6.-darcy'slaw&k_w2-lecture6.pdf

7/25/2019 6.-Darcy'sLaw&K_W2-Lecture6.pdf

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Reservoir Engineering

1

Course (

1

st

Ed.)
http://www.about.me/AlamiNiamailto:[email protected]://www.about.me/AlamiNiamailto:[email protected]


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1. Rock Properties

:

A. Porosity

B. Saturation

C. Wettability

D.

Capillary Pressure

E.

Transition Zone


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1. Darcy Law: Linear Flow Model

2. Permeability Measurements

3. Darcy Law: Radial Flow Model

4. Permeability

-

Averaging Techniques

5. Effective Permea

bilities

6.

Rock Compressibility

7.

Homogeneous and Heterogeneous Reservoirs

8.

Two

-

Phase Permeability


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Darcys Law

Permeability is a property of the porous mediumthat measures the capacity and ability of theformation to transmit fluids.

The rock permeability, k, is a very important rockproperty because it controls the directionalmovement and the flow rate of the reservoir fluidsin the formation.

This rock characterization was first defined

mathematically by Henry Darcy in 1856. In fact, theequation that defines permeability in terms ofmeasurable quantities is called Darcys Law.

2013

H. AlamiNia Reservoir Engineering

1

Course: Darcy

s Law & Permeability 5


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Darcys Equation

If a horizontal linear flow of an incompressible fluid isestablished through a core sample of length L and across-section of area A (includes the area of the rockmaterial as well as the area of the pore channels), then

the governing fluid flow equation is defined as

Where = apparent fluid flowing velocity, cm/sec, k =proportionality constant, or permeability, Darcys, =viscosity of the flowing fluid, cp, dp/dL = pressure dropper unit length, atm/cm

2013

H. AlamiNia Reservoir Engineering

1

Course: Darcy

s Law & Permeability 6


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Darcys Equation Cont.)

The velocity, , is notthe actual velocity ofthe flowing fluid butIs the apparent velocity

determined by dividingthe flow rate by thecross-sectional areaacross which fluid isflowing.

Substituting therelationship, q/A, inplace of and solvingfor q results in:

Pressure vs. Distance in a Linear Flow2013 H. AlamiNia Reservoir Engineering 1 Course: Darcys Law & Permeability 7


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Permeability Unit

With a flow rate of one cubic centimeter persecond across a cross-sectional area of one squarecentimeter with a fluid of one centipoise viscosityand a pressure gradient at one atmosphere percentimeter of length, it is obvious that k is unity.

For the units described above, k has been arbitrarilyassigned a unit called Darcy in honor of the manresponsible for the development of the theory of flow

through porous media.

2013 H. AlamiNia Reservoir Engineering 1 Course: Darcys Law & Permeability 8


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Darcy Unit

One Darcy is a relatively high permeability as thepermeabilities of most reservoir rocks are less thanone Darcy.

In order to avoid the use of fractions in describingpermeabilities, the term millidarcy is used.

The negative sign is necessary as the pressureincreases in one direction while the lengthincreases in the opposite direction.



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Linear Flow Model

The Equation can beintegrated when thegeometry of the systemthrough which fluidflows is known.



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Conventional Linear Flow Equation

The volumetric flow rate, q, is constant for liquidsbecause the density does not change significantlywith pressure.

Since p1 is greater than p2, the pressure terms can be

rearranged, which will eliminate the negative term in theequation.



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Permeability Determination

in Laboratory

Dry gas is usually used (air, N2, He) in permeabilitydetermination because of its convenience,availability, and to minimize fluid-rock reaction.

The measurement of the permeability should berestricted to the low (laminar/viscous) flow rateregion, where the pressure remains proportional toflow rate within the experimental error.



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Using Dry Gas in Measuring K

For high flow rates, Darcys equation as expressedby q=kA (p1-p2)/L is inappropriate to describe therelationship of flow rate and pressure drop.

In using dry gas in measuring the permeability, thegas volumetric flow rate q varies with pressurebecause the gas is a highly compressible fluid.Therefore, the value of q at the average pressure in

the core must be used in the Equation.



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Using Dry Gas in Measuring K Cont.)

Assuming the used gases follow the ideal gasbehavior (at low pressures), p1V1=p2V2=pmVm sop1q1=p2q2=pmqm with pm= (p1+p2)/2

The gas flow rate is usually measured at base(atmospheric) pressure Pb and, therefore, the term Qgsc(gas flow rate at standard conditions) is introduced, soQgscpb=qmpm



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The Klinkenberg Effect in

Gas Permeability Measurements

Klinkenberg (1941)discovered thatpermeabilitymeasurements made

with air as the flowingfluid showed differentresults frompermeabilitymeasurements made

with a liquid as theflowing fluid.



