp/dL)
(2.15)
where the symbols have previously defined. It can be observe that, all other parameters remaining invariable, k
L 1 Le
(2.16)
Hence, permeability decreases as tortuosity increases.
20
CHAPTER TWO
THEORY
From the preceding discussion, it is evident that both the electric conductivity and the permeability of a porous medium are determined by the effective length of the path of flow of ions. The greater this length, the lower the conductivity and permeability. An empirical equation relating these two physical properties of the porous medium can be obtained by combining Eqs. (2.11), (2.12) and (2.16). This relation is of the form (Bassiouni 1994). B
F Ak where (A) and (B) are constants for specific formation.
(2.17)
Various experimental studies corroborate the relationship given by Eq. 2.17. One of the earliest was conducted by Archie (1942), who examined a group of core plugs taken from the producing zones. He concluded that the formation factor varies, among other properties, with the permeability of the reservoir rock, as shown in Fig. (2.14). In an extensive statistical study, Carthers (in Bassiouni 1994) also observed that a relation exists between permeability and formation resistivity factor. This relationship is
k (4.0 10 8 )/F 3.65 for limestone rocks and
(2.18)
(2.19) k (7.0 10 8 )/F 4.5 for sandstones rocks. In Eqs. 2.18 and 2.19, permeability is given in millidarcies.
Fig. (2-14): Relation of permeability to the formation factor for several producing zones (after Archie 1942). 21
CHAPTER TWO
THEORY
2.4.5- Acoustic Wave Velocity and Permeability Relation : The shape of the acoustic signals that used in sonic log has been noted to be changed with the permeability of beds (Doverkin 1994). Theoretical and experimental research has shown that acoustic attenuation follows a law of the following forms (Cheng 1996, Xiaoming 1996, Staal 1977): 1/3
3 S
2π k ρ f f (2.20) δ 1.2*10 μ φ =attenuation in dB/cm, φ =porosity, S=specific surface area in cm2/cm3of rock, k=permeability in md, f=fluid density gm/cc, f =frequency in Hz, =viscosity in cp. It has also shown that attenuation governs the shape of the signals received. This shape can be characterized by an index (Ic) based on the amplitude of the successive arches of the signals (Fig. 2-15): v v3 Ic 2 (2.21) v1 The index of the amplitudes signals (Ic) are related so much with the vertical permeability (kv) as shown in Fig. (2-16) (Lebreton et al. 1978) V2 V1
2 V3
Fig. (2-15):Diagram of an acoustic signal in sonic logging (after Lebreton et al. 1978).
Fig. (2-16): Ic- kv relationship in a well (after Lebreton et al. 1978).
22
CHAPTER TWO
THEORY
2.4.6. Relation of the Nuclear Magnetic Resonance (NMR) to Permeability : Specific permeability of sandstone was first related to the Nuclear Magnetic Resonance (NMR) measurable parameters (the spin-lattice relaxation time (T1) and free fluid index (FFI)), by Seevers (1966). The prediction of the Korringa et al. model (in Seevers 1966), which describes the mechanism of the spin-lattice relaxation of protons of a hydrogenous liquid in a porous medium, was combined with Kozeny model, as: T1 TB k A. FFI TB T1
2
(2.22)
where, T1 is the spin lattice relaxation time observed from the saturated pore sample, and TB is the bulk relaxation time (i.e. the T1 of the saturating liquid, free from the surface effect ), measured separately in a glass container. The parameter (A) in Eq. (2.22) is given by, A
h rs , in which (h) T
is the thickness of the modified layer of liquid in a pore, with a relaxation rate (rs), and (T) is the tortuosity of the porous medium (for detailed explanation of these quantities (see Seevers 1966). By measuring permeability (k) in the left hand side of Eq. (2.22) and the parameters in the right hand side simultaneously, on suites of sandstone samples from different fields; (A) can be determined as an empirical constant. 2.5. THE MAJOR METHODS FOR PERMEABILITY
DETERMINATION: 2.5.1. Core Analyses Method The laboratory methods of determining the permeability coefficient (k) can be performed using Darcy’s law q
k A dp μ dl
(2.23)
23
CHAPTER TWO
THEORY
The permeability (k) in Darcy consists in a flow of a fluid of known viscosity (air, water, or oil) through cores of known pressure differentials and measuring the resulting rates of flow. Fig. (2-17) illustrates the basic arrangement for measuring permeability. P1
P2
A
Fluid of viscosity
L
Fig. (2-17): Basic arrangement for the permeability measurement.
