Iraqi Journal of Chemical and Petroleum Engineering Vol.15 No.3 (September 2014) 51-60 ISSN: 1997-4884 Analytical Model for Detection the Tilt in Originally Oil Water Contacts Jalal Abdulwahid Al-Sudani Petroleum Engineering Department-College of Engineering- University of Baghdad Abstract Many carbonate reservoirs in the world show a tilted in originally oil-water contact (OOWC) which requires a special consideration in the selection of the capillary pressure curves and an understanding of reservoir fluids distribution while initializing the reservoir simulation models. An analytical model for predicting the capillary pressure across the interface that separates two immiscible fluids was derived from reservoir pressure transient analysis. The model reflected the entire interaction between the reservoir-aquifer fluids and rock properties measured under downhole reservoir conditions. This model retained the natural coupling of oil reservoirs with the aquifer zone and treated them as an explicit-region composite system; thus the exact solutions of diffusivity equation could be used explicitly for each region. The reservoir-aquifer zones were linked by a capillary transition zone that reflected the pressure difference across the free water level. The principle of superposition theorem was applied to perform this link across the free water level to estimate the reflected aquifer pressure drop behavior that holds the fluid contacts in their equilibrium positions. The results of originally oil water contact positions generated by the proposed model were compared with data obtained from a carbonate oil field; the results given by the model showed full agreement with the actual field data. Keywords: Capillary pressure, Tilted oil water contact Introduction Many carbonate reservoirs in the world show a tilted in OOWC; an inaccurate determination for this tilting causes erroneous results in estimating the original oil in place (OOIP) in reservoir simulation process, and provides inadequate water flooding schemes while field development process, would affect reservoir recovery process and reserve prediction. The actual cause for the tilt of OOWC is still under investigation. The difficulties in studying the OOWC tilt related to the lack of flank wells intersecting the OOWC. In addition, most of the available information coming from edge wells in areas of high cristal production resulting in dynamic oil water contact (OWC), not OOWC. This OOWC differs from the dynamic OWC that is related to the pressure gradient caused by oil Iraqi Journal of Chemical and Petroleum Engineering University of Baghdad College of Engineering Analytical Model for Detection the Tilt in Originally Oil Water Contacts 52 IJCPE Vol.15 No.3 (September 2014) -Available online at: www.iasj.net production and subsequent pressure drop in the producing area which vary along the oil-water interface and causing the water to move with varying speeds in the different parts of the reservoir; this causes a distortion in the OWC. Previously, a wide explanation for the tilt was a huge regional dynamic aquifer to what recently has established the cause for the regional temperature gradient variation [1]. Several theories were presented in the past to investigate the tilted OOWC, but in fact none were able to account successfully for the origin of this phenomenon. However, Stenger [2] shows that the geothermal gradient is a controlling parameter of the aquifer salinity and this affects the gradual change in oil-water densities. Shawket G.Ghaidan et al. [3] stated that capillary pressure reflects the interaction of rock and the fluids filling these pores. The measurements of capillary pressure must ideally perform in reservoir conditions to reflect the actual field analysis, but unfortunately one is extremely difficult and is usually done by injecting mercury at ambient conditions. However, the experience showed that either capillary pressure measured in the laboratory or capillary pressure converted to reservoir conditions may not match the log derived Sw - Pc data which also has an accuracy of +/- 10% in best and may be as high as +/- 20% . [3]. Anderson [4] also confirmed that the capillary pressure - saturation relationship depends on the interaction of wettability, pore structure, initial saturation, and saturation history. No simple relationship exists that relates the capillary pressure determined at two different wettabilities. Therefore, the most accurate measurements are made with cores that have native reservoir wettability. Majid