Chapter 1 Full

 Petroleum reservoir rock may be composed can range from very loose and

unconsolidated sand to a very hard and dense sandstone, limestone, or dolomite

 The grains may be bonded together with a number of materials, the most common

of which are silica, calcite, or clay.

 Two main categories of core analysis tests that are:

- Routine core analysis tests  Porosity  Permeability  Saturation

- Special tests  Overburden pressure  Capillary pressure  Relative permeability  Wettability

 Surface and interfacial tension

To provide an understanding of:  The concepts of rock matrix and porosity  The

difference between original (primary) and induced (secondary) porosity

 The difference between total and effective porosity  Laboratory methods of porosity determination  Determination of porosity from well logs

Definition: Porosity is the fraction of the bulk volume of a material (rock) that is occupied by pores (voids ). Porosity is an intensive property describing the fluid storage

capacity of rock

Porosity is a static property – it can be measured in the

absence of flow

• Rock matrix is the grains of sandstone,

limestone, dolomite, and/or shale that do not make up the supporting structure. • Matrix is the non-pore space • Pore space is filled with fluids. •water •oil •natural gas

Rock matrix

Pore space

Rock matrix

Water

Oil and/or gas

 Vma is often showed as Vg (grain volume)

 Vb = Vg + Vp

Rock-forming Source of process material

IGNEOUS

SEDIMENTARY

METAMORPHIC

Molten materials in deep crust and upper mantle

Weathering and erosion of rocks exposed at surface

Rocks under high temperatures and pressures in deep crust

Crystallization (Solidification of melt)

Sedimentation, burial and lithification

Recrystallization due to heat, pressure, or chemically active fluids

The three major rock types are sedimentary, igneous, and metamorphic rocks. Their classification is based on their origins.

Sedimentary rocks are formed from particles derived from igneous, metamorphic or other sedimentary rocks by weathering and erosion. Sedimentary rocks provide the hydrocarbon source rocks and most of the oil and gas reservoir rocks. Igneous rocks are formed from molten material which is either ejected from the earth during volcanic activity (e.g., lava flows, and ash falls), or which crystallizes from a magma that is injected into existing rock and cools slowly, giving rise rocks such as granites. Igneous rocks are of minor importance for oil exploration. Rarely, hydrocarbon is produced from fractured igneous rocks. Metamorphic rocks are formed by subjecting any of the three rock types to high temperatures and pressures, that alter the character of the existing rock. Common examples of metamorphic rocks are marble derived from limestone and slate derived from shale. Due to the high temperature and pressures there is very little organic matter or hydrocarbons in metamorphic rocks.

• Clastics •Carbonates •Evaporites

Grain-Size Classification for Clastic Sediments Name

Boulder Cobble Pebble Granule Very Coarse Sand Coarse Sand Medium Sand Fine Sand Very Fine Sand Coarse Silt Medium Silt Fine Silt Very Fine Silt Clay

Millimeters

Micrometers

4,096

256 64 4 2 1 0.5 0.25 0.125 0.062 0.031 0.016 0.008 0.004

500 250 125 62 31 16 8 4 (modified from Blatt, 1982)

Average Detrital Mineral Composition of Shale and Sandstone Mineral Composition

Shale

Sandstone

Clay Minerals

60 (%)

Quartz

30

65

4

10-15

<5

15

3

<1

<3

<1

Feldspar Rock Fragments Carbonate Organic Matter, Hematite, and Other Minerals

5 (%)

(modified from Blatt, 1982)

MAJOR COMPONENTS OF SANDSTONE

Matrix

Sand (and Silt) Size Detrital Grains Silt and Clay Size Detrital Material Cement

Material Precipitated Post-Depositionally, During Burial. Cements Fill Pores and Replace Framework Grains Pores

Voids Among the Above Components

COMPONENTS OF SANDSTONE “matrix” CONSISTS OF QUARTZ, FELDSPAR, CEMENT AND CLAY

PORE (QUARTZ)

CEMENT

(CLAY)

(FELDSPAR)

0.25 mm

PRIMARY (ORIGINAL) POROSITY

•Developed at deposition •Typified by: Intergranular pores of clastics or carbonates, AND Intercrystalline and fenestral pores of carbonates •Usually more uniform than induced porosity SECONDARY (INDUCED) POROSITY

 

Developed by geologic processes after deposition (diagenetic processes) Examples: Grain dissolution in sandstones or carbonates, Vugs and solution cavities in carbonates, Fracture development in some sandstones, shales, and carbonates

PRIMARY

 Particle sphericity and angularity  Packing

 Sorting (variable grain sizes)

SECONDARY (DIAGENETIC)

 Cementing materials

 Overburden stress (compaction)  Vugs, dissolution, and fractures

 In the geology section, we show core photographs with examples of porosity.

For now, it is useful to note these effects:

 Porosity increases as angularity of particles increases.

 Porosity increases as the range of particle size decreases. In contrast, porosity

decreases as the volume of interstitial and cementing material increases.

 Porosity decreases as the compaction increases (greater depth generally

means higher overburden stresses, higher compaction forces, and lower porosity)

 Vugs and fractures will contribute to porosity, but to understand their affect

on effective porosity requires careful study of cores and special logging measurements.

Porosity

ROUNDNESS AND SPHERICITY OF CLASTIC GRAINS

High

Low

Very Angular Angular

SubSubWellRounded Rounded Angular Rounded

ROUNDNESS

Porosity

GRAIN PACKING IN SANDSTONE Line of Traverse (using microscope)

4 Types of Grain Contacts Packing Proximity

Sutured Contact

A measure of the extent to which sedimentary particles are in contact with their neighbors

Long Contact

Packing Density

Tangential Contact

Cement

Matrix (clays, etc.)

A measure of the extent to which sedimentary particles occupy the rock volume Concavo-Convex Contact

This Example Packing Proximity = 40% Packing Density = 0.8 (modified from Blatt, 1982)

CUBIC PACKING OF SPHERES Porosity = 48%

 Bulk volume = (2r)3 = 8r3

4  r3 3 volume  Pore volume = bulk volume - matrix  Matrix volume =

Pore Volume Porosity  Bulk Volume Bulk Volume  Matrix Volume  Bulk Volume 8 r3  4 / 3  r3    1  47.6% 3 2 3  8r

RHOMBIC PACKING OF SPHERES Porosity = 27 %

Sorting (variable grain sizes)

Packing of Two Sizes of Spheres Porosity = 14%

Grain-Size Sorting in Sandstone

Very Well Sorted

Well Sorted

Moderately Sorted SORTING

Poorly Sorted

Very Poorly Sorted

TYPES OF TEXTURAL CHANGES SENSED BY THE NAKED EYE AS BEDDING Sand Shale

Slow Current

Fast Current

Change of Composition

Change of Size River

Eolian Beach Fluvial

Change of Shape

Change of Orientation

Change of Packing

PROGRESSIVE DESTRUCTION OF BEDDING THROUGH BIOTURBATION Regular Layers

Mottles (Distinct)

Irregular Layers

Mottles (Indistinct)

Homogeneous Deposits

Bioturbated Sandstone (Whole Core)

FACTORS THAT AFFECT SECONDARY (DIAGENETIC) POROSITY  Cementing materials

 Overburden stress (compaction)  Vugs, dissolution, and fractures

 Caused when a rigid rock is strained beyond its elastic limit-it

cracks

 The forces cauing it to break are in a constant direction. Hence

all the fractures are also aligned.

