numerical simulation1.doc

7/28/2019 Numerical Simulation1.doc

1/17

NUMERICAL SIMULATION

IntroductionMost reservoir engineering, production, formation evaluation, and drilling issues are

ultimately reduced to the nature and spatial distribution of rock properties. This

observation provides the incentive for this course; about 90% of the insight into suchproperties can be captured through fairly simple theoretical and mathematical relations.

This course reviews fundamental principles dealing with rock properties and introduces

advanced concepts about displacements, residual phase saturations, and effective

properties. We will rely strongly on conceptual and simplified mathematical models.

Readers interested in more depth on these aspects should consult the references. As will

be emphasized in this Introduction, the entire course will focus on the petrophysical

properties that go into the input of a numerical simulator.

Course ObjectivesWhen you finish you should:

Understand permeability and its origins,

Be able to illustrate capillary pressure and how to use it,

Find out what factors affect relative permeability,

Illustrate heterogeneity measures,

Understand the difference between heterogeneity and correlation, and

Show the benefits of statistical assignments.

In each case, we will attempt to make some specific points which should be useful in

evaluating data, making recommendations about laboratory procedures and in usinginformation to its best extent. As the outline shows, we will concentrate on single-phase

and water/oil properties although we will refer to gas/oil and more exotic properties

from time to time.

Numerical SimulationSince this course deals with the quantities that go into the input of a simulator, its

appropriate to spend a few minutes talking about simulation in general.

Most knowledge progresses through the building, testing, and reformulating of models

or representations of reality. This statement is true regardless of the field of study.

Models -- Four basic types

Conceptual -- e.g., Fluvial deltaic

Physical -- e.g., Hele-Shaw cells, corefloods

Numerical -- State of the art for engineers

Philosophical -- Whether we know it or not

Geologists are quite comfortable with the idea of model building even though their

models (like models in many other fields) are rarely numerical. Physical models were

very common in the early days of the petroleum industry, but these have beensupplanted by numerical models.


2/17

Actually, numerical

models have had a long

pedigree in hydrocarbon

production prediction as

the following illustrates.

Numerical models --A short history

Tank type --

Emphasis on fluids

o Volumetrics

o Primary

depletion

Streamline --

Emphasis on

patterns and rates,

permeability

anisotropy

o Areal sweep

o Oil in place

Simulators -- All of

the above plus

heterogeneity

Tank type models are still

useful today in determining primary drive mechanisms, oil and gas originally in placeand even in the initial stages of an involved simulation study.

Streamline models, invented in the mid 1950s, have experienced a resurgence in

interest because, in part, of the ability to assist in fluid flow visualization and the ability

to solve very large problems. The image to the right is a classical illustration of the

streamlines in a 5-spot pattern.

This figure also illustrates one of the best uses of streamline modeling: identifying the

amount of fluid that will pass outside of a given area.

What we mean by a numerical model orsimulator in this course is something which

represents the behavior of one system through the use of another (Webster's Unabridged

Dictionary)

First system -- Reservoir and process

Second system -- A model

A model or numerical simulator is a sequence of numerical operations whose output

represents the behavior of a particular process in a particular reservoir. It is, therefore,

an integrating tool for combining, in their proper weight, all of the factors that influence

production in a reservoir.


3/17

Models are nothing more

than solutions to

conservation equations

that are coupled to

several

phenomenological lawsthat are needed to make

the number of equations

and unknowns equation.

Listed below are the

basic equations and laws

associated with our

models.

Basic Equations

Conservation of

o Mass

o Energy

Empirical laws

o Darcy

o Capillary pressure

o Phase behavior

o Fick

o Reaction rates

For the purposes of this course, Darcy's law (covered in Module 1) and the capillarypressure and relative permeability behavior (covered in Module 2) are the most

important ones.

Even though the number and type of equations can be said briefly, there is enormous

complexity in the laws as the image to the right illustrates.

This is a portion of a table from Lake (1989) that shows the equations solved by a

simulator. (Don't be concerned about reading the details in the table. The idea here is to

impress rather than inform.) In general, there is one partial differential equation to be

solved for each component and these must be solved in three spatial dimensions.

