review of basic steps in derivation of flow equations sig4042 reservoir simulation

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW

EQUATIONS

SIG4042 Reservoir Simulation

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Conservation of Momentum

Black Oil Model

Conservation of Mass

Constitutive Equations

Questions

Flow Equation

Boundary and Initial Conditions

Multiphase Flow

Non-horizontal Flow

Mutlidimensional Flow

Coordinate Systems

Boundary and Initial Conditions of Multiphase Systems

Nomenclature

Introduction

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Contents



Flow Equation

Black Oil Model



Multiphase Flow

Non-horizontal Flow


Coordinate Systems

Boundary and Initial Conditions of Multiphase

Systems

Questions

Introduction

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Black Oil Model



Questions

Flow Equation


Multiphase Flow

Non-horizontal Flow


Coordinate Systems


Nomenclature

Introduction

IntroductionIntroduction

Flow equations for flow in porous materials are based on a set of mass, momentum and energy conservation equations, and constitutive equations for fluids and the porous material.

For simplicity, we will in the following assume isothermal conditions, so that we not have to involve an energy conservation equation. However, in cases of changing reservoir temperature, such as in the case of cold water injection into a warmer reservoir, this may be of importance.

Equations are described for linear, one-dimensional systems, but can easily be extended to two and three dimensions, and to other coordinate systems.

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Introduction

Conservation of MassConservation of mass

Again we will consider the following one dimensional slab of porous material:

fluid

x

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Flow Equation


Multiphase Flow

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Introduction

Conservation of MassMass conservation may be formulated across a control element of the

slab, with one fluid of density , flowing through it at a velocity u:

x

u

The mass balance for the control element is then written as:

element theinside

mass of change of Rate

Dx+at xelement

theofout Mass

at xelement

theinto Mass

Continue

uA x uA

xxt

Ax

or

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Introduction


uA x uA

xxt

Ax

Dividing equation above by x, and taking the limit as x goes to zero, we get the conservation of mass, or continuity equation:

t

AuAx

t

ux

For constant cross sectional area, the continuity equation simplifies to:

Continue

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Flow Equation


Multiphase Flow

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Nomenclature

Introduction

Conservation of MomentumConservation of momentum

Conservation of momentum is governed by the Navier-Stokes equation, but is normally simplified for low velocity flow in porous materials to be described by the semi-empirical Darcy's equation, which for one dimensional, horizontal flow is:

x

Pku

Continue

Alternative equations are the Forchheimer equation, for high velocity flow:

nuk

ux

P

where n is proposed by Muscat to be 2.

The Brinkman equation, which applies to both porous and non-porous flow:

2

2

x

u

ku

x

P

Brinkman's equation reverts to Darcy's equation for flow in porous media, since the last term then normally is negligible, and to Stoke's equation for channel flow because the Darcy part of the equation then may be neglected.

In the following, we assume that Darcy's equation is valid for flow in

porous media.

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Introduction

Constitutive EquationsConstitutive equation for porous materials

To include pressure dependency in the porosity, we use the definition of rock compressibility:

Tr P

c

1

Keeping the temperature constant, the expression may be written:

rcdP

d

Normally, we may assume that the bulk volume of the porous material is constant, i.e. the bulk compressibility is zero. This is not always true, as witnessed by the subsidence in the Ekofisk area.

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Introduction

Constitutive EquationsConstitutive equations for fluids

Recall the familiar fluid compressibility definition, which applies to any fluid at constant temperature:

Tf P

V

Vc

1

Equally familiar is the gas equation, which for an ideal gas is:

PV nRTFor a real gas includes the deviation factor, Z:

PV nZRTThe gas density may be expressed as:

g gSP

Z

ZS

PS

where the subscript S denotes surface (standard) conditions.

Continue

These equations are frequently used in reservoir engineering applications.

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Flow Equation


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Introduction

Description of Black Oil model

For reservoir simulation purposes, we normally use either so-called Black Oil fluid description, or compositional fluid description. For now, we will consider the Black Oil model, and get back to compositional model later on.

Black Oil Model

conditions standardat volume

conditionsreservoir at volumeB

conditions standardat oil of volume

conditions standardat oil from evolved gas of volumesoR

Continue

The density of oil at reservoir conditions is then, in terms of these parameters and the densities of oil and gas, defined as:

o oS gsRso

Bo

Continue

The standard Black Oil model includes Formation Volume Factor, B, for each fluid, and Solution Gas-Oil Ratio, Rso, for the gas dissolved in oil, in addition to viscosity and density for each fluid. A modified model may also include oil dispersed in gas, rs, and gas dissolved in water, Rsw. The definitions of formation volume factors and solution gas-oil ratio are:

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Black Oil Model



Questions

Flow Equation


Multiphase Flow

Non-horizontal Flow


Coordinate Systems


Nomenclature

Introduction

Black Oil ModelTypical pressure dependencies of the standard Black Oil parameters are(click on buttons):

