Material Balance Equations Author: Jon Kleppe, NTNU Assistant producer: Vidar W. Moxness The Statfjord area in the North Sea. Source: Statoil

Material Balance Equations INTRODUCTION Introduction To 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 goals Basic understanding of material balance The handout “Material Balance Equations” can be downloaded from here:here MODELLING APPLICATION SUMMARY SaturationBlock diagram Material conservation Graph AGraph B Equations Water influence Initial gascap Introduction Modelling Application Summary matbal.pdf Plot 1Plot 2 Plot 3

Material Balance Equations Block diagram of a producing reservoir The 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. INTRODUCTION MODELLING APPLICATION SUMMARY Block diagram Material conservation Graph A B Equations Saturation Click to display symbols used

Material Balance Equations From the block diagram we get the expression below, which is the basis for the material balance formulas. Principle of material conservation INTRODUCTION Block diagram Material conservation Graph A B Equations Saturation Note that “fluids produced” include all influence on the reservoir: Production Injection Aquifer influx Amount offluids present in the reservoir initially (st. vol.) Amount of fluids produced (st. vol.) Amount offluids remaining in the reservoir finally (st. vol.)                              APPLICATION SUMMARY MODELLING

Material Balance Equations P BoBo B o vs. P P BgBg B g vs. P P BwBw B w vs. P Formation Volume Factor in the Black Oil model Click to display symbols used INTRODUCTION Block diagram Material conservation Graph A B Equations Saturation The formation volume factors (FVF) tell how much the oil, gas and water is compressed at a given pressure. B o = reservoir volume of oil / standard volume of oil B g = reservoir volume of gas / standard volume of gas B w = reservoir volume of water / standard volume of water The graphs below show how the FVF of oil, gas and water develop vs pressure. Click on the buttons to show the graphs. APPLICATION SUMMARY MODELLING

Material Balance Equations P R so R so vs. P Solution Gas-Oil Ratio in the Black Oil model INTRODUCTION Block diagram Material conservation Graph A B Equations Saturation Click to display symbols used 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. R s = standard volume gas / standard volume oil Click on the button below to see the typical pressure dependency of the solution gas-oil ratio in the black oil model. APPLICATION SUMMARY MODELLING

Material Balance Equations Where: production terms are oil and solution gas expansion terms are gas cap expansion terms are and rock and water compression/expansion terms are The complete black oil material balance equation The final material balance relationships is given below. How these expressions are derived can be studied in the Material Balance pdf document.pdf document INTRODUCTION Block diagram Material conservation Graph A B Equations Saturation Click to display symbols used matbal.pdf APPLICATION SUMMARY MODELLING

Material Balance Equations Saturation and pressure development Click to display symbols used View 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. INTRODUCTION Block diagram Material conservation Graph A B Equations Saturation APPLICATION SUMMARY MODELLING

Material Balance Equations Application of Material Balance Click to display symbols used In 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. INTRODUCTION MODELLING SUMMARY APPLICATION Initial gascap Plot 1 Plot 2 Water influence Plot 3

Material Balance Equations Application of Material Balance Initial gas cap (Havlena and Odeh approach) Click to display symbols used General 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 1 Large version Plot 2 INTRODUCTION MODELLING APPLICATION Initial gascap Plot 1 Plot 2 Water influence Plot 3 SUMMARY

Material Balance Equations Application of Material Balance Initial gas cap (Havlena and Odeh approach) Click to display symbols used 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. Assuming no water influence, gas injection and rock or water compression/expansion. Return Large version Plot 2 (2) INTRODUCTION MODELLING APPLICATION Initial gascap Plot 1 Plot 2 Water influence Plot 3 SUMMARY

Material Balance Equations Application of Material Balance Initial gas cap (Havlena and Odeh approach) Click to display symbols used 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 1 Return (3) INTRODUCTION MODELLING APPLICATION Initial gascap Plot 1 Plot 2 Water influence Plot 3 SUMMARY

Material Balance Equations Application of Material Balance Water influence (Havlena and Odeh approach) Click to display symbols used In 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 B w =1. Neglecting E f,w due to it’s small influence and assuming no initial gascap. (1) (4) (5) (6) Large version Plot 3 (7) Water influx model for radial aquifer shape: INTRODUCTION MODELLING APPLICATION Initial gascap Plot 1 Plot 2 Water influence Plot 3 SUMMARY

Material Balance Equations Application of Material Balance Water influence (Havlena and Odeh approach) Click to display symbols used 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. Return (6) INTRODUCTION MODELLING APPLICATION Initial gascap Plot 1 Plot 2 Water influence Plot 3 SUMMARY

Material Balance Equations Summary MODELLING: 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 diagram INTRODUCTION MODELLING APPLICATION SUMMARY Saturation & pressure

Material Balance Equations Jon Kleppe. Material balance. L.P. Dake Fundamentals of reservoir engineering, Elsevier, Amsterdam, 443 pp. L.P. Dake The practice of reservoir engineering, Elsevier, Amsterdam, 534 pp. Svein M. Skjæveland (ed.) & Jon Kleppe (ed.) SPOR monograph : recent advances in improved oil recovery methods for North Sea sandstone reservoirs Norwegian Petroleum Directorate, Stavanger. 335 pp. References INTRODUCTION MODELLING APPLICATION SUMMARY

Material Balance Equations About this module Title: 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: -- INTRODUCTION MODELLING APPLICATION SUMMARY

Material Balance Equations On every page, you will find the title at the top, and a menu with the main chapters in bold to the left. These are hyperlinks which enable you choose the chapters in whichever order you wish to view them. Keep in mind that the module is set up in the order the author believes is most appropriate for study. These chapters are also represented with an illustration on the introduction slide linked to the appropriate chapter. The chapter you are currently viewing in is shown with this marker:, while the subchapter (when applicable) is highlighted in orange. Within the main frame (the white area), you’ll find text and illustrations as well as animations and videos etc. Many pictures have enlargement buttons near them. Help Navigation tools in the module INTRODUCTION MODELLING APPLICATION SUMMARY Previous picture in an animation or sequence of pictures. Next picture in an animation or sequence of pictures. At bottom of the slide you’ll find a few standardised buttons which occur on every page (some may not be present in the module): shows the list of references. shows information about the module (e.g. author and assistant producer). shows a list of frequently asked questions if there are any. takes you to previously viewed slide. is linked to the previous chapter and slide, respectively. is linked to the next chapter and slide, respectively. you may turn off the sound, or turn it on (when available). you have figured it out! will end your session with the current module. If you have any problems, please let us know by sending an to Please include the title of module and description of the problem. We will respond as quickly as

Material Balance Equations BgBg Formation volume factor for gas (res.vol./st.vol.)SgSg Gas saturation BoBo Formation volume factor for oil (res.vol./st.vol.)SoSo Oil saturation BwBw Formation volume factor for water (res.vol./st.vol.)SwSw Water saturation CrCr Pore compressibility (pressure -1 )TTemperature CwCw Water compressibility (pressure -1 )VbVb Bulk volume (res.vol.) PP P 2 -P 1 VpVp Pore volume (res.vol.) E f,w Rock and water expansion/compression termWeWe Cumulative aquifer influx (st.vol.) EgEg Gas cap expansion termWiWi Cumulative water injected (st.vol.) EoEo Oil & solution gas expansion termWpWp Cumulative water produced (st.vol.) GiGi Cumulative gas injected (st.vol.)RDensity (mass/vol.) GpGp Cumulative gas produced (st.vol.)  Porosity mInitial gas cap size (res.vol. of gas cap)/(res.vol. of oil zone) NOriginal oil in place (st.vol.) NpNp Cumulative oil produced (st.vol.) PPressure PbPb Bubblepoint Pressure RpRp Cumulative producing gas-oil ratio (st.vol./st.vol.) = G p /N p R so Solution gas-oil ratio (st.vol. gas/st.vol. oil) Symbols used in material balance equations INTRODUCTION MODELLING SUMMARY Click to return to calculation APPLICATION