These equations are far too complicated to be solved analytically (directly). Instead,

what is done, when solving for a reservoir, is to divide it up into a number of cells or

grid blocks and the equations solved on these much smaller volumes. This division is,

of course, arbitrary but it is the only way to proceed. It would be no surprise to find that

the division into grid blocks leads to a number of artifacts in the simulator results. But it

should equally be of no surprise to realize that these artifacts tend to disappear as the

number of blocks increase; with current simulation technology and with newer

simulation technology the artifacts seem to comprise less of a problem than simply not

knowing the simulator input to begin with.


4/17

It has historically not been possible to write a single simulator to deal with all possible

cases of flow. The following lists some of the simulator classifications.

Simulators -- Classified by:

Dimensionality (1-D,...,3-D) Numerical algorithm, e.g.,

o Finite difference

o IMPES

o Implicit

o Direct solvers

Vectorization

Physical properties

o Single-phase (aquifers or gas)

o Black oil

o

Compositionalo Thermal

o Dual porosity

It is good to know what each of these terms mean, but we won't discuss them here.

What is discussed in this course is the petrophysical and geological parameters that are

common inputs to all types of simulators.

Although there are some variations, the following is a typical simulator structure:

Simulator structure

1. Nonrecurrent data

a. Fluid properties

b. Reservoir properties

c. Block or cell assignments

2. Recurrent data

a. Well properties

b. Output control

3. Calculation

4. Output

A simulator run progresses through the nonrecurrent data simulator directly and only

once. The items between the recurrent data and output may be processed several times

as well conditions and/or the manner of calculation changes. This course is mainly

concerned with the block assignments under the recurrent data, but we provide a brief

summary of the other types for completeness.

Typical Inputs

1(a) FluidsThe fluid input, summarized below, ranges from the basic properties of

viscosity, solubility and volume relations, through much more complicated


5/17

pressure-volume-temperature relationships embodied in and equation of state

simulator.

Viscosities

Densities

Surface tensions

Phase behavior

The properties are normally input via table or equation as functions of pressure

and, for thermal simulators, functions of temperature. Compositional simulators

will calculate these properties as functions of pressure and fluid composition.

1(b) ReservoirThe reservoir input deals with properties that define the gross volume of the

fluids in a reservoir.

Depth (top surface)

Thickness (bottom surface)

Lateral extent

Original contacts

For complicated geometries, as are now commonly modeled, the depth and

thicknesses can be input by cell--a subject we discuss below. The discussion in

Module 2 about capillary pressure shows how the original contacts used to

estimate original hydrocarbon in place.

1(c) Blocks

What this course deals with the most are the block-by-block petrophysicalassignments.

Initial pressure and saturations

Porosity

Permeability (three values)

Relative permeability

Capillary pressure

2(a) WellsWell properties can be among the most important to a simulation run because it

is through these that the operational decisions about how to deplete a field arequantified.

Location

Completion interval

Productivity (e.g. skin factor)

Status (e.g. producer)

How operated

The bullet location can include some quite complicated geometries for

horizontal or deviated wells. Many times a great deal of the simulator is devoted

to modeling effects in the wellbore and it is becoming common for the well

model to be hooked up to separate models of surface facilities.


6/17

These things constitute the great bulk of the input: a typical simulation run might have

more than 10,000 cells so it is easy to see how this type of data can run into the millions

of values. This input, furthermore, constitutes the main way that the geology is input

into a simulator. The lack of quantitative detail in the geological description coupled

with the sheer volume of data means that this type of input has historically been among

the most overlooked in simulator input. This neglect forms one of the prime motivationsfor this course.

Difficulties with numerical simulationThough it is the only one discussed here, the assignment of grid block properties is only

one of several difficulties with numerical simulation. Some of these are:

Truncation errors

Stability

Grid orientation effects

Property assignments

Scale adjustment

Property assignments to cells away from measurements (usually at wells) form the

culmination of the material in Module 3. The last item, adjustment for scale or scaling

up, is truly a simulator data input also. It involves the fact that the size of the blocks in a

typical simulation run is invariably much larger than the size of the measurements. The

following table illustrates some of the typical grid block sizes.

