discussion on some aspects of compositional simulation

WHAT IS SIMULATION?WHAT IS SIMULATION? a process of inferringthe behaviour of a hcreservoir from the

performance of a modela model

CoreScaled Down Scaled Down

Physical ModelPhysical Model

p

p

pp

rp

p

pp

rp

pp

rp

BB

kk

B

kk

B

kk

pS

t=

zz

+yy

+xx

Mathematical ModelMathematical Model

Engineering/Simulation Engineering/Simulation ModelModel

p

p

pp

rp

p

pp

rp

pp

rp

BB

kk

B

kk

B

kk

pS

t=

zz

+yy

+xx

Grid BlockGrid Block(~100mx100mx10m)(~100mx100mx10m)

Permeability, k Permeability, k Porosity, Porosity, Thickness, hThickness, hElevation, dElevation, d Saturation, SSaturation, S Pressure, pPressure, pCompressibility, c Compressibility, c Fluid PVTFluid PVT RelPerms, kRelPerms, krr

Well data, q, pWell data, q, p

Reservoir Parametersto be assigned

Geologyand

Geophysics

Petro-physics

RCA+SCAL

Geo- Mechanics

WellTests

TracerTesting

DataDataAcquisitionAcquisition

Data Processing Data Processing and Integrationand IntegrationReservoir ModelReservoir Model

Up-scaling/Downscaling

SimulationSimulationModelModel

Up-scaling

Usually Millions of Grid Blocks

CORE SCALECORE SCALE

Validation/Validation/History MatchHistory Match

Technical Scenario

Economical Scenario

Reservoir Study

PRE-SIMULATION ERAPRE-SIMULATION ERA

Log AnalystGeologist

ProductionAnalyst

Geo-physicist

Reserv.Eng.

SurfaceFacilityDesign

Interaction of peopleIntegration of expertise

Petroleum ReservoirPetroleum ReservoirPerformance EvaluationPerformance Evaluation

viavia

POST-SIMULATION ERAPOST-SIMULATION ERA

Log Analyst

Geologist

Production

Analyst

Geo-physicis

t

Reserv.Eng.

Surface

FacilityDesign

Surface FacilityDesign

Reserv.Eng.

Geologist Geo-

physicist

Production

Analyst

Log Analyst

CLASSIFICATION OF SIMULATORS BASED ON FLUID DESCRIPTION

•Black Oil Simulators

•Compositional Simulators.

Black Oil Simulators:

•These types of simulators treat hydrocarbons as two components; gas and oil.

•They are applicable to dissolved gas, medium gravity (~20-35o API) oil-bearing reservoirs under moderate reservoir pressures (2000-6000 psi) and temperatures (200-280oF).

•They can be applied to almost all conventional water flooding simulation studies.

Black Oil Simulators:

•If the oil formation factor is less than two, they can safely be applied to solution-gas drive, gas cap extension or gas injection studies.

•Black oil simulators can also be used for some cases where the formation volume factor is greater than 2.

That is possible if oil and gas formation volume factors, gas in solution, and oil and gas viscosity as a function of pressure can be determined accurately by calculation or experiment.

The underlying assumptions of Black Oil Models can be summarized as follows:

•The reservoir system should be characterized adequately with three components, namely oil, water and gas.

•The formulation is based on the concept of a solution gas-oil ratio, by which the amount of gas dissolved in oil is represented merely as a function of pressure and temperature.

•The injected gas combines with reservoir oil, in the same way as in-place reservoir gas, disregarding the compositional differences between the gas phases.

The defects of the black oil model can be summarized as follows:

•It assumes that the properties measured in the lab completely describe the behavior of the fluid system throughout the project life.

•Compositional variations are ignored.

Compositional Models

• This class of simulators has been in use since the late 1960's and became particularly important in the mid 1970's.

•In such models, a balance is kept of

all hydrocarbon component specified,

the distribution between vapor and liquid phases of component, and

the transfer from node to node of each phase and its associated components.

the simulation method is capable of handling:

•Enhanced Oil Recovery by CO2 or enriched gas injection

•Natural depletion or injection of gases such as nitrogen or residue gas into gas condensate reservoirs

•Natural depletion or gas injection into volatile oil reservoirs

•Re-evaporation of residual oil by injecting residual gas

compositional simulators are designed to handle the following two classes of process:

•Depletion and/or gas cycling of volatile oil and gas condensate reservoirs

•Multiple contact miscibility (MCM) by miscible flooding.

