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ENPE 523 Introduction to Reservoir Engineering

Mingzhe Dong

Chapter 1 Review of Flow in Porous Media and Core Analysis In this chapter, fluid flow through porous materials is reviewed. First, the macroscopic pore structure parameters determined by the pore structure of the medium are introduced. They are porosity, permeability, specific surface area, pore-size distribution, and formation resistivity factor. In the second part of this chapter, four core analysis methods are introduced. They are fluids saturations, porosity, capillary pressure curves and relative permeability curves.

1.1 Fundamentals of Porous Rock Properties The dictionary definition of porous is "full of holes or pores." A porous medium is defined as a material containing pores (voids). Such materials are widely encountered in everyday life. The natural soil that makes agriculture possible is a porous material. Voids in the soil absorb moisture and nutrients that support plant growth. Most of the natural rocks are also porous to varying degrees. Other everyday examples of porous materials are biological tissues (such as skin, bones and wood), sand, snow, concrete and ceramics, cotton, wool, glass-wool, paper, cloth and foamed plastics, like Styrofoam. The properties of porous media play important roles in many areas of science and industry, such as: agriculture, water filtration, geosciences (seismic imaging, ground water hydrology, petroleum geology), biology, engineering (petroleum reservoir engineering, geo-mechanics, catalysis and reactor design, mechanics of cement concrete and other construction materials. Understanding the behavior of porous rocks is central to petroleum reservoir engineering and we will explore the nature and properties of porous media from the perspective of petroleum engineering. 1.1.1 Porosity Sedimentary rocks are source and reservoir rocks for gas and oil. 99.9% of oil and gas were found in sedimentary rocks. All of the sedimentary rocks are important to the study of petrophysics and reservoir engineering. Therefore, flow in porous media, from perspective of petroleum engineering, is focused on sedimentary rocks. Loose sediments become hard rocks in the subsurface by the process of cementation and compaction: 1) Cementation – salts precipitate out of the subsurface water to form coatings on the sediment grains and bridge the loose sediment grains together. 2) Compaction – sediments are solidified under high pressure exerted by overlying rocks.

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ENPE 523 Introduction to Reservoir Engineering

Mingzhe Dong

Three components of the Sedimentary Rocks (clastic) are: 1) Sediment grains – frame or matrix 2) Natural cements coating and bonding the grains together 3) Pore space – filled with fluids (oil, water, gas)

Grain Oil

Water

Cements Figure 1.1 Parts of sedimentary rock (clastic) The two most common petroleum reservoir rocks are: 1. Sandstones Sandstone rocks are sedimentary rocks made from small grains of the minerals quartz and feldspars (KAlSi3O8, NaAlSi3O8, CaAlSi2O8). (They are often used as building materials for stone houses.) 2. Carbonates (limestones and dolomites) Carbonate rocks are a class of sedimentary rocks composed primarily of carbonate minerals. The two major types are limestone and dolomite, composed of calcite (CaCO3) and the mineral dolomite (CaMg(CO3)2) respectively. The porosity I of a porous rock is the volume fraction of its pore space. It is a measure of the storage capacity of reservoir fluids. The porosity of a porous medium is expressed as

I

Vp

(1.1)

Vb

where Vb = the bulk volume of the porous medium Vp = the pore volume. The pore volume of a reservoir rock is the void space in it. The porosity varies between zero and unity, depending on the type of porous material and the way it was formed. In porous media, especially in consolidated media, such as reservoir rocks, there are two types of pores or voids. One is the pore space, called “interconnected” or “effective” pore 2 Winter 2016

Chapter 1

ENPE 523 Introduction to Reservoir Engineering

Mingzhe Dong

space. The other is “dead” pore space that consists of “isolated” or “non-interconnected” pores or voids dispersed in the medium. Only the effective or interconnected pore space contributes to the flow of fluids through the porous medium. Therefore, there are two types of porosity, namely: Absolution porosity I a Total pore volume Bulk volume Effective Porosity I

Ia

I

(1.2)

Interconnected pore volume Bulk volume

(1.3)

Effective porosity is a measure of the void space that is interconnected and can participate in flow process. Oil and gas can only be produced from such interconnected pores. The following table shows the range of porosity values of sedimentary rocks: Porosity 0–5 5 – 10 10 – 15 15 – 20 20+

