Society of Petroleum Engineers Distinguished Lecturer Program - PowerPoint Presentation

Society of Petroleum Engineers Distinguished Lecturer Programwww.spe.org/dl USING FRACTALS TO DETERMINE A RESERVOIR’S HYDROCARBON DISTRIBUTION Steve Cuddy

Outline How we determine a reservoir’s hydrocarbon distributionWhy fractals make this easyDemonstrated using several case studies

Why we need a reservoir modelThe 3D reservoir model is required to calculate hydrocarbon in place and for dynamic modelling The model requires fluid contacts, net reservoir cut-off and a water saturation vs. height function Limited core and electrical log data available at the well locations

What are Fractals?

Fractals on the Small ScaleSnowflakesRoman Cauliflower

Fractals on the Big ScaleRiver channelsHimalayas

Fractals on the Really Big ScaleThe cosmic microwave background is scale invariantIf we zoom in the patterns are indistinguishableThese patterns give rise to galactic superclusters Galactic superclusters are built up from galaxiesThe universe is fractal Prof. Brian Cox – ‘Forces of Nature’ 2016

What are Fractals?A fractal is a never-ending patternFractals are infinitely complex patterns that look the same at every scaleThey are created by simple repeating processBenoit B. Mandelbrot setOther names for fractals are Self-similarity Scale invariance

Why Fractals are UsefulFractals are objects where their parts are similar to the whole except for scaleA simple repeating process can create a complex objectMany complex objects can be described by fractals Mathematically simple

How to verify if something is fractalCoastlines show more detail, the closer you zoom inThe length Great Britain’s coastline (N) depends on the length of your ruler (r)Ruler size decreasing in size (i.e. 1/r) North Sea

Coastline Fractals As the ruler shrinks the measured coastline increasesIf the coastline is fractal the relationship between r and N is linear when plotted using log scalesD = fractal dimension = gradient of the lineLog (r)Log (N) Ruler size decreasing in size (i.e. 1/r) North Sea

Pixel Size (smaller) Porosity (No. pixels)Fractals in reservoir rocksThin sections of reservoir rocks are imaged with a scanning electron microscope (SEM )For different magnifications the number of pixels representing porosity are countedBerea sandstone

The Bulk Volume of Water (BVW)Bulk Volume of Water = Porosity x Water SaturationB V W = % volume of water in a unit volume of reservoirThis is what is measured by electrical logs and by core analysis

The Free Water Level (FWL)FWL is the horizontal surface of zero capillary pressure

Hydrocarbon Water ContactThe HWC is the height where the pore entry pressure is sufficient to allow hydrocarbon to start invading the formation pores This depends on the local porosity & permeability It is a surface of variable height 0 Water Saturation 1 Height above FWL Hydrocarbon Water Contact Free Water Level

Used to initialize the 3D reservoir modelTells us how water saturation varies as a function of the height above the Free Water Level (FWL)Tells us how the formation porosity is split between hydrocarbon and waterTells us the shape of the transition zoneThe reservoir model needs a Sw vs. Height Function 0 Water Saturation (%) 100 Height above FWL (Feet)FWL Water Hydrocarbon Water saturation vs. Height Function

What a Good Saturation Height Function RequiresThree independent sources of fluid distribution data are consistent- Formation pressure data - Electrical log data- Core dataMust account for varying permeability and fluid contactsMust upscale correctlyShould be easy to apply

The Structure and Electrical Properties of WaterThe water molecule is made up of 2 atoms of hydrogen and 1 atom of oxygenThe water molecule is polarized with distinct negative (oxygen) and positive (hydrogen) endsThis causes water molecules to be strongly attracted to each other and to reservoir rocks The electrostatic force is 1036 times greater than the gravitational force H20

Buoyancy Forces in Reservoir FluidsWater is in the reservoir firstWhen hydrocarbons migrate into a trap, the buoyancy force exerted by the lighter oil (or gas) will push the water that was previously in the pore space downwardHowever, not all of the water is displaced; some of it will be held by capillary forces within the poresNarrower capillaries, pores with smaller pore throats, with the larger surface area, hold onto the water the strongest

Forces Acting on Reservoir FluidsThe water at a given height in a reservoir is determined by the balance between the capillary forces holding the water up to the force of gravity pulling the water downThe oil (or gas) is the mobile phase and only enters the leftover space in the reservoir poresConsequently a given part of the pore space within the reservoir will contain both oil and waterThe percentage of water in the pore space is called the water saturation (Sw)Height Sw vs. Height Function OilWaterFree Water Level Water Saturation

Capillary Pressure holds the Water upWhen two fluids meet in a capillary tube there is a difference in pressure across their interface. This "Capillary Pressure" is caused by the preferential wetting of the capillary walls by the water and gives rise to the familiar curved meniscus and causes the water to rise up the capillaryWater Air

Capillary Pressure and Pore Size The smallest pores (throats) hold on to the most water Hydrocarbon requires more pressure to enter small pores

Fractals describe the rock pore networkThe rock pore space can be described by the fractal formula Where: V Pore space in rock volume r Radius of the rock capillaries Df Fractal dimension (non-integer constant) This reduces to the fractal Swh function Where: BVW Volume of capillary bound water in the rock Height above the free water level & Constants  

Water Saturation vs. Height Data0 Water Saturation (%) 100400 HeightAboveFWL (feet) 0Source – Southern North Sea Gas fieldWhat do we see in the well data?

Classical Water Saturation vs. Height Curves0 Water Saturation (%) 100400 HeightAboveFWL (feet) 0 Increasing Porosity < Porosity bands >

Problems with Classical Swh Functions Sufficient data are required for each porosity band Defining the pore entry pressure (threshold height) can be difficult Visually and mathematically unconvincing0 Water Saturation (%) 100400 HeightAboveFWL (feet) 0 Increasing Porosity < Porosity bands >

Water Saturation vs. Height Data0 Water Saturation (%) 100400 HeightAboveFWL (feet) 0Source – Southern North Sea Gas fieldOnly net reservoir data is plotted

Bulk Volume of Water vs. Height Data0 Bulk Volume of Water (%) 15400 HeightAboveFWL (feet) 0 Matrix Oil Water < Porosity > < BVW > Only net reservoir data is plotted

BVW is Independent of Rock Properties0 Bulk Volume of Water (%) 15400 HeightAboveFWL (feet) 0The bulk volume of water is independent of rock properties Can be verified by simply plotting facies-type, porosity or permeability on the z-axis (colour) on the cross-plot

Required for upscaling reservoir model parametersNet Reservoir - The portion of reservoir rock which is capable of storing hydrocarbon - Relatively easy to pick - Usually based on a porosity cutoff Net Pay - “The portion of reservoir rock which will produce commercial quantities of hydrocarbon”- Often used to select perforation intervals - Very difficult to pick - Depends on the oil price?Net Reservoir Cut-off

What the fractal function tells us about net reservoir Bulk Volume of Water = Function (Height above the FWL) The BVW fractal function gives the net reservoir cutoffIn this example: porosity > 9 porosity units

Net Reservoir Cut-off Net reservoir porosity cut-offHeight above the FWLFree Water Level Net reservoir is defined as the rock capable of holding hydrocarbon The net cut-off is required for averaging porosity, permeability and water saturations in the reservoir model The net reservoir cut-off varies as a function of height above the FWL

The Net Reservoir cut-off varies as a function of height above the free water level (FWL)Reservoir high above the FWL has low saturations of capillary bound water and hydrocarbon enters the smaller pores Reservoir just above the FWL, with higher porosities, contains high saturations of capillary bound water and there is a no room available for hydrocarbonsNet Reservoir ExampleNet Porosity 25 pu 0 Sand Shale Gas FWL

Where:= Bulk Volume Water (Sw*Phi) = Height above FWL, = Constants  The Fractal Water Saturation vs. Height FunctionDerived from the fractal nature of reservoir rocks Based on the bulk volume of water (BVW)Independent of facies type, porosity and permeabilityTwo parameters completely describe your reservoir Bulk of Volume of Water Height above the FWL Free Water Level >

The Fractal Function is easily calculated BVW = a H bThe BVW function is a straight line when plotted on log scaleslog BVW= log a + b log HOnly 2 valid core or electrical log points required to calculate the constants ‘a’ & ‘b’

Sw vs. Height Modelling Required to initialise the reservoir modelLondon Petrophysical Society- North Sea supplied dataVery difficult data setHeterogeneous formationSw increases with height!Log Derived Water Saturations 1 0 Porosity 25 pu 0 Permeability 0.01 mD 1000

Good match in all litho-facies typesPermeability not required Defined by only 2 parametersDo we always need resistivity logs?Fractal & Log Derived Water Saturations 1 0 Porosity 25 pu 0 Porosity 25 pu 0 Permeability 0.01 mD 1000 Where: = Bulk Volume Water (Sw*Phi) = Height above FWL , = Constants   Sw vs. Height modelling Results

Accurate core water saturationsWell drilled with oil base mud doped to identify any mud filtrate contaminationCored above the FWL where the capillary bound water is immobile Only cores centres sampledThe core confirms the water saturations determined by fractal functionCore Water Saturations

Comparison between resistivity and fractal derived water saturationsSwept zone showing residual oil saturationsBy-passed hydrocarbon The resistivity log is incorrect in thin beds, close to bed boundaries and where there are conductive shalesThe fractal function ignores thin beds, bed boundaries and shalesThe Differential Reservoir Model

North Sea Oil FieldTwo wells that don’t intercept the FWL BVW trend identifies the FWL and confirms the wells are probably in the same compartment Picking the Free Water Level0 Bulk Volume of Water (%) 2510,750 Depth (ftTVDss) 10,350'Well 1Well 2 FWL=10,730 ftTVDss

Using the Fractal Function to Identify the Hydrocarbon to Water Contact S Where: = Height above FWL , = Constants The fractal Swh function gives the hydrocarbon water contact as a function of porosity   Hydrocarbon water contact Water Saturation (%) Height above FWL (Feet) 5pu Free Water Level

Depth is the most important downhole measurementTrue vertical depth subsea can be +/- 30 feetIdentification of the FWL normalises well depthsUsing the Fractal Function for Depth control0 Bulk Volume of Water (%) 25 Depth (ftTVDss)Well 1Well 2 FWL

Field Fluid Type Porosity Perm (pu) (mD)  Gas Permian Fluvial 9 0.2   Oil M. Jurassic Deltaic 13 3   Oil Devonian Lacustrine 14 7   Gas Permian Aeolian 14 0.9   Oil Palaeocene Turbidite 20 21   Gas Permian 20 341 Aeolian   Gas Condensate L. Cretaceous 24 847 Turbidite   Oil U. Jurassic Turbidite 21 570   Oil Palaeocene Turbidite 21 24   Oil Palaeocene Turbidite 22 27   Gas Palaeocene Turbidite 32 2207 Field average data 0 Porosity ( p.u .) 35 Permeability (mD) Log and core data from 11 North Sea fields compared

Functions from different fieldsLower Sw for same porosity and Height above FWLTransition zones comparedWhich is the best Swh function?The shape of the transition zone is related to pore geometry rather than porosity or permeability aloneFractal Functions quantify the pore geometryCase Study Results Bulk Volume of Water Height above the FWL Best transition zone to the left SwH Functions from different fields

Comparison between Log and Core BVW FunctionsThe Fractal Water Saturation vs. Height Function is linear on log-log scales Electrical log and core functions are the same irrespective to whether they were determined from logs or core dataThis confirms the fractal distribution of reservoir capillaries Bulk Volume of Water Log derivedSwH Functions Bulk Volume of WaterCore derived SwH Functions Height above the FWL Capillary Pressure

Conclusions – fractals and hydrocarbon volumes Swh function derived from the fractal nature of reservoir rocksCan be derived from electrical log or core dataUsing simple linear regression of a log-log plotLogs and core give the same function. They QC each otherThis confirms fractal distribution of reservoir capillariesDefines the Net Reservoir Cut-off and the shape of the Transition Zone Determines Free Water Level and Hydrocarbon Water ContactIndependent of rock characteristicsFacies type, porosity and permeabilityYou can forget about thin beds, bed boundary effects and shaliness Simple implementation in your reservoir model

0 Water Saturation (%) 100400 Height AboveFWL (feet) 0 0 Bulk Volume of Water (%) 15 Matrix Oil Water < Porosity > < BVW > Key Conclusion Forget water saturation. Think Bulk Volume of Water

USING FRACTALS TO DETERMINE A RESERVOIR’S HYDROCARBON DISTRIBUTIONQuestions?Steve.Cuddy@btinternet.com

Water Saturation vs. Height Data0 Water Saturation (%) 100400 HeightAboveFWL (feet) 0Source – Southern North Sea Gas fieldWhat do we see in the well data?

Bulk Volume of Water vs. Height Data0 Bulk Volume of Water (%) 15400 HeightAboveFWL (feet) 0 Matrix Oil Water < Porosity > < BVW >

What’s Benoit B. Mandelbrot middle name?

Upscaling Water SaturationsSw-Height functions (SWHF) are used to initialize the 3D reservoir modelIt is essential that the SWHF predicted water saturations upscale accuratelyFrom ½ foot to the cell size of the reservoir modelThis is done by integrating the Sw-Height function Unlike other parameters, such as porosity, water saturation must be pore volume averaged

Upscaling Water Saturations = average water saturation = average porosity = average bulk volume of water “A function that predicts BVW from height is especially appropriate to this application” Paul Worthington

Irreducible Water Saturation (Swirr)Is the lowest water saturation that can be achieved in a core plugThis is achieved by flowing hydrocarbon through a sample or spinning the sample in a centrifuge 0 Water Saturation (%) 100Height above FWL Swirr ? Swh profile This depends on the drive pressure or the centrifuge speed Water saturation therefore depends on the height above the free water level A minimum Swirr does not exist The transition zone extends indefinitely The Fractal Swh function determines Swirr as a function of height and porosity

Uncertainty ModellingPartial differentiation of the saturation equation allows us the derive the upside (P10) downside (P90) and most likely (P50) fractal Swh functionsIncludes the uncertainty in porosity, Rt, Rw, m, n etc.These are used to calculate upside and downside volumes of hydrocarbon in place Bulk Volume of Water Height above FWL P10 P50 P90

Capillary Pressure EquationThe height of the water in a capillary depends on the capillary pressure - which is determined by the radius of the capillary and the fluid types Where: Pc capillary pressure r capillary radius interfacial tension contact angle  

Gravity pulls the Water DownThe force of gravity on the column of water is determined by the difference between the water and oil densities and is called the buoyancy pressure Pb Where: P b buoyancy pressure due to gravity water density oil density g acceleration of gravity height above the free water level (FWL)  

Forces Acting on Reservoir FluidsThe capillary-bound water comprises a continuous column of water within the oil leg, with a hydrostatic pressure gradient  The oil is located in the remaining pore space also as a continuous phase and will have a lower pressure gradient  Although oil and water can coexist in the same localized volume of rock, the pressures acting on the two fluids are different  The intersection of the pressure gradients indicates the free water level (FWL) FWL Heigh t Buoyancy Pressure Oil pressure gradient Pressure Hydrostatic water gradient

Forces Acting on Reservoir FluidsThe buoyancy pressure (the difference in pressure between the oil and water phases) increases with height above the FWLAs the buoyancy pressure increases with height above the FWL, the oil phase will displace more water from increasingly smaller pore volumes  Therefore water saturation will tend to decrease with height above the FWL FWL Heigh t Buoyancy Pressure Oil pressure gradient Pressure Hydrostatic water gradient