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Second Credit Seminar Report On “A study of fluid displacement in porous media by water and CO2 injection” Submitted byDeepti Mishra Guided by Dr. M. N. Mehta Dr. V. H. Pradhan Applied Mathematics & Humanities Department S. V. National Institute of Technology, SURAT– 395 0072010

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Introduction to enhanced oil recovery Enhanced oil recovery (EOR) is a generic term for techniques for increasing the amount of crude oil that can be extracted from an oil field . EOR can 30-60% or more of the reservoir original oil can be extracted compared to 20-40% using primary and secondary recovery. EOR is also called improved oil recovery or tertiary recovery. EOR is achieved by water injection, gas injection, chemical injection, microbial injection. In the middle of 70’s years, Alkanes type gaseous are utilize to boost the oil recovery for enhance oil recovery.   Since 80’s the nitrogen and carbon dioxide gaseous were selected to replace Alkanes gaseous and to be pumped down into oil wells for enhancing oil extraction.

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Darcy’s law Mathematically, it can be written as,   (1.1)   Limitation of Darcy’s law There is both an upper limit and a lower limit to the validity of Darcy’s law. The Reynold number R for fluid flow through porous media is defined as   (1.2)   The upper limit to the validity of Darcy’s law is given by R ≤ A Where A is a number , its value is given between 1 and 10. The limitation of Darcy’s law is due to turbulence and due to molecular effects.

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Generalization of Darcy’s law The generalized Darcy’s law may be written as   (1.3) The alternative form of equation (1.3) in terms of pressure is (1.4) So Darcy's law gives an indication of how fast the oil can move through the reservoir under the influence of gravity. q= Q/A= Ki (1.5) where q is the volume flux or Darcy velocity. The law is used more in hydrogeology as the movement of petroleum products is often dependant on gas pressure as opposed to gravity

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Buckley leverett equation The approach uses fractional flow theory and is based on the following assumptions: Flow is linear and horizontal Water is injected into an oil reservoir Oil and water are both incompressible Oil and water are immiscible Gravity and Capillary pressure effects are negligible The Buckley Leverett frontal advance equation: (1.18) The derivative is the velocity of the moving plane with water saturation , and the derivative is the slope of the fractional flow curve .

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Introduction to oil reservoir The oil and gas bearing structure is typically a porous rock such as sandstone or washed out limestone. About 100 to 200 million years ago Oil and gas deposits form as organic material is transformed by high temperature and pressure into hydrocarbons. To form an oil reservoir , porous rock needs to be covered by a non porous layer such as salt, shale, chalk or mud rock that can prevent the hydrocarbons from leaking out of the structure. As rock structures become folded and uplifted as a result of tectonic movements, the hydrocarbons migrates out of the deposits and upward in porous rocks and collects in crests under the non permeable rock, with gas at the top, the oil and fossil water at the bottom.

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Introduction to primary secondary and tertiary recovery Before proceeding with a discussion of EOR, it is important to have a basic understanding of primary and secondary recovery. Primary recovery: Primary production is the first oil out, the “easy” oil. This is caused due to the naturally occurring reservoir pressure. It is the least expensive method of extraction , since it uses natural forces to “ move” the oil, it typically recovers only 10 to 15 % of the original oil in place (OOIP).

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Secondary recovery: It is used when there is insufficient underground pressure to move the remaining oil. The most common technique , water flooding ,utilizes injector wells to introduce large volume of water under pressure into the hydrocarbon zone. As the water flows through the formation toward the producing wellbore , it sweeps some of the oil it encounters along with it . On reaching the surface , the oil is separated out for sale and the water is reinjected. Tertiary recovery: Gas injection appears to an ideal choice . Two types of flow are possible in this case, immiscible and miscible . Gas, air or water is injected into an reservoir to increase its productivity. Water flooding can be done where water is injected in the oil reservoir. CO2 can also be injected . This process is called Solvent drive as shown in below figure.

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Method used for oil recovery Chemical injection The injection of various chemical usually as dilute solution have been used to improve oil recovery . Microbial injection Microbial injection is part of microbial enhanced oil recovery. Presently it is rarely used because of its higher cost and development in this field are more recent than other techniques.   Thermal injection Various method are used to heat the crude oil in the formation to reduce its viscosity and or vaporize part of oil. Method include cyclic steam injection, steam drive ,and in situ combustion.

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Gas injection Gases used include CO2, natural gas or nitrogen. It reduces the viscosity of the crude oil by mixes with it. Oil displacement depends on the phase behavior of the mixture of that gas and the crude, which are strongly dependent on reservoir temperature pressure and crude oil composition. This mechanism range from oil swelling and viscosity reduction for injection of immiscible fluids to completely miscible displacement in high pressure application.

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Types of displacement Miscible displacement It occurs under suitable reservoir pressure and oil density condition . Gas injected (carbon dioxide )will mix thoroughly with the oil within the reservoir. The