Qualitative evaluation negligible poor fair good very good

The porosity of a reservoir may vary vertically or/and horizontally. The average porosity of a reservoir can be calculated by using the following formulas: Arithmetic average

I

Thickness weighted average I

¦I

i

(1.4)

n

¦I h ¦h ¦I A ¦A ¦I Ah ¦ Ah i i

(1.5)

i

Areal weighted average

I

i

i

(1.6)

i i

(1.7)

i

Volumetric weighted average I

i

i i

where n hi

Ii

= the total number of core samples = the thickness of core sample i or reservoir area i = the porosity of core sample i or reservoir area i 3

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Chapter 1

ENPE 523 Introduction to Reservoir Engineering Ai

Mingzhe Dong

= the area of the reservoir area i

If a reservoir rock shows large variations in porosity vertically but not horizontally in bedding planes, the arithmetic average porosity or the thickness weighted average porosity can be used as the average porosity; if the porosity of a reservoir changes greatly from one portion to another portion, the average porosity can be calculated from the areal or volumetric weighted average porosity.

1.1.2 Fluid Saturations Saturation is defined as the fraction, or percent, of the pore volume (Vp) occupied by a particular fluid (oil, gas, or water): Oil, gas, and water saturation are: Vg Vo Vw , , (1.8) Sg Sw Vp Vp Vp Where Vo, Vg and Vw are oil, gas, and water volumes, respectively. The saturation of each individual phase ranges between zero and 1.0. The sum of the saturations is 1.0:

So

So + Sg + Sw = 1.0, and the sum of pore spaces occupied by the fluids is total pore space: Vo + Vg + Vw = Vp.

1.1.3 Permeability and Darcy’s Law Permeability is the property of a porous medium that allows a fluid to flow through it. The darcy permeability k is calculated by applying Darcy’s law (Darcy, 1856) to a slow (creeping), one dimensional, horizontal, steady flow of a Newtonian fluid: k A 'P , (1.10) P L where q = volumetric flow rate (cm3/sec) A = cross-sectional area of the sample normal to the flow direction (cm2) L = length of the sample in the flow direction (cm) ǻ3 = hydrostatic pressure drop (atm) ȝ = viscosity of the fluid (cP). q

Using these units in Darcy’s law results in the practical unit of permeability darcy (D). One darcy is equal to 0.987 (ȝP)2 in SI units. One darcy is a relatively high permeability, and for tight porous materials the unit millidarcy (mD) is used. 1D = 1,000 mD. 4 Winter 2016

Chapter 1

ENPE 523 Introduction to Reservoir Engineering

Mingzhe Dong

If the flow is under gravity drainage, the volumetric flow rate is given by: q

kA

P

Ug

(1.11)

Where ȡ g

= the density of the fluid = the gravitational constant

When the flow is under both gravity and hydrostatic pressure drop, Darcy’s law can be expressed in the following form: kA ') , P L :KHUHɎLVWKHIOXLGSRWHQWLDOGHILQHGDVIROORZV

(1.12)

q

)

P  Ug z ,

(1.13)

so ')

'P  U g 'z .

(1.14)

The pressure increment ǻɎ is the pressure drop measured in the fluid flowing through the column, P LVWKHK\GURVWDWLFSUHVVXUHȡLVWKHIOXLGGHQVLW\g is the gravitational constant and z is the distance upward measured from an arbitrary datum level. Permeability is the property of a medium of allowing fluids to pass through it without change in the structure of the medium or displacement of its part. By general convention, the following terms are applied to permeability values of oil and gas reservoirs: Range of permeability values of oil and gas reservoirs Permeability (mD) < 1.0 – 15 15 – 50 50 – 250 250 – 1000 > 1000

Qualitative description poor or fair moderate good very good excellent

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ENPE 523 Introduction to Reservoir Engineering