Such measurements are generally, made on dry cores under conditions that do not duplicate field conditions of mineral hydration and of overburden pressure. Accordingly, there is a considerable doubt as to the validity of the absolute values of permeability so obtained. Darcy’s law is only valid when there is no chemical reaction between the fluid and rock, and only when one fluid phase completely, fills the pores. The situation is far more complex for mixed oil and gas phases, although a Darcy-type equation is assumes to apply. Flow rate depends on the ratio of permeability to viscosity. Thus, gas reservoirs may be able to flow at commercial rates for permeabilities of only a few millidarcies, where oil reservoirs needs minimal permeabilities in the order of tens of millidarcies (Scheidgger 1963, Muskat 1949). 2.5.2. Well Test Analyses: Well test provides an excellent technique for determining permeability if the test approaches the conditions assumed in the mathematical model. An important advantage of permeability determined by this method is that it represents the effective permeability of large volume of reservoir rock under
24
CHAPTER TWO
THEORY
reservoir conditions (John lee 1984). Unfortunately, such measurements are usually, not made in every well (Blekhman 2001). The following types of pressure transient analyses explain briefly, how permeability can be evaluated: 2.5.2. A- Pressure Buildup Test The test is conducted by producing a well at a constant rate for some time, shutting the well in (usually at the surface), allowing the pressure to build in the well bore, and recording the pressure (usually down hole) in the well bore as a function of time. From these data, it is frequently possible to estimate formation permeability, current drainage area and formation pressure. The analysis of well test method is basically based on line source solution, for diffusivity equation through porous media, and largely on a plotting method, suggested by Horner (John lee 1984). If the well has produced for a time (tp) at a rate (q) before shut-in and if we call the time elapsed since shut-in Δt , then by superposition: Pws Pi 162.6
q μ B tp Δt log kh Δt
(2.24)
The form of Eq. (2.24) suggests that shut in borehole pressure, Pws recorded during a pressure build up test should plot as a straight-line function of log (
tp Δt ). Further, the slope (m) of this straight line should be: Δt m 162.6
qμ B kh
(2.25)
Thus, formation permeability (k) can be determined from a build up test by measuring the slope m.
25
CHAPTER TWO
THEORY
2.5.2. B- Pressure Drawdown Test A pressure drawdown is conduct by producing a well for enough periods, starting ideally with uniform pressure in the reservoir. The rate and pressure are recorded as functions of time. An idealized constant-rate drawdown test in an infinite acting reservoir modeled by the logarithmic approximation to the Ei function solution: qμ B log(t) (2.26) kh If the conditions assumed in this model is valid, a plot of (Pwf) versus Pwf Pi 162.6
log (t) will be a straight line with slope (m), given by: qμ B kh Thus, effective formation permeability can be estimated from this slope. For
m=162.6
more details about well test analyses subject you can see reference (27). 2.5.3. Well Logging Methods: The attempts to use well logs to estimate formation permeability have resulted in empirical correlations, which may not be universally applicable but have good accuracy for derived regional. A review of the literature shows that logging methods proposed for estimating rock permeability fall into one of the following categories:2.5.3.1. Irreducible Saturation Methods:A general correlation between rock permeability, porosity and surface area (Swi =f (specific surface area)) may be writing as shown below:k= a
φc
(2.31)
Swi b
Where, Swi=connate (irreducible) water saturation, a, b, & c=constants to be determined. Many scientists were dealing with this subject such as:
26
CHAPTER TWO
THEORY
Wyllie & Rose (1950) discussed some theories about quantitative evaluation of the physical characteristics of reservoir rocks. They have expanded an empirical proposed by Tixier (1949) and showed that the order of magnitude of formation permeability may be obtains from the relationship: 1 k Cons. 1 2 2 Pc F m Sw
(2.32)
where Cons. =
21.2 * δ 2 ts
,
Pc=capillary pressure (psi), δ =interfacial tension
dyn/cm2, ts=constant (2-2.5). If Pc is unobtainable (i.e., in the absence of an oil-water contact in the reservoir), Wyllie & Rose (1950) suggested another correlation between irreducible water saturation, permeability and formation factor, which is: 1 k2
c
3 Swi
(2.33)
where, c is a constant equal 250 for oil bearing, and 79 for gas. Timur (1968), Investigated the relationships between permeability, porosity and residual water saturation in a three different oil fields. He tested several relations for k, φ & Swi, by statistical technique to find the standard error of estimate and correlation coefficient for each field, and then for all fields. He found the best estimation of permeability through the empirical equation: k 0.136
φ 4.4 Swi 2
(2.34)
Equation (2.34) is applicable for a clean consolidated sandstone formation with a medium porosity in an oil-bearing reservoir.