 Important source of permeability in low porosity carbonate

reservoirs.

 Classed either being vertical or horizontal. But can also appear

in any angle. Can penetrate from oil column down into the water, as it have very high permeability, can cause production problems.

 Defined as non-connected pore space  Do not contribute to the producible fluid total.  Caused by dissolution of soluble material such as shell

fragments after rock has been formed

 Have irregular shapes.  Not connected to each other in any producible manner and

hence do not contribute to formation productivity.

 Fractures

 Vugs

DIAGENESIS Diagenesis is the PostDepositional Chemical and Mechanical Changes that

Carbonate Cemented

Occur in Sedimentary Rocks Some Diagenetic Effects Include

Oil Stained

Compaction Precipitation of Cement Dissolution of Framework Grains and Cement The Effects of Diagenesis May Enhance or Degrade Reservoir Quality

Whole Core Misoa Formation, Venezuela

Photo by W. Ayers

DUAL POROSITY IN SANDSTONE 1. 2.

Primary and secondary “matrix” porosity system Fracture porosity system

FRACTURE

DISSOLUTION PORE

PORE CEMENT

(QUARTZ) (CLAY)

(FELDSPAR)

0.25 mm

POROSITY IN SANDSTONE Porosity in Sandstone Typically is Lower Than That of Idealized Packed Spheres Owing to:

Pore Quartz Grain

Variation in Grain Size Variation in Grain Shape Cementation Mechanical and Chemical Compaction

Scanning Electron Micrograph Norphlet Sandstone, Offshore Alabama, USA Photomicrograph by R.L. Kugler

POROSITY IN SANDSTONE Pore Throat

Pores Provide the Volume to Store Hydrocarbons Pore Throats Restrict Flow through pores

Scanning Electron Micrograph Norphlet Formation, Offshore Alabama, USA

INTERGRANULAR PORE AND MICROPOROSITY

Intergranular Pore

Microporosity

Kaolinite

Quartz Detrital Grain

Backscattered Electron Micrograph Carter Sandstone, Black Warrior Basin, Alabama, USA

Intergranular Pores Contain Hydrocarbon Fluids Micropores Contain Irreducible Water

(Photograph by R.L. Kugler)

Clay Minerals in Sandstone Reservoirs, Authigenic Kaolinite Secondary Electron Micrograph

Significant Permeability Reduction High Irreducible Water Saturation

Migration of Fines Problem

Carter Sandstone North Blowhorn Creek Oil Unit Black Warrior Basin, Alabama, USA

(Photograph by R.L. Kugler)

DISSOLUTION POROSITY Partially Dissolved Feldspar Pore Quartz Detrital Grain Thin Section Micrograph - Plane Polarized Light Avile Sandstone, Neuquen Basin, Argentina

Dissolution of Framework Grains (Feldspar, for Example) and Cement may Enhance the Interconnected Pore System This is Secondary Porosity Photo by R.L. Kugler

DISSOLUTION POROSITY

Partially Dissolved Feldspar Scanning Electron Micrograph Tordillo Formation, Neuquen Basin, Argentina

Dissolution Pores May be Isolated and not Contribute to the Effective Pore System

Photo by R.L. Kugler

CARBONATES POROSITY TYPES Interparticle

Pores Between Particles or Grains

Intraparticle

Pores Within Individual Particles or Grains

Intercrystal

Pores Between Crystals

Moldic

Pores Formed by Dissolution of an Individual Grain or Crystal in the Rock

Fenestral

Primary Pores Larger Than Grain-Supported Interstices

Fracture

Formed by a Planar Break in the Rock

Vug

Large Pores Formed by Indiscriminate Dissolution of Cements and Grains

Idealized Carbonate Porosity Types

Interparticle

Intraparticle

Intercrystal

Moldic

Fabric

Selective Fenestral

Shelter

Growth-Framework

Non-Fabric Selective Fracture

Channel

Vug

Breccia

Boring

Burrow

Fabric Selective or Not Fabric Selective (modified from Choquette and Pray, 1970)

Shrinkage

CARBONATE POROSITY - EXAMPLE Moldic Pores Dolomite Moldic Pore

• Due to dissolution and collapse of ooids (allochemical particles) • Isolated pores

• Low effective porosity Calcite Thin section micrograph - plane-polarized light Smackover Formation, Alabama

• Low permeability Blue areas are pores. (Photograph by D.C. Kopaska-Merkel)

CARBONATE POROSITY - EXAMPLE Moldic and Interparticle Pores Interparticle Pores

• Combination pore system

• Moldic pores formed through dissolution of ooids (allochemical particles) • Connected pores Moldic Pore

• High effective porosity • High permeability Thin section micrograph Smackover Formation, Alabama Black areas are pores. (Photograph by D.C. Kopaska-Merkel)

 Some void spaces become isolated due to excessive

cementation thus many void spaces are interconnected cementation, thus many void spaces are interconnected and others are isolated.

 This leads to the following classification: 1)

Absolute (total) porosity

2)

Effective porosity

 Absolute porosity is the ratio between the total pore volume

(interconnected pores and isolated ones) and the bulk volume:

 Effective porosity is the ratio between the interconnected pore

volume and the bulk volume:

 Effective porosity indicates the percentage of the total volume

of reservoir rock where the void space is connected by flow channels.

 If the porosity of a rock sample was determined by saturating

the rock sample 100% with a fluid of known density and then determining, by weighing, the increased weight due to the saturating fluid, what would this yield?

 If the porosity of a rock sample was determined by saturating

the rock sample 100% with a fluid of known density and then determining, by weighing, the increased weight due to the saturating fluid, what would this yield?

Effective porosity measurement

 If the rock sample were crushed with a mortar and pestle to

determine the actual volume of the solids in the core sample, what would this yield?

 If the rock sample were crushed with a mortar and pestle to

determine the actual volume of the solids in the core sample, what would this yield?

Absolute porosity measurement

 Isolated Void Space  This sandstone would not be an acceptable reservoir rock,

regardless of the value of its porosity and the hydrocarbon saturations, because each void is isolated from the other void spaces.

 This sandstone has

a high absolute porosity but a zero effective porosity

 Interconnected Void Space  This sandstone would be an acceptable

reservoir rock because of the interconnected pore spaces and hydrocarbon saturation.

 This sandstone has a high

absolute porosity and a high effective porosity



Total porosity, t =



Effective porosity, e =

Total Pore Volume Bulk Volume

Interconnected Pore Space Bulk Volume •

Effective porosity – of great importance; contains the mobile fluid

 Very clean sandstones : e  t  Poorly to moderately well -cemented

intergranular materials: t  e

 Highly cemented materials and most

carbonates: e < t

1.