Typical Grid Block Sizes

Process

DX

(meters)

DZ

(meters)min max min max

Undersaturated oil reservoir 500 500 7 9

Gas reservoir 1500 1500 70 229

Simulation of well test 244 671 6 61

Retrograde gas reservoir 236 457 34 61

Solution gas drive reservoir 610 610 3 18

Where DX and DZ are the lengths of the sides of the grid blocks. From Haldorsen

(1983).

Rather than speaking of grid blocks we should be speaking of grid pancakes. And the

size of these blocks are those of a modest sized building.

NotationMuch of the following material comes from Chapters 3 and 6 of a text on enhanced oil

recovery (Lake, 1989). This manual adopts certain conventions from that work,

particularly regarding subscript conventions of the various fluids.

SUBSCRIPT CONVENTIONS

1 Water or aqueous phase


7/17

2 Oil or oleic phase

3 Gas phase

nw Nonwetting

w Wetting

r Residual

R Remaining

Other symbols are Society of Petroleum Engineers or industry standards. Thus, Swr

means the wetting phase residual saturation and S2R means the remaining oil saturation.

We will attempt to maintain consistency in definitions and to define each as near to the

first usage as possible.

References

1. Haldorsen, Helge Hove, Reservoir Characterization Procedures for Numerical

Simulation, Ph.D. Dissertation, The University of Texas at Austin, May 1983

2. Lake, Larry W.,Enhanced Oil Recovery, Prentice Hall, 1989


8/17

IntroductionIn this module we discuss the following properties. The properties are all engineering

properties (as opposed to geologic or geophysical) because they are needed in flow

calculations. Most specifically, they form a central part of the input to numerical

simulators. Indeed, it is the single-phase properties and their distributions that allow

geology and geophysics to be inserted into such models.

Single phase petrophysical properties

Porosity

Permeability

Permeability-Porosity Relationships

Tensorial Character of Permeability

The single-phase porosity is an important predictor of hydrocarbon in place; the

permeability as an indicator of production rate. Their understanding provides a basis for

more advanced concepts which we will treat in later modules. Of course the great

advantage in treating single-phase flow separately is that it delays the need to consider

multiphase flow, which is itself a complicated topic.

The objectives for Module 1 are given below. As always, the emphasis will be on

conceptual models which generally give correct trends, but are not always quantitative

predictors. Such models can frequently be corrected for real-world effects, but they are

always good aids to understanding.

Module 1 Objectives Understand a Representative Elementary Volume (REV)

Illustrate importance of specific surface area and tortuosity to permeability

Give effects of grain size distribution on permeability

Examine relationship between permeability and porosity Illustrate the origins of anisotropy

As stated in the Introduction, the emphasis here is on transport properties, those

properties which govern fluid flow. The transport properties form the lion's share of the

input to a numerical simulator although they play other roles as well.

Representative Elementary Volume (REV)

We will follow the same basic procedure in developing each single-phase property.


9/17

First, the property will be described mathematically in terms of simplified physical

laws, usually based on incompressible, steady-state flow in even simpler geometries.

The simple geometry represents the smallest element of the permeable medium -- the

connected pore or the microscopic

scale. Second, the petrophysical

property for the connected pore isthen modified to account for the

actual permeable medium local

geometry -- variable pore (or particle) cross-sections, variable flow lengths (tortuosity),

and multiple connections of one pore with another. This step translates the microscopic

relation from the actual permeable medium flow domain to that of the locally-

continuous representative elementary volume (REV). Unfortunately, this step

constitutes a great deal of art, and is invariably restricted to fairly simple idealizations

about the local pore geometry.

The very idea of a REV is difficult. Consider the following figure.

The plot shows the trend in a particular medium property (porosity is easiest to

visualize) measured over some volume V as V shrinks to zero. For large V the curve

changes smoothly but distinctly; we call this change heterogeneity and discuss itseparately in Module 3. For intermediate V, the curve becomes stable at some fairly

precise value of porosity and then becomes erratic at an even smaller V where pore-

scale variations become important. The REV is the value of V (indicated in the figure

above) above which the fluctuations are negligible.