Advantages of Compositional Simulation

Can account for effects of

• Surface Tension and or Interfacial Tension (IFT)

– Especially the effect of composition on IFT

– The effect of IFT on residual oil saturation and on relative permeabilities.

Problems and Difficulties with Problems and Difficulties with Compositional SimulationCompositional Simulation

• When Calculations of phase composition is in plait - point region, difficulties occur.

– First: the K - values and physical property correlations as well as EOS are less accurate in plait point.

– Second: in the plait point region experimental data are lacking, thus it is difficult to tune the EOS in this region.

• Numerical dispersion.

Ternary Diagram

Plait Point: tie line length = 0

extension of critical tie line

tie lines connect liquidand vapor phases inequilibrium

Vapor Phase Composition: y

Liquid Phase Composition: x

100% methane

100% C4100% C12

40% C1, 50% C4, 10% C12

20% C1, 20% C4, 60% C12

z

Problems and Difficulties with Compositional Simulation

• Model viscous fingering– Complete mixing of fluids within a

grid block assumed. Approximations by Barker and Fayers - works but not correct.

• Computer Time Requirements.

Comparison of Black Oil and Compositional Models

Key Difference: PVT

Compositional: Flash = EOS

Black Oil: Table Vs. Pressure


Black Oil

(2 components, volatile oil)

Gas

Oil + Solution Gas (Rs)


Compositional

Components I =

(nc Components)

yi

123.nc

123.nc

Components I =

xi

V

L


B

B

Rf p

o

g

s

( ) Table Values

Assumes Composition of Gas and Oil Phases constant with Pressure and time

Black Oil


Assumes EOS represents fluids at all, T, P, Composition

),,,( ii

i

ii

yxTPfx

yK

Compositional

Table look-up or EOS Flash


Unknowns (3 Phase System)

3 variables per grid block

P

S

S

w

g

Black Oil


Unknowns ( 3 Phase System)

(Nc+2) variables per grid block

PSwzi, i = 1, .... nc (molar density)

Compositional

Nomenclaturez = elevation in vertical direction (l.c.)

zi = Feed mole fraction of component i (l.c.)

Z = Compressibility, Zl = compressibility of phase l

Zc critical compressibility (u.c.)

Ki = K-value (equilibrium ratio) of component i (u.c.)

kij = Binary interaction coefficient between components i and j (l.c.)

ko kg = Relative permeability of the oil and gas phases (l.c.)

Relationships

/NN=V n/N=Ny

/N=N=L n/N=Nx

+Vy=Lx z/N Nz

N=N NN

NN

iVyiiVViVi

LixiiLLiLi

iiiii

i

iVi

iL

ii

VL

Relationships

LNN

VNN

L

V

L V 1

1 i i i

iii yxz (EQ 38)

(EQ 37)

The Continuity The Continuity EquationEquation

The The Equation Equation of Stateof State

The Equation The Equation of Flowof Flow

ConstitutiveConstitutiveEquationsEquations

Simulation Equations

C o n t i n u u mD o m a in

R e p r e s e n t a t i v eE le m e n t a r yV o lu m e

V o lu m e

1

0

CONTINUUM CONCEPT and SCALECONTINUUM CONCEPT and SCALE

xV zV

x

z

y

yV yy VV

xx VV yy VV zz VV

x

P=

x

P

0x

Lim

t=

z

V-

y

V-

x

V- zyx

1-D Horizontal Flow

uk P

xx

3D - 1 Phase Flow with Gravity

uk

Pg

gc

where gc is a conversion constant, note z is positive down

Permeability Tensor

z

2

y

2

x

2

k

Cos+

k

Cos+

k

Cos=

k

1

P1

P2

k

Angle

x

Y

U

kv

Permeability is second rank symmetrictensor

zz zyzx

yz yyyx

xz xyxx

kk k

kk k

kkk

= [k]

where kxy = kyx, kxz = kzx and kyz = kzy.

zz

yy

xx

k 0 0

0 k 0

0 0 k

SimplificationSimplification

If the principal axes of permeability coincides with the axes of the co-ordinates system, the cross terms disappear.

Figure FPM-4: Reference co-ordinatescompared with spatially varying principle

axes of permeability

Reference Co-ordinates

PermeabilityAxes

General Compositional Component Balance

Balance on mass or moles of nc components subscript i (moles = mass/MW)

3 Phases

V (g)

L(o)

W

yi = mole fraction of component iin Vapor Phasexi = mole fraction of component i inLiquid (hydrocarbon Phase)

ci = mole fraction of component i in waterPhase

Note: for general treatment all components can dissolvein water phase


Component Balance - 1 - D, No sources or sinks for component (no elevation difference)

iggooiww

w

iw

wwgi

g

ggoi

o

oo

zSScSt

x

Pc

k

x

Py

k

x

Px

k

x

)

Where Zi = Total hydrocarbon (feed) mole fraction of component i


If we Have Nc components (index i) and 3 phases (index l)

Components Unknowns

3

3

33

3

4Nc

2

xi, yi, ci, zi

pl

sl

kl (rel. Perm.)