27
CHAPTER TWO
THEORY
Coats & Dumanoir (1973) developed an approach to improve log derived-permeability. The permeability index was derived as a function of porosity, formation resistivity at irreducible water saturation (Rtirr), hydrocarbon density ( ρh ) and rock type (w). The basic relationships:
φ 2w k 4 w Rw Rtirr c
(2.35)
c 23 465 ρh - 188 ρh 2
(2.36)
(log(Rw Rtirr))2 w 3.75 φ 2 2
(2.37)
Coats and Dumanoir’s formula provides a consistent result in a wide variety of lithology if the basic stipulations listed must have been meeting: a-The well-known Archie’s equation: a F m
(2.38)
φ
Where a=formation factor coefficient & m=cementation exponent. If (a) is constant, it should be equal one (when porosity is 100%, F must be equal one (Ro= Rw) at φ =100%), therefore, equation (2.38) becomes: F
1
(2.39)
φm
b-With the support of core and log studies, a common exponent, w, has been adopted for both the saturation exponent (n) and cementation exponent (m). Coats (13) set m & n equal and called them both as w, thus m=n=w. c-Within the limits of reasonable judgment, for most reservoir systems were observed, a simple equilateral hyperbola provides a very reasonable correlation between irreducible water saturation and porosity data. Thus, the product of ( φ. Sw ), tends to be constant. φ. Swi = constant
(2.40)
28
CHAPTER TWO
THEORY
It is worth to mention that a single equilateral hyperbola can be expected to fit data from reservoir having a thick transition zone or variable lithology. d-Experiences indicate that getting the most coherent pattern from the ( φ. Swi ) plot, the formation would be at irreducible water saturation, be reasonably homogeneous, have a constant hydrocarbon type. It consists primarily of intergranullar porosity, and originally water wet. The textural parameter w is the exponent in the equation that relate bulk volume of water ( φ. Swi ) to the resistivity ratio (Rw/Rtirr) assuming that m and n are equal: Swi n
F.Rw Rtirr
(2.41)
By combining 2.38 & 2.40, tend to, Swi n φ m
Rw Rtirr
Let n=m=w, so Rw Rtirr
(2.42)
log(Rw/Rti rr) log(Swi φ)
(2.43)
(Swi φ) w
Thus,
w
Gomez (1977) discussed some considerations for the possible use of the parameter (a) and (m) as a formation evaluation’s tool through well logs, his conclusion that computed (a) and (m) from well logs can be used for detecting permeable zones, as follow: φm φ 2 1 k ( ) a 1 φ Swi 2
(2.44)
Where, the above parameters had previously defined. Raymer (1981) has studied the possible correction for log-derived permeability relationships that proposed by Timur, Wyllie and Coats, to the
29
CHAPTER TWO
THEORY
effect of elevation and hydrocarbon density. A correction factor, (c1) modified the relationships as follows: k
a φb
(2.45)
(c1.Swi) c
Where c1 is a correction factor applied to the log derived irreducible water saturation value, it is a function of the capillary pressure. Yonthan (1981) like Timur, Wyllie & Rose, recommended to use equation (2.31) in formation evaluation’s studies. He referred to some errors, which could happen by this relation taken in consideration, that core-derived values are obtained at room temperature and pressure, from discreted and selected small size samples, whereas by log-derived from continuous measurements of indirect parameters involved a large rock volume. Kapadia & Menzie (1985) presented a technique for the determining of the coefficient of permeability variation, V, from quantitative well log analysis without the requirement of extensive coring programs or extensive laboratory effort. The method is an application of the improved empirical relationships proposed by Coats and Dumanoir (1973). 2.5.3.2. Methods Based on Acoustic Log: Permeability of rock formation has been also detected using acoustic data log. Many studies dealt with such a subject, such as: Staal and Robinson (1977) showed that the acoustic wave train attenuation, that corresponds in velocity to the Stoneley wave (which commonly called tube wave is the last packet of energy to arrive at the receiver), it is strongly related to formation permeability. Acoustic energy was measured in a time of a 1400 µs to 2310 µs after the first arrival time. This energy was normalized by dividing it by the
30
CHAPTER TWO
THEORY
energy of the first arrival wave. The normalized energy shows a strong relation to core permeability when plotted together at the same depth level. Lebreton et al. (1978) studied the influence of permeability on acoustic waveform. They noticed a relationship between formation permeability and a new parameter chosen in the compressional wave (Ic), where, Ic