Hassanizadeh et al. [5] stated that although the processes that determine the distribution of fluid phases in porous media are extremely complicated. The main theoretical and practical tool currently used to quantify the capillary pressure function is an empirical relationship between capillary pressure and saturation and as such, it lacks a firm theoretical foundation. This simple empirical model is implicitly assumed to account for all effects and processes that influence the equilibrium distribution of fluids, such as surface tension, presence of fluid–fluid interfaces, wettability of solid surfaces, grain size distribution, and microscale heterogeneities. All of these effects are essentially lumped into the capillary pressure and saturation relationship. In fact, there is ample theoretical and experimental evidence that this simple relationship is not unique, but it depends on the flow dynamics; it depends on both the history and the rate of change of saturation. The dependence of capillary pressure– saturation curves on the history of flow is known as capillary pressure hysteresis this is a well-known effect and has been the subject of extensive investigations. The dependence of capillary curves on the rate of change of saturation is due to dynamic effects. Theories Used in Initialization of OOWC Due to the variance of the theories made to explain the reasons of tilted OOWC, Hsueh et al. [1] suggested four different methods that could be used in initializing the OOWC as follows: 1- Initialization with capillary pressure correction: This method provides a correction term for capillary pressure added to the general definition of capillary pressure to balance the difference in oil column Jalal Abdulwahid Al-Sudani -Available online at: www.iasj.net IJCPE Vol.15 No.3 (September 2014) 53 thickness with respect to the free water level. 2- Initialization with specified water saturation: This method suggests that if there is a large saturation contrast in the vertical direction, it may cause instability at time zero of reservoir saturation initialization; this instability may extend over the course of the history match. 3- Initialization by equilibrium region: This method is simple in application; the reservoir area is divided into two equilibrium regions with two different OWC. The disadvantage of this method is that it is not applicable to account for more than one tilt direction. 4- Initialization by dynamic injection- production method: This method suggests using pseudo injection/production wells located at the sides of the tilt to artificially create the tilted OWC. The pseudo injection well is located at the side of high water column of the tilt, while the pseudo production well is located at the lower water column of the tilt. Image Theory The most powerful techniques in reservoir engineering is the principle of image theorem that can be applied to removing the restrictions that have been imposed on various forms of solution. The principle of image theorem says that the response of the system to a number of events is exactly equal to the sum of the responses to each of the events as if they were present by themselves. It is well known that the capillary pressure holds the fluid interfaces at their equilibrium positions unless a pressure disturbance may occur at the contacted free fluid interfaces [6]. Thus, the new theory of measuring the dynamic capillary pressure under reservoir conditions suggests creating reservoir pressure drop pulse that is usually performed throughout conventional well test analysis. Therefore, it is essential to formulate dynamic capillary pressure model in an appropriate way due to its importance in reservoir simulation and engineering applications; in order to reduce the complexity, the formulation of an accurate analytical dynamic capillary pressure model in which the created reservoir pressure drops moving between two different fluid mediums (oil and water) could be analyzed by treating each fluid-phase zone as explicit single-phase zone that takes its own fluid and rock properties, as already adopted by Layne et al. [7]; hence, the reservoir-aquifer zones are linked across the free water level (zero capillary pressure) by a capillary transition zone that is balancing the pressure difference across the free water level. Hence, the free water level could be considered as a no-flow boundary and the reservoir may behave as an infinite acting reservoir as well as the free water level still not affected by the created reservoir pressure drops. Based on