Core samples (measure two of: Vb, Vp, or Vm) - RCA -SCAL

2. Openhole wireline logs

MATRIX DENSITIES (ΡM) OF TYPICAL PURE COMPONENTS OF RESERVOIR ROCK

APPLICABILITY AND ACCURACY OF MATRIX MEASUREMENT TECHNIQUES  Known or assumed matrix density  Accurate only if matrix density is known and not assumed  Core samples are often mixtures of several components with varying

matrix densities, so density must be measured

LABORATORY METHODS OF POROSITY DETERMINATION To determine porosity, measure 2 of 3 basic parameters: 1. Bulk volume

(Vb)

2. Matrix volume (Vm) • Assumed matrix (grain) density • Displacement method • Boyles Law

3. Pore volume

(Vp)

MATRIX VOLUME FROM DISPLACEMENT METHOD  Reduce sample to particle size  Measure matrix volume of particles by  Volumetric method  Archimedes method (gravimetric measurement)

CALCULATE THE POROSITY OF A CORE SAMPLE USING THE DISPLACEMENT METHOD AND MATRIX VOLUME The core sample from previous example was stripped of the paraffin coat, crushed to grain size, and immersed in a container with liquid. The volume of liquid displaced by the grains was 7.7 cm3. Calculate the matrix volume and the core porosity. Is this effective porosity or total porosity? (It is total porosity)

Bulk Volume, Vb = 9.9 cm3 Matrix Volume, Vma = 7.7 cm3

Vb  Vma = 9.9 cm3 – 7.7 cm3 = 0.22 or 22% Porosity     Vb Vb 9.9 cm3 Vp

LABORATORY METHODS OF POROSITY DETERMINATION To determine porosity, measure 2 of 3 basic parameters: 1. Bulk volume

(Vb)

2. Matrix volume (Vm) • Assumed matrix (grain) density • Displacement method • Boyles Law (Gas Expansion)

3. Pore volume

(Vp)

MATRIX VOLUME FROM GAS EXPANSION METHOD  Involves compression of gas into pores  Uses Boyle’s law

p1 V1  p2 V2

GAS EXPANSION METHOD TO CALCULATE VMA  Initial conditions, with volumes of 2 cells known

 Place core in second cell, evacuate gas (air) from second cell  Open valve

GAS EXPANSION METHOD TO CALCULATE VMA P1

Initial conditions

Core

V1

Cell 1

Valve closed

Evacuate Cell 2

GAS EXPANSION METHOD TO CALCULATE VMA P1

Final conditions

P2

Core

Cell 1

Valve open

Cell 2

GAS EXPANSION METHOD TO CALCULATE VMA  Vf = Volume of Cell 1 + Volume of

Cell 2 - Matrix Volume of Core

 Vt = Volume of Cell 1 + Volume of

Cell 2

 Vm =

Vt - Vf

APPLICABILITY AND ACCURACY OF MATRIX MEASUREMENT TECHNIQUES  Displacement method - Very accurate when

core sample is crushed without destroying individual matrix grains  Gas expansion method - Very accurate,

especially for samples with low porosities Neither method requires a prior knowledge of core properties

LABORATORY METHODS OF POROSITY DETERMINATION To determine porosity, measure 2 of 3 basic parameters: 1. Bulk volume (Vb) 2. Matrix volume (Vm) 3. Pore volume (Vp)

LABORATORY METHODS OF POROSITY DETERMINATION

Pore volume determination (Effective) 1.

Gravimetric (Archimedes) Vp =

2. •

Wsat - Wdry fluid

Boyle’s Law: (Gas expansion)

p1 V1  p2 V2

PORE VOLUME FROM SATURATION METHOD  Measures the difference between the weight of a core sample

saturated with a single fluid and the dry weight of the core  Pore volume,

Vp 

Wsat  Wdry

f

ARCHIMEDES METHOD OF CALCULATING POROSITY A CORE SAMPLE Using the gravimetric method with the following data, calculate the pore and bulk volumes and the porosity. Is this porosity total or effective? Dry weight of sample, Wdry = 427.3 g Weight of sample saturated with water, Wsat = 448.6 g Density of water (f ) = 1.0 g/cm3 Weight of saturated sample submerged in water, Wsub = 269.6 g

ARCHIMEDES METHOD OF CALCULATING POROSITY A CORE SAMPLE

Wsat – Wdry Vp = f

448.6 – 427.3 g 1.0 g/cm3

=

Wsat – Wsub Vb = f

Porosity   

=

Vp Vb

448.6 – 269.6 g 1.0 g/cm3

=

21.3 cm3 179.0

cm3

= 21.3 cm3

= 179.0 cm3

=

0.12 or 12%

APPLICABILITY AND ACCURACY OF PORE VOLUME MEASUREMENT TECHNIQUES Saturation (Archimedes) method  Accurate in better quality rocks if effective pore

spaces can be completely saturated  In poorer quality rocks, difficult to completely saturate sample  Saturating fluid may react with minerals in the core (e.g., swelling clays)

LABORATORY METHODS OF POROSITY DETERMINATION

Pore volume determination (Effective) 1.

2. •

Gravimetric (Archimedes) Wsat - Wdry Vp = fluid Boyle’s Law: (Gas expansion)

p1 V1  p2 V2

PORE VOLUME FROM GAS EXPANSION METHOD Initial conditions P1

Core

V1

Cell 1

Valve closed

Cell 2

PORE VOLUME FROM GAS EXPANSION METHOD Final conditions P1 P2

Core

Cell 1

Valve open

Cell 2

SUMMARY To determine porosity, measure 2 of 3 basic parameters: 1. Bulk volume 2. Matrix volume 3. Pore volume

 use in determining the original hydrocarbon volume in place.

 In case of large variation in the porosity vertically and no or small

variation horizontally or parallel to the planes, then the arithmetic average or thickness-weighted average porosity is used:

 Due to the change in sedimentation or depositional conditions Due

to the change in sedimentation or depositional conditions can cause porosity in one portion of the reservoir to be greatly different from that in another area, so the areal-weighted average or the volume-weight average can be used:

CORES  Allow direct measurement of reservoir

properties  Used to correlate indirect measurements, such

as wireline/LWD logs  Used to test compatibility of injection fluids  Used to predict borehole stability  Used to estimate probability of formation

failure and sand production

SOME KEY FORMULAS

Vb  Vma Porosity  φ   Vb Vb Vp

V V V V   (V ) V  (1   )(V ) m  (  )(V ) b

p

m

m

p

b

b

 Definition: Fraction, or percent, of the pore volume occupied by

a particular fluid (oil, gas, or water).

 Is an intensive property

 all saturation values are based on

gross reservoir volume.

pore volume and not on the

 By definition, the sum of the saturations is 100%,

 Fundamental relationships - Pore volume is occupied by fluids ( water, oil, and/or gas)

- For two phase case, only one of the two saturations is

independent, the other must make the sum of saturations equal to unity (1)

- For three phase case, only two saturations are independent.

 Fundamental relationships - Mass of fluids in the pore volume is comprised of water, oil

and/or gas.

- At laboratory condition it is often assumed that gas density is

negligible

 Concepts: Typical petroleum accumulation scenario

- Pores initially saturated with water ( Sw=1) - Hydrocarbons migrate up dip into traps due to lower density

than water (gravity force)

- Hydrocarbons (oil and/or gas) distributed such that gravity

and capillary forces are in equilibrium

- minimum interstitial water saturation remains in HC zone, even after accumulation occurs a)

Irreducible wetting phase saturation – water wet, drainage accumulation process

b)

Residual non-wetting phase saturation – oil wet, imbibition accumulation process

 Methods for determination of reservoir fluid saturations.