P

Bw

P

Bg

Bw vs. P Bg vs. P

P

Bo

Bo vs. P

P

w

w vs. P

P

g

g vs. P

P

o

o vs. P

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Black Oil Model



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Flow Equation


Multiphase Flow

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Introduction

Black Oil ModelTypical pressure dependency of the Solution Gas-Oil Ratio in Black Oil model is(click on button):

P

Rso

Rso vs. P

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Black Oil Model



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Flow Equation


Multiphase Flow

Non-horizontal Flow


Coordinate Systems


Nomenclature

Introduction

Flow EquationFlow equation

For single phase flow, in a one-dimensional, horizontal system, assuming Darcy's equation to be applicable and that the cross sectional area is constant, the flow equation becomes:

Btx

P

B

k

x

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Flow Equation


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Introduction

Boundary and Initial ConditionsDirichlet conditions

When pressure conditions are specified, we normally would specify the pressures at the end faces of the system in question. Applied to the simple linear system described above, we may have the following two pressure BC's at the ends:

R

L

PtLxP

PtxP

0,

0,0

For reservoir flow, a pressure condition will normally be specified as a bottom-hole pressure of a production or injection well, at some position of the reservoir. Strictly speaking, this is not a boundary condition, but the treatment of this type of condition is similar to the treatment of a boundary pressure condition.

More

Boundary conditions

We have two types of BC's: pressure conditions (Dirichlet conditions) and rate conditions (Neumann conditions). The most common boundary conditions in reservoirs, including sources/sinks, are discussed in the following.

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Boundary and Initial ConditionsNeumann condition

Alternatively, we would specify the flow rates at the end faces of the system in question. Using Darcy's equation at the ends of the simple system above, the conditions become:

For reservoir flow, a rate condition may be specified as a production or injection rate of a well, at some position of the reservoir, or it is specified as a zero-rate across a sealed boundary or fault, or between non-communicating layers.

More

0

x

L x

PkAQ

LxL x

PkAQ

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Boundary and Initial ConditionsInitial condition (IC)

The initial condition specifies the initial state of the primary variables of the system. For the simple case above, a constant initial pressure may be specified as:

The initial pressure may be a function of postition. For non-horizontal systems, hydrostatic pressure equilibrium is normally computed based on a reference pressure and fluid densities:

Continue

P(x, t 0) P0

P(z,t 0) Pref (z zref )g

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Multiphase FlowMultiphase flow

A continuity equation may be written for each fluid phase flowing:

The corresponding Darcy equations for each phase are:

Continue

llll St

ux

x

Pkku l

l

rll

gwol ,,

gwol ,,

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Black Oil Model



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Flow Equation


Multiphase Flow

Non-horizontal Flow


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Nomenclature

Introduction

Multiphase FlowThe continuity equation for gas has to be modified to include solution

gas as well as free gas, so that the oil equation only includes the part of the oil remaining liquid at the surface:

Where oL represents the part of the oil remaining liquid at the surface (in the stock tank), and oG the part that is gas at the surface.

Continue

o oS gSRso

Bo

oL oG

ooGggooGgg SSt

uux

ooLooL St

ux

Thus, the oil and gas continuity equations become:

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Black Oil Model



Questions

Flow Equation


Multiphase Flow

Non-horizontal Flow


Coordinate Systems


Nomenclature

Introduction

Multiphase FlowAfter substitution for Darcy's equations and Black Oil fluid properties,

and including well rate terms, the flow equations become:

The oil equation could be further modified to include dispersed oil in the gas, if any, similarly to the inclusion of solution gas in the oil equation.

Continue

o

oso

g

gosog

o

o

roso

g

gg

rg

B

SR

B

S

tqRq

x

P

oB

kkR

x

P

B

kk

x

Where:wocow PPP

ogcog PPP

Sl

lo, w, g 1

w

ww

w

ww

rw

B

S

tq

x

P

B

kk

x

o

oo

o

oo

ro

B

S

tq

x

P

B

kk

x

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Non-horizontal FlowNon-horizontal flow

For one-dimensional, inclined flow:

the Darcy equation becomes:

Continue

x

Du

dx

dDg

x

Pku

or, in terms of dip angle, , and hydrostatic gradient:

sinx

Pku

where =g is the hydrostatic gradient of the fluid.

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Black Oil Model



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Flow Equation


Multiphase Flow

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Introduction

Multidimensional FlowMultidimensional flow

The continuity equation for one-phase, three-dimensional flow in cartesian coordinates, is:

The corresponding Darcy equations are:

More

t

uz

uy

ux zyx

x

D

x

Pku x

x

y

D

y

Pku y

y

z

D

z

Pku z

z

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Black Oil Model



Questions

Flow Equation


Multiphase Flow

Non-horizontal Flow


Coordinate Systems


Nomenclature

Introduction

Coordinate SystemsCoordinate systems

Normally, we use either a rectangular coordinate system or a cylindrical coordinate system in reservoir simulation (click the buttons):

x

Rectangularcoordinates

y

z

r

Cylindricalcoordinates

z

r

Sphericalcoordinates

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Flow Equation


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Introduction Boundary and Initial Conditions of Multiphase Systems

Boundary conditions of multiphase systems

The pressure and rate BC's discussed above apply to multiphase systems. However, for a production well in a reservoir, we normally specify either an oil production rate at the surface, or a total liquid rate at the surface. Thus, the rate(s) not specified must be computed from Darcy's equation. The production is subjected to maximum allowed GOR or WC, or both. We will discuss these conditions later.