this principle, the pressure difference across the free water level must remain constant while the pressure transient flow period which reflects the dynamic capillary pressure across the free water level representing the initial transition zone thickness of height depends on the density difference between the oil and water. Thus, the exact solution could be generated using the principle of image theorem throughout reflecting the created reservoir pressure drops at the free water level to estimate the reflected pressure drops in a manner exactly identical to the procedure used to estimate the reservoir pressure drops for a well bounded by a no-flow boundary; therefore, this solution is valid only during the pressure transient Analytical Model for Detection the Tilt in Originally Oil Water Contacts 54 IJCPE Vol.15 No.3 (September 2014) -Available online at: www.iasj.net flow period. Hence, both of the created and the reflected pressure drops should exhibit transient flow behavior. Since, based on the principle of image theorem, the exact solution to estimate the pressure difference across the free water level is exactly equal to the sum of the responses for the reservoir and aquifer zones, taking into consideration that the solution of Van- Everdingen and Hurst [8] for infinite- radial flow will represent the created reservoir pressure drops encountered due to oil production activity, and Keith Coats [9] exact solution to represent the imposed reflected pressure drops encountered at free water level. The aquifer will respond to the created reservoir pressure drops by an imposed reflected pressure drops that depend on its own water and rock properties. Hence, both of the created and the imposed reflected pressure drops should exhibit transient flow behavior, as well as the created reservoir pressure drops have not reached the free water levels yet. Estimation Dynamic Capillary Pressure The main theoretical and practical tool currently used to quantify the capillary pressure function is only an empirical simple function that lacks meeting all parameters affecting capillary pressure-saturation relationship as given by Eq. 1. …(1) Where are the pressures exerted at the free water level by both of reservoir fluid and aquifer water zones, respectively. Adding and subtracting the initial reservoir pressure ( ) to the right hand side of Eq. 1 and arranging it which could be rewritten as follows; …(2) The term represents the created reservoir pressure drop behavior during transient flow period, which could be estimated using Van- Everdingen and Hurst [8] for infinite radial flow solution; while, represents the imposed reflected pressure drops by the aquifer zone, which could be estimated using Keith [9], for two dimensional flow solution. The numerical results for a computer program prepared to estimate the pressure difference across the free water level for many different hypothetical systems show constant pressure difference behavior between the created and the reflected pressure drops during the pressure transient solutions as shown in Fig. 1; however, the values of pressure difference depend entirely between the contacted rock and fluid properties (Darcy’s law). Fig. 1, Schematic drawing shows the constant difference between the created and reflected pressure drops denoting the dynamic capillary pressure across the free water level As commonly used in reservoir performance analysis, the pressure drop behavior could be represented in dimensionless form. Thus, the capillary pressure term could also be converted to dimensionless capillary pressure form. Therefore, Eq. 2 can be Jalal Abdulwahid Al-Sudani -Available online at: www.iasj.net IJCPE Vol.15 No.3 (September 2014) 55 expressed in dimensionless form as follows: ...(3) Where is the dimensionless dynamic capillary pressure. Hence, the dynamic capillary pressure is zero at the free water levels; thus Equation 2 could be written at the water free level as follows; …(4) Using Van-Everdingen and Hurst [8], for infinite radial flow solution for the created reservoir pressure drops expressed by the following expression: …(5) Where represents the dimensionless time provided by the following expression as given by John Lee et al. [10] and Tarik Ahmed [11]: …(6) Where ( and ) are the reservoir porosity and permeability ( ) are the reservoir fluid viscosity and compressibility, respectively, is the producing well radius, and ( ) is the producing time. While, the reflected pressure drops by the aquifer zone exhibit two dimensional flow in radial and