The existence of the REV is tenuous, because it has never been identified in real media

despite some fairly serious efforts (Goggin et al., 1988). There probably are several

plateaus in which case the REV would be defined as the volume at the smallest of these.

The size of the REV is related to how locally correlated the property is on the

microscopic scale. Our best guess is the REV is somewhere around 100-1000 grain

diameters--large enough for averages to be statistically meaningful but small enough to

Approach to Property Development

Describe on pore (microscopic) scale

Translate to local (REV) scale


10/17

avoid heterogeneity. There is a school of thought based on the concept of continuous

variation (fractals) that admits no such plateaus as V decreases.

Magic Numbers1. Typical REV size = 100 - 1000 grain diameters (sandstones)

But, despite all the hurdles, the notion of the REV is absolutely essential for reservoir

engineering work because it allows us to use continuous mathematics (limits taken to

zero are really taken to the REV). We pay for this through the introduction of various

non-standard (and perhaps ill-defined) physical properties some of which we are about

to cover in this module.

Porosity

Porosity is the ratio of void or pore volume to macroscopic or bulk volume; the rock or

solid phase volume is the bulk volume less the pore volume. Porosity is a static

property--can be measured in the absence of flow. For most naturally-occurring media

the porosity is between 0.10 to 0.40 although on occasion values outside this range havebeen observed. Many times porosity is reported as a percent, but it should always be

used in calculations as a fraction. From these typical values the rock "phase" clearly

occupies the largest volume in any medium.

The porosity of a permeable medium is a strong function of the variance of the local

pore or grain size distribution, and a weak function of the average pore size itself. For

sandstones the porosity is usually determined by the sedimentological processes under

which the medium was originally deposited. For carbonate media, on the other hand, the

porosity is mainly the result of changes that took place after deposition.

The pore space as well as the porosity can be divided into an interconnected or effectiveporosity that is available to fluid flow and a disconnected porosity that is unavailable to


11/17

fluid flow. Certain of the enhanced oil recovery processes exhibit behavior whereby

some of the effective porosity (dead end pores) is shielded from the displacing agent. At

the other end of the flow scale is the fracture porosity that expresses the fraction of the

volume of a particular medium that is tied up in large-scale voids. One could easily

argue that a volume that contains large-scale voids is

below the REV for that medium. In what follows, theword porosity references to a volume that is

substantially larger than the smallest discrete feature

in the rock.

Permeability

Permeability is also a basic permeable medium

property that, unlike porosity, cannot be defined apart

from fluid flow. We use it most commonly to

estimate production/injection rates from which comes

estimates of economic lives.

Definition - Permeability is the proportionality "constant" between the fluid flow rate

and an applied pressure or potential gradient. The figure on the right illustrates this for

flow of a single incompressible fluid.

The last line of this figure is one for of Darcy's law, which for single-phase, one

dimensional flow, constitutes a definition of permeability. The other terms in the

equation are the superficial velocity (the flow rate) and the fluid viscosity .

Darcy's law contains the superficial or Darcy velocity. This is the volumetric flow ratedivided by the cross-sectional area normal to flow (q/A).

Velocities in Reservoir Engineering Superficial velocity

o u = Flow rate/area normal to flow Interstitial velocity

o

v = Superficial velocity/porosity

The interstitial or frontvelocity is the rate a fluid particle actually moves through the

medium. The two velocities are frequently confused.

To illustrate some features of the permeability definition more clearly, write Darcys

law as


12/17

This simple form of Darcy's law is as important for what it does notsay as it is for what

it does say. For example, if we increase the rate q, we find that the pressure drop P

increases apace; k is not a function of either flow rate or P. Doubling (for example) the

fluid viscosity results in a doubling ofP and no change in k. Permeability is not a

function of the fluid viscosity, nor, indeed, of the fluid identity. Increasing L for the

same P results in a decrease in q so that k is again unchanged. Permeability is not a

function of the size of the sample. We say that k is an intrinsic function of the medium,

or rather of the medium's properties. We delve into this in the remainder of this module.