v2 v3 , v1
and v1, v2 & v3, being the amplitudes of the first three half cycles of compressional wave. From the acoustic signals recorded in several wells, Lebreton et al. suggested the following correlation: kv β (2.46) μ Where, kv is vertical permeability, is rock wetting fluid viscosity & Ic α log
α , β constants for a given equipment and well.
Williams et al. (1984) have developed a new long spaced acoustic logging tool, which is proposed to record superior quality acoustic data and extracting tube wave amplitude and velocity data. The energy loss of the tube wave, as it propagates along a formation, should be related to the permeability of that interval, hence, core permeabilities are compared to both tube wave interval time and energy ratio, it is confirmed that both correlate with formation permeability. Williams et al. (1984) introduced examples, one of them is from well drilled through Cretaceous carbonate sequence; showing that improvement of data extracted from Long space Acoustic Log (LASAL) for this well under adverse bore hole conditions for tube wave attenuation to core permeability. Hsui et al. (1985), and Dvorkin et al. (1994) investigated the theoretical relationship between tube wave attenuation and permeability using two different models. The first one is a simple model of a borehole with absorbing wall and the second is a borehole with a Biot porous medium 31
CHAPTER TWO
THEORY
in the formation, Hsui et al. have also investigated the relative influence of intrinsic formation attenuation (an elasticity) and permeability on the attenuation of tube waves. Hsui et al. found that in rocks with low to medium permeability (less than 100 md); intrinsic attenuation is the major contributor to tube wave attenuation. In high permeability formation (>100md) rocks, fluid flow associated with in-situ permeability is as important intrinsic alternation in controlling tube wave attenuation. In either case, if one can estimate the intrinsic formation attenuation from the other parts of the full waveform (Pwave or pseudo-Rayleigh wave) an estimation of formation permeability can be obtained. Dovrkin et al. introduced a poroelasticity model that corporate the two mechanisms of solid/fluid interaction in rocks: the Biot mechanism and Squirt flow mechanism. This combined Biot /Squirt (BISQ) model relates compressional velocity and attenuation to the elastic constants of the drained Skelstone and of the solid phase porosity, permeability and compressibility. Xiamong and Cheng
(1996) described a fast algorithm (called
inversion) for estimating formation permeability from Stoneley wave logs. The procedure used a simplified Biot-Rosenbaum model. The input to the inversion are the Stoneley wave spectral amplitudes at each receiver position, the bore-hole fluid properties (velocity and density), the bore-hole caliper log, the formation density and porosity, and the compressional, and shear velocities for the interval of the interest. Xiamong and Cheng (1996) method also used a reference depth of known permeability and compared amplitude’s variations at other depth relative to the reference depth. The inversion procedure can be formulated for data acquisition configurations:
32
CHAPTER TWO
THEORY
A-Array, in the array configuration, the amplitude decay of the wave from a cross the array can be used in the inversion to calculate permeability. This method requires that each receiver in the array should have the same frequency response. B- Iso-offset, for the iso-offset Stoneley wave data at depth (z), the recorded wave spectrum includes several contributions, as given in the following: A( f , z) S( f ) E( f , z) P(p(z), f ) exp[ik(p)d z] Q( f ) R( f )
(2.47)