vertical directions using the solution of Keith Coats [9], which can be written as follows: √ √ [ ] …(7) Where represents the dimensionless time provided by the following expression which is written by using aquifer zone properties as follows: …(8) is dimensionless aquifer size, which is expressed as follows [9, 11]: √ √ …(9) is a constant that depends on aquifer size as given by Keith Coats [9]. The final form of Keith Coats [9] model given by (7) was arranged by Al-Sudani [12] to be applicable in all reservoir sizes as follows: …(10) Where is the dimensionless aquifer pressure drops at free water level. …(11) Substituting Eqs. 5, 6, 8 and 10 in Eq. 3 will generate the following expression: [ ( ) ] [ ( ) ] …(12) The values of as given by Keith Coats [9] behave as shown in the following expression: …(13) Substituting Eq. 13 in 12 yields the following: [ ( ) ] [ ( ) ] …(14) Analytical Model for Detection the Tilt in Originally Oil Water Contacts 56 IJCPE Vol.15 No.3 (September 2014) -Available online at: www.iasj.net Simplifying Eq. 14 yields the following: ( ) …(15) And; ( ) …(16) Where is the effect of aquifer size on capillary transition zone that is expressed by: …(17) For further simplification, it is assumed that the displaced and displacing fluid zones have the same porosities; thus, Eq. 16 could be rewritten as follows: ( ) …(18) Eq. 18 provides general expression for dimensionless dynamic capillary pressure term performed under reservoir conditions. The expression can be converted to its dimensional form throughout multiplying the dimensionless values by (ΔP/PD): …(19) Thus, the dynamic capillary pressure term could be written as follows: …(20) However, the transition zone thickness is the function of the pressure difference across the free water level and the density difference between oil and water, which is commonly expressed as given by [1 and 5] as follows: …(21) Where, …(22) While and represent the initial transition zone thickness and the dynamic capillary pressure, respectively and is conversion parameter that equals = . Application of this method requires only one conventional drawdown well test analysis and some simple laboratory measurements for oil and water viscosity and compressibility at initial reservoir conditions to estimate accurately the dynamic capillary pressure rather than performing some complicated laboratory measurements for capillary pressure which in due course cannot be done at reservoir conditions. Results and Discussion The presented model given in Eq. 20 which is derived from the simple definition of capillary pressure shown in Eq. 1 involves all rock and fluid properties for both of reservoir and aquifer zones. Moreover, all of the properties can be measured under reservoir conditions as the reservoir permeability can be obtained from well test analysis; while, the aquifer’s zone permeability can be estimated from injectivity test or using Voigt’s [13] model. Then, both of reservoir and aquifer viscosity and compressibility can be estimated from PVT data at actual reservoir pressure and temperature. Since, the developed model reflects the real interaction between all rock and fluid properties, and will eliminate the logging analysis and coring works in determining the fluid contact positions. Figs 2, 3 and 4 were built for a hypothetical reservoir-aquifer system as given in Table 1; the effects of Jalal Abdulwahid Al-Sudani -Available online at: www.iasj.net IJCPE Vol.15 No.3 (September 2014) 57 reservoir and aquifer permeabilities, reservoir thickness and aquifer size properties were verified to take full knowledge for the controlling parameters on dynamic capillary pressure. Investigating these figures show the emphasized effect for reservoir characteristics in increasing the dynamic capillary pressure which led for thicker transition zone than that observed for aquifer characteristics. Table 1: Reservoir-Aquifer hypothetical system variables Fig. 2 Fig. 3 Fig. 4 , B/D 5000 5000 5000 , md Variable 600 600 , md Variable 120 120 , cp 2 2 2 , cp 1 1 1 , Ib/ft 3 52.1 52.1 52.1 , lb/ft 3 63.2 63.2 63.2 , ft 90 Variable 90 , ft 80 80 80 ,RB/STB 1.2 1.2 1.2 ,RB/STB 1.01 1.01 1.01 , Psi -1 8*10 -6 8*10 -6 8*10 -6 , Psi -1 2*10 -6 2*10 -6 2*10 -6 , ft 0.25 0.25 0.25 0.2 0.2 0.2 0.2 0.2 0.2 S 0.0 0.0 0.0 0.005 0.005 Variable 0.2 0.2 0.2 It is noticed that the aquifer permeability has an adverse effect on dynamic capillary pressure compared with the effect of reservoir permeability as shown in Fig. 2; this conclusion gives an accurate thorough explanation