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PETROLEUM RESERVOIR SIMULATION

Introduction to year 3 courseDr RICHARD WHEATONENG69222/09/2014211Details of 2014 2015 courseMarking: Coursework - December 2014 January 2015 = 40%, submitted 11/12/2014Examination - June 2015 = 60%

WEEKS 1 - 12Basic Rock and Fluid properties: week 1Reserves and Drive mechanisms: week 2Petroleum Economics : weeks 3-4Field Appraisal and Development Planning: weeks 5-7Analytical Methods: weeks 8-10Welltest analysis: weeks 11-12

All in Lecture Theatre BB 4.05

22Details of 2014-2015 courseWEEKS 13-25

Numerical SimulationBasic structure of numerical models: & finite difference methods week 13Input data structure: week 14Conceptual modelling: week 15Full Field modelling and History matching: week 16Unconventional Resources: weeks 17- 18Booking Reserves: week 19-20Revision: weeks 21-2532Course AidsOn Moodle

Lecture slidesNotes by me on each of the above topicsThese will cover everything in the course

Suggestions on background reading

Excel spreadsheets covering a number of simulation and analytical tools

Eclipse, Petrel and CMG information42Central role of the Reservoir Engineer

52Central role of the Reservoir Engineer

62BASIC ROCK AND FLUID PROPERTIES

There are three fundamental types of properties of a hydrocarbon reservoir.

The rock properties of porosity, permeability and rock all dependent on solid grain/particle arrangements and packing

The wettability properties capillary pressure, phase saturation and relative permeability, these are dependent on interfacial forces between the solid and the water and hydrocarbon phases

The initial ingress of hydrocarbons into the reservoir trap and the thermodynamics of the resulting reservoir mixture composition.

72Porosity

= void volume/((grain + void volume)

where Vp = pore space volume andVb = bulk volume (grain + void volume)

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Porosity

It will depend on the average shape of the solid grains and the way they are packed together. This in turn will depend on the way the rock was formed, either from sedimentation over time -for example solid grains of sand deposited gradually on river beds (clastics) or growth and decay of biological materials (carbonates).

This initial distribution of solids is then often disturbed by subsequent events which re-arranges the solids distribution affecting the porosity (digenesis) .

Typical porosity values are in the range 5 - 30 %. 15% would be a very typical value.

In reservoir engineering. We are normally only interested in interconnected porosity that is the volume of connected pores to total bulk rock volume.

29Porosity

Hydrocarbon pore volume (HCPV) is the total reservoir volume that can be filled with hydrocarbons HCPV = Vb.. (1- Swc) where V= bulk rock reservoir volume and Swc = connate (or irreducible) water saturation as a fraction of pore space.As pressure decreases with hydrocarbon production, rock particles will tend to pack closer together so that porosity will decrease somewhat as a function of pressure. This is known as rock compressibility cr where

where Vp = Vb. = pore volume

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10Porosity

Porosity is very variable in its nature, changing over quite small distances within a reservoir, and even if two samples have the same porosity, it does not mean that they will have the same absolute permeability or the same wettability characteristics which in turn means that they can have very different capillary pressure and relative permeability properties.

The key factors here are the average pore geometry and the polar -non-polar nature of the rock itself.

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Net to Gross (NTG)Some regions of a reservoir are often considered to have such low porosity and transport potential as to be effectively non-reservoir. They are therefore left out of estimated reservoir volume and considered as 'dead' rock. So for example net thickness of a formation is = average gross thickness x NTG. 212Permeability

We can derive the Darcy equation from first principals if various simplifying assumptions are made.From first principals - with Conservation of Momentum, we can consider a volume element (V) of fluid moving through porous material. V will have a different position at time t + t compared with that at time t and it may also have a different volume; see below:

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For a volume element if we assume a steady state situation and ignore inertial effects (i.e flow rates are relatively small) we can derive the equation from conservation of momentumPermeabilityRepresents a balance of average forces within a volume element, arising during the steady-state flow of a single phase fluid through fractured material. The first term on the left-hand side represents the force due to any pressure gradient, whilst The second term represents frictional forces due to viscosity of the fluid. The first right-hand term describes frictional forces due to the solid rock matrix (i.e friction between moving fluid and the rock matrix) which will be much greater than those due to viscous effects within the fluid itself; we therefore neglect the second term on the left-hand side, the second term on the right hand side represents body forces (i.e. gravity). We therefore obtain the following relationship

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PermeabilityThis can be simplified to

if we assume that k = Kg where Kg is a geometric constant and d is a 'characteristic length' for the porous material and = porosity. Rearranged this gives the standard form of the Darcy equation

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Laboratory Determination of PermeabilitySingle phase absolute permeability is measured on core in a steel cylinder where pressures P1 and P2 are measured for a given gas flow rate Q

For a gas: from Darcy's law for horizontal flow

For an incompressible liquid: for horizontal flow

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Permeability from Well test analysisFor a constant production flow rate Q, permeability can be estimated from average formation thickness h, fluid viscosity , bottom hole pressure Pw, initial reservoir pressure Pe at an assumed undisturbed (still at initial conditions) distance re from the well. wellbore radius is rw using the equation217