- Direct Measurement  Core Analysis of samples obtained from formation of interest

in original state and measure saturations directly is ideal

- Indirect Measurement  Capillary Pressure Measurement  Well log analysis ( electrical conductivity primarily on water

saturation)

 Factors affecting fluid saturations in cores:

- Flushing of core by filtrate from drilling fluids (especially for

overbalanced drilling)

-water filtrate – water based mud - oil emulsion mud - oil filtrate

- oil based mud - inverted oil emulsion mud

- gas filtrate

- air drilling - foam drilling

 Factors affecting fluid saturations in cores:

- Changes in pressure and temperature as core sample is

brought from bottomhole conditions to surface conditions - Example: Oil zone at minimum interstitial water saturation, water based drilling mud

 Application of core saturations: Water Based Mud - Presence of oil zone - Original oil/gas contact - Original oil/water contact

 Application of core saturations: Oil based mud - Fairly accurate minimum interstitial water saturation

- Original oil/ water contact

 Other Application of Core Saturations

- Correlation of indirect methods

 Connate (interstitial) water saturation, Swc

- reduces the amount of space available between oil and gas - Not uniformly distributed throughout reservoir, varies with

permeability, lithology and height above free water level

- Define the maximum water saturation at which the water phase will

remain immobile.

 Critical oil saturation, Soc

- For the oil phase to flow, the saturation of the oil must exceed a certain value - the oil remains in the pores and, for all practical purposes, will not

flow.

 Residual oil saturation, Sor - During the displacing process of the crude oil system from the porous

media by water or gas injection (or encroachment), there will be some remaining oil left that is quantitatively characterized by a saturation value that is larger than the critical oil saturation

- Associated with the non wetting phase when it is being displaced by a wetting phase.  Movable oil saturation, Som

- fraction of pore volume occupied by movable oil

 Critical gas saturation, Sgc - As the reservoir pressure declines below the bubble-point pressure, gas

evolves from the oil phase and consequently the saturation of the gas increases as the reservoir pressure declines

 gas phase remains immobile until its saturation exceeds a certain

saturation which gas begins to move

 Average saturation of each reservoir fluid is calculated from

the following equations:

 Factors affecting overburden stress:

a) Depth b) Nature of the structure c) Consolidation of formation d) Geologic age e) History of the rock  The weight of the overburden simply applies a

compressive force to the reservoir

 The pressure in the rock pore spaces does not

normally approach the overburden pressure.

 Definition: The pressure difference between overburden and internal pore pressure  During pressure depletion operations, the internal

pore pressure decreases and, therefore, the effective overburden pressure increases

 The increase of effective overburden pressure causes: a) The bulk volume of the reservoir rock is reduced. b) Sand grains within the pore spaces expand.

 Rock-matrix compressibility, cr

- defined as the fractional change in volume of the solid rock material (grains) with a unit change in pressure.

 Rock-bulk compressibility, cB

- Defined as the fractional change in volume of the bulk volume of the rock with a unit change in pressure.

 Pore compressibility, cp

- defined as the fractional change in pore volume of the rock with a unit change in pressure

 In terms of porosity

 The

formation compressibility cf is the term

commonly used to describe the total compressibility of the formation and is set equal to cp

 Formation compressibility plays important role in

understanding reservoir performance.

 As reservoir pressure decreases, external stresses (overburden

stress) tend to compact the rock and reduce pore volume. This results in reduction in porosity which helps expel more fluid out of res rock. It also can close out or reduce size of pores and pore throats resulting in reduction in permeability. The significance of this effect depends on value of formation compressibility.

 Formation compressibility range from 3 × 10−6 to 25 × 10−6

psi−1

 Can also be written as:

 Geertsma (1957) suggested that the bulk

compressibility cB is related to the pore compressibility cp as expressed:

 Geertsma has stated that in a reservoir only

the vertical component of hydraulic stress is constant and that the stress components in the horizontal plane are characterized by the boundary condition that there is no bulk deformation in those directions.

 For those boundary conditions, he developed

the following approximation for sandstones:

 The reduction in the pore volume due to

pressure decline can also be expressed in terms of the changes in the reservoir porosity

 By integrating:

 Will obtain:

 Note that:

 Using the expansion series and truncating

the series after two terms:

 Total reservoir

compressibility ct in

transient flow equation and MBE:

 For undersaturated oil reservoirs, the

reservoir pressure is above the bubble- point pressure, i.e., no initial gas cap:

 Hall (1953) correlated the pore

compressibility with porosity

 Hall (1953) correlated the pore

compressibility with porosity

 Newman (1973) to develop a correlation

between the formation compressibility and porosity.

Formation compressibility vs Initial Porosity

 Yale et. al provided correlation to calculate

formation compressibility for various type of sands.

 Provide correlation for formation

compressibility which is function of pressure.

 As reservoir pressure decrease, formation

compressibility decreases.

 Unconsolidated sands, if formation pressure

reduced, grain particles due to overburden pressure will be rearranged permanently. If pressure of formation increased, formation will not necessarily go back to its original configurations.

 Overburden stress can be approximated by 1.0 psi/ft or integrating

density log

 Definition a property of the porous medium and is a measure of the capacity of the medium to transmit fluids  OR a measure of the fluid conductivity of the

particular material

 Permeability is an INTENSIVE property of a

porous medium (e.g. reservoir rock)  The rock permeability, k, is a very important rock property because it controls the directional movement and the flow rate of the reservoir fluids in the formation.

 Definition a property of the porous medium and is a measure of the capacity of the medium to transmit fluids  OR a measure of the fluid conductivity of the

particular material

 Permeability is an INTENSIVE property of a

porous medium (e.g. reservoir rock)  The rock permeability, k, is a very important rock property because it controls the directional movement and the flow rate of the reservoir fluids in the formation.

The quality of the reservoir, as it relates to permeability can be classified as follows k < 1 md 1 < k < 10 md 10 < k < 50 md 50 < k < 250 md 250 md < k

poor fair moderate good very good

This scale changes with time, for example 30 years ago k< 50 was considered poor.

Core analysis Well test analysis (flow testing) - RFT (repeat formation tester) provides small well tests Production data -production logging measures fluid flow into well Log data -MRI (magnetic resonance imaging) logs calibrated via core analysis

Examples, Typical PermeabilityPorosity Relationship

From Tiab and Donaldson, 1996

A

h1-h2

q

A

(Sand Pack Length) L

•Flow is Steady State •q = KA (h1-h2)/L •K is a constant of proportionality

•h1>h2 for downward flow

q

h1 h2

A

h1-h2

q

A

(Sand Pack Length) L

•Flow is Steady State •q = KA (h1-h2)/L •K is a constant of proportionality

•h1>h2 for downward flow

q

h1 h2

Independent of direction of flow

 Water velocity is proportional to the manometric level, the flow

velocity is proportional to:

where Δz is the elevation in the gravitational field. (Δz accounts for the inclined flow direction relative to horizontal flow.)  If sand filter is made longer, a reduced flow velocity is expected

and similarly if the water is replaced by a fluid of higher viscosity, a reduced flow velocity is expected.

 It is an intrinsic property of a reservoir rock

that indicates the flow capacity of the reservoir.  Reservoir engineers use permeability, reservoir pressure, and a few other parameters to estimate oil and gas productivity.  Petrophysicists use core permeability values to help calibrate permeability derived from well log data.

 Darcy developed a fluid flow equation that

has since become one of the standard mathematical tools of the petroleum engineer.