for the variation in oil water contact positions than previously acknowledgment, which only relates to the phenomenon for reservoir permeability variation. The reason for the shortage explanation of tilt OOWC, can be attributed for unavailable exact theoretical model in the literature to estimate the dynamic capillary pressure that involve all influencing reservoir and aquifer properties. While Figs 3 and 4, respectively, show the adverse effect for reservoir thickness and aquifer sizes on dynamic capillary pressure. This can be attributed for the reduction in reservoir pressure props. Hence, it can be stated that same effects for both reservoir and aquifer’s fluids properties (viscosity and compressibility) on dynamic capillary pressure, as both of them affect the created and imposed reflected pressure drops. Fig. 2: Reservoir and Aquifer permeability effects on capillary pressure Fig. 3: Effect of reservoir thickness on capillary pressure Meanwhile, the tilt in originally oil- water contact has been studied for Iraqi carbonate oil field producing from two main reservoirs under different reservoir-aquifer properties as given in Appendix-A; the generated results Analytical Model for Detection the Tilt in Originally Oil Water Contacts 58 IJCPE Vol.15 No.3 (September 2014) -Available online at: www.iasj.net from the current model were compared with the actual field data that showed very identical results. Fig. 4: Effect of Dimensionless aquifer size on capillary pressure Conclusions The presented model provides the real measurements and full explanation without any limitation for the tilts in OOWC under actual reservoir conditions. This method uses only the conventional well test analysis performed in reservoir. Moreover, the theory has a lot of advantages which could be stated as follows; 1- The analysis provides the actual measurements of the tilt in OOWC. 2- It is directly applied using reservoir pressure transient analysis and no experimental laboratory works needed, since it is a cost effective procedure when performing some complicated experiments. 3- This method could be applied in any position within the entire reservoir area, since it could provide the tilt of OOWC in all directions. 4- The model will assist in accurately evaluating the originally hydrocarbon in place throughout determining the transition zone intervals and saturation distribution. 5- The model can be considered an easy tool in estimating the OOWC in reservoir simulation studies. Nomenclature Symobile Difintion Unit Bo Oil formation volume factor RB/STB BOPD Barrel oil per day. Total Aquifer compressibility Psi -1 Total reservoir compressibility Psi -1 FWL Free Water Level Conversion constant =32.2 Oil reservoir column thickness, ft Transition zone thickness, ft Aquifer zone permeability in horizontal direction. Reservoir zone permeability Aquifer zone permeability in vertical direction. Absolute aquifer zone permeability Dimensionless Aquifer size. Capillary pressure, Psi Dimensionless capillary pressure Initial reservoir pressure, Psi Aquifer pressure at FWL Psi Reservoir pressure at FWL Psi Dimensionless pressure drop Dimensionless reservoir pressure drop Dimensionless reflected pressure drop occurs in the free water level Modified dimensionless reflected pressure drop occurs in the free water level Oil flow rate, STB/Day Well radius, ft Jalal Abdulwahid Al-Sudani -Available online at: www.iasj.net IJCPE Vol.15 No.3 (September 2014) 59 t Production time, days Dimensionless time Dimensionless time calculated using aquifer zone properties Dimensionless time calculated using reservoir zone properties Aquifer size effect on capillary zone. Greek Symbols μo Oil viscosity Cp μw Water viscosity Cp ρo Oil density Ib.m/ft 3 ρw Water density Ib.m/ft 3 Density difference Ib.m/ft 3 Reservoir porosity Aquifer porosity References 1- Hsueh P.T., Pham T.R. and Bu- Hulaigah E.H.”A Review of different initialization and history matching a reservoir model with tilted oil water contact”.SPE-53373 – 1999. 2- Stenger B.A.” Regional Temperature Gradient: A Key to Tilted OOWC”. SPE-53197, 1999. 3- Shawkat G. Ghaidan, Bertrand M. Thiebot and Douglas A Boyd.”Modeling and Validation of Initial Water Saturation in the transition Zone of Carbonate Oil Reservoirs”. SPE- 88756, 2004. 4- Anderson,W.G.: “Wettability Literature Survey – Part 4: Effects of Wettability on Capillary Pressure”, JPT (Oct. 1987) 1283- 1300. 