Darcy equationIn field units the Darcy equation will be:

wherek in milliDarcies (mD) u is in RB/day/ft2dp/dx is in psi/ft is in centipoise (cP) is specific gravity (dimensionless)

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Permeability variation in a reservoirPermeability measured from core is obviously very local this changes continually across a reservoir depending on depositional and subsequent re-arrangement effects. A reservoir can be divided into what are called 'Flow Units which have common permeability (and hence flow) characteristics. We saw above that:k = Kg The permeability of each flow unit will depend on the average characteristic length (d), geometic constant (KG) and porosity in that unit.If we have a set of hydraulic unit type systems with common geometric features and that KG. d2 is approximately a constant for each with respect to porosity, we will have a range of permeability - porosity relationships like that shown below:

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Wettability

When two immiscible fluids are in contact with a solid surface, one will tend to spread or adhere to the solid more than the other. This is the results of a balance of intermolecular forces and surface energies between fluids and the solid. This is shown below where vector forces are balanced at the oil-water-solid contact point giving the relationshipos - ws = ow cos cwhere os = the interfacial tension between oil and solidws = the interfacial tension between water and solidow = the interfacial tension between oil and water andc = the contact angle between water and oil at contact point measured through the water

220Wettability

Where c < 90o a system is known as 'water wet' so that water will tend to spread on the solid surfaceand where c > 90o a system is known as 'oil wet wet' where oil will spread on the solid surface

Wettability will control the distribution of oil and water in pore space. In water wet systems oil will tend to be found in the centres of pores whilst in oil wet systems oil will be retained around the solid grains. This will of course have a fundamental effect on oil recovery in waterflooding

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WettabilityWettability is fundamental in determining capillary pressure and relative permeabilityHysteresisThe history of the porous rock (in terms of the history of the phases -water-oil or gas) that have occupied the pore space) will have a strong effect on its wettability, this is known as 'hysteresis'

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Capillary PressureCapillary pressure is the pressure difference existing across an interface between two immiscible fluids so for an oil-water systemPcow = po pw It will depend on the average water/oil/rock contact angle () and the average pore space radius (r) Therefore capillary pressure is a function of both average wettability and average pore size

If we take an example section of pore space:

223Capillary PressureIt can be shown that the capillary pressure between the wetting and non-wetting phases is given by:Pcnw-w = 2.KG nw Cos()/rwwhere KG is a geometric constant dependent on the average geometry of the pore space, nw is the oil/water interfacial tension and rw is the average radius of wetting phase occupied pores.

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Capillary PressureThis wetting phase radius rw will be a function of the saturation of the wetting phase. The smaller the wetting phase saturation the more it will be concentrated in smaller pores so that rw will be small and hence the capillary pressure will be larger. The smaller the interfacial tension between the two phases the smaller will be the capillary pressure.A drainage (decreasing wetting phase) capillary pressure curve is shown below. Pb is the threshold pressure = the minimum pressure required to initiate drainage displacement. Capillary pressure increases as wetting phase saturation decreases225

Reservoir Saturation with depthThe major importance of capillary pressure is its effect on the distribution of phases in the reservoir with depth.The reservoir will have initially been filled with water (Sw = 100%). Oil migrating up from below (oil having a lower density than water) will have gradually displaced water - a drainage process 226

Reservoirs with a Gas Cap

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Relative Permeability

If we have more than one fluid flowing simultaneously through a porous medium each has its own effective permeability that will depend on the saturation of each fluidke = k. kr Where ke = effective permeabilityk = absolute permeabilitykr = relative permeability

For phase where kr = f(S)Relative permeability is a very important parameter in reservoir engineering but it is unfortunately a very difficult function to measure in the laboratory and then to relate to what is likely to happen in the reservoir.

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Oil-water systemsAt connate (or irreducible) water saturation (Swc), oil relative permeability is at its maximum (krom). As water saturation increases (imbibition) oil relative permeability decreases and water relative permeability increases until no more oil can be displaced by water at which point oil saturation = Sor (irreducible oil saturation) and water saturation Sw = 1 -Sor. At this point water relative permeability is at a maximum (krwm). The broken lines outside this region from Sw=Swc to Sw = 1-Sor are not reached in the reservoir but can correspond to some laboratory experiments. The situation in the reservoir during production will normally be an imbibition as water either from an aquifer or from water injection displaces oil driving it towards the producing wells229

Water wet and oil wet systemsThe crossover point for the oil and water relative permeability curves is indicative of the water wet or oil wet nature of the porous material. Where the crossover occurs with Sw < 0.5 we will have an oil wet system. When crossover occurs with Sw > 0.5 this would normally be considered a water wet system.

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Correlations for oil/water relative permeabilities231

Rangekromaxkrwmax

SwcSocNonwMax0.90.60.30.43.03.0Min0.80.30.10.22.02.0average0.850.450.20.32.52.5Spreadsheet for oil-water rel perms232

Excel spreadsheet: oil-water rel perms-sv32revisionLook at effects of all parameters on relative permeability 233