See a picture

O ILOIL

GAS

WATER

Z

GOC

WOC

AQUIFER

BC:

1) Pbh = constant

2) Q = constant

BC:

1) Pbh = constant

2) Qinj = constant

BC:

k = 0 0

x

PkAq

BC:

q = 0

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Flow Equation


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Introduction Boundary and Initial Conditions of Multiphase Systems

Initial conditions of multiphase systems

In addition to specification of initial pressures, we also need to specify initial saturations in a multiphase system. This requires knowledge of water-oil contact (WOC) and gas-oil contact (GOC). Assuming that the reservoir is in equilibrium, we may compute initial phase pressures based on contact levels and densities. Then, equilibrium saturations may be interpolated from the capillary pressure curves. Alternatively, the initial saturations are based on measured logging data.

O IL OIL

GAS

WATER

Z

GOC

WOC

See a picture

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Flow Equation


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Introduction

Questions1) Write the mass balance equation (one-dimentional, one-phase).

Next2) Write the most common relationship between velocity and

pressure, and write an alternative relationship used for high

fluid velocities.

3) Write the expression for the relationship between porosity

and pressure.

4) List 3 commonly used expressions for relating fluid density to

pressure.

5) Describe briefly Black Oil model.

6) Sketch typical dependencies of the standard Black Oil

parameters.

7) Write Darcy equation for one-dimentional, inclined flow.

8) Write continuity equation for one-phase, three-dimensional

flow in cartesian coordinates.

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Introduction

A - area, m2

B - formation volume factor

c - compressibility, 1/Pa

k - permeability, m2

kr - relative permeability, m2

L - lenght, m

N - number of grid blocks

n - number of moles

O(...) - discretization error

P - pressure, Pa

Pc - capillary pressure, Pa

Q, q - flow rate, Sm3/d

R - gas constant

Rso - solution gas-oil ratio

r - radius, m

S - saturation

T - temperature, K

t - time, s

u - Darcy velocity, m/s

V - volume, m3

x - distance, m

x, y, z - spatial coordinate

Z - deviation factor

- angle

x - lenght of grid block, m

t - time step, s

- porosity

- viscosity, Pa·s

- density, kg/m3

Subscripts:

0 - initiall value

e - end of reservoir

f - fluid

g - gas

i - block number

L - left side

l - liquid

o - oil

R - right side

S - surface (standard) conditions

r - rock

w - water

w - well

Nomenclature

Back to presentation

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General information

About the author

Title: Review of Basic Steps in Derivation of Flow Equations

Teacher(s): Jon Kleppe

Assistant(s): Szczepan Polak

Abstract: Review of basic steps in derivation of flow equations for flow in porous materials based on set of mass, momentum and energy conservation equations, and constitutive equations for fluid and porous material in isothermical conditions.

Keywords: conservation equations, constitutive equations, black oil model, multiphase flow, non-horizontal flow, multidimensional flow, coordinate systems

Topic discipline: Reservoir Engineering -> Reservoir Simulation

Level: 4

Prerequisites: Good knowledge of reservoir engineering

Learning goals: Learn basic principles of Reservoir Simulation

Size in megabytes: 0.9

Software requirements: -

Estimated time to complete: 60 minutes

Copyright information: The author has copyright to the module

Thor A. Thorsen

incling sound, video, animation files

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for example Geophysics -> Processing

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0 is very easy – 4 is most difficult

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in minutes

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Individual learning or project-based learning

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Title with reference to version number starting with 1.0

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Flash-player, etc.

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With link to an web-site with contact information, taks and publications

Jonny Hesthammer

Ideally, this link should be to the authors web homepage as this will ensure continuous update of adresses, publications etc. However, for those who have not yet created their own homepage we provide a separate page in this document for author information. The current link is to that page and it must be changed (right click with mouse on action button and select "edit hyperlink") to the relevant web link.

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FAQ

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References

Aziz, K. and Settari, A.: Petroleum Reservoir Simulation, Applied Science Publishers LTD, London (1979)

Mattax, C.C. and Kyte, R.L.: Reservoir Simulation, Monograph Series, SPE, Richardson, TX (1990)

Skjæveland, S.M. and Kleppe J.: Recent Advances in Improved Oil Recovery Methods for North Sea Sandstone Reservoirs, SPOR Monograph, Norvegian Petroleum Directoriate, Stavanger 1992

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Flow Equation


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Introduction

Summary

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Introduction

About the AuthorName

Jon Kleppe

Position

Professor at Department of Petroleum Engineering and Applied Geophysics at NTNU

Address: NTNU S.P. Andersensvei 15A 7491 Trondheim

E-mail: [email protected]

Phone: +47 73 59 49 33

Web: http://iptibm3.ipt.ntnu.no/

~kleppe/

review of basic steps in derivation of flow equations sig4042 reservoir simulation

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