 If a horizontal linear flow of an

incompressible fluid is established through a core sample of length L and a cross-section of area A, then the governing fluid flow equation is defined as

 Apparent velocity determined by dividing the flow

rate by the cross-sectional area across which fluid is flowing.

q  A  Substituting the relationship, q/A, in place of ν and

solving for q results in:

Condition must exist during measurement of permeability: a) Horizontal flow. b) Incompressible fluid. c) 100% fluid saturation in the porous medium. d) Stationary flow current, i.e. constant cross-section in flow direction. e) Laminar flow current (satisfied in most liquid flow cases). f) No chemical exchange or - reactions between fluid and porous medium

Absolute permeability of rock

 Permeability is found by integrating the linear flow equation

where the permeability is experimentally determined using the formula:

 where the flow rate

q and the pressure difference Δp are the

measured data. Permeability is found by plotting the measured data as shown.

 A cylindrical core sample is properly cleaned and all remains

of hydrocarbons are removed from the pore space. The core is saturated with water and then flushed horizontally. The core length is 15cm, it’s diameter is 5 cm and the water viscosity is 1.0 cp.

 The average or representative permeability is

mD

k = 0.1 D or 100

1.

Measure inlet and outlet pressures (P1 and P2) at several different flow rates

2.

Graph ratio of flow rate to area (q/A) versus the pressure function (P1 - P2) / L

3.

For laminar flow, data follow a straight line with slope of k/μ

4.

At very high flow rates, turbulent flow is indicated by a deviation from straight line

 By convention the unit for the permeability is called the

Darcy.

‘The permeability is 1 Darcy if a fluid with viscosity of 1cp is flowing at a rate of 1 cm3/s through a porous medium with a cross-section of 1cm2, creating a pressure difference of 1 atm/cm.”  It follows from these evaluations that,

 Instead of the unit 1 Darcy, the 1/1000 fraction is used, which then is

called millidarcy (mD).

 Darcy’s “K” was determined to be a

combination of

k, permeability of the sand pack (porous medium, e.g. reservoir rock)  , viscosity of the liquid 

 Permeability is a derived dimension  From

Darcy’s equation, the permeability is length squared

dimension

of

qμ L  L3 P  T L 1 1  k ;     2    L2 A Δp  T 1 1 L P 

 

 This is not the same as area, even though for

example, it is m2 in SI units

 In Darcy and SI Units, this equation is coherent  Oilfield units are non-coherent, a unit conversion constant is required

 Permeability is a derived dimension based on

Darcy’s Equation

k = (q  L) / (A Dp)  The unit of permeability is the Darcy [d]  The oilfield unit is millidarcy [md]  The Darcy is defined from Darcy’s Equation, where: q [cm3/s]  [cp] L [cm] A [cm2] Dp [atm]

 Dry gas is usually used (air, N2, He) in permeability

determination because of its convenience, availability, and to minimize fluid-rock reaction.

 In dry gas, the gas volumetric flow rate q varies

with pressure because the gas is a highly compressible fluid.

 Measurement of the permeability should be

restricted to the low (laminar/viscous) flow rate region, where the pressure remains proportional to flow rate within the experimental error

 For high flow rates, Darcy’s equation is

inappropriate to describe the relationship of flow rate and pressure drop.

 Considering mass flow of gas qρ,

where ρ is the density of the gas at certain pressure  It follows from the perfect gas law (pV =

nRT) that

 which when substituted into the Darcy equation equation

yields:

 Taking into account the invariant quantity,

q

 Finally obtains,

where p = (p1 + p2)/2 is a mean (average) pressure in the core during the measurements.

 Combining the invariant mass flow; qr = qr

and the results generated from the perfect gas law; r p =r p

where q is the mean or average flow rate.

A gas permeability test has been carried out on a core sample, 1in in diameter and length. The core has been cleaned and dried and mounted in a Hassler core holder

 The gas is injected and the pressure, p1 measured, at one end of the

core sample, while the gas rate, q2 is measured at the other end, at atmospheric pressure, i.e., p2 = 1atm.

 The gas permeability could be estimated

 Given the pressure p1 and the gas rate q2, the mean pressure in the

core sample, p and the pressure drop across the core, Dp, are calculated from the equation above. The gas permeability k is found as a function of the mean core pressure.

Absolute gas permeability of the core sample is therefore found as the asymptotic value of permeability, when p ! ¥ or more conveniently, when 1/p ! 00 Taken from the table is plotted and the absolute permeability is found kliquid = 3.0mD.

1. Cut core plugs from whole core or use sample from whole core 2. Clean core and extract reservoir fluids, then dry the core 3. Flow a fluid through core at several flow rates 4. Record inlet and outlet pressures for each

 Gas volumetric flow rate q varies with pressure because the

gas is a highly compressible fluid.

 Assuming the used gases follow the ideal gas behavior (at low

pressures),

 Mean pressure, Pm expressed as:

 Gas flow rate is usually measured at base

(atmospheric) pressure Pb

 Subsituting Darcy’s Law into above expression :

 Equation summarised:

 Routine core analysis is generally concerned

with plug samples drilled parallel to bedding planes and, hence, parallel to direction of flow in the reservoir. These yield horizontal permeabilities (kh).

 Measured permeability on plugs that are

drilled perpendicular to bedding planes are referred to as vertical permeability (kv).

 Core sample may not be representative of

the reservoir rock because of reservoir heterogeneity.

 Core recovery may be incomplete.  Permeability of the core may be altered

when it is cut, or when it is cleaned and dried in preparation for analysis. This problem is likely to occur when the rock contains reactive clays.

 Sampling process may be biased. There is a

temptation to select the best parts of the core for analysis.

a)

Overburden pressure

- experiments have shown that the permeability is even more

dependent on the overburden pressure than the porosity

b) Interaction between the fluid and the porous medium - To avoid this effect, gases (helium, nitrogen, carbon-dioxide

and air) are often used for permeability measurements.

- The use of gases introduce other problems, such as turbulent

flow behaviour, increased uncertainty in gas rate measurements and at low pressure, the Klinkenberg effect.

 Klinkenberg (1941) discovered that

permeability measurements made with air as the flowing fluid showed different results from permeability measurements made with a liquid as the flowing fluid

 Klinkenberg postulated,that liquids had a

zero velocity at the sand grain surface, while gases exhibited some finite velocity at the sand grain surface.

 One of the conditions for the validity of

Darcy’s law the requirement of laminar flow, i.e. that the fluid behaves "classically" with respect to intermolecular interactions in the gas.

 At low gas pressure, in combination with

small (diameter) pore channels, this condition is broken.

 At low p, gas molecules are often so far apart, that

they slip through the pore channels almost without interactions (no friction loss) and hence, yield a increased flow velocity or flow rate.

 At higher pressures, the gas molecules are closer

together and interact more strongly as molecules in a liquid.

 Compared to laminar flow, at a constant pressure

difference, the Klinkenberg dominated flow will yield a higher gas rate than laminar flow,

 The gases exhibited

slippage at the sand

grain surface. This slippage resulted in a higher flow rate for the gas at a given pressure differential.

 For a given porous medium as the mean

pressure increased the calculated permeability decreased.

 Klinkenberg developed a method to correct gas

permeability measured at low mean flowing pressure to equivalent liquid permeability.

 A plot of measured permeability versus 1/Pm is

extrapolated to the point where 1/Pm = 0 (Pm = infinity). This permeability approximates the liquid permeability.