Where S (f) =source of spectrum, R (f) receiver response, f denotes frequency and Q (f) is the amplitude reduction caused by the intrinsic attenuation along the wave path form source to the receiver. The actual processing procedure for permeability estimation illustrated in Fig. (2.18), where the waveform data first input into the computer disk, if the compressional and/or shear wave velocity, information is missing in the log data, the waveform data are processed to obtain the velocity logs. Cheng and Cheng (1996) estimated formation velocity, permeability and shear wave anisotropy using acoustic logs. Field data sets collected by an array monopole acoustic logging tool and a shear wave tool are processed and interpreted. The compressional (P) and shear (S) wave velocities of the formation are determined by threshold detection with cross-correlation correcting from the full waveform and the shear-wave log, respectively. The array monopole acoustic logging data are processed using the extend Prony’s method to estimate the borehole Stoneley wave phase velocity and attenuation as a function of frequency. The estimated permeabilities from low frequency Stoneley wave velocity and attenuation are in a good agreement with core measurements.
33
CHAPTER TWO
THEORY
In addition, it is proven that the formation permeability is not the cause the discrepancy. Input Waveform Data
Event Tracking to Pick Stoneley Arrivals
Process Waveform Data to Obtain Velocity Logs
Wave Field Separation to Obtain Transmitted Wave Field
Combine Other Log Data for Input Parameter
Permeability Estimation Program
Fig. (2.18): Procedure of permeability estimation from Stoneley wave logs.
2.5.3.3. Nuclear Magnetic Log Method The nuclear magnetism log (NML) provides a measure of the free fluid contents (FFI) and the relaxation time (T1) for the fluid in the porous rocks. These two parameters are related greatly with permeability and effective porosity. The historical development in Nuclear Magnetic Log (NML) is: Hull and Coolidge (1960) presented field examples from various operating area which demonstrate the capability of the nuclear magnetism log to operate under most drilling and formation conditions. Based on these examples, they showed that the free fluid index of NML provides a superior means of reservoir rock definition, yields a measure of formation permeability and productivity. Seevers (1966) derived a mathematical and physical model which relates permeability to (FFI) and (T1). When Kozeny-Carman equation was combined with Seevers’s equation (nuclear spin polarization of liquid in porous media) one gets: T1.TB k A.φ. TB T1 34
(2.48)
CHAPTER TWO
THEORY
h.rs 2 Where, A , TB is the bulk relaxation time for the liquid (µsec), T1 is 2
the observed relaxation time (µsec). This relationship is valid for a uniform pore system of sandstone and water zone. Timur (1968) studied the relation between effective porosity and permeability of sandstone through out (NML) principles in the same fields as (Timur 1968, part J), where the free fluid index (FFI), spin relaxation time (T1) and permeability (k), were measured for more than 150 samples. Seevers’s model modified by Timur into the following form:
α ,β
(2.49)
TI .TB Where, α FFI , A & s are empirical constants. Again, Timur TB T1 modified another correlation in the following form:
k B .β t Where, β
(2.50)
φ 4.4 1.4.FFI 3.2 10 4 1 φ
2
β and t are empirical constant.
α , β can be computed from laboratory measurement or logging data, and (A,
B, s & t) can be calculated by maximum likelihood regression analysis techniques. Parra et al. (2001) used digital processing of optical macroscopic (OM); X-ray computed thermograph (CT) images, and the petrography to characterize the pore space of the pore system of vuggy carbonates in south Florida. The results of this analysis provided supportive information to evaluate nuclear magnetism resonance (NMR) well log. The measurements
35
CHAPTER TWO
THEORY
were established empirical equation to extract permeability from (NMR) well logs as follows: k 1014. FZI BVI 0.0946 1 φ where, FZI BVI 1 1 1.05 φ
2
φ3
1 φ 2
(2.51)
0.337
is flow zone indicator, BVI is bound
fluid volume. 2.5.3.4. Methods Based On Resistivity Log Data The historical development of the method based on resistivity data logs is as follow: Tixier’s (1949) empirical or so-called resistivity gradient method, it represents the earlier trial to relate permeability to resistivity log data. This method is applicable only when permeability of the transition zone is a proximately constant. Tixier noted that, in transition zone, a small height is necessary to go from 100% water saturation to residual saturation in highly porous and permeabl formation, whereas a great high required in low permeability formations. He gave the permeability as a function of the following parameters:
kC
Where, a
a2
(2.52)
(γ w γ o ) 2
1 ΔRt ΔRt , Resistivity gradient, Ro = Resistivity of 100% Ro Δh Δh
water saturated formation, γ = Specific gravity, w & o are subscripts of water and oil respectively.