Mingzhe Dong

1.1.4 Gas permeability and Klinkenberg Effect Gas Permeability In using gas in measuring the permeability, the gas volumetric flow rate varies with pressure. Therefore, the value of q at the average pressure in the core must be used. Assuming the used gases follow the ideal gas behavior, Darcy’s law is in the following form:

qgb qgb A L P1 P2 Pb ȝg

2 kA( p12  p 2 ) 2Pg Lpb

(1.15)

gas volumetric flow rate at base pressure (cm3/sec) (p1 or p2) cross-sectional area of the sample (cm2) length of the sample in the flow direction (cm) inlet (upstream) pressure (atm) outlet (downstream) pressure (atm) base pressure (atm) gas viscosity at mean pressure (cP)

The Klinkenberg Effect The permeability of a porous medium sample measured by flowing air is always greater than the permeability obtained when a liquid is the flowing fluid. This is because that gases exhibit slippage at the pore wall surface. The gas slippage results in a higher flow rate for the gas at a given pressure differential. For a given porous medium, the calculated permeability decreased as the mean pressure (pm) increased. If a plot of calculated permeability versus 1/pm is extrapolated to the point of 1/pm = 0 (or pm = infinite), this permeability would be approximately equal to the liquid permeability (see Figure 1.1). The straight-line relationship in Fig.1.1 can be expressed as:

kg

§ 1 · ¸¸ k L  c¨¨ p © m¹

(1.16)

where kg = calculated gas permeability pm = mean pressure (pm = (p1+p2)/2) kL = Klinkenberg permeability c = slope of the line The magnitude of the Klinkenberg effect varies with the core permeability and the type of the gas used in the experiment.

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Mingzhe Dong

Fig. 1.1 The Klinkenberg effect in gas permeability measurement

1.1.5 Pore size and pore size distribution The distribution of pore size is somewhat arbitrary. If D ( De ) is defined as the pore volume per unit interval of pore diameter De, then there is: f

³D ( D )dD e

e

Vp .

(1.17)

0

Mercury Porosimetry In mercury porosimetry the volume of mercury penetrating the sample is measured as a function of the pressure imposed on the mercury. The pore size is calculated from this pressure by Laplace’s equation of capillarity and, using the bundle of capillary tube model of pore structure, the volume of mercury is assigned to this pore size. Mercury porosimetry and adsorption isotherms are widely used for determination of pore size distribution. These and other methods of pore size distribution have been reviewed in depth by Dullien (1992). The principal shortcoming of most methods is the pore model used. The pore structure of porous media does not resemble a bundle of capillary tubes. Rather, the pores resemble irregularly shaped particles (Dabbas and Rumpf, 1966). In the case of interconnected pore space the “body” of each pore is connected to the bodies of adjacent pores via “necks” or “throats”. The “sizes” of both the pore bodies and the pore throats play an important role in determining various macroscopic properties, such as permeability, capillary pressure curves of porous media, etc. Both the body and the throat sizes can be “measured” using computer reconstruction of pore structure from photomicrographs of serial sections of the sample (Kwiecien, 1988).

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ENPE 523 Introduction to Reservoir Engineering

Mingzhe Dong

Mercury porosimetry is widely used to characterize the pore structure of porous samples. In mercury porosimetry, gas is evacuated from the test cell in which there is a test porous sample, and mercury is then transferred into the sample cell under vacuum and pressure is applied to force mercury into the sample. During measurement, applied pressure p and intruded volume of mercury, V, are recorded. As a result of analysis, an intrusion-extrusion curve is obtained (Fig.1.2). Parameters describing the pore structure of the sample can be calculated from the data obtained.

Figure 1.2 Mercury porosimetry measurement curves

Mercury porosimetry is based on the following equation˖

p

PC

4V cosT De

(1-18)

where p is the pressure applied to mercury, and Pc is the capillary pressure at the mercury meniscus and its direction is opposite to the injection, r is the equivalent radius of the pore ZKHUHPHUFXU\LQWUXGHVı is the surface tension of mercury and șis the contact angle of the mercury on the surface of a solid sample. Generally used values for surface tension and contact angle of mercury are 480 mN/m and 140-180°, respectively. Total pore volume (Vp) is the total intruded volume of mercury at the highest pressure applied. Volume pore size distribution, D ( De ) , is defined as the pore volume per unit interval of pore diameter (De) by Equation 1-19:

D ( De )

Pc dV De dPc

(1-19)

Volume pore size distribution is based on a model of cylindrical pores.