In truth, all of the above negations are only approximately true. k does depend on the

fluid identity, water vs. oil, or fresh water vs. brine, in may cases. Similarly, k also

depends on the scale of the measurement L. That is why we use the term absolute

permeability with caution When k depends on rate or P, the phenomena are referred

to as non-Darcy effects. Such deviations are important in many cases, but we shall

continue the exposition for the ideal case to focus on rock properties.

We will use the following slide as organization for our treatment of permeability.

(Absolute) Permeability . . . The Basic Flow Property of a Medium

Depends on porosity ando Grain size

o Local heterogeneity (sorting)o Cement amount and type

Is direction dependent (tensorial)

Depends strongly on position (heterogeneity)

The following paragraphs briefly discuss each of these items. Heterogeneity (aside from

the pore scale) is treated in Module 3.

Carman-Kozeny Equation


13/17

We derive most of our

insights from the

Carman-Kozeny (CK)

equation because it lends

itself to more physical

reasoning than some ofthe other equations

(Dullien, 1991). As we

shall see, the CK

equation introduces a

direct dependence

between porosity and permeability, but also, through the concepts of specific surface

area and tortuosity, leads to an explanation of how k depends on local rock texture.

To develop the CK equation the local pore model is the capillary tube, probably the

most common such model in permeable media studies.

Consider the single-phase, steady-state, laminar flow of a constant viscosity Newtonian

fluid through a horizontal capillary of radius R and length Lt as shown above. These

conditions will lead to a parabolic velocity profile in the tube. The maximum velocity at

the tube centerline is twice the mean velocity; the minimum velocity is zero (no slip) at

the wall.

Laminar flow is worthy of separate discussion both because it is an important

fundamental concept and because its application to flow through permeable media is so

iffy. Laminar flow just means that fluid flow elements don't cross. For the condition of

laminar flow in a tube this means that the fluid elements are slipping past each other inexactly the same manner as the layers in a telescope do as it is being extended. The

simplicity of laminar flow is very appealing because it is easy to visualize and, hence

understand. But it is extremely rare in practice, being limited to very slow flows, very

viscous fluids or flows in simple geometries. These conditions are combined in the

familiar Reynolds number. We always have very slow flow rates in our applications

(about 10 cm/day is typical) and many times we are dealing with viscous fluids. But the

flow is rarely truly laminar because the local geometry of the rock grains and its

surfaces are so irregular that fluid flow lines will cross except at very small flow rates.

Magic Numbers1. Typical REV size = 100- 1000 grain diameters (sandstones)

2. Typical reservoir flow rate = 1 ft/day

It is misleading to think of laminar flow as being measured solely by low Reynolds

number. The idea here is correct -- the Reynolds number is the ratio ofinertial to

viscous forces and laminar flow always occurs at a sufficiently low Reynolds number.

However, the specific Reynolds number limit depends on the flow geometry; for

example, the laminar-turbulent transition occurs at about 2100 for flow in a tube and at

about 600 for flow in a slit. We have no way of knowing where it would occur in flowthrough permeable media, but we think it is about 0.1 to 0.5 (the length scale being the


14/17

grain diameter) for most of the flows we are interested in. Such flows are sometimes

characterized as creeping flows.

With some manipulations of a force balance on the fluid flowing through the tube wearrive at the following relationship between the average velocity and the tube

dimensions and the pressure drop.

Average Velocity of a Fluid in Laminar Flow

Hagen-Poiseuille Equation

Single-phase, horizontal flow

Laminar flow

No entrance or exit effects

Capillary tube model

The basic equation for REV translation

This is the celebrated Hagen-Poiseuille equation for laminar flow in a tube. In order forthis equation to apply, the tube must be long enough for the flow to be free of entrance

or exit effects. This condition certainly does not hold in a permeable medium pore, but

the simplicity of the equation as well as its similarity to Darcy's law encourages us to

proceed.

As simple as it is, the Hagen-Poiseuille equation tells us two things about permeability,

which in this equation is the R2/8 term. First, k has units of square length (L2). The

historical unit is Darcys (D) which is nearly equal to the SI unit ofm2 (= 10-12 m2).