 Magnitude of the Klinkenberg effect varies with the core

permeability and the type of the gas used in the experiment

 Slope Ac ≅ is a function of the following factors:

• Absolute permeability k, i.e., permeability of medium to a single phase completely filling the pores of the medium kL.  Type of the gas used in measuring the permeability, e.g., air.  Average radius of the rock capillaries.

Klinkenberg expressed the slope c by the following relationship:

 Magnitude of the Klinkenberg effect varies with the core

permeability and the type of the gas used in the experiment

 In many cases, the reservoir contains

distinct layers, blocks, or concentric rings of varying permeabilities.

 Because smaller-scale heterogeneities

always exist, core permeabilities must be averaged to represent the flow characteristics of the entire reservoir or individual reservoir layers (units).

 In many cases, the reservoir contains

distinct layers, blocks, or concentric rings of varying permeabilities.

 Because smaller-scale heterogeneities

always exist, core permeabilities must be averaged to represent the flow characteristics of the entire reservoir or individual reservoir layers (units).

 Three simple permeability-averaging

techniques:

a) Weighted-average permeability b)

Harmonic-average permeability

c) Geometric-average permeability

a) Weighted-average permeability  to determine the average permeability of

layered-parallel beds with different permeabilities

 Total flow rate from the entire system:

 Total flow rate qt is equal to the sum of the flow rates through

each layer or:

 Combining the above expressions gives:

 average absolute permeability for a parallel-layered system:

 If similar layered system with variable layers width:

b) Harmonic-Average Permeability  Permeability variations can occur laterally

in a reservoir as well as in the vicinity of a well bore

 For a steady-state flow, the flow rate is

constant and the total pressure drop Δp is equal to the sum of the pressure drops across each bed, or Δp = Δp1 + Δp2 + Δp3

 Substituting for the pressure drop by

applying Darcy’s equation

 Canceling the identical terms and simplifying gives:

 Expressed in a more generalized form to give

 In the radial system, the above averaging methodology can be

applied

 In the radial system, the above averaging methodology can be

applied

c) Geometric-Average Permeability  Warren and Price (1961) illustrated

experimentally that the most probable behavior of a heterogeneous formation approaches that of a uniform system having a permeability that is equal to the geometric average.

 If the thicknesses (hi) of all core samples are the same:

 It is possible to correlate connate water

content with the permeability of the sample in a given reservoir and to a certain extent between reservoirs.

 Calhoun (1976) suggested that in an ideal

pore configuration of uniform structure, the irreducible connate water would be independent of permeability, lower permeabilities being obtained merely by a scaled reduction in particle size

 Experience indicates a general relationship

between reservoir porosity (φ) and irreducible water saturation (Swc) provided the rock type and/or the grain size does not vary over the zone of interest. This relationship is defined by the equation

 where C is a constant for a particular rock

type and/or grain size.

 Several investigators suggest that the

constant C that describes the rock type can be correlated with the absolute permeability of the rock

a) The Timur Equation  - Timur (1968)

proposed the following

expression for estimating the permeability from connate water saturation and porosity:

b) The Morris-Biggs Equation  Morris and Biggs (1967) presented the

following two expressions for Estimating the permeability if oil and gas reservoirs:

For an oil reservoir:

For a gas reservoir

 In a gas-oil systems, the direction of displacement is

particularly important, as the process can represent a drainage process, such as gas drive (gas displacing oil immiscibily) or an imbibition process, such as:

1. Movement of an oil zone into receding depleting gas cap. 2. Movement of an aquifer into receding depleting gas cap.  The relation is based on a definition of liquid saturation:

 Effective permeability is the permeability of

a rock to one fluid in a two phase system.

 For example, the effective permeability of oil

in an oil-water system (Ko) will be less than absolute permeability.

 In the same rock and fluid system, the

effective permeability of water (Kw) could be higher or lower than Ko.

Definition: ratio of the effective permeability of a fluid at a given saturation to some base permeability.  Base permeability is typically defined as –absolute

permeability (Ka), –air permeability (Kair), or effective permeability to non-wetting phase at irreducible wetting phase saturation, for example Ko @ Sw = SWir.  Because the definition of base permeability varies, the definition used must always be confirmed before applying relative permeability data •noted along with tables and figures presenting relative permeability data.

 Darcy’s Equation rearranged as Darcy velocity

(volumetric flux)

vs = q/A = (k/) (Dp/L)  This equation applies for any L,  as L0

vs = q/A = -(k/) (dp/ds)

where, vs Darcy velocity, (volumetric flux) s distance along flow path (0s  L), in the direction of decreasing pressure (note negative sign)  The differential form is

Darcy’s Law

Brine of µ = 1 Cp is flowing at a rate of 0.3

bbl/day under a 30 psi pressure differential into a core plug of

0.1 ft long and 0.0215 ft2 cross section. Calculate the absolute penneability in oil field units.

Can you calculate in Darcy units ?

 An extenslon of the Darcy's law can be obtained for an inclined

reservoir) In this case the pressure difference (Pi - P2) is not the only driving force in .the tilted reservoir as gravitational force also contributes. Note that the gravitational force is always vertically downward where as pressure could be at any direction In practice, we introduce a new parameter  called fluid potential, ɸ The fluid potential at any point in the

reservoir is defined as :

 The generalized form of Darcy’s Law includes pressure

and gravity terms to account for horizontal or nonhorizontal flow

qs k  dp g dz  vs     A   ds c ds   The gravity term has dimension of pressure / length

 Flow potential includes both pressure and gravity terms,

simplifying Darcy’s Law

q k  dΦ  vs    A μ  ds 

  = p - gZ/c ; Z+; Z is elevation measured from a datum (+ve

below datum, and –ve above datum)   has dimension of pressure

  = p - gZ/c  Z+  Z is elevation measured from a datum

  has dimension of pressure  Oilfield Units  c = (144 in2/ft2)(32.17 lbmft/lbfs2)

LABORATORY DETERMATION OF PERMEABILITY

Some slides in this section are from NExT PERF Short Course Notes, 1999. Some slides appear to have been obtained from unknown primary sources that were not cited by NExT. Note that some slides have a notes section.

LABORATORY METHODS FOR DETERMINING ABSOLUTE PERMEABILITY 1” or 1 1/2”

Plug Most Common

Full Diameter Heterogeneous

Whole Core

Heterogeneous

Slab Taken for •Photography •Description •Archival

WHOLE-CORE METHOD  Uses selected pieces from the full or whole

core  Core sizes 2 1/2 to 5 1/2 inches in diameter  Several inches to several feet long

 Most applicable approach for very

heterogeneous formations.  Additional expense limits the practical number

of tests.

CORE PLUG METHOD  Most commonly applied method.  Uses small cylindrical core samples

 3/4 inch to 1 1/2 inch diameter  1 to a few inches long  May not apply to heterogeneous formations.

I

Different Lithologies Require Careful Selection of Suitable Core Plugs or Require Whole-Core Analyses

IIa

IIb Unacceptable

kH

~1 ft

?

