material balance eqs

Material Balance EquationsAuthor: Jon Kleppe, NTNUAssistant producer: Vidar W. Moxness

Material Balance Equations

INTRODUCTIONIntroductionTo illustrate the simplest possible model we can have for analysis of reservoir behavior, we will start with derivation of so-called Material Balance Equations. This type of model excludes fluid flow inside the reservoir, and considers fluid and rock expansion/compression effects only, in addition, of course, to fluid injection and production.

This module is meant to be an extra help to the lectures in Reservoir recovery techniques by giving examples to the curriculum covered by the handout Material Balance Equations.

The structure of the model is shown below.Learning goalsBasic understanding of material balance

The handout Material Balance Equations can bedownloaded from here:MODELLINGAPPLICATIONSUMMARY SaturationBlockdiagramMaterialconservationGraph AGraph BEquationsWaterinfluenceInitialgascapIntroductionModellingApplicationSummarymatbal.pdfPlot 1Plot 2Plot 3


Block diagram of a producing reservoirThe essence of material balance is described in the block diagram below.

From the initial stage oil, gas & water is produced. At the same time gas & water is (re)injected into the reservoir to maintain pressure. There is also an influx from the aquifer below the reservoir.Due to change in pressure, the pore volume as well as the fraction of the volume occupied by gas, oil & water will change.INTRODUCTIONMODELLINGAPPLICATIONSUMMARY Block diagramMaterial conservationGraph A BEquationsSaturationClick to display symbols used


From the block diagram we get the expression below, which is the basis for the material balance formulas.Principle of material conservationINTRODUCTIONBlock diagramMaterial conservationGraph A BEquationsSaturationNote that fluids produced include all influence on the reservoir: Production Injection Aquifer influx

APPLICATIONSUMMARY MODELLING


PBoBo vs. PPBgBg vs. PPBwBw vs. PFormation Volume Factor in the Black Oil model

Click to display symbols usedINTRODUCTIONBlock diagramMaterial conservationGraph A BEquationsSaturationThe formation volume factors (FVF) tell how much the oil, gas and water is compressed at a given pressure.

Bo = reservoir volume of oil / standard volume of oil

Bg = reservoir volume of gas / standard volume of gas

Bw = reservoir volume of water / standard volume of waterThe graphs below show how the FVF of oil, gas and water develop vs pressure. Click on the buttons to show the graphs.APPLICATIONSUMMARY MODELLING


PRsoRso vs. PSolution Gas-Oil Ratio in the Black Oil modelINTRODUCTIONBlock diagramMaterial conservationGraph A BEquationsSaturationClick to display symbols usedThe Rso plot shows how the solution gas ratio develops vs pressure. When the pressure reaches the bubblepointpressure, it is no longer possible to solve more gas into the oil. Thus the gradient of the curve becomes zero.

Rs = standard volume gas / standard volume oilClick on the button below to see the typical pressure dependency of the solution gas-oil ratio in the black oil model.APPLICATIONSUMMARY MODELLING


Where: production terms areoil and solution gas expansion terms aregas cap expansion terms areand rock and water compression/expansion terms areThe complete black oil material balance equationThe final material balance relationships is given below. How these expressions are derived can be studied in the Material Balance pdf document.INTRODUCTIONBlock diagramMaterial conservationGraph A BEquationsSaturationClick to display symbols usedmatbal.pdfAPPLICATIONSUMMARY MODELLING


Saturation and pressure developmentClick to display symbols usedView the animations below to see how the pressure and oil-, gas- and water-saturation typically develops in a reservoir initially above the bubblepoint develops versus time. Also included is how pressure might develop versus time.The plot to the left shows how the saturations and the pressure in the reservoir develop vs time in a reservoir if there is small or no water injection.

The plot to the right shows the same for a reservoir with large water injecton.INTRODUCTIONBlock diagramMaterial conservationGraph A BEquationsSaturationAPPLICATIONSUMMARY MODELLING


Application of Material BalanceClick to display symbols usedIn material balance calculations there are in most cases many uncertainties with regard to reservoir parametres. Uncertain values may for instance include the size of the initial gascap, the initial amount of oil in the reservoir and the influx of the aquifer.

In the following pages ways of finding some of these values will be explained.The animation below shows a producing reservoir with gas and water injection.INTRODUCTIONMODELLINGSUMMARY APPLICATIONInitial gascap Plot 1 Plot 2Water influence Plot 3


Application of Material BalanceInitial gas cap (Havlena and Odeh approach)Click to display symbols usedGeneral mass balance formula:(1)(2)(3)For gascap reservoirs the value of m is in most cases uncertain. The value of N can however usually be defined well through producing wells. In this case a good approach will be to plot F as a function of (Eo+mEg) for an assumed value of m. (eq. 2) For the correct value of m the slope will be a straight line passing through origo with a slope of N. For a too large value of m, the plot will deviate down and for a too small value it will deviate up.

If both the value of m and N are uncertain one should plot F/Eo as a function of Eg/Eo. This plot should be linear and will intercept the y axis at a value of N and have a slope of mN. (eq. 3)Assuming no water influence, gas injection and rock or water compression/expansion.Large version Plot 1Large version Plot 2INTRODUCTIONMODELLINGAPPLICATIONInitial gascap Plot 1 Plot 2Water influence Plot 3SUMMARY


Application of Material BalanceInitial gas cap (Havlena and Odeh approach)Click to display symbols usedFor gascap reservoirs the value of m is in most cases uncertain. The value of N can however usually be defined well through producing wells. In this case a good approach will be to plot F as a function of (Eo+mEg) for an assumed value of m. (eq. 2) For the correct value of m the slope will be a straight line passing through origo with a slope of N.For a too large value of m, the plot will deviate down and for a too small value it will deviate up.

Assuming no water influence, gas injection and rock or water compression/expansion.ReturnLarge version Plot 2(2)INTRODUCTIONMODELLINGAPPLICATIONInitial gascap Plot 1 Plot 2Water influence Plot 3SUMMARY


Application of Material BalanceInitial gas cap (Havlena and Odeh approach)Click to display symbols usedIf both the value of m and N are uncertain one should plot F/Eo as a function of Eg/Eo. This plot should be linear and will intercept the y axis at a value of N and have a slope of mN. (eq. 3)Assuming no water influence, gas injection and rock or water compression/expansion.Large version Plot 1Return(3)INTRODUCTIONMODELLINGAPPLICATIONInitial gascap Plot 1 Plot 2Water influence Plot 3SUMMARY


Application of Material BalanceWater influence (Havlena and Odeh approach)Click to display symbols usedIn water drive reservoirs the biggest uncertainty is in most cases the water influx, We. To find this we plot F/Eo vs We/Eo. In this plot We must be calculated with a known model. (e.g. eq. 7)

For a correct model of We we will get a straight line. For the wrong model the plot will deviate from a straight line as shown in plot 3.General mass balance formula:Assuming no water or gas injection and Bw=1.Neglecting Ef,w due to its small influence and assuming no initial gascap.(1)(4)(5)(6)Large version Plot 3(7)Water influx model for radial aquifer shape:INTRODUCTIONMODELLINGAPPLICATIONInitial gascap Plot 1 Plot 2Water influence Plot 3SUMMARY


Application of Material BalanceWater influence (Havlena and Odeh approach)Click to display symbols usedFor a correct model of We we will get a straight line. For the wrong model the plot will deviate from a straight line as shown in plot 3.Return(6)INTRODUCTIONMODELLINGAPPLICATIONInitial gascap Plot 1 Plot 2Water influence Plot 3SUMMARY


SummaryMODELLING: Block diagram: Material balance equations are based on a model with a know start- and end-point. Between the two stages oil, gas & water is produced and gas & water is (re)injected into the reservoir to maintain pressure. There is also an influx from the aquifer below the reservoir. Due to change in pressure, the pore volume as well as the fraction of the volume occupied by gas, oil & water will change.

Material conservation: Amounts of fluids in the reservoir at stage one is equal to the amount of fluids at stage two plus the amount of fluids produced.

Graph A: The formation volume factors (FVF) tell how much the oil, gas and water is compressed at a given pressure.

Graph B: The Rso plot shows how the solution gas ratio develops vs pressure. When the pressure reaches the bubblepointpressure, it is no longer possible to solve more gas into the oil. Thus the gradient of the curve becomes zero.

Equations: The material balance equations consist of a general part, oil and solution gas expansion terms, gas cap expansion terms and rock and water compression/expansion terms

Saturation: Pressure and saturations change versus time, depending on production/injection. See figure to the right.

APPLICATION: Initial gascap: In a gas drive reservoirs m may be calculated by plotting F as a function of (Eo+mEg). For the correct value of m the plot will be a straight line. Alternatively m & N may be calculated by plotting F/Eo vs Eg/Eo. The curve will intercept the y axis at a value of N and have a slope of m*N.

Water influence: In a water drive reservoir the water influx, We, can be recovered by plotting F/Eo vs We/Eo. In this plot We must be calculated with a known model.Block diagramINTRODUCTIONMODELLINGAPPLICATIONSUMMARY Saturation & pressure


Jon Kleppe. Material balance. http://www.ipt.ntnu.no/~kleppe/SIG4038/02/matbal.pdf

L.P. Dake 1978. Fundamentals of reservoir engineering, Elsevier, Amsterdam, 443 pp.

L.P. Dake 1994. The practice of reservoir engineering, Elsevier, Amsterdam, 534 pp.

Svein M. Skjveland (ed.) & Jon Kleppe (ed.) 1992. SPOR monograph : recent advances in improved oil recovery methods for North Sea sandstone reservoirs Norwegian Petroleum Directorate, Stavanger. 335 pp.ReferencesINTRODUCTIONMODELLINGAPPLICATIONSUMMARY


About this moduleTitle: Material Balance Equations

Author: Prof. Jon Kleppe

Assistant producer: Vidar W. Moxness

Size: 0.8 mb

Publication date: 24. July 2002

Abstract: The module describes the basics of material balance calculations.

Software required: PowerPoint XP/XP Viewer

Prerequisites: none

Level: 1 4 (four requires most experience)

Estimated time to complete: --INTRODUCTIONMODELLINGAPPLICATIONSUMMARY


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Symbols used in material balance equationsINTRODUCTIONMODELLINGSUMMARY Click to returnto calculationAPPLICATION

BgFormation volume factor for gas (res.vol./st.vol.)SgGas saturationBoFormation volume factor for oil (res.vol./st.vol.)SoOil saturationBwFormation volume factor for water (res.vol./st.vol.)SwWater saturationCrPore compressibility (pressure-1) TTemperatureCwWater compressibility (pressure-1) VbBulk volume (res.vol.)DPP2-P1VpPore volume (res.vol.)Ef,wRock and water expansion/compression termWeCumulative aquifer influx (st.vol.)EgGas cap expansion termWiCumulative water injected (st.vol.)EoOil & solution gas expansion termWpCumulative water produced (st.vol.)GiCumulative gas injected (st.vol.)RDensity (mass/vol.)GpCumulative gas produced (st.vol.)fPorositymInitial gas cap size (res.vol. of gas cap)/(res.vol. of oil zone)NOriginal oil in place (st.vol.)NpCumulative oil produced (st.vol.)PPressurePbBubblepoint PressureRpCumulative producing gas-oil ratio (st.vol./st.vol.) = Gp/NpRsoSolution gas-oil ratio (st.vol. gas/st.vol. oil)

material balance eqs

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