L,V

l

l

Need 4Nc + 17 Independent Relationshipsto solve this system

Relationships From

1. Differential equations

2. Conservation principles

3. PVT data

4. Relative permeability data

5. Phase equilibrium

6. Flash relationships

7. Capillary pressure

Relationships From

1. PDE for each component (Nc equations)

S S Sg o w

1 (1 equation)

mole fractions in each phase = 1

(3 equations)

x = 1

y = 1

c = 1

i

i

i

2.

Nc+4

Relationships From3. From PVT data

iww

igg

ioo

cTPf

yTPf

xTPf

,,

,,

,,

(3 equations)

iww

igg

io

cTPf

yTPf

xTPf

,,

,,

,,

(3 equations)

Nc+10

Relationships From

4. Relative Permeability Data (3 equations)

k f S S S

k f S S S

k f S S S

o g o w

g g o w

w g o w

, ,

, ,

, ,

Nc+13

Relationships From

L + V = 1 {molar fractions of liquid and vapor} (1 equation)

equations) (N ciii VyLxz

5. Phase Equilibrium - Flash - Thermodynamic principles give the distribution of a component between Liquid (L) and Vapor (V) phases

2Nc+14

Relationships From

g Vz K

V Ki i

ii

nc

1

1 10

1

(Flash, 1 equation)*

* To be discussed in Flash and EOS sections

6. Flash Equation can be developed frombasic principles

2Nc+15

Generalized Model

(2Nc equations)

,,,

,,

iowigo

igw

i

i

iiigwi

i

iiigoi

i

KK

K

c

x

cyPTfKc

y

yxPTfKx

y

6. Flash Relationships, continued

Not independent

4Nc+15

Generalized Model

7. Capillary Pressure (2 equations)

P P P f S S S

P P P f S S S

cgo g o g o w

cow o w g o w

, ,

, ,

Total 4Nc + 17 Equations

4Nc+17

Simplification for Most Compositional Models

Mole fractions of hydrocarbon components in the water are zero

c i wi 0 1, , cw

xw 0 0, yw

No water component dissolves in either the liquid (L) or vapor (V).

In the simplified model, phase behavior involves only the hydrocarbon components in the L and V phases.

3-D, Nc components, 3 phases - simplified Model

Water component

wwwww

rww S

tq

kk

(EQ 39)

Differential Equation (PDE)

NumericalApproximation

Solve UsingA Computer

Objective [Model of our process]

“FINITE DIFFERENCE”

“A FORTRAN PROGRAM”

Numerical Solution of Partial Differential Equations

Specification of Relative Permeability for Compositional Simulators

•In the case of immiscible displacement processes, relative permeability curves show considerable curvature.

•However, for compositional studies, such as multiple contact miscibility, as the fluid approaches to the critical point, the interfacial tension approaches to zero causing:

residual gas and oil saturations to approach zero,relative permeability curves to approach straight line.

•Therefore, gas oil-relative permeability is no longer constant and a relationship must be found to govern this change.

Irre

duci

ble

Sat

urat

ion Res

idua

lS

atur

atio

nkro

krg

Definition - Surface Tension is the stress at the surface between a liquid and a vapor caused by differences between the molecular force in the vapor and those in the liquid and by the imbalance of these forces at the interface.

Surface tension generally expressed as a pressure difference in a capillary tube or capillary pressure

= surface tension (dynes/cm)

Know:

= contact angle between liquid and solid

r = radius of tube, cm

cc rg

P cos2

Surface Tension Effects

• Parameter called Parachor is predicted from structure of molecules or computed for pure substances or for mixtures from surface - tension measurements


.......