Suitable

kV

Or FullDiameter

kH

kV

III

IV

V

kV Full Diameter `4” - 9”

kH

Matrix Only Fracture k and ? Whole Core Analysis (2-3 ft)

Whole Core Photograph, Misoa “C” Sandstone, Venezuela

Photo by W. Ayers

LAB PROCEDURE FOR MEASURING PERMEABILITY  Cut core plugs from whole core or use sample

from whole core  Clean core and extract reservoir fluids, then

dry the core  Flow a fluid through core at several flow rates  Record inlet and outlet pressures for each rate

PERM PLUG METHOD LIQUID FLOW  Measure inlet and outlet pressures (p1 and p2)

at several different flow rates

 Graph ratio of flow rate to area (q/A) versus

the pressure function (p1 - p2)/L  For laminar flow, data follow a straight line with

slope of k/  At very high flow rates, turbulent flow is

indicated by a deviation from straight line through origin

LABORATORY DETERMINATION OF ABSOLUTE PERMEABILITY, LIQUID FLOW Darcy Flow

Non-Darcy Flow

q A

Slope =

k 

0 0

(p1 - p2) L

ISSUES AFFECTING LABORATORY MEASUREMENTS OF PERMEABILITY  Core Handling, Cleaning, and Sampling  Fluid-Rock Interactions

 Pressure Changes  Rock Heterogeneities (Fractures)

 Gas Velocity Effects (Klinkenberg)

CORE HANDLING PROCESSES AFFECT PERMEABILITY MEASUREMENTS  Core Handling  Cleaning  Drying (Clay Damage)  Storage (Freezing)  Sampling

FLUID-ROCK INTERACTIONS AFFECT MEASUREMENTS OF PERMEABILITY  Fresh water may cause clay swelling, reducing

permeability  Tests may cause fines migration, plugging

pore throats and reducing permeability  Reservoir or synthetic reservoir fluids are

generally preferred

PRESSURES AFFECT LABORATORY MEASUREMENTS OF PERMEABILITY  Core alterations resulting from loss of

Confining Pressure during core recovery  Core testing may be conducted by applying a

range of net overburden pressures

CORE HETEROGENEITIES AFFECT MEASUREMENTS OF PERMEABILITY  Naturally-fractured reservoirs  Core plugs represent matrix permeability  Total system permeability (matrix + fractures) is higher  Core Mineralogy problems (Salts, Gypsum)

EXAMPLE CORE REPORT

Factors Affecting Permeability Determination Non-Darcy Flow

Some figures in this section are from “Fundamentals of Core Analysis,” Core Laboratories, 1989. Some slides in this section are from NExT PERF Short Course Notes, 1999. Some slides appear to have been obtained from unknown primary sources that were not cited by NExT. Note that some slides have a notes section.

Air Permeability Measurement  Measurement of permeability in the laboratory is most

commonly done with air

 Convenient and inexpensive  Problem: low values of mean flowing pressure  downstream pressure, patm  upstream pressure, just a few psi higher than patm

 At low mean flowing pressure, gas slippage occurs  Diameter of flow path through porous media approaches the “mean free path” of gas molecules  mean free path is a function of molecule size  mean free path is a function of gas density

 Increasing mean flowing pressure results in less slippage  as pmean, we obtain absolute (equivalent liquid) permeability

Non-Darcy Flow - Gas Slippage  Liquid flow and gas flow at high mean flowing pressure is

laminar

 Darcy’s Law is valid  flow velocity at walls is zero

 At low mean flowing pressure gas slippage occurs  Non-Darcy flow is observed  flow at walls is not zero

 Klinkenberg developed a method to correct gas permeability

measured at low mean flowing pressure to equivalent liquid permeability

Non-Darcy Flow - Klinkenberg Effect  As pmean, gas permeability approaches absolute

permeability

Non-Darcy Flow - Klinkenberg Effect  Klinkenberg correction for kair depends on mean flowing

pressure

 correction ratio shown is for pmean = 1 atm

Non-Darcy Flow - Klinkenberg Effect  Klinkenberg correction for kair is more important for low

absolute permeability

NON-DARCY FLOW - HIGH FLOW RATES  In the field, gas wells exhibit non-Darcy flow at high flow

rates

 At high flow velocity, inertial effects and turbulence become

important, and cause non-Darcy flow  inertial effect

Non-Darcy Flow - Turbulence  Recalling Darcy’s equation for gas flow, (zg )=Constant

q g, sc

k A  Tsc     L  T p sc 

 1   2zμ g 

 2  p1  p 22  



)

 For laboratory flow experiments we can assume T=Tsc and z=1

q g, sc

k  μg



 A  p12  p 22   2L  psc 

)

 For Darcy flow, plotting (qg,sc psc)/A vs. (p12-p22)/(2L) results in

straight line.

 line passes through origin [when qg,sc =0, then (p12-p22)=0]  slope = k/ g  behavior departs from straight line under turbulent flow conditions

(high flow velocity)

NON-DARCY FLOW - TURBULENCE

Darcy flow

Non-Darcy flow

q psc A

Slope =

k 

0 0

(p12- p22) 2L

Non-Darcy Flow - Forchheimer Equation  Forchheimer proposed a flow equation to account for the

non-linear effect of turbulence by adding a second order term

 qg   dp μ g  q g      β ρ g   ds k A A

2

 Note that unit corrections factors would be required for non-coherent

unit systems.  As flow rate decreases, we approach Darcy’s Law (2nd order term approaches zero)

Non-Darcy Flow - Forchheimer Plot  Based on Forchheimer’s Equation a plotting

method was developed to determine absolute permeability even with Non-Darcy effects  (1/kgas) vs. qg,sc  kgas determined from Darcy’s Law (incorrectly assuming Darcy flow) and is a function of qg,sc

(1/kgas), (1/md)

 intercept = (1/kabs); absolute permeability

Slope = [(bg,sc)/(gA)] Intercept = [1/kabs] qg,sc

Non-Darcy Flow - Forchheimer Equation  Non-Darcy Coefficient, b, is an empirically determined

function of absolute permeability  For Travis Peak (Texas)

NOB=Net Overburden

Conversion Factors for Oilfield Units

Need for Unit Conversions  Petroleum Engineers must be able to work with various

unit systems

 International scope of industry  Unit systems used varies geographically  Team members may not all be located in same geographical location  Joint ventures between companies

 Particular units may be required at your location  Legislated units for reporting and regulatory compliance  Company protocol

Oilfield Units  Oilfield units are non-coherent  Newton’s 2nd Law (F=ma)  SI: Force (Newton) is a derived unit to make equation coherent  USCS: Mass (slugs) is a derived unit to make equation coherent  AES, Oilfield Units: A unit conversion constant required (F=ma/gc )

 Darcy’s Law  Darcy units: Permeability is a derived unit to make equation coherent  SI: coherent (permeability unit is m2 )  Oilfield Units: A unit conversion constant is required  The constant may include geometry terms (integrated form)

 For gas flow, the constant may include standard temperature and pressure,

even for Darcy and SI units

q  C k dΦ vs   A μ ds

Learning Objectives  Deriving unit conversion constants  Given  A physical relationship expressed as an equation, using coherent units or

with a correct conversion constant supplied  and appropriate unit conversion factors between unit systems  Find

 The required unit conversion constant (including its units) to express the

equation in a different unit system

 Correctly apply Darcy Equations for incompressible fluid

and real gas, using oilfield units

Darcy’s Law - Darcy Units  Linear (1-D) flow of an incompressible fluid

kA Δp) q μL  where,  q cm3/s  k darcies  Acm2  Dp

atm

  cp  L cm

 The Darcy a derived unit of permeability, defined to make this

equation coherent (in Darcy units)