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ENPE 523 Introduction to Reservoir Engineering

Mingzhe Dong

1.1.6 Specific surface The specific surface of a porous medium S is the internal surface area of the voids and pores per unit mass. The specific surface based on the bulk volume of the porous material is designated by Sbv and the one based on the solid volume is denoted by Sgv, i.e., Sbv = (Surface area of the pores of a sample)/(bulk volume of the sample) Sgv = (Surface area of the pores of a sample)/(Solid volume of the sample) and Spv = (Surface area of the pores of a sample)/(Pore volume of the sample) Specific surface plays an important role in a variety of applications of porous media such as adsorption, catalysis and ion exchange processes. For fluid flow in porous media, the specific surface is an important parameter in models of conductivity or permeability of a porous medium. For example, two porous samples with the same porosity may have very different permeabilities because of different specific surface areas. The one with a higher specific surface area has a smaller average pore size and lower permeability. 1.1.7 Resistivity Factor The electrical resistivity of a fluid-saturated porous medium is a measure of its ability to impede the flow of electric current though it. Dry porous samples of zero electrical conductivity exhibit infinite resistivity. The resistivity of porous media containing an aqueous phase is a function of porosity, aqueous phase saturation and the salinity of the aqueous phase (Prison, 1963). Measurements using different porous materials show that the resistivity is also dependent on pore structure. A “resistivity factor”, often called “formation resistivity factor”, is defined as Ro , (1.20) Rw where Ro is the electrical resistance of the porous sample saturated with an ionic solution (see Figure 1.3a), and Rw is the bulk resistance of the same ionic solution occupying the same space as the porous sample (same cross-section area and length) (Figure 1.3b). In the case of nonconductive solids Ro is always greater that Rw and therefore, F is always greater than unity. FR

(a) (b) Figure 1.3 Measurements of Ro (a) and Rw (b) (Tiad and Donaldson, 1999). 9 Winter 2016

Chapter 1

ENPE 523 Introduction to Reservoir Engineering

Mingzhe Dong

Many attempts have been made to correlate the formation resistivity factor with porosity I and a “cementation exponent” m. The first such known relationship was suggested by Archie (1942): FR

I m.

(1.21)

The cementation factor is a function of the shape and distribution of pores. It is determined from a plot of the formation resistivity factor FR vs. porosity I on a log-log graph. Such a plot generally can be fit by a straight line with a slope of m. Humble formula can be used to correlate better the formation resistivity factor with porosity: FR

a

(1.22)

Im

where the values of a and m depend on the types of rocks. Therefore, they are determined from laboratory measurements. Archie formula and Humble formula are plotted in the following two Figures for various values of a and m: 1. Plot of Archie formula with various values of m (From Core Laboratories)

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ENPE 523 Introduction to Reservoir Engineering

Mingzhe Dong

2. Plot of Humble formula with various values of a and m (From Core Laboratories)

A great deal of information on the formation resistivity factors for various reservoir rocks has been presented by Tiab and Donaldson (1999). For detailed discussions of the correlations of FR with the “tortuosity factor” of porous media readers are referred to the review by Dullien (1992).

1.1.8 Permeability-Porosity Relationships The permeability-porosity relationship schematically shown in Figure 1.4 is also often seen for actual formations. For a reservoir formation which may be considered uniform and homogeneous, there may not be a specifically defined trend line between permeability and porosity values. It is also possible to have a very high porosity with a low permeability. The reverse can also be true, i.e., high permeability with a low porosity, such as in micro-factrued carbonates. However, a very useful correlation between them within one formation can be found. k

Porosity

Figure 1.4 Permeability-porosity relationships for some formations

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Chapter 1

ENPE 523 Introduction to Reservoir Engineering

Mingzhe Dong

Figure 1.5 shows semi-log plots of permeability versus porosity for various grained sandstones. Figure 1.6 shows typical permeability and porosity trends for various rock types. Such a relationship is very useful in the understanding of fluid flow through porous media.

Figure 1.5 Permeability-porosity relationships for various grained sandstones (Core Laboratories)

Figure 1.6 Permeability-porosity relationships for various rock types (Core Laboratories)

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Chapter 1

ENPE 523 Introduction to Reservoir Engineering

Mingzhe Dong

Kozeny Model Kozeny correlated the permeability with porosity and specific surface area for the simple tube bundle porous samples as...