//

//

//

333

222

1114/1

VVLL

VVLL

VVLL

MdyMdxP

MdyMdxP

MdyMdxPWeinaug and Katz

Where

P = Parachor for any component of mixture

xi = mole fraction in liquid phase

dL = density of liquid phase, g/cc

ML = molecular weight, liquid phase

yi = mole fraction in vapor phase

dV = density of vapor phase, g/cc

MV = molecular weight, vapor phase

1,2,3 .... component number

in dynes/cm

Weinaug and Katz

0

0

1000

400

300

200

100

400300200100

900

800

700

600

500

Molecular Weight

Parachors for hydrocarbons.(Katz, Monroe, and Trainer)

Par

acho

r

Crude Oil

n - Paraffin

Heptane plus

Gasoline


Surface Tension Effects Component Parachor

n-Pentane

n-Hexane

n-Heptanen-OctaneEthylene

AcetylenePropyleneHydrogen

Nitrogen

EthanePropaneIsobutanen-ButaneIsopentane

Methane 77.0

189.9

225.0

231.5

271.0

312.5

351.5100.1

88.6

139.9

34 approx.41 approx.

108.0

150.3181.5

In compositional model can have

Oil

W

Gas

W

direct transition fromoil to gas from onetime step to another

t=t1+tt=t1

Treatment of Near Critical Oil and Gas Relative Permeabilities

Question: What should hydrocarbon relative permeability be?

rwrg1

rwro1

K and K :Δtt

K and K :t


Temperature

Pre

ssur

eReservoir Temperature

Phase Envelopes

Tcrit

Gas

Tcrit

Oil

At t1 we have an oil, at t1 + dt we have a gas.

t1

t1 + dt


Answer - There must be continuity between

hydrocarbon relative permeability Kro when

system is an oil, Krg when system is a gas, for

any Sw.


Solution - Define the water - hydrocarbon rel. perm. Krh when have single phase HC system

Krh

Krh Kro as system oilKrh Krg as system gas

To set up interpolation - define pseudo-critical temp. Tcrit

(Li correlation)

compcc

compccc

crit

ZV

ZVT

T


Then define: f = Tcrit / Tres

f = 1 res.temp = crit temp.

f > 1 for oil Tcrit > Tres

f < 1 for gas Tcrit < Tres

System completely oil when fo = 1.25

completely gas when fg = 0.75


Define interpolating function

E= (f-fg) / (fo-fg) in region fg < f < fo

E = 0 or 1 outside range

0

.75 1.251

E

f

1


Hydrocarbon - water rel. perm. defined

rgwrowrhw KEKEK 1Continuous User input

Krow Krgw


3 phases: oil, gas, water

(default Eclipse method)

owcwwcw

grgowrhwrg

gwcwwcw

orowwrhwro

SSSSSh

SKhShKK

SSSSSg

SKgSgKK

/ where

1

/ where

1


Capillary Number Model

The CN has two effects on the gas and oil relative permeability

As CN increases

•Residual saturation reduces

•Relative permeability curve changes from user-specified (immiscible) to miscible curve (internally generated)

jrvjjcj

jrvjcj

ggcj

PKKSN

orL

PKKN

or

vN

2/12

Three models the calculate CN for phase j (oil, gas)

= gas-oil surface tension


Once the phase CN has been calculated

Calculate Normalized Capillary Number (NCN)

cj

cbjcnj N

NN

Where Ncbj is Base Capillary Number (BCN) = lower threshold value below which the CN has no effect of the phase relative permeabilities.

BCN should be determined experimentally or E300 will estimate


Effect of CN

Kro

Krg

Soil

Immiscible

Soil

Kro

Krg

Miscible

CN at or below BCN Very high CN


Effect of CN on Residual Saturation

As CN rises above the BCN, the residual saturation is reduced

Use saturation scaling parameter Xj

Must know from Special Core Analysis Srbj , a phase residual saturation and parameter mj

cnjj Nmj

rbjjrbj

eX

where

SXS

1

If mj is set to zero phase residual saturation in zero and independent of CN.


Dia is the activity-corrected diffusion coefficient.

ii

iai xf

DD

ln/ln Diffusion

Thus

Diffusion

Diffusion coefficients

Use diffuse flow relationship:

c is the total molar concentration, given by

c = 1 / Vm

Vm the molar volume of mixture.

Ji is the molar flux of component i per unit area

Di is the diffusion coefficient of component i

is the mole fraction gradient of component i

d

xcDJ i

ii

d

xi

Diffusion

At high pressures, concentration gradient is not most accurate predictor for diffusion.

Component chemical potential should be used. Can then include effect of gravity.

M = Molecular Weight

G = Gravity Constant

h = elevation

00 ln hhGMfRT ii

Diffusion

For isothermal, horizontal flow and using , RT, let Di

a be defined by

d

fxcDJ i

iaii

ln

d

x

xx

f

d

f i

ii

ii

1

ln

lnlnExpanding So that

d

x

x

fDcJ i

i

iaii

ln

ln

d

xcDJ i

iicomparing

DiffusionTwo possibilities exist in E300:

1. Use normal diffusion coefficients and mole fractions as the driving force Normal diffusion coefficients are entered using DIFFCOIL

and DIFFCGAS.