Second, we see that k is a reflection of the size of the channels or holes through which

fluids are flowing. A k = 1 m2 rock, for example, nominally quite permeable, has anapproximate hole size of a very small 1 m Productive naturally-occurring media have

permeabilities ranging from 0.1 mD to 20 D for liquid flow and down to 10 D for gas

flow.

Magic Numbers1. Typical REV size = 100- 1000 grain diameters (sandstones)2. Typical reservoir flow rate = 1 ft/day

3. 1 Darcy = 1 m/s (hydraulic conductivity)Handy conversion factor: wg= 0.433 psi/ft


15/17

To convert to Darcy's law we eliminate the length from the Hagen-Poiseuille equation

with the concept of tortuosity and replace the radius with the hydraulic radius. The

equation is then solved for the pressure difference which is inserted in Darcy's law to

give the following for permeability (Lake, 1989).

The Carman-Kozeny equation in terms of specific surface area

This is the most

fundamental form of the

CK equation because it

deals with quantities that

can be defined for nearly

any permeable medium.

The tortuosity is the

squared ratio of the mean

flow path length to the

medium length. It isrelated to the formation

resistivity factor and

tabulated in Pirson

(1979 ). The specific

surface area av, with

units of inverse length, is

the internal surface of the

medium per unit volume,

an intrinsic and highly characteristic property of the medium. Unfortunately, it is not

routinely measured.

The CK equation also illustrates the rather elementary observation that zero porosity

rocks will also have zero permeability. Unfortunately, this is about as far as we can go

with this equation without more work for it is certainly true that there is not a one-to-

one correspondence between porosity and permeability.

The physical basis for the dependency on surface area in this equation is that all of the

pressure drop (flow resistance) is caused through viscous interactions between adjacent

lamella of fluids as they "slip" past each other. The slippage is, in turn, caused by the

pore walls where there is no slip. Thus, if we are considering locally creeping flows the

entire surface area of a particle should enter into the flow resistance. This is suggested

by the image to the right.


16/17

Even at low fluid rates, however, flow can separate in the wake of a particle as shown

on the right panel. This is particularly likely to be the case because the approach

velocity is unlikely to be anything but uniform. If separation occurs, the surface area to

be used in the CK equation should be the exposed area rather than the total area. In the

formulas encountered thus far using the exposed area has only the effect of reducing a v

However, for nonspherical particles it can provide at least a partial explanation for thedirectional dependence of permeability. At even higher rates an appreciable amount of

energy begins to be expelled in the flow reversal region which manifests itself in

turbulence corrections.

The most evident prediction of the above equation is that permeability decreases as

specific surface area increases. This accounts for the observation that media composed

of clay minerals, which have large specific surface areas, also have low permeabilities.The table to the right gives order-of-magnitude estimates of specific surface area for

clays and rocks.

The above values were based on nitrogen adsorption and tend to seriously underestimate

permeability when used in the CK equation (that is, the adsorption area is greater than

the dynamic area). Nevertheless, it is clear that clays have much more surface area than

do the outcrop Berea and Torpedo sands.

For the same reason, it is a common observation that permeability correlates inversely

with the clay content of sands. Normally, a sand is not thought of productive to oil if the

clay content is more than 35%. We will go into the connection between cementingmaterial and permeability in more depth below.

The connection between clay content and permeability also suggests why gamma ray

log response is sometimes useful as an indicator of permeability. This particular log is

measuring the radioactive decay of unstable species bound up in the clays. The larger

the response, the higher the clay content and, hence, the lower the permeability. Since

the specific surface area varies considerably with clay type and the gamma ray response

is rather unspecific for clay type, such a relationship can be at most qualitative. The

gamma ray response for clean sandstones is quite small.

Bibliografia

ttp://www.spe.org/learning/demo_sm/1frame.htm

Typical Specific Surface Areas

Material aV(cm-1)

Berea 20

Torpedo 80

Kaolinite (clay) 500

Smectite (clay) 1300

Illite (clay) 2800

Adapted from Faris et al. (1985)


17/17

numerical simulation1.doc

Documents