5- Hassanizadeh SM, Oubol Oung, Sabine Manthey;” Laboratory Experiments and Simulations on the Significance of Non-Equilibrium Effect in the Capillary Pressure- Saturation Relationship”, Unsaturated Soils: Experimental Studies, Springer Proceedings in Physics Volume 93, 2005, pp 3-14. 6- Al-Khalifah, A.J.A.; Odeh, A.S.: "Well Test Analysis in Oil Reservoirs with Gas Cap and/or Water Aquifer". SPE-19842, paper presented at the 64 th annual technical conference and exhibition of SPE held in San Antonio,Tx, Oct.8-11, 1989. 7- Layne M.A., Numbere D.T. and Koederitz L.F.: "Analytical Solution to Predict the Performance of Gas Reservoirs under Water Drive" SPE-26669, Paper Presented at 68 th Annual Technical Conference and Exhibition of SPE Held in Houston, Texas 3-6 Oct. 1993. 8- Van-Everdingen, A.F. and Hurst, W.; "The Application of Laplace Transformation to Flow Problems in Reservoirs", Trans. AIME (1949) Vol. 186, 305. 9- Keith H. Coats; "A Mathematical Model Water Movement about Bottom-Water Drive Reservoirs.” SPE-160, paper presented at 36 th annual fall meeting of SPE, Oct.8- 11, 1961, in Dallas. 10- John-Lee, John B. Rollins, John P. Spivey “Pressure Transient Testing” SPE Textbook Series, Texas.2003. 11- Tarik Ahmed, Paul D. McKinney; "Advance Reservoir Engineering" Elsevier, 2005, Chapters (1 and 2). 12- J.A. Al-Sudani,” Prediction the Future Performance of reservoir Fluids Movement under Combined Drive Mechanism” Ph.D. Thesis submitted to Uinv. Of Baghdad, Pet. Eng. Dept. May, 2009. 13- Voigt, H.D. and Astl, A.; "Determination of Permeability from Responses in Aquifer Wells". SPE-16996, 1987. http://link.springer.com/search?facet-author=%22S.+Majid+Hassanizadeh%22 http://link.springer.com/search?facet-author=%22Oubol+Oung%22 http://link.springer.com/search?facet-author=%22Sabine+Manthey%22 http://link.springer.com/book/10.1007/b138014 http://link.springer.com/book/10.1007/b138014 http://link.springer.com/bookseries/361 http://link.springer.com/bookseries/361 Analytical Model for Detection the Tilt in Originally Oil Water Contacts 60 IJCPE Vol.15 No.3 (September 2014) -Available online at: www.iasj.net APPENDIX-A: Field Example: It is an Iraqi carbonate oil field given by Al-Sudani [12] produced from two main reservoirs under bottom water drive aquifer. The reservoir and aquifer zones properties are listed as shown in Tables A-1 and A-2, respectively. Other reservoir and aquifer fluids properties are as follows; (Co=8.95*10 - 6 Psi -1 , µo=0.65 Cp, Bo=1.46 RB/STB) and (Cw=2.5*10 -6 Psi -1 , µw=0.705 Cp, Bw=1.02 RB/STB), app. oil density=53.4 Ib/ft 3 , approximate water density=63 Ib/ft 3 . Approximate dimensionless aquifer size = 0.005. Four wells penetrating the main reservoirs were selected in this test representing the west and east sides of the reservoir; the flow rates of these wells are listed in Table A-3. Required: Estimation of Capillary transition zone thickness to predict the tilt in OOWC. Table A-1: Main reservoir characteristics Unit A D Net thickness (ft) 25 125 0.21 0.24 K- md 580 1115 Table A-2: Aquifer characteristics West East Unit A D A D Kw – md 1381 2297 305 1165 0.32 0.23 0.14 0.2 Thick- ft 92 100 43 110 Table A-3: Well Production of Oil in units A and D (BOPD) East Region West Region Reservoir 4100 8300 A 5155 7875 D Solution: A computer program on Excel sheet was prepared to be used in estimating dynamic capillary pressures and the transition zone thickness based on the derived analytical model represented by Eqs. 18, 20 and 21; the results are listed as shown in Table A-4; These results obtained by the new model show high tilted OOWC that exists between west-east regions for units (A and D); the results confirm this carbonate field of OWC characteristic and may provide the best explanation for the tilted OOWC in some carbonate reservo irs which is still under investigation. Table A-4: Calculated values of capillary pressure and transition zone thickness (A); East (A); West (D); East (D); West 1.15379 4.63114 2.7346 4.2977 , Psi 43.716 355.221 13.553 32.539 , ft 20.3836 165.630 6.3195 15.1722 The results of this field data exactly show the titled in originally oil-water contact from west to east, especially in reservoir unit (A) due to the high variation in rock and fluid properties with the aquifer, compared with what is observed in reservoir unit (D). The results confirm the actual tilts in OOWC from west to east in both units; therefore, the new methodology may explain the reasons beyond this phenomenon which is still under investigation by some authors such as Hsueh [1]. Jalal Abdulwahid Al-Sudani -Available online at: www.iasj.net IJCPE Vol.15 No.3 (September 2014) 61