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Radial Flow Model

Darcy Equation (q=-kA/dp/dL) can be expandedto describe flow in anyporous medium where

the geometry of thesystem is not toocomplex to integrate.For example, the flow into

a well bore is not linear,but is more often radial.

Figure illustrates the typeof flow that is typical ofthat occurring near aproducing well.



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Darcys Equation for Radial Flow

For a radial flow, Darcys equation in a differentialform can be written as:

dL has been replaced by dr, as the length term has nowbecome a radius term.

The minus sign is no longer required for the radial

system as the radius increases in the same direction asthe pressure. In other words, as the radius increasesgoing away from the well bore, the pressure alsoincreases.



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Darcys Equation for Radial Flow

Cont.)

At any point in thereservoir, the cross-sectional area acrosswhich flow occurs willbe the surface area of acylinder, which is 2rh.

Since the cross-sectional area is relatedto r, then A must beincluded within theintegral sign as follows:



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Remarks for Darcys Equation

Radial Flow)

The above equationassumes that thereservoir ishomogeneous and is

completely saturatedwith a single liquidphase, where:

q = flow rate, reservoircm3/sec

k = absolutepermeability, Darcy

h = thickness, cmre = drainage radius, cm

rw = well bore radius,cm

pe = pressure at

drainage radius, atmpwf = bottom-hole

flowing pressure

= viscosity, cp



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Averaging Absolute Permeabilities

Three simple permeability-averaging techniquesare commonly used to determine an appropriateaverage permeability to represent an equivalenthomogeneous system.

These are:

Weighted-average permeability

Harmonic-average permeability

Geometric-average permeability



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Linear Flow through Layered Beds



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Weighted-Average Permeability

Linear)

The average absolutepermeability for aparallel-layered systemcan be expressed in thefollowing form:



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Linear Flow through Layered Beds

with Variable Area Linear)

Figure shows a similarlayered system withvariable layers width.

Assuming no cross-flowbetween the layers, the

average permeability canbe approximated to give:

(Aj = cross-sectional areaof layer j, wj = width oflayer j)



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Harmonic-Average Permeability

Linear)

For a steady-state flow,the flow rate isconstant and the totalpressure drop p isequal to the sum of thepressure drops acrosseach bed, or

p = p1 + p2 + p3



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Harmonic-Average Permeability

Radial)

The relationship can beused as a basis forestimating a number ofuseful quantities in

production work. Forexample, the effects ofmud invasion, acidizing,or well shooting can beestimated from it.



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Geometric-Average Permeability

Warren and Price (1961)illustrated experimentallythat the most probablebehavior of aheterogeneous formation

approaches that of auniform system having apermeability that is equalto the geometric average.Where ki = permeability

of core sample i

hi = thickness of coresample in = total number of

samples

If the thicknesses (hi)

of all core samples arethe same



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Effective Permeability Definitions

As the saturation of a particular phase decreases,the permeability to that phase also decreases.

The measured permeability is referred to as the effectivepermeability and is a relative measure of the

conductance of the porous medium for one fluid whenthe medium is saturated with more than one fluid. (kg,ko, kw)

The sum of the effective permeabilities is always

less than or equal to the absolute permeability, i.e.kg + ko + kw k



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Effective Permeability in Darcy

s Law

The effective permeability is used mathematicallyin Darcys Law in place of the absolute permeability.



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Relative Permeability

Relative permeability is defined as the ratio of theeffective permeability to a given fluid at a definitesaturation to the permeability at 100% saturation.

The relative permeability to a fluid will vary from avalue of zero at some low saturation of that fluid toa value of 1.0 at 100% saturation of that fluid.



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Rock Compressibility Types

Geertsma (1957) points out that there are threedifferent types of compressibility that must bedistinguished in rocks:

Rock-matrix compressibility, cr

Rock-bulk compressibility, cB

Pore compressibility, cp



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Formation Compressibility

For most petroleum reservoirs, the rock and bulkcompressibility are considered small in comparison withthe pore compressibility cp.

The formation compressibility cf (range from 3 106

to 25 106 psi1) is the term commonly used todescribe the total compressibility of the formation andis set equal to cp, i.e.:

In general, the formation compressibility cf is the sameorder of magnitude as the compressibility of the oil andwater and, therefore, cannot be regulated.



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Total Reservoir Compressibility

The total reservoir compressibility ct is extensivelyused in

the transient flow equation and

the material balance equation.



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Gross Pay Thickness

A fundamental prerequisite to reservoirperformance prediction is a satisfactory knowledgeof the volume of oil originally in place.

The reservoir is necessarily confined to certaingeologic and fluid boundaries, i.e., GOC, WOC, andGWC, so accuracy is imperative.

Within the confines of such boundaries, oil is containedin what is commonly referred to as Gross Pay.



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Net Pay Thickness

Net Pay is that part of the reservoir thickness thatcontributes to oil recovery and is defined byimposing the following criteria:

Lower limit of porosity

Lower limit of permeability

Upper limit of water saturation

All available measurements performed on reservoirsamples and in wells, such as core analysis and welllogs, are extensively used in evaluating thereservoir net thickness.



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Reservoir Heterogeneity

The heterogeneity of reservoirs is, for the mostpart, dependent upon the depositionalenvironments and subsequent events.

It is important to recognize that there are nohomogeneous reservoirs, only varying degrees ofheterogeneity.

The reservoir heterogeneity is then defined as a

variation in reservoir properties as a function ofspace.



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Homogeneous Vs. Heterogeneous

Reservoirs

Ideally, if the reservoir is homogeneous, measuringa reservoir property at any location will allow us tofully describe the reservoir.

The task of reservoir description is very simple for

homogeneous reservoirs.

On the other hand, if the reservoir isheterogeneous, the reservoir properties vary as afunction of a spatial location.

These properties may include permeability, porosity,thickness, saturation, faults and fractures, rock facies,and rock characteristics.



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Heterogeneous Reservoirs

For a proper reservoir description, we need topredict the variation in these reservoir properties asa function of spatial locations.

There are essentially two types of heterogeneity:Vertical heterogeneityAreal heterogeneity

Geostatistical methods are used extensively in the

petroleum industry to quantitatively describe thetwo types of the reservoir heterogeneity.



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Two-Phase Relative Permeability

When a wetting and a nonwetting phase flowtogether in a reservoir rock, each phase followsseparate and distinct paths.Since the wetting phase occupies the smaller pore

openings at small saturations, and these pore openingsdo not contribute materially to flow, it follows that thepresence of a small wetting phase saturation will affectthe nonwetting phase permeability only to a limitedextent.

Since the nonwetting phase occupies the central orlarger pore openings that contribute materially to fluidflow through the reservoir, however, a small nonwettingphase saturation will drastically reduce the wettingphase permeability.



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Typical Two-Phase Flow Behavior

Figurepresents atypical set ofrelative

permeabilitycurves for awater-oilsystem withthe water

beingconsideredthe wettingphase.



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Hysteresis Effects in Relative

Permeability



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Effective Phase Saturation

Most of the Two-phase Relative Permeabilitycorrelations use the effective phase saturation as acorrelating parameter. The effective phasesaturation is defined by the following set of

relationships:



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Relative Permeability Ratio

The relative (or effective) permeability ratio lends itselfmore readily to analysis and to the correlation of flowperformances than does relative permeability itself.

The relative permeability ratio expresses the ability of a

reservoir to permit flow of one fluid as related to itsability to permit flow of another fluid under the samecircumstances.

The two most useful permeability ratios are krg/kro

and krw/kro.

The relative permeability ratio may vary in magnitudefrom zero to infinity.



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Relative Permeability Ratio Plot

In describing two-phase flow mathematically, it isalways the relative permeability ratio (e.g., krg/kroor kro/krw) that is used in the flow equations.

Because the wide range of the relative permeabilityratio values, the permeability ratio is usually plottedon the log scale of semilog paper as a function ofthe saturation.

The central or the main portion of the curve isquite linear.



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Krg/Kro as a Function of Saturation

Figure showsa plot ofkrg/kro versusgas

saturation.



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Krg/Kro as a Function of Saturation

Cont.)

It has become common usage to express thecentral straight-line portion of the relationship inthe following analytical form:

The constants a and b may be determined by selectingthe coordinate of two different points on the straight-line portion of the curve and substituting in the

Equation. The resulting two equations can be solvedsimultaneously for the constants a and b.



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1. Ahmed, T. (2006). Reservoir engineering

handbook (Gulf Professional Publishing). Ch4and Ch5


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1. Reservoir Characteristics

A. Reservoir Fluid Types According To CompressibilityB. Types of Flow Regimes

C. Types of Reservoir Geometries

D. Darcys Law Remarks

2. SS Linear Flow and Tilted Reservoirs


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6.-darcy'slaw&k_w2-lecture6.pdf

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