Darcy’s Law - Oilfield Units  Linear (1-D) flow of an incompressible fluid

CkA Δp) q μL  where,  q bbl/D  k millidarcies  Aft2  Dp

psia

  cp

 L ft

 The approach demonstrated will be to convert each term back to

Darcy units, restoring the coherent equation, then collecting the conversion factors to obtain the oilfield unit constant, C

Darcy’s Law - Oilfield Units q [cm3/s] = q [bbl/D] · 5.61458 [ft3/bbl] · (30.48)3 [cm3/ft3] · (1/86400) [D/s] = 1.84013 [(cm3/s)/(bbl/D)] · q [bbl/D] k [d] = k [md] · (1/1000) [d/md] A [cm2] = A [ft2] · (30.48)2 [cm2/ft2] Dp [atm] = Dp [psia] · (1/14.6959) [atm/psia] L [cm] = L [ft] · 30.48 [cm/ft]

k[md]  0.001[d/md ] A[ft 2 ]  929.03[cm 2 /ft 2 ] q[bbl/D]  1.84013[(cm /s)/(bbl/D )]  μ[cp] L[ft]  30.48[cm/ft] 3

 Δp[psia]  0.068046[a tm/psia] )

Darcy’s Law - Oilfield Units  Collecting the constants and canceling

0.001127 k[md] A[ft 2 ] Δp[psia] ) q[bbl/D]  μ[cp] L[ft]  The unit of the constant is defined from the above equation

[bbl/D] [cp]  [ft] C  0.0011271 [md][ft 2 ][psia]

 We were able to cancel leaving the units of C as shown above because,

[cm3 /s]  [cp]  [cm] 1[d]  1 [cm2 ][atm]

Some figures were taken from Amyx, Bass and Whiting, Petroleum Reservoir Engineering (1960).

 If permeability is not a constant function of space

(heterogeneity), we can calculate the average permeability  Common, simple flow cases are considered here  Linear, Parallel (cores, horizontal permeability)  Linear, Serial (cores, vertical permeability)  Radial, Parallel (reservoirs, horizontal layers)  Radial, Serial (reservoir, damage or stimulation)

 Average permeability should represent the correct flow

capacity

 For a specified flow rate, average permeability results in same

pressure drop (and vice versa)

 Review Integral Averages (Self Study, e.g. Average Velocity)

 Review, Darcy’s Law:  horizontal flow (=p)

L

A q

q k  dp  vs     A μ  ds  kA q ds   dp μ

1

L

2

p2

kA q  ds   dp  μ p1 0 kA p1  p2 ) q μL

 We can determine how

pressure varies along the flow path, p(x), by considering an arbitrary point, 0x  L  Integral from 0x

x

kA q  ds   μ 0

p(x)

 dp

p1

qμ x p(x)  p1  kA OR, equivalently L

A

 We q could alsoLintegrate 2

xL

1

p2

kA q  ds   dp  μ p(x) x q μ (L  x) p(x)  p2  kA

 Pressure profile is a

linear function for homogeneous properties

p1

p

 slope depends on flow rate

A q

1

L

2

p(x) p2

0

0

x

x

L

 There are three simple permeability-averaging techniques that

are commonly used to determine an appropriate average permeability to represent

 an equivalent homogeneous system. These are:  • Weighted-average permeability  • Harmonic-average permeability  • Geometric-average permeability

• Permeability varies across several horizontal layers (k1,k2,k3)  Discrete changes in permeability

h  h1  h 2  h 3   h i  Same pressure drop for each layer

p1 - p2  Δp  Δp1  Δp2  Δp3  Total flow rate is summation of flow rate for all layers

q  q1  q 2  q 3   q i  Average permeability results in correct total flow rate

kwh q Δp ; A  w  h μL

• Substituting,

kwh k1 w h1 k2 w h2 k3 w h3 q Δp  Δp  Δp  Δp μL μL μL μL • Rearranging,

k  k

i

 hi

h

• Average permeability reflects flow capacity of all layers

• Permeability varies across several vertical layers (k1,k2,k3)  Discrete changes in permeability

L  L1  L2  L3   Li  Same flow rate passes through each layer

q  q1  q 2  q3  Total pressure drop is summation of pressure drop across layers

p1  p2  Δp1  Δp2  Δp3   Δpi  Average permeability results in correct total pressure drop

qμ L p1 - p2  ; A  wh kwh

• Substituting,

qμ L q μ L1 q μ L2 q μ L3 p1 - p2     k w h k1 w h k 2 w h k 3 w h

L k Li k i

• If k1>k2>k3, then – Linear pressure profile in each layer

p1

k

p

• Rearranging,

p2

0

0

x

L

 Review, Darcy’s Law:  horizontal flow (=p)

q k  dp  vs     A μ  ds  q k dr  dp 2π rh μ rw

q

rw

re

1 2π kh q  dr  r μ re

pw

 dp

pe

2π kh p e  p w ) q μ ln(re /rw )

 We can determine how

pressure varies along the flow path, p(r), by considering an arbitrary point, rwr  re  Integral from r  rw

OR  Integral from rer

rw

pw

1 2π kh q  dr  dp  r μ p(r) r q μ ln(r/rw ) p(r)  p w  2π k h OR, equivalently r

1 2π kh q  dr  r μ re

p(r)

 dp

pe

q μ ln(re /r) p(r)  p e  2π k h

 Pressure profile is a

linear function of ln(r) for homogeneous properties

pe

 slope depends on flow rate

p

p(r)

q

pw

0

rw

re

rw

r

ln(r) 

re

• Permeability varies across several (3) horizontal layers (k1,k2,k3)  Discrete changes in permeability

h  h1  h 2  h 3   h i  Same pressure drop for each layer

pe - pw  Δp  Δp1  Δp2  Δp3  Total flow rate is summation of flow rate for all layers

q  q1  q 2  q 3   q i  Average permeability results in correct total flow rate

2π k h q Δp μ ln(re /rw )

• Substituting,

2π k h q Δp μ ln(re /rw ) 2π k1 h1 2π k 2 h 2 2π k 3 h 3  Δp  Δp  Δp μ ln(re /rw ) μ ln(re /rw ) μ ln(re /rw ) • Rearranging,

k  k

i

 hi

h

• Average permeability reflects flow capacity of all layers

• Permeability varies across two vertical concentric cylindrical layers [k(rwrr2) = k1, k(r2rre = k2]

R1 of this figure is r2 of equations

 Discrete changes in permeability

re  rw  Δr1  Δr2   Δri  Same flow rate passes through each layer

q  q1  q 2  Total pressure drop is summation of pressure drop across layers

pe  p w  Δp1  Δp2   Δpi

 Average permeability results in correct total pressure drop

q μ ln(re /rw ) pe - p w  2π k h

• Substituting (rw=r1, r2 ,re=r3),

q μ ln(re /rw ) q μ ln(r2 /rw ) q μ ln(re /r2 ) pe - p w    2π k h 2π k1 h 2π k 2 h • Rearranging,

ln(re /rw ) k (ln(ri 1/ri )  ki All Layers

A hydrocarbon reservoir is characterized by five distinct formation segments that are connected in series. Each segment has the same formation thickness. The length and permeability of each section of the five bed reservoir are given below: