MODIFICATION OF THE DYKSTRA-PARSONS METHOD TO INCORPORATE BUCKLEY-LEVERETT DISPLACEMENT THEORY FOR WATERFLOODS. A Thesis RUSTAM RAUF GASIMOV

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1 MODIFICATION OF THE DYKSTRA-PARSONS METHOD TO INCORPORATE BUCKLEY-LEVERETT DISPLACEMENT THEORY FOR WATERFLOODS A Thesis by RUSTAM RAUF GASIMOV Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE August 2005 Major Subject: Petroleum Engineering

2 ii - MODIFICATION OF THE DYKSTRA-PARSONS METHOD TO INCORPORATE BUCKLEY-LEVERETT DISPLACEMENT THEORY FOR WATERFLOODS A Thesis by RUSTAM RAUF GASIMOV Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Approved by: Chair of Committee, Daulat D. Mamora Committee Members, Bryan J. Maggard Robert R. Berg Head of Department, Stephen A. Holditch August 2005 Major Subject: Petroleum Engineering

3 iii - ABSTRACT Modification of the Dykstra-Parsons Method to Incorporate Buckley-Leverett Displacement Theory for Waterfloods. (August 2004) Rustam Rauf Gasimov, B.S., Azerbaijan State Oil Academy Chair of Advisory Committee: Dr. Daulat D. Mamora The Dykstra-Parsons model describes layer 1-D oil displacement by ater in multilayered reservoirs. The main assumptions of the model are: piston-like displacement of oil by ater, no crossflo beteen the layers, all layers are individually homogeneous, constant total injection rate, and injector-producer pressure drop for all layers is the same. Main drabacks of Dykstra-Parsons method are that it does not take into account Buckley-Leverett displacement and the possibility of different oil-ater relative permeability for each layer. A ne analytical model for layer 1-D oil displacement by ater in multilayered reservoir has been developed that incorporates Buckley-Leverett displacement and different oilater relative permeability and ater injection rate for each layer (layer injection rate varying ith time). The ne model employs an extensive iterative procedure, thus requiring a computer program. To verify the ne model, calculations ere performed for a to-layered reservoir and the results compared against that of numerical simulation. Cases ere run, in hich layer thickness, permeability, oil-ater relative permeability and total ater injection rate ere varied. Main results for the cases studied are as follos. First, cumulative oil production up to 20 years based on the ne model and simulation are in good agreement. Second, model ater breakthrough times in the layer ith the highest permeability-thickness product

4 iv - (kh) are in good agreement ith simulation results. Hoever, breakthrough times for the layer ith the loest kh may differ quite significantly from simulation results. This is probably due to the assumption in the model that in each layer the pressure gradient is uniform behind the front, ahead of the front, and throughout the layer after ater breakthrough. Third, the main attractive feature of the ne model is the ability to use different oil-ater relative permeability for each layer. Hoever, further research is recommended to improve calculation of layer ater injection rate by a more accurate method of determining pressure gradients beteen injector and producer.

5 v - DEDICATION I ish to dedicate my thesis: To my daughter, Nezrin, the person I love the most, and my ife, Sabina, for her love and support. To my parents, Rauf and Zulfiya, for all their support, encouragement, sacrifice, and especially for their unconditional love; I love you both.

6 vi - ACKNOWLEDGEMENTS I ish to express my sincere gratitude and appreciation to: Dr. Daulat D. Mamora, chair of my advisory committee, for his continued help and support throughout my research. It as great to ork ith him. Dr. J. Bryan Maggard, member of my advisory committee, for his enthusiastic and active participation and guidance during my investigation. Dr. Robert Berg, member of my advisory committee, for his encouragement to my research, his motivation to continue studying and his alays-good mood. Finally, I ant to express my gratitude and appreciation to all my colleagues in Texas A&M University: Marylena Garcia, Anar Azimov, and Adedayo Oyerinde.

7 vii - TABLE OF CONTENTS Page ABSTRACT DEDICATION.... ACKNOWLEDGEMENTS... TABLE OF CONTENTS... iii v vi vii LIST OF FIGURES... ix LIST OF TABLES xv CHAPTER I INTRODUCTION Buckley-Leverett Model Dykstra-Parsons Model Problem Description Objectives... 5 II LITERATURE REVIEW Buckley-Leverett Frontal Advance Theory Stiles Method Dykstra-Parsons Approach III NEW ANALYTICAL METHOD Calculation Procedure Case Case Case Case Case Case Case Case Case

8 viii - CHAPTER Page IV SIMULATION MODEL OVERVIEW V SUMMARY, CONCLUSIONS AND RECOMMENDATIONS Summary Conclusions Recommendations NOMENCLATURE REFERENCES APPENDIX A APPENDIX B APPENDIX C VITA

9 ix - LIST OF FIGURES FIGURE Page 1.1 Oil production: comparison of results based on Dykstra-Parsons model, and numerical simulation for 2-layered model, i = 800 STB/D Typical fractional flo curve as a function of ater saturation Water saturation distribution as a function of distance, before breakthrough in the producing ell Water saturation distribution at, and after breakthrough in the producing ell Schematic piston-like displacement in a layer in the Dykstra-Parsons model Corey type relative permeability curves for Case Fractional flo curve for Case Schematic representation of the aterflood process at the moment of breakthrough in layer Comparison of oil production rate of ne analytical model vs. simulation (Case 1) Comparison of ater injection rate of ne analytical model vs. simulation by layers (Case 1) Comparison of cumulative oil produced calculated ith ne analytical model vs. simulation (Case 1) Oil production rates by layers, ne analytical model vs. simulation (Case 1) Water production rate by layers, ne analytical model vs. simulation (Case 1) Total ater production rate comparison of ne analytical model vs. simulation (Case 1)... 36

10 x - FIGURE Page 3.10 Sets of relative permeabilities for layers one and to (Case 2) Fractional flo curves for layers one and to respectively (Case 2) Oil production rate of ne analytical model vs. simulation, different sets of relative permeabilities are applied (Case 2) Water injection rates by layers of ne analytical model vs. simulation, different sets of relative permeabilities are applied(case 2 ) Cumulative oil produced comparison of ne analytical model vs. simulation, to sets of relative permeability are provided for each layer (Case 2) Oil production rate by layers comparison of ne analytical model vs. simulation, to sets of relative permeability are provided for each layer (Case 2) Water production rate by layers, comparison of ne analytical model vs. simulation, to sets of relative permeability are provided for each layer (Case 2) Total ater production rate, comparison of ne analytical model vs. simulation, to sets of relative permeability are provided for each layer (Case 2) Oil production comparison of ne analytical model to the simulation, variable kh is applied (Case 3) Water injection rate comparison of ne analytical model versus simulation, variable kh is applied (Case 3) Cumulative oil produced comparison of ne analytical model to the simulation, variable kh is applied (Case 3) Oil production layer by layer comparison of ne analytical model to the simulation, variable kh is applied (Case 3) Water production layer by layer comparison of ne analytical model to the simulation, variable kh is applied (Case 3) Total ater production comparison of ne analytical model to the simulation, variable kh is applied (Case 3)... 53

11 xi - FIGURE Page 3.24 Total oil production comparison of ne analytical model versus simulation, variable kh and relative permeability sets are applied (Case 4) Water injection layer by layer comparison of ne analytical model to simulation, variable kh and relative permeability sets are applied (Case 4) Cumulative oil produced, comparison of ne analytical model to simulation, variable kh and relative permeability sets are applied (Case 4) Oil production layer by layer, comparison of ne analytical model to simulation, variable kh and relative permeability sets are applied (Case 4) Water production layer by layer, comparison of ne analytical model to simulation, variable kh and relative permeability sets are applied (Case 4) Water production rate, comparison of ne analytical model to the simulation, variable kh and relative permeability sets are applied (Case 4) Oil production rate, comparison of ne analytical model to the simulation, injection rate of 1600 STB/D is applied (Case 5) Water injection rate by layers, comparison of ne analytical model to simulation, injection rate of 1600 STB/D is applied (Case 5) Cumulative oil production, comparison of ne analytical model to simulation, injection rate of 1600 STB/D is applied (Case 5) Oil production rate on layer basis, comparison of ne analytical model to simulation, injection rate of 1600 STB/D is applied (Case 5) Water production on layer basis, comparison of ne analytical model to simulation, injection rate of 1600 STB/D is applied (Case 5) Water production rate, comparison of ne analytical model to simulation, injection rate of 1600 STB/D is applied (Case 5) Oil production rate, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 6).. 66

12 xii - FIGURE Page 3.37 Water injection rate by layers, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 6) Cumulative oil produced, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 6) Oil production rate by layers, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 6) Water production rate by layers, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 6) Total ater production rate, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 6) Oil production rate, comparison of ne analytical model to simulation, different kh and injection rate of 1600 STB/D are applied (Case 7) Water injection rate by layers, comparison of ne analytical model to simulation, different kh and injection rate of 1600 STB/D are applied (Case 7) Cumulative oil produced, comparison of ne analytical model to simulation, different kh and injection rate of 1600 STB/D are applied Oil production by layers, comparison of ne analytical model to simulation, different kh and injection rate of 1600 STB/D are applied (Case 7) Water production by layers, comparison of ne analytical model to simulation, different kh and injection rate of 1600 STB/D are applied (Case 7) Total ater production, comparison of ne analytical model to simulation, different kh and injection rate of 1600 STB/D are applied (Case 7) Total oil production rate, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 8).. 78

13 xiii - FIGURE Page 3.49 Water injection rate by layers, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 8) Cumulative oil, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 8) Oil production by layers, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 8) Water production by layers, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 8) Water production rate, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 8) Oil production rate, comparison of ne analytical model to simulation, for 2 identical layers (Case 9) Water injection rate by layers, comparison of ne analytical model to simulation, for 2 identical layers (Case 9) Cumulative oil produced, comparison of ne analytical model to simulation, for 2 identical layers (Case 9) Oil production by layers, comparison of ne analytical model to simulation, for 2 identical layers (Case 9) Water production by layers, comparison of ne analytical model to simulation, for 2 identical layers (Case 9) Water production rate, comparison of ne analytical model to simulation, for 2 identical layers (Case 9) Simulation results indicate cumulative oil production ith and ithout grid refinement is practically identical Simulation results for Case 1 at 274 days, shoing earlier ater breakthrough in layer that has a higher kh... 93

14 xiv - FIGURE Page 4.3 Simulation results for Case 2 at 274 days, shoing faster aterflood displacement in loer layer, hich has different oil-ater relative permeability Simulation results for Case 3 at 274 days, shoing faster ater front propagation in upper layer that has a higher kh value Simulation results for Case 4 at 274 days, shoing faster oil displacement in upper layer that has a higher kh value Simulation results for Case 5 at 274 days, shoing that flood front advanced more than that in Case 1, as ater injection rate as doubled Simulation results for Case 6 at 274 days, shoing similarity to Case 2 except that loer layer broke through Simulation results for Case 7 at 274 days, shoing faster front propagation than that in Case 3 due to increased injection rate Simulation results for Case 8 at 274 days, shoing faster front propagation than that in Case 4 due to increased injection rate Simulation results for Case 9 at 274 days, shoing identical displacement in both layers, as both layers have identical reservoir properties Injector-producer pressure drop for simulation model on example of Case

15 xv - TABLE LIST OF TABLES Page 3.1 Reservoir Properties for Case Reservoir Properties of Layer 1 for Case Reservoir Properties of Layer 2 for Case Height and Permeability Variation in Layers 1 and Summary of Reservoir Properties for Each Case Oil-Water Relative Permeability Set Parameters... 92

16 1 - CHAPTER I 2. INTRODUCTION 1.1 Buckley-Leverett Model In 1941, Leverett 1 in his pioneering paper presented the concept of fractional flo. Beginning ith the Darcy s la for ater and oil 1-D flo, he formulated the folloing fractional flo equation: f kkro Pc 1+ g ρ sinα qtµ o x =,.....(1.1) µ kro 1+ µ k o r here f is the fractional flo of ater, q t is the total flo rate of oil and ater, k ro and k r are relative permeabilities of oil and ater respectively, µ o and µ are viscosities of oil and ater respectively, P c x is the capillary pressure gradient, ρ is the density difference ρ ρ ), α is the reservoir dip angle, and g is the gravitational constant. ( o For the case here the reservoir is horizontal ( α = 0 ), Eq. 1.1 reduces to: This thesis follos the style of the Journal of Petroleum Technology.

17 2 - f 1 = µ 1+ µ o k k ro r....(1.2) In 1946, Buckley and Leverett 2 presented the frontal advance equation. Applying mass balance to a small element ithin the continuous porous medium, they expressed the difference at hich the displacing fluid enters this element and the rate at hich it leaves it in terms of the accumulation of the displacing fluid. This led to a description of the saturation profile of the displacing fluid as a function of time and distance from the injection point. The most remarkable outcome of their displacement theory as the presence of a shock front. The frontal advance equation obtained as: x t S = qt f A φ S,......(1.3) t here q t is a total volumetric liquid rate, equal to q + q o, A is the cross-sectional area of flo, φ is porosity, S is ater saturation. 1.2 Dykstra-Parsons Model An early paper by Dykstra and Parsons 3 presented a correlation beteen aterflood recovery and both mobility ratio and permeability distribution. This correlation as based on calculations applied to a layered linear model ith no crossflo. This first ork on vertical stratification ith inclusion of mobility ratios other than unity as presented in the ork of Dykstra and Parsons ho have developed an approach for handling stratified reservoirs, hich allos calculating aterflood performance in multi-

18 3 - layered systems. But their method requires the assumption that the saturation behind the flood front is uniform, i.e. only ater moves behind the aterflood front. There are other assumptions involved such as: linear flo, incompressible fluid, piston-like displacement, no cross flo, homogeneous layers, constant injection rate, and the pressure drop ( P) beteen injector and producer across all layers is the same. Governing equation for Dykstra-Parsons front propagation is as follos: x x j n = M M 2 k + k j n ( 1 M ) 2 [( 1 M ) ],..(1.4) here M is the end point mobility ratio, x n is the distance of front propagation of the layer in hich ater just broke through, hich is equal to L the total layer length; x j is the distance of ater front of the next layer to be flooded after layer n. Generalizing Eq. 1.4 for N-number of layers, the coverage (vertical seep efficiency) can be obtained: C n = n N + N n+ 1 M M 2 k + k n ( 1 M ) j N 2 [( 1 M ) ],...(1.5) here Cn is the vertical coverage after n layers have been flooded, n is the layer in hich ater just broke through.

19 4-1.3 Problem Description For many years analytical models have been used to estimate performance of aterflood projects. The Buckley-Leverett frontal advance theory and Dykstra-Parsons method for stratified reservoirs have been used for this purpose, but not in combination for stratified reservoirs ith different kh and oil-ater relative permeability. The Dykstra-Parsons method has a major draback in that it assumes the displacement of oil by ater is piston-like. As illustration, I have compared oil production rate estimated by Dykstra-Parsons method against that from numerical simulation (GeoQuest Eclipse 100). For the comparison, I used a 2-layered reservoir ith the folloing parameters: length L-1200 ft., idth -400 ft., height h-35 ft. each layer, ith total injection rate i t -800 STB/D. The results are shon in Fig. 1.1 The folloing observation can be made Oil production rate, STB/D Dykstra-Parsons method Simulation Time, yr Figure 1.1 Oil production: comparison of results based on Dykstra-Parsons model, and numerical simulation for 2-layered model, i = 800 STB/D.

20 5 - First, the ater breakthrough time based on simulation is significantly earlier compared to that from Dykstra-Parsons method. Second, cumulative oil produced at the moment of breakthrough in layer 1, is more for the Dykstra-Parsons analytical model compared to simulation. This is because Dykstra-Parsons model assumes that at breakthrough, all moveable oil has been sept from layer 1, hereas in the simulation model at breakthrough, there is still moveable oil behind the front. 1.4 Objectives The goal of this research is to modify the Dykstra-Parsons method for 1-D oil displacement by ater in such a manner that it ould be possible to incorporate the Buckley-Leverett frontal advance theory. This ould require modeling fractional flo behind the aterflood front instead of assuming piston-like displacement. By incorporating Buckley-Leverett displacement, a more accurate analytical model of oil displacement by ater is expected. Permeability-thickness and oil-ater relative permeability ill be different for each layer, ith no crossflo beteen the layers. The analytical model results (injection rate, ater and oil production rate) ill be compared against simulation results to ensure the validity of the analytical model.

21 6 - CHAPTER II 2. LITERATURE REVIEW 2.1 Buckley-Leverett Frontal Advance Theory Since the original paper of Dykstra-Parsons, a great number of papers have suggested some modifications to the basic approach. The literature revie gives the reader an overvie of these modifications. Buckley and Leverett (1946): The Buckley-Leverett frontal advance theory considers the mechanism of oil displacement by ater in a linear 1-D system. An equation as developed for calculating the frontal advance rate. In the Buckley-Leverett approach oil displacement occurred under so-called diffuse flo condition, hich means that fluid saturations at any point in the linear displacement path are uniformly distributed ith respect to thickness. The fractional flo of ater, at any point in the reservoir, is defined as f =,......(2.1) q o q + q here q is ater flo rate, and qo is oil flo rate. Using Darcy s la for linear one dimensional flo of oil and ater, considering the displacement in a horizontal reservoir, and neglecting the capillary pressure gradient e get the folloing expression:

22 7 - f 1 = µ 1+ k r k µ ro o....(2.2) Provided the oil displacement occurs at a constant temperature then the oil and ater viscosities have fixed values and Eq. 2.2 is strictly a function of the ater saturation. This is illustrated in Fig. 2.1 for typical oil-ater relative permeability and properties. 1 f 0 0 S c S (fraction) 1-S or Figure 2.1 Typical fractional flo curve as a function of ater saturation. In their paper Buckley and Leverett presented hat is recognized as the basic equation describing immiscible displacement in one dimension. For ater displacing oil, the equation describes the velocity of a plane of constant ater saturation traveling through

23 8 - the linear system. Assuming the diffuse flo conditions and conservation of mass of ater floing through volume element Adx : x S W df i =,....(2.3) Aφ ds S here Wi is the cumulative ater injected and it is assumed, as an initial condition, that W i = 0 hen t = 0. There is a mathematical difficulty encountered in applying this technique, hich exists due to the nature of the fractional flo curve creating a saturation discontinuity or a shock front. In 1952 Welge 4 presented the simplified method to the frontal advance equation. This method consists of integrating the saturation distribution over the distance from the injection point to the front, obtaining the average ater saturation behind the front S. Water injection Liquid production 1-S or S S S f S c 0 0 x 1 x x 2 Figure 2.2 Water saturation distribution as a function of distance, before breakthrough in the producing ell 5.

24 9 - Fig. 2.2 presents ater saturation profile as a function of distance. Applying the simple material balance: W i = x Aφ ( S S ),......(2.4) 2 c here S is average ater saturation behind the front, x 1 is distance in the reservoir totally flooded by ater, x2 is distance of aterflood front location. Eqs. 2.3 and 2.4 yield the folloing solution to S : S S c = Wi = x Aφ 2 df ds 1 S f....(2.5) The expression for the average ater saturation behind the front can also be obtained by direct integration of the saturation profile as S (1 S or ) x x2 x1 S dx =. (2.6) x And since x S α df ds S the Eq. 2.6 can be expressed as S = (1 S or df ) ds 1 Sor df ds + Sf x2 x1 S df d ds. (2.7) After rearranging Eq. 2.7,

25 10 - S df ( f S ) f Sf = S f (2.8) ds Note that for f = 0, Eq. 2.8 reduces to Eq Cumulative oil production at the breakthrough can be expressed by folloing equation: N pdbt ( S S c ) 1 = WiDbt = qidtbt = =,......(2.9) df S bt ds here N pdbt is dimensionless cumulative oil produced at the moment of breakthrough, W idbt is dimensionless cumulative ater injected at the moment of breakthrough. Eq. 2.5 is true only for the aterflood before and at the point of breakthrough. 1-S or S S bt S e S bt = S f S c 0 0 x L Figure 2.3 Water saturation distributions at, and after breakthrough in the producing ell 5.

26 11 - From the Fig. 2.3, S e is the current value of the ater saturation at the producing ell after ater breakthrough. Water saturation by the Welge technique gives: S 1 = S e + ( 1 f e )...(2.10) df S ds e Folloing Eq. 2.9 oil recovery after ater breakthrough can be expressed as: N pd c ( S e S c ) + ( f e ) WiD = S S = 1....(2.11) 2.2 Stiles Method Stiles 6 (1949): This method for predicting the performance of aterflood operations basically involves accounting for permeability variations, vertical distribution of flo capacity kh. Most important assumption as that ithin the reservoir of various permeabilities injected ater seeps first the zones of higher permeability and that first breakthrough occurs in these layers. The different flood-front positions in liquid-filled, linear layers having different permeabilities, each layer insulated from the others. Stiles assumes that the rate of ater injected into each layer depends only upon the kh of that layer. This is equivalent to assuming a mobility ratio of unity. Also it is assumed that fluid flo is linear and the distance of penetration of the flood front is proportional to permeability-thickness product. The Stiles method assumes that there is piston-like displacement of oil, so that after ater breakthrough in a layer, only ater is produced from that layer. After ater breakthrough, the producing WOR is found as follos: WOR κ k µ r o = Bo,.....(2.12) 1 κ µ kro

27 12 - here κ is the fraction of the total flo capacity represented by layers having ater breakthrough. In addition, the Stiles method assumes a unit mobility ratio. In his ork Stiles rearranged the layers depending on their permeability in descending manner. Later Johnson 7 developed a graphical approach that simplified the consideration of layer permeability and porosity variations. Layer properties ere chosen such that each had equal flo capacities so that the volumetric injection rate into each layer as the same. 2.3 Dykstra-Parsons Approach Dykstra and Parsons (1950): An early paper presented a correlation beteen aterflood recovery and both mobility ratio and permeability distribution. This correlation as based on calculations applied to a layered linear model ith no crossflo. More than 200 flood pot tests ere made on more than 40 California core samples in hich initial fluid saturations, mobility ratios, producing WOR s, and fractional oil recoveries ere measured. The permeability distribution as measured by the coefficient of permeability variation. The correlations presented by Dykstra-Parsons related oil recovery at producing WOR s of 1, 5, 25, and 100 as a fraction of the oil initially in place to the permeability variation, mobility ratio, and the connate-ater and flood-ater saturations. The values obtained assume a linear flood since they are based upon linear flo tests. The Dykstra-Parsons method considers the effect of vertical variations of horizontal permeabilities for the aterflood performance calculation. Similar to the Stiles method, permeabilities are arranged in descending order. Folloing is a full list of assumptions for Dykstra-Parsons approach. (1) Linear flo

28 13 - (2) Incompressible displacement (3) Piston-like displacement (4) Each layer is a homogenous layer (5) No crossflo beteen layers (6) Pressure drop for all layers is the same (7) Constant ater injection rate (8) Velocity of the front is proportional to absolute permeability and end point mobility ratio of the layer As there is a piston-like displacement in each layer, flo velocity of oil and ater in any layer can be expressed as: v o ko dp =,.....(2.13) µ dx o v k dp =,....(2.14) µ dx here k o is effective oil permeability and shos the sample of piston-like displacement. k effective ater permeability. Fig. 2.4

29 14 - P 1 P 2 x 1 L Figure 2.4 Schematic piston-like displacement in a layer in the Dykstra-Parsons model. From assumption 6, P = P 1 + P 2....(2.15) Subsequently Eqs and 2.14 can be presented as: v k P x 1 =,......(2.16) µ 1 v o k P o 2 =....(2.17) µ o ( L x ) 1 Assuming incompressible flo, v = v substitute in Eq. 2.15: v Rearranging Eq. 2.18, o. After rearranging Eqs and 2.17 and ( L x ) 1 µ o 1 o ko x µ + v k = P....(2.18)

30 15 - v o = µ k x P µ + ( L ) o 1 x 1 ko. (2.19) Effective permeability for oil and ater can be expressed as k o = kkro, k = kkr, hich on substituting into Eq yields: v o = µ k r k1 P µ o x1 + k ro ( L x ) 1. (2.20) Using assumption that k r and k ro are the same for all layers: v v oi o = k k i 1 µ kr µ kr µ o x1 + kro µ o xi + k ro ( L x ) 1 ( L x ) i...(2.21) The end point mobility ratio is defined as: M ep k re o =....(2.22) µ µ k roe Eq may be rearranged and integrated ith respect to x to give the folloing expression: 2 xi xi ki ( 1 M ) + 2M ( 1+ M ) = 0 ep L ep L k 1 ep....(2.23) Eq is a quadratic equation, therefore solving for L x i :

31 16 - x i L = M ep ± M 2 ep k + k 1 ( M 1) ep i ( 1 M ) ep (2.24) Generalizing Dykstra-Parsons Eq for any to layers ith k > k, and n is the layer, in hich ater just broke through: n j x x j n = M ep M 2 ep k + k ( M 1) ep j n ( 1 M ) ep (2.25) Finally expression for the coverage can be obtained: C n = n + N n+ 1 N x x j n = n + N n+ 1 M ep M N 2 ep k + k ( M 1) ep j n ( 1 M ) ep (2.26) And after rearrangement: C n = n + ( N n) M ep M 1 ep 1 M 1 ep N n+ 1 M N ep M 2 ep k + k j n ( 1 M ) ep 2 0.5,...(2.27) here N is the total number of layers. Kufus and Lynch 8 (1959): Kufus and Lynch in their paper presented ork hich can incorporate Buckley-Leverett theory in the Dykstra-Parsons calculations. Important assumptions Kufus and Lynch have made ere that all layers have same relative permeability curves to oil and ater and ater injection rate in each layer is constant value and dependent only on the absolute permeability and on fraction of average ater relative permeability to average fractional flo in the current layer, hich is made

32 17 - similar to Dykstra-Parsons model.. The data presented in the paper ere valid only for viscosity ratio of unity. And as in Dykstra-Parsons it as assumed that relative permeabilities to oil and ater ere same for all layers. Mobility ratio as represented by folloing equation: M µ o k = k µ ' ro f r av,... (2.28) here k' ro is the oil relative permeability ahead of the aterflood front. Using computation procedure the major parameters can be calculated. Hiatt 9 (1958): Hiatt presented a detailed prediction method concerned ith the vertical coverage or vertical seep efficiency attained by a aterflood in a stratified reservoir. Using a Buckley-Leverett type of displacement, he considered, for the first time, crossflo beteen layers. The method is applicable to any mobility ratio, but is difficult to apply. 12 Warren and Cosgrove 10 (1964): presented an extension of Hiatt s original ork. They considered both mobility ratio and crossflo effects in a reservoir hose permeabilities ere log-normally distributed. No initial gas saturation as alloed, and piston-like displacement of oil by ater as assumed. The displacement process in each layer is represented by a sharp pseudointerface as in the Dykstra-Parsons model. Reznik 11 et al. (1984): In this ork the original Dykstra-Parsons discrete solution has been extended to continuous, real time basis. Work has been made considering to injection constraints: pressure and rate. This analytical model assumes piston-like displacement. The purpose of the paper as to extend the analytical, but discrete, stratification model of Dykstra-Parsons to analytically continuous space-time solutions. The Reznik et al. ork retained the piston-like displacement assumption.

33 18 - CHAPTER III 3. NEW ANALYTICAL METHOD The main drabacks of the Dykstra-Parsons method are that (1) oil displacement by ater is piston-like, and (2) relative permeability end-point values are the same for all layers. Applying Buckley-Leverett theory to each layer is also not correct because it ould mean that ater injection rate is (1) constant for each layer, and (2) proportional to the kh of each layer. Thus a ne analytical model has been developed ith the folloing simplifying main assumptions: (1) Pressure drop for all layers is the same. (2) Total ater injection rate is constant. (3) Oil-ater relative permeabilities may vary for each layer. (4) Water injection rate in each layer may vary. 3.1 Calculation procedure The equations and steps used in the ne analytical method are as follos. For simplicity, the method has been applied to a 2-layered system ith no cross-flo. Step 1-Calculate oil-ater relative permeabilities For relative permeability calculation Corey 13 type relative permeability curves for oil and ater have been used.

34 19 - For oil k ro = k roe ( So Sor ) (1 S c S or ) n o, (3.1) here n o is Corey exponent for oil For ater k r = k re ( S S c ) (1 S c S or ) n,... (3.2) here n is Corey exponent for ater Using Corey equation the folloing relative permeability curves shon on Fig. 3.1 ere obtained: kro1 kr1 kro, kr, fraction S, fraction Figure 3.1 Corey type relative permeability curves for Case 1.

35 20 - Step 2-Fractional flo calculations After obtaining relative permeabilities for oil and ater, the fractional flo curve is to be found. Using definition of fractional flo, f (Eq. 2.2), and substituting for k ro and k r from Eqs. 3.1 and 3.2, e obtain: f = µ 1+ µ o k k roe re no n ( So Sor ) ( 1 S c Sor ) n no ( S S ) (1 S S ) 1 c c or......(3.3) Applying Welge technique: average saturation behind the aterflood front S, fractional flo at the ater breakthrough f bt, and ater saturation at the breakthrough S bt are df found. One necessary step is to calculate the fractional flo derivative ds. In order to perform this operation ith the more precision; e must take derivative of Eq After necessary mathematical derivation the folloing equation should be used: df ds = ( ) + ( ) ( ) 2 no n f f 1 S S or S S c..(3.4) Fractional flo curve is shon on Fig. 3.2.

36 21-1 S av f bt S bt 0.7 f, (fraction) S, (fraction) Figure 3.2 Fractional flo curve for Case 1. Step 3-Estimate ater injected in layer 1 at the moment of breakthrough After obtaining the fractional flo, e calculate cumulative ater injection in layer 1 at the moment of breakthrough. As e kno the cumulative ater injection in first layer at the moment of breakthrough W 1 can be calculated using Buckley-Leverett theory: i bt i bt ( S bt S c ) W 1 = PV,......(3.5) here PV = Lhφ / is the pore volume of the first layer. Step 4-Estimate ater injection rate in layer 1 Although total injection rate is constant, ater injection rate for each layer is going to change ith time as the relative permeabilities of ater and oil are going to change.

37 22 - Because of that e can not use the folloing approach in calculating the ater injection rate for layer 1: k1h i 1 = i t.....(3.6) kh But Eq. 3.6 can be used as an initial estimate or guess in the iterative procedure. Step 5-Calculation of the time of breakthrough After obtaining value of ater injection rate in layer 1 using Eq. 3.6, the folloing steps should be taken: Using Eq. 3.5 estimate ater injection rate in layer 1, after hich calculate time of breakthrough W i1bt t bt =... (3.7) i 1 Step 6-Calculation of total cumulative ater injected In this step total cumulative ater injected at the time of breakthrough is calculated, W = i t....(3.8) itbt t bt Step 7-Calculation of ater injected in layer 2 Since the total cumulative ater injection and cumulative ater injection in layer 1 are available, from material balance the cumulative ater injection in layer 2 can be obtained. Wi 2 Witbt Wi 1bt =... (3.9) Step 8-Calculation of average ater saturation in layer 2, pore volume displaced by ater in layer 2 and location of aterflood in layer 2

38 23 - Described process occurs at Buckley-Leverett frontal displacement, so the average ater saturation of second layer behind the front before breakthrough is constant and equal to first layer average ater saturation behind the front at the moment of breakthrough. W 1 = S,... (3.10) i2 S 2 + S c2 = + PVx f ' 2 c2 here f ' 2 is constant and equal to f ' 1bt, PV x is pore volume of layer 2 displaced by ater. From Eq e can obtain PV : x PV = W x f i2 ' 2. (3.11) Main point of this calculation is to find x the location of aterflood front in layer 2. It can be done using folloing expression PV x = xhφ / (3.12) The importance of x value is crucial for the calculations after the layer 1 broke through as it is only controlling parameter specifying at hich step after layer 1 broke through layer 2 is going to break through. Fig. 3.3 shos aterflood process at the moment of ater breakthrough in layer 1. Step 9-Recalculation of ater injection rate in layer 1 We need to develop different approach for calculating i 1 ; as it has been assumed the pressure gradient across all layers is the same

39 24 - P 1 L P 2 P Figure 3.3 Schematic representation of the aterflood process at the moment of breakthrough in layer 1. From this assumption the folloing expression can be derived: P 1 = P2 + P '... (3.13) From Darcy s la, ater injection rate in layer 1 can be expressed as i 1 ck1k = µ r1 P1,...(3.14) L here k r1 is the average ater relative permeability in layer 1. Similarly for ater injection rate in layer 2, i 2 ck2 k = µ r2 P x 2 2,....(3.15)

40 25 - here ater. P 2 x is pressure gradient of the region in layer 2, hich has been displaced by Using Darcy s la again for oil flo in layer 2 q o 2 ck2k = µ ro2 o P'..... (3.16) ( L x ) 2 For incompressible flo the folloing expression: i B qo2 2 = B ; applying Eq to Eqs e obtain o i Lµ i x µ i L x µ ( 2 2 ) o = +....(3.17) k 1 k r1 k 2 k r2 k 2 k ro2 Knoing that total ater injection rate is constant, simple material balance expression follos: i = i i...(3.18) 2 t 1 Substituting Eq in Eq gives the folloing: i Lµ 1 x µ ( L x ) µ =..(3.19) k k 1 r1 ( ) 2 2 o i + t i1 k2 k k r2 2k ro2 And solving for i 1 : i 1 = x 2µ ( L x2 ) µ o it + k2 k k2k r2 ro2.....(3.20) Lµ x2µ ( L x2 ) µ o + + k1kr 1 k k 2 kr2 2k ro2

41 26 - Eq may be rearranged to give: i 1 it =.. (3.21) Lk2k ro2µ k r2 1+ k k ( x µ k + ( L x ) µ k ) 1 r1 2 ro2 2 o r2 Step 10-Repeat Steps 5-9 until iterated ater injection rate in layer 1 is obtained. At this point of calculation e use the estimated value of ater injection rate in layer 1. Using Eq. 3.21, here relative permeabilities calculated using the Corey type curves, e can obtain a value of ater injection rate in layer 1, and compare it to the estimated value. In case of inconsistency, iterate until the true value of i 1 is reached. Step 11-Calculation of cumulative oil produced The N p value at the time of breakthrough can be calculated using Buckley-Leverett approach W itbt N p =....(3.22) Bo Also there is slightly different method to calculate N p value, using Dykstra-Parsons method using the vertical seep efficiency or so-called coverage factor C n Lhφ N p = ( Soi S c ) C B o n...(3.23) Substituting Eq in Eq the folloing expression for C n could be obtained C n PV ( S S c ) =,....(3.24) PV (1 S S ) c or

42 27 - here Eq is a general expression for coverage factor after breakthrough. Hoever in current case second layer haven t reached the producer yet, in hich case coverage factor must be divided in to parts C 1 and C 2, here C 1 PV1 ( S 1 S c1) =,......(3.25) PV (1 S S ) t c1 or1 and C 2 PVx ( S 2 S c2 ) =..(3.26) PV (1 S S ) t c2 or 2 And finally N p calculation: N MOVt = ( C 1 C2 ),... (3.27) B p + o here MOVt is total moveable oil in the reservoir. Step 12-Calculation after the breakthrough in layer 1 and subsequently in layer 2 Second part of procedure starts after the 1 st layer breakthrough but before the 2 nd layer breakthrough. It is necessary to specify the saturation change step in the first layer, for hich the folloing expression can be used: S = ( 1 Sor S bt ) here N is the number of steps to be defined. N, (3.28) During the course of the calculation procedure S is going to be calculated using Eq here S e = S + S. Basically all calculation steps ill remain unchanged except

43 28 - the several equations such as: calculation of cumulative ater injected in layer 1, after breakthrough W i1 1 df...(3.29) ds S e Another difference beteen the 1 st stage of procedure and the 2 nd stage is N p calculation, as the equation has to account for produced ater from layer 1. In order to calculate produced ater at each saturation change, the cumulative oil production from the first layer N p1 has to be calculated: ( Soi ( 1 S ) PV N p1 =..(3.30) B o After N p1 and W i1 are calculated, ater produced can be calculated as follos: W p1 Wi 1B N p1 = B...(3.31) o In the procedure the N p 1, W i 1 and W p 1 are used to calculate their corresponding cumulative amounts. Finally last part of the calculation procedure interprets behavior of the reservoir hen the second layer breaks through and beyond. Because of change in process, calculation steps must contain the N p 2 calculation, hich is analogical to N p 1 and mass balance must account for the produced ater from 2 nd layer W p 2. The method presented here differs from Buckley-Leverett original solution by calculating ater injection rate in specific layer on each saturation change, hereas for Buckley-Leverett method applied by Craig 14, ater injection rate in each layer is constant and depends only on the kh of each layer.

44 29 - In order to plot changing ater injection rates in layer 1 and layer 2 before breakthrough, cumulative ater injected in layer 1 at the moment of breakthrough W 1 must be calculated using Eq Then divide W 1 by the number of steps needed. As the upper i bt limit is knon there are no further complications: considering Eqs aterflood performance can be obtained. Only change ill include deriving the ater injection rate in layer 1 before the breakthrough i 1, and it can be found by folloing expression: i bt i 1 it =,...(3.32) Lk 2k k roe2 r2µ x1µ kroe k k k x µ k + 1 r1 roe1 2 roe2 ( L x1 ) µ o kr 1 ( L x2 ) µ k 2 o r here x 1 is the distance of the front in layer 1. All programming ork has been done in Microsoft VBA and Excel and can be found in APPENDIX B. Nine cases have been studied in hich injection rate and reservoir parameters are varied. Results based on the ne analytical model are compared against simulation results to verify the validity of the ne model. Brief descriptions of each of the nine cases follo. 3.2 Case 1 Current research based on the implementing Buckley-Leverett theory to the to phase homogeneous, horizontal reservoir consisting of the to non-communicating layers ith the different absolute permeabilities. Major assumptions are the constant total injection P rate i t, constant pressure gradient across all layers, incompressible and immiscible L displacement and no capillary or gravity forces. Parameters for case 1 are shon in Table 3.1.

45 30 - TABLE 3.1 RESERVOIR PROPERTIES FOR CASE 1 Reservoir properties Reservoir length, L Reservoir idth, Reservoir height, h Value 1200, ft 400, ft 70, ft Reservoir porosity, φ 25 % First layer permeability, k 1 Second layer permeability, k 2 500, md 350, md End point relative permeability of oil, k 0.85 roe End point relative permeability of ater, k 0.35 re Initial oil saturation, Connate ater saturation, S 80% oi S 20% c Residual oil saturation, S 20% or Oil viscosity, µ 8, cp o ater viscosity, µ 0.9, cp Total ater injection rate, i 800, STB/D t Oil formation volume factor, B o Water formation volume factor, B 1.25, RB/STB 1, RB/STB The height of layer 1 is equal to the height of layer 2 in case 1.

46 31 - Belo are oil production results of ne analytical model compared to simulation model: Oil production rate, STB/D Simulation Ne analytical model Time, yr Figure 3.4 Comparison of oil production rate of ne analytical model vs. simulation (Case 1). From Fig. 3.4, it can be seen that oil production rates based on the ne model and simulation are practically identical. Hoever after breakthrough in layer 1, oil production rate is about 50 STB/D higher based on simulation. Breakthrough time for layer 1 is almost identical based on the ne model and simulation. Hoever, there is significant difference in the second layer breakthrough times. This difference is probably caused by the method used in calculating ater injection rate in each layer. Fig. 3.5 presents the ater injection rate by layer based on the ne model and compared against simulation results. It can be seen that layer injection rate before breakthrough in layer based on the ne model and simulation is in very good agreement. Hoever,

47 32 - after breakthrough in layer 2, layer injection rate is about 25 STB/D higher in layer 1 and about 25 STB/D loer in layer 2 based on the ne model compared to simulation Injection rate, STB/D Simulation, layer 1 Simulation, layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.5 Comparison of ater injection rate of ne analytical model vs. simulation by layers (Case 1). Fig. 3.6 shos cumulative oil production versus time. It can be seen that cumulative oil production based on the ne model and simulation is in close agreement.

48 33-700, ,000 Cumulative oil produced, STB 500, , , ,000 Ne analytical model Simulation 100, Time, yr Figure 3.6 Comparison of cumulative oil produced calculated ith ne analytical model vs. simulation (Case 1). Fig. 3.7 shos oil production rate by layer. It can be seen that oil production rates for both layers before breakthrough in layer 1 based on the ne model is very similar to that based on simulation. Hoever after breakthrough in layer 1, oil production rate in layer 2 is higher based on simulation. Nevertheless, after breakthrough in layer 2, oil production rate for both layers based on the ne model are in close agreement ith simulation results.

49 Oil production rate, STB/D Ne analytical model, layer1 Ne analytical model, layer 2 Simulation, layer 1 Simulation, layer Time, yr Figure 3.7 Oil production rates by layers, ne analytical model vs. simulation (Case 1). Fig. 3.8 presents ater production rate by layer. Because of higher ater injection rate in layer 1 based on the ne model, it can be seen that ater production rate in layer 1 is also higher, and vice-versa for layer 2. Hoever, as can be noted from Fig. 3.9, the total ater production rate based on the ne model is in good agreement ith simulation results after breakthrough in layer 2.

50 Water production rate, STB/D Simulation, layer 1 Simulation, layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.8 Water production rate by layers, ne analytical model vs. simulation (Case 1).

51 Water production rate, STB/D Ne analytical model Simulation Time, yr Figure 3.9 Total ater production rate comparison of ne analytical model vs. simulation (Case 1). 3.3 Case 2 One of the goals of this study as to apply different sets of relative permeability to different layers and compare calculated results to that of simulation. Case 2 is identical to Case 1 except the oil-ater relative permeability set for layer 2 has been changed as follos: k roe2 is 0.9 and kre2 is 0.5 fraction, residual oil saturation S or 2 is 35 %, and S c2 is 20 %. Parameters for Case 2 are shon in Tables

52 37 - TABLE 3.2 RESERVOIR PROPERTIES OF LAYER 1 FOR CASE 2. Layer 1 Characteristics Layer 1 length, L Layer 1 idth, Layer 1 height, h Value 1200, ft 400, ft 35, ft Layer 1 porosity, φ 25 % Layer 1 permeability, k 1 500, md End point relative permeability of oil, k 0.85 roe End point relative permeability of ater, k 0.35 re Initial oil saturation, Connate ater saturation, S 80% oi S 20% c Residual oil saturation, S 20% or Oil viscosity, µ 8, cp o ater viscosity, µ 0.9, cp ater injection rate in layer 1, i variable, STB/D 1 Oil formation volume factor, B o 1.25, RB/STB Water formation volume factor, B 1, RB/STB

53 38 - TABLE 3.3 RESERVOIR PROPERTIES OF LAYER 2 FOR CASE 2. Layer 2 Characteristics Layer 2 length, L Layer 2 idth, Layer 2 height, h Value 1200, ft 400, ft 35, ft Layer 2 porosity, φ 25 % Layer 2 permeability, k 2 350, md End point relative permeability of oil, k 0.9 roe End point relative permeability of ater, k 0.5 re Initial oil saturation, Connate ater saturation, S 70% oi S 30% c Residual oil saturation, S 35% or Oil viscosity, µ 8, cp o ater viscosity, µ 0.9, cp ater injection rate in layer 2, i variable, STB/D 2 Oil formation volume factor, B o 1.25, RB/STB Water formation volume factor, B 1, RB/STB To sets of oil-ater relative permeability are shon in Fig

54 kro2 kr2 kro1 kr S, fraction Figure 3.10 Sets of relative permeabilities for layers 1 and 2 (Case 2). Based on the relative permeability data to fractional flo curves have to be created, as shon in Fig

55 f1 f2 f, fraction S, fraction Figure 3.11 Fractional flo curves for layers 1 and 2 respectively (Case 2). From Fig. 3.12, it can be seen that oil production rates based on the ne model and simulation are in very good agreement. Hoever after breakthrough in layer 1, oil production rate is about 70 STB/D higher based on simulation. Breakthrough time for layer 1 is almost identical based on the ne model and simulation. Note that there is significant difference in the second layer breakthrough times. This difference might be caused by the method used in calculating ater injection rate in each layer.

56 Oil production rate, STB/D Simulation Ne analytical model Time, yr Figure 3.12 Oil production rate of ne analytical model vs. simulation, different sets of relative permeabilities are applied (Case 2). Fig presents the ater injection rate by layer based on the ne model and simulation results. It can be noted, that layer injection rate before breakthrough is in very good agreement. Nevertheless, after breakthrough in layer 2, layer injection rate is about 60 STB/D higher in layer 1, and about 60 STB/D loer in layer 2 according to the ne model compared against simulation.

57 Injection rate, STB/D Simulation, layer 1 Simulation, layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.13 Water injection rates by layers of ne analytical model vs. simulation, different sets of relative permeabilities are applied (Case 2). Fig shos cumulative oil production versus time. We can see that for the first four years of production cumulative oil produced is in good agreement for ne model versus simulation. Hoever for the next 20 years of production ne model shos quite significant difference against that of simulation.

58 43-700, ,000 Cumulative oil produced, STB 500, , , ,000 Ne analytical model Simulation 100, Time, yr Figure 3.14 Cumulative oil produced comparison of ne analytical model vs. simulation, to sets of relative permeability are provided for each layer (Case 2). Fig shos oil production rate by layer. It can be seen that oil production rates for both layers before breakthrough in layer 2 (as due to difference in oil-ater relative permeability layer 2 breaks through first) based on the ne model is very similar to that based on simulation. Hoever after breakthrough in layer 2, oil production rate in layer 1 is higher based on simulation. Nevertheless, after breakthrough in layer 1, oil production rate for both layers based on the ne model are in close agreement ith simulation results.

59 Oil production rate, STB/D Ne analytical model, layer1 Ne analytical model, layer 2 Simulation, layer 1 Simulation, layer Time, yr Figure 3.15 Oil production rate by layers comparison of ne analytical model vs. simulation, to sets of relative permeability are provided for each layer (Case 2). Fig presents ater production rate by layer. Because of higher ater injection rate in layer 2 based on the ne model, it can be seen that ater production rate in layer 2 is also higher, and vice-versa for layer 1. Hoever, as can be noted from Fig. 3.17, the total ater production rate based on the ne model is in good agreement ith simulation results after breakthrough in layer 1.

60 Water production rate, STB/D Simulation, layer 1 Simulation, layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.16 Water production rate by layers, comparison of ne analytical model vs. simulation, to sets of relative permeability are provided for each layer (Case2).

61 Water production rate, STB/D Ne analytical model Simulation Time, yr Figure 3.17 Total ater production rate, comparison of ne analytical model vs. simulation, to sets of relative permeability are provided for each layer (Case 2). 3.4 Case 3 Next case represents the variation of the case 1 ith different set of kh. Table 3.4 contains the changes made to the model:

62 47 - TABLE 3.4 HEIGHT AND PERMEABILITY VARIATION IN LAYERS 1 AND 2. Characteristics Absolute permeability of layer 1, k 1 Height of layer 1, h 1 Absolute permeability of layer 2, k 2 Height of layer 2, h 2 Value 500, md 50, ft 100, md 25, ft From Fig. 3.18, it can be seen that oil production rates based on the ne model and simulation are in good agreement. Hoever after breakthrough in layer 1, oil production rate is about 30 STB/D loer based on simulation. Breakthrough time for layer 1 is very close based on the ne model and simulation. Note that there is significant difference of 2.5 years in the second layer breakthrough times. This difference is probably caused by the method used in calculating ater injection rate in each layer.

63 Oil production rate, STB/D Simulation Ne analytical model Time, yr Figure 3.18 Oil production comparison of ne analytical model to the simulation, variable kh is applied (Case 3). Fig presents the ater injection rate by layer based on the ne model and simulation results. It can be noted, that there is constant difference of 30 STB/D in layer injection rate before and after breakthrough.

64 Injection rate, STB/D Simulation, layer 1 Simulation, layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.19 Water injection rate comparison of ne analytical model versus simulation, variable kh is applied (Case 3). Fig shos cumulative oil production versus time. We can see that overall cumulative oil production is in good agreement for ne model versus simulation. Hoever for period of time from 2 nd year of production to 10 th year there is significant difference of ne model against that of simulation.

65 50-700, ,000 Cumulative oil produced, STB 500, , , ,000 Ne analytical model Simulation 100, Time, yr Figure 3.20 Cumulative oil produced comparison of ne analytical model to the simulation, variable kh is applied (Case 3). Fig shos oil production rate by layer. It can be seen that oil production rate for both layers before breakthrough in layer 1 based on the ne model gives a difference of 30 STB/D to that based on simulation, here simulation production rate is higher. Hoever after breakthrough in layer 2, oil production rate in layer 2 is higher based on ne model. Nevertheless, after breakthrough in layer 1, oil production rate for both layers based on the ne model are in close agreement ith simulation results.

66 Oil production rate, STB/D Ne analytical model, layer1 Ne analytical model, layer 2 Simulation, layer 1 Simulation, layer Time, yr Figure 3.21 Oil production layer by layer comparison of ne analytical model to the simulation, variable kh is applied (Case 3). Fig presents ater production rate by layer. Because of higher ater injection rate in layer 1 based on the simulation, it can be seen that ater production rate in layer 1 is also higher, and vice-versa for layer 2. Hoever, as can be noted from Fig. 3.23, the total ater production rate based on the ne model is in close agreement ith simulation results after breakthrough in layer 2 based on the simulation.

67 Water production rate, STB/D D Simulation layer 1 1-D Simulation layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.22 Water production layer by layer comparison of ne analytical model to the simulation, variable kh is applied (Case 3).

68 Water production rate, STB/D Ne analytical model Simulation Time, yr Figure 3.23 Total ater production comparison of ne analytical model to the simulation, variable kh is applied. 3.5 Case 4 Case 4 is obtained by using kh parameters of Case 3 applied to the different oil-ater relative permeability set of Case 2. From Fig. 3.24, it can be noted that oil production rates based on the ne model and simulation are in good agreement. Hoever after breakthrough in layer 1, oil production rate is about 50 STB/D higher based on simulation. Breakthrough time for layer 1 is very close based on the ne model and simulation. Note that there is significant difference in

69 54 - the second layer breakthrough times. This difference is probably caused by the method used in calculating ater injection rate in each layer Oil production rate, STB/D Simulation Ne analytical model Time, yr Figure 3.24 Total oil production comparison of ne analytical model versus simulation, variable kh and relative permeability sets are applied (Case 4). Fig presents the ater injection rate by layer based on the ne model and simulation results. In this case the most significant difference of 100 STB/D among all cases can be seen. We might note that simulation results for layer 1 sho higher values than that of the ne model.

70 Injection rate, STB/D Simulation, layer 1 Simulation, layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.25 Water injection layer by layer comparison of ne analytical model to the simulation, variable kh and relative permeability sets are applied (Case 4). In spite of the difference in Fig. 3.25, in Fig. 3.26, hich shos cumulative oil production versus time, it can be seen that cumulative oil production based on the ne model and simulation is in close agreement.

71 56-700, ,000 Cumulative oil produced, STB 500, , , ,000 Ne analytical model Simulation 100, Time, yr Figure 3.26 Cumulative oil produced, comparison of ne analytical model to simulation, variable kh and relative permeability sets are applied (Case 4). Fig shos oil production rate by layer. It can be seen that oil production rate for both layers before breakthrough in layer 1 based on the ne model gives a significant difference of 70 STB/D to that based on simulation, here simulation production rate is higher for layer 1. Hoever after breakthrough in layer 2, oil production rate in layer 2 is higher based on ne model. Hoever, after breakthrough in layer 1, oil production rate for both layers based on the ne model are in close agreement ith simulation results.

72 Oil production, STB/D Ne analytical model, layer 1 Ne analytical model, layer 2 Simulation, layer 1 Simulation, layer Time, yr Figure 3.27 Oil production layer by layer, comparison of ne analytical model to simulation, variable kh and relative permeability sets are applied (Case 4). Fig presents ater production rate by layer. Because of higher ater injection rate in layer 1 based on the simulation, it can be seen that ater production rate in layer 1 is also higher, and vice-versa for layer 2. The difference is about 110 STB/D. Hoever, as can be noted from Fig. 3.29, the total ater production rate based on the ne model is in close agreement ith simulation results.

73 Water production rate, STB/D Simulation, layer 1 Simulation, layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.28 Water production layer by layer, comparison of ne analytical model to simulation, variable kh and relative permeability sets are applied (Case 4).

74 Water production rate, STB/D Ne analytical method Simulation Time, yr Figure 3.29 Water production rate, comparison of ne analytical model to the simulation, variable kh and relative permeability sets are applied (Case 4). 3.6 Case 5 In folloing case I kept all parameters of Case 1 unchanged except total injection rate, hich I increased tice to a value of 1600 STB/D. Fig shos the oil production rate based on the ne model versus simulation results. As it can be seen from Fig it shos exactly the same behavior as in Fig. 3.4 but ith doubled production rate and halved time of breakthrough.

75 Oil production rate, STB/D D Simulation Ne analytical model Time, yr Figure 3.30 Oil production rate, comparison of ne analytical model to the simulation, injection rate of 1600 STB/D is applied (Case 5). Fig presents the ater injection rate by layer comparison of the ne model against that of simulation. Similarly it can be seen from Fig that it shos exactly the same behavior as in Fig. 3.5 but ith doubled injection rate and halved time of breakthrough.

76 61 - Injection rate, STB/D Simulation, layer 1 Simulation, layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.31 Water injection rate by layers, comparison of ne analytical model to simulation, injection rate of 1600 STB/D is applied (Case 5). Fig presents the cumulative oil production comparison of the ne model against that of simulation. Similarly it can be seen from Fig that it shos exactly the same behavior as in Fig. 3.6.

77 62-700, ,000 Cumulative oil produced, STB 500, , , ,000 Ne analytical model Simulation 100, Time, yr Figure 3.32 Cumulative oil production, comparison of ne analytical model to simulation, injection rate of 1600 STB/D is applied (Case 5). Fig shos oil production rate by layer. It can be seen that oil production rates for both layers before breakthrough in layer 1 based on the ne model is very similar to that based on simulation. Hoever after breakthrough in layer 1, oil production rate in layer 2 is higher based on simulation. Nevertheless, after breakthrough in layer 2, oil production rate for both layers based on the ne model are in close agreement ith simulation results. Fig is similar to Fig. 3.7.

78 Oil production rate, STB/D Ne analytical model, layer1 Ne analytical model, layer 2 Simulation, layer 1 Simulation, layer Time, yr Figure 3.33 Oil production rate on layer basis, comparison of ne analytical model to simulation, injection rate of 1600 STB/D is applied (Case 5). Fig presents ater production rate by layer. The results are similar to Fig. 3.8 results of Case 1. Hoever, as can be noted from Fig. 3.35, the total ater production rate based on the ne model has significant difference in breakthrough time comparing to Fig. 3.9.

79 Water production rate, STB/D Simulation, layer 1 Simulation, layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.34 Water production on layer basis, comparison of ne analytical model to simulation, injection rate of 1600 STB/D is applied (Case 5).

80 Water production rate, STB/D Ne analytical model Simulation Time, yr Figure 3.35 Water production rate, comparison of ne analytical model to simulation, injection rate of 1600 STB/D is applied (Case 5). 3.7 Case 6 Case 6 represents the variation of Case 2 here injection rate value has been increased to the 1600 STB/D of ater. Fig shos the oil production rate based on the ne model versus simulation results. As can be seen from Fig it shos exactly the same behavior as in Fig from Case 2 but ith doubled production rate and halved time of breakthrough.

81 Oil production rate, STB/D Simulation Ne analytical model Time, yr Figure 3.36 Oil production rate, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 6). Fig presents the ater injection rate by layer comparison of the ne model against that of simulation. Similarly it can be seen from Fig that it shos exactly the same behavior as in Fig of Case 2, but ith doubled injection rate.

82 Injection rate, STB/D Simulation, layer 1 Simulation, layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.37 Water injection rate by layers, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 6). Fig presents the cumulative oil production comparison of the ne model against that of simulation. Similarly it can be seen from Fig that it shos exactly the same behavior as in Fig

83 68-700, ,000 Cumulative oil produced, STB 500, , , ,000 Ne analytical model Simulation 100, Time, yr Figure 3.38 Cumulative oil produced, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 6). Fig shos oil production rate by layer. It can be seen that Fig oil production rates are similar to Fig oil production rates.

84 Oil production rate, STB/D Ne analytical model, layer1 Ne analytical model, layer 2 Simulation, layer 1 Simulation, layer Time, yr Figure 3.39 Oil production rate by layers, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 6). Fig presents ater production rate by layer. The results are similar to Fig results of Case 2. Hoever, as can be noted from Fig. 3.41, the total ater production rate based on the ne model is in good agreement to that of simulation.

85 Water production rate, STB/D Simulation, layer 1 Simulation, layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.40 Water production rate by layers, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 6).

86 Water production rate, STB/D Modified Dykstra-Parsons 1-D Simulation Time, yr Figure 3.41 Total ater production rate, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 6).

87 Case 7 In this case it as decided to use increased injection rate of 1600 STB/D on the Case 3, model ith single relative permeability set and ith substantial difference in kh. Fig shos the oil production rate based on the ne model versus simulation results. As it can be seen from Fig it shos exactly the same behavior as in Fig from Case 3, but ith doubled production rate and halved time of breakthrough Oil production rate, STB/D Simulation Ne analytical model Time, yr Figure 3.42 Oil production rate, comparison of ne analytical model to simulation, different kh and injection rate of 1600 STB/D are applied (Case 7). Fig presents the ater injection rate by layer comparison of the ne model against that of simulation. Similarly it can be seen from Fig that it shos exactly the same behavior as in Fig of Case 3, but ith doubled injection rate.