36
CHAPTER TWO
THEORY
Johnson (1960) gave an empirical correlation of apparent residual oil saturation (Shr) (computed from well log data, where data from well drilled by salt mud system) and permeabilities from whole core analyses has been useful in evaluation in San Andres formation in west Texas. Johnson’s model has the following form: k f (Rxo , φ, Ros, Rmf) f (Shr)
(2.53)
Where, Rxo is the resistivity of flushed zone, Rmf is the resistivity of mud filtrate, Ros is the residual oil saturation. In an oil reservoir, the degree of flushing of the Rxo zone by the mud filtrate and, therefore the residual oil saturation in that zone, is largely dependent upon the permeability of the reservoir rock. It is expected that as the permeability decreases the amount of residual oil (remaining in the Rxo (flushed) zone) increases. Sutton (1961) studied the capability of using electrical and sonic logs in the Delaware sand in the Ford and Geraldine field as an formation evaluation tools, where permeability can be estimated in this field from the sonic velocity logs. This is primarily a result of the rather good correlation which exists between porosity and permeability. Because of this, Sutton suggested that the permeability could also be estimated from the microlaterolog. In some zones, permeability is controlled by the amount of shale presence. Pirson (1963) proposed a qualitative and quick method for detecting permeable zone, by set of logs (caliper log, Sp log, and microlog), when a positive separation happens in microlog (R”2 >R1”x
” 1
) and there is an
indication of mud cake presence (caliper log) with negative deflection in Sp curve (sand base line), these represent permeable zone.
37
CHAPTER TWO
THEORY
Rodrignez & Pirson (1968) showed advantages of the continuous dipmeter as a tool for studies in directional sedimentation and directional tectonics. They were also noted that the strongest grain orientation is parallel with the direction of maximum permeability in bedding planes. Rodrignez and Pirson attacked across their studies to the directional variations of directional resistivity in X-Y plane of sedimentary rock, and they concluded that the electrical resistivity should correlate closely with permeability variations as modified Kozeny equation in the following form: 10 8 φ2 k 2 Ro/Rw Sw 2 (1 φ) 2 Where, the symbols have been previously defined.
(2.54)
Ershaqi et al. (1978) developed a method for estimation of permeability profile in a geothermal well from a Dual Induction-Laterolog, a porosity log, and drilling data. The procedure is based on modeling the invasion of the drilling mud filtrate into the formation and using a history matching technique to arrive the permeability profile. A numerical simulator of well bore hydraulics during drilling is used to predict the invasion profiles for an assumed permeability profile, where the radius (r) of invasion is a function:
r f (mc, Δp, t, φ, k)
(2.55)
Where, Δp Is the pressure differential between well bore and formation.
mc=Mud composition. t = Exposure time. Predicted invasion is compared with those obtained by the dualinduction laterolog to indicate whether, or not the assumed permeability profile is reasonable. The process is then repeated until a satisfactory match
38
CHAPTER TWO
THEORY
is obtained. Nevertheless, this method is not believed to be applicable to oil and gas reservoirs. 2.5.3.5. Wire Line Formation Tester The wireline formation tester was used successfully for many years to recover fluid samples and record pressure measurements. The information from such surveys, when combined with conventional drill stem or production test results, usually adequately described the behavior of homogeneous reservoirs. The authors dealing with this subject are historically arranged: Raymer and Freeman (1984) provided an approach of combining pressure measurements made with wireline formation tester tool with a conventional well log, where water saturation and porosity are obtained; with in-situ determination of capillary pressure profile and a pore-size distribution profile; hence the mean effective pore radius and the rock permeability can also be obtained. For determining permeability two relations were proposed, one for permeability, which is relating to mean effective pore throat radius ( rp ) :
k 37 * φ . rp 2
(2.56)
, and the other is relating permeability to the displacement pressure (Pd): γ 2φ3 (2.57) Pd 2 Where, γ is surface tension in dynes/cm2, and Pd is a capillary pressure (psi) k 9.4
corresponding to about 95% water saturation. In low-permeability rocks, the displacement pressure can be determined from wireline sufficient accuracy and vice versa.
39
CHAPTER TWO
THEORY
Yarborough (1984) made a study of the effectiveness of some well site applications of wireline formation testing tool, with emphasis on “quicklook” permeability indicators in sandstone reservoir. An example is shown where a correlation was established between effective permeability, as derived from drawdown and buildup analyses of the wire line formation tester data, and functions of log-derived porosity for the water bearing section of the reservoir. Yarborough noted that this technique is possible when permeabilities are in the range of 0.5 to 10 millidarcies and formation damage by the drilling process is small. If the above conditions are met, a “minimum” effective permeability line can be established. Badaam et al. (1998) used test results of modern tester tool, the Modular Formation Dynamic Tester or (Multi probe Formation Tester). The tester provides a capability to conduct controlled local production, a vertical interference test along an open hole and permits a flexible configuration, (more information about tool description and operation can be getting in Badaam et al. 1998). For layered and heterogeneous formations, this configuration with high-resolution well bore images is used to delineate inter-layer communication and determine layer permeabilities along the well bore. A vertical interference test across stylolite zones were carried out in a number of wells using the wireline multiprobe formation tester as a part of the reservoir characterization for the pilot gas of injection area in a Middle East carbonate reservoir. For each test; the pressure and flow rate at the sink and probe; and the pressures at the horizontal and two vertical observation probes were recorded as a function of time. The interpretation of the pressure measurements from these tests, and open-hole logs were used to identify vertical and horizontal layer permeabilities. 40
CHAPTER TWO
THEORY
Kerchner et al. (2000) described a project undertaken by Conoco and Schlumberger companies. The project objectives were to: *develop a field-specific method to reliably, and estimate permeability using wire line logging methods. *demonstrate how this information impact economic and engineering consideration related to the hydraulic stimulation program. *describe the importance of knowledge of permeability in post-audit analyses. *recommend best- practice guidelines for use of a particular method to obtain permeability. The primary goals of this project was to develop a continuous permeability measurement derived from wire line logging analysis using modern logging tools such as a typical suit of AIT(Array Induction Tool), LDT(Litho-Density Tool), CNL (Compensated Neutron Log), Gamma Ray, and also using an extensive logging program. The CMR (Combinable Magnetic Resonance) tool was run for pore size distribution to relate it to permeability and the ECS (Element Capture Spectroscopy tool for clay typing to be used in petrophysical analysis and completion design. 2.5.3.6. Thermal Decay Time (TDT) Log Englik and Helchie (1971) developed a new qualitative method based on (TDT) log in open and cased hole for detecting permeable zones in fractured and vuggy formations. Englik and Helchie showed that if filtration mud (during drilling operation) have a high salinity than the formation water then the neutron life time log with porosity data may be used together to derive qualitative estimation of the invasion profile and determine the
41
CHAPTER TWO
THEORY
permeable zone. The comparison between permeability (invasion) shown on the logs and those from the cores are in reasonably good agreement. TDT log method required the use of salt base or boron treated mud during the drilling operations and will give a reasonable result information of low porosity and deep invasion. 2.5.3.7. Other Techniques: Brown & Hussein (1977) presented the results of a study made to find the best method for estimation of formation permeability from well logs of the Shaybah field in Saudi Arabia. A case history was presented showing how an optimum empirical relation to predict permeability can be developed for a particular formation and area. The optimum permeability, prediction relationship for Shaybah, field included a term of capillary pressure (Pc) in addition to water saturation (Sw) and porosity ( φ ) where:
φ0.86 k 57 Pc0.89 Sw1.26
(2.58)
This equation is fitted to data from a carbonate reservoir of lower Cretaceous age in shaybah field. Raiga-Clemenceaus (1977) introduced a paper about the variation of cementation exponent (m) with permeability, and studied the previous relations of porosity and formation (F), and the influence of cementation exponent variation. His laboratory work covered most of the physical characteristics including porosity, permeability, formation resistivity factor, tortuosity, cross sectional area index, and packing index. He concluded that the linear relation of the formation factor and porosity with a constant cementation exponent has certain application.
42
CHAPTER TWO
THEORY
Clemenceaus proposed an equation to derive the variable cementation exponent (m) from the permeability value: m 1.28
2 2 log(k)
(2.59)
Salih et al. (1993) suggested an experimental relationship between permeability, water saturation, and rock resistivity, which could be used for interpreting any of these three parameters, when the other two are known. Their study is based on electrical measurements of core samples of different permeabilities. Brine saturation, Sw, and corresponding rock resistivity, Rt, were used together to express the permeability of the rock. In order to eliminate the brine resistivity effect, the rock resistivity was represented by an apparent formation factor, Fa, which is the ratio of the partially saturated rock resistivity, Rt, to the brine resistivity, Rw. It was observed that the apparent formation factor versus water saturation curves for samples of different permeabilities, where ranged similarity to capillary pressure curves. Therefore, permeability was expressed as a function of the apparent formation factor and the brine saturation: 1 , Sw) (2.60) Fa When fractional brine saturation is 1.0, permeability becomes a function of k f(
formation factor only. Mohaghegh et al. (1995) tested the applicability of the two most promising methods for permeability prediction from well logs, namely, multiple regression and virtual measurements. A heterogeneous formation West Virginia was designated for this test. Eight wells that had both geophysical well log and core analysis were chosen. The procedure of the test is as follow:
43
CHAPTER TWO
THEORY
1. Seven of the eight wells are chosen to develop the regression and neural method. 2. The developed models will be applied to the eight well. Using the eight well’s log data a permeability profile for the well will be predicted. 3. The predicted permeability profile will be compared with actual laboratory measurements of the permeability for this well. The technique that performs better under these circumstances should be the superior method. 4. Step 1 through 3 will be repeated by substituting the eight well with one of the seven wells. This is to ensure the robustness of the methods. Cuddy (1998) and Brown et al. (2000) were presented a new mathematics technique for permeability prediction by well logs. Cuddy used “Fuzzy Logic” as a new interpretation technique to predict permeability and litho-facieses in uncored wells. Fuzzy logic is simply an application of recognized statistical techniques. Whereas conventional techniques deal with absolutes, the new methods carry the inherent error term through the calculation rather than ignoring or minimizing it. This retains the information associated with errors and gives surprisingly better results. The new techniques quantify these errors and use them, together with the measurement, to improve the prediction. In addition, the method uses basic log data sets such as gamma ray and porosity rather than depending on new logging technology. Brown et al. (2000)presented a genetic algorithm (GA) that predicts permeability in reservoirs based on electronic log data and a training sample of borehole core data. The GA was constructed using a genetic functional framework developed at the Robert Gordon University. An empirical approach is adapted to the task of determining permeability from combining of electronic survey represented by sonic and gamma ray log, and core sample data.
44
CHAPTER THREE
TECHNIQUES OF PERMEABILITY EVALUATION
CHAPTER THREE TECHNIQUES OF PERMEABILITY EVALUATION INTRODUCTION: Several different equations and techniques exist for determining permeability from logs. However, none of these is universally applicable from field to field, well to well, or even zone to zone without adjusting constants or exponents, or other compensations. To understand why this is necessary a review was undertaken of the parameters, which affect logcalculated permeabilities. 3.1. POROSITY CORRELATION METHOD :In some fields, a linear or bilinear relationship is obtain between the logarithm of the permeability and porosity. A schematic example is given in Fig. (3-1). A regression line:Log k=a φ +b
(3.1)
found for low porosities. Another one:(3.2) Log k=a φ +b is obtained for high porosities generally with a> a and b