2. Use activity corrected diffusion coefficients and chemical potential as the driving force. Activity corrected diffusion coefficients are entered using

DIFFAOIL and DIFFAGAS.

DiffusionAt low pressures, the two coefficients are equal,

as fixiP

With both oil and gas present, the molar concentrations include the saturation and porosity, so that

igioi JJJ

dxDbSJ

d

RTGhMfxDbSJ

iiomooio

iii

aio

mooio

/or

/ln

with

DiffusionNote that separate oil and gas diffusion

coefficients may be defined, Dio and Dig.

Often only gas coefficients is non-zero.

DiffusionThe diffusive flow between grid blocks becomes:

RTGhMfbSxDTF iim

ooia

ioDdiff

io /ln

RTGhMfbSyDTF iim

ggia

igDdiff

ig /ln

Combinations such as (xiSobom) are defined on the cell

and treated using up-steam weighting.

Grid Orientation• Yanosik and McCracken reported on a nine-point finite

difference scheme that would reduce grid orientation effects in high mobility ratio displacements.

i,j

i,j-1

i

j

i+1,j

i+1,j-1i-1,j-1

i+1,j+1i,j+1

i-1,j

i-1,j+1

normal 5-pointtransmissibilities

diagonal transmissibilitiesadded for 9-point

Diagonal and Parallel Grids for Class Problem

As expected from the saturation distribution shapes shown the diagonal grid has injected fluid breakthrough at a later time and recovers more oil than the parallel grid model.

Water-Cut

time

parallel

diagonal

Carbon Dioxide Solution in the Aqueous Phase

When CO2 is injected into a oil reservoir a significant amount of the CO2 will dissolveinto the water.

In a WAG injection the solution of the CO2

can be a very important factor.

To model this in the compositional model,fugacities for the CO2 in the three phasesmust be calculated for the 3 phase flash.


The basic model is a fugacity function for aqueous CO2 which is constructed to match solubility data in the form:

PaPf COA

CO 22

Function (P) is constructed by considering a pureCO2/aqueous mixture - gas phase fugacity is obtained from EOS.


Phase equilibrium between aqueous CO2 and hydrocarbon phases is defined by the conditions that fugacity values are equal.

In E300:

CO2SOL turns on the option.

SOLUBILITY keyword allows the input of non-default properties.


SOLUBILITY

Pressure Rs(CO2) Bwater Viscosity Compress- Saturated ibility mole Psia Mscf/stb RB/Mscf CP 1/Psi fraction

14.6959 0.0021 1.02802 0.30000 0.00000272 0.00028895 787.3909 0.0836 1.04759 0.30000 0.00000272 0.01120103 1560.0858 0.1268 1.05784 0.30000 0.00000272 0.01689270 2332.7808 0.1511 1.06357 0.30000 0.00000272 0.02006145 3105.4757 0.1656 1.06700 0.30000 0.00000272 0.02195732 3878.1707 0.1752 1.06923 0.30000 0.00000272 0.02318996 4650.8656 0.1820 1.07083 0.30000 0.00000272 0.02407221 5423.5606 0.1874 1.07210 0.30000 0.00000272 0.02477146 6196.2555 0.1921 1.07320 0.30000 0.00000272 0.02538009 6968.9505 0.1967 1.07426 0.30000 0.00000272 0.02596283 7741.6454 0.2012 1.07532 0.30000 0.00000272 0.02654488 8514.3404 0.2057 1.07637 0.30000 0.00000272 0.02712623 9287.0353 0.2103 1.07743 0.30000 0.00000272 0.02770689 10059.7303 0.2148 1.07848 0.30000 0.00000272 0.02828686 10832.4252 0.2193 1.07953 0.30000 0.00000272 0.02886613 11605.1202 0.2238 1.08058 0.30000 0.00000272 0.02944471 12377.8151 0.2284 1.08164 0.30000 0.00000272 0.03002261 13150.5101 0.2329 1.08269 0.30000 0.00000272 0.03059981 13923.2050 0.2374 1.08374 0.30000 0.00000272 0.03117633 14695.9000 0.2420 1.08478 0.30000 0.00000272 0.03175217

--------------------- ----------- ----------- ----------- -----------

discussion on some aspects of compositional simulation

Documents

components gas

reservoir oil

gas phases

dissolved gas

solution gasoil ratio

residue gas

gas viscosity

injected gas