88 Injection rate, STB/D D Simulation layer 1 1-D Simulation layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3-43 Water injection rate by layers, comparison of ne analytical model to simulation, different kh and injection rate of 1600 STB/D are applied (Case 7). Fig presents the cumulative oil production comparison of the ne model against that of simulation. Similarly it can be seen from Fig that it shos exactly the same behavior as in Fig

89 74-700, ,000 Cumulative oil produced, STB 500, , , ,000 Ne analytical model Simulation 100, Time, yr Figure 3.44 Cumulative oil produced, comparison of ne analytical model to simulation, different kh and injection rate of 1600 STB/D are applied (Case 7). Fig shos oil production rate by layer. It can be seen that Fig oil production rates are similar to Fig oil production rates of Case 3.

90 Oil production rate, STB/D Ne analytical model, layer1 Ne analytical model, layer 2 Simulation, layer 1 Simulation, layer Time, yr Figure 3.45 Oil production by layers, comparison of ne analytical model to simulation, different kh and injection rate of 1600 STB/D are applied (Case 7). Fig presents ater production rate by layer. The results are similar to Fig results of Case 3. Hoever, as can be noted from Fig. 3.47, the total ater production rate based on the ne model is in good agreement to that of simulation after breakthrough in layer 2.

91 Water production rate, STB/D Simulation, layer 1 Simulation, layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.46 Water production by layers, comparison of ne analytical model to simulation, different kh and injection rate of 1600 STB/D are applied (Case 7).

92 Water production rate, STB/D Ne analytical model Simulation Time, yr Figure 3.47 Total ater production, comparison of ne analytical model to simulation, different kh and injection rate of 1600 STB/D are applied (Case 7). 3.9 Case 8 Current case is the same as Case 4 except total injection rate ill be changed to 1600 STB/D. Fig shos the oil production rate based on the ne model versus simulation results. As can be seen from Fig. 3.48, it shos exactly the same behavior as in Fig from Case 4, but ith doubled production rate and halved time of breakthrough.

93 Oil production rate, STB/D Simulation Ne analytical model Time, yr Figure 3.48 Total oil production rate, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 8). Fig presents the ater injection rate by layer comparison of the ne model against that of simulation. Similarly it can be seen from Fig that it shos exactly the same behavior as in Fig As can be seen from Fig.3.49, ater injection rate comparison of analytical versus numerical models shos the most significant difference by analogy to Case 4.

94 Injection rate, STB/D Simulation, layer 1 Simulation, layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.49 Water injection rate by layers, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 8). Fig presents the cumulative oil production comparison of the ne model against that of simulation. Similarly it can be seen from Fig that it shos exactly the same behavior as in Fig

95 80-700, ,000 Cumulative oil produced, STB 500, , , ,000 Ne analytical model Simulation 100, Time, yr Figure 3.50 Cumulative oil, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 8). Fig shos oil production rate by layer. It can be seen that Fig oil production rates are similar to Fig oil production rates of Case 4.

96 Oil production rate, STB/D Ne analytical model, layer 1 Ne analytical model, layer 2 Simulation, layer 1 Simulation, layer Time, yr Figure 3.51 Oil production by layers, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 8). Fig presents ater production rate by layer. The results are similar to Fig results of Case 4. Hoever, as can be noted from Fig. 3.53, the total ater production rate based on the ne model is in good agreement to that of simulation after breakthrough in layer 2.

97 Water production rate, STB/D Simulation, layer 1 Simulation, layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.52 Water production by layers, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 8).

98 Water production rate, STB/D Ne analytical model Simulation Time, yr Figure 3.53 Water production rate, comparison of ne analytical model to simulation, 2 relative permeability sets and injection rate of 1600 STB/D are applied (Case 8) Case 9 In the Case 9, the reservoir parameters are identical for both layers. This case as run basically for validation of the analytical model program. Fig presents oil production rate of the ne model compared to simulation, as it can be seen from the Fig the results are practically identical.

99 Oil production rate, STB/D Simulation Ne analytical model Time, yr Figure 3.54 Oil production rate, comparison of ne analytical model to simulation, for 2 identical layers (Case 9). Fig presents the ater injection rate by layer comparison of the ne model against that of simulation. Water injection rate for each layer shos the good match.

100 Injection rate, STB/D Simulation, layer 1 Simulation, layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.55 Water injection rate by layers, comparison of ne analytical model to simulation, for 2 identical layers (Case 9). Fig presents the cumulative oil production comparison of the ne model against that of simulation, shoing a very good agreement.

101 86-700, ,000 Cumulative oil produced, STB 500, , , ,000 Ne analytical model Simulation 100, Time, yr Figure 3.56 Cumulative oil produced, comparison of ne analytical model to simulation, for 2 identical layers (Case 9). Fig is comparison of the oil production rate by layer of the ne model versus simulation. Very close agreement achieved on the Fig as ell.

102 Oil production rate, STB/D Ne analytical model, layer1 Ne analytical model, layer 2 Simulation, layer 1 Simulation, layer Time, yr Figure 3.57 Oil production by layers, comparison of ne analytical model to simulation, for 2 identical layers (Case 9). So far analytical model shoed very close results compared ith numerical simulation model. Fig shos ater production rate by layer.

103 Water production rate, STB/D Simulation, layer 1 Simulation, layer 2 Ne analytical model, layer 1 Ne analytical model, layer Time, yr Figure 3.58 Water production by layers, comparison of ne analytical model to simulation, for 2 identical layers (Case 9). Fig shos total ater production rate of the ne model compared to that of simulation.

104 Water production rate, STB/D Ne analytical model Simulation Time, yr Figure 3.59 Water production rate, comparison of ne analytical model to simulation, for 2 identical layers (Case 9).

105 90 - CHAPTER IV 3. SIMULATION MODEL OVERVIEW For comparison of the ne analytical model to numerical simulation, I used GeoQuest Eclipse 100 softare as the simulator. A 2-layer simulation grid model as used, ith no crossflo beteen the layers, and 1-D displacement in each layer. By constructing simulation model in this manner, both gravity and capillary pressure effects are ignored. Using data provided for each of the nine cases in chapter 3, modifications to the reservoir properties ere made. Hoever for all nine cases, the grid dimensions ere kept the same. As an initial step, a simple 2-layer numerical simulation model as created. The model had 1x100x2 grid blocks, ith variable grid in the y-direction. Initial time step t as 36.5 days. For sensitivity purposes the grid in the y-direction as varied from 200 grid blocks to 400 grid blocks or from 1x100x2 to 1x200x2. The initial time step t as reduced to 3.65 days. The result of the refinement is shon in Fig. 4.1, indicating practically identical results. Thus, since each simulation run takes only about to minutes, the finer grid 1x200x2 model as used for the study.

106 91-700,000 Cumulative oil produced, STB/D 600, , , , , ,000 1x200x2 simulation model 1x100x2 simulation model Time, yr Figure 4.1 Simulation results indicate cumulative oil production ith and ithout grid refinement is practically identical. The nine cases studied represent different reservoir parameters for the 2-layered reservoir, as summarized in Tables 4.1 and 4.2.

107 92 - TABLE 4.1 SUMMARY OF RESERVOIR PROPERTIES FOR EACH CASE. Case i t, STB/D h 1, ft* h 2, ft k 1, md* k 2, md Relative permeability set** * Subscripts 1 and 2 denote layer number (1 = upper, 2 = loer) ** Relative permeability set data listed in Table 4.2. TABLE 4.2 OIL-WATER RELATIVE PERMEABILITY SET PARAMETERS. Parameters Set 1 Set 2 S c, fraction S or, fraction k roe, fraction k re, fraction n o n 2.8 2

108 93 - The effect of varying the parameters for each of the nine cases is significant, as shon in Figs These figures sho the oil saturation profile in each layer at 274 days since injection, from hich e can see the different aterflood advancement. The fact that these advancements differ significantly for each case is desired to fully test the validity of the ne analytical model. Figure 4.2 Simulation results for Case 1 at 274 days, shoing earlier ater breakthrough in layer that has a higher kh.

109 94 - Figure 4.3 Simulation results for Case 2 at 274 days, shoing faster aterflood displacement in loer layer, hich has different oil-ater relative permeability. Figure 4.4 Simulation results for Case 3 at 274 days, shoing faster ater front propagation in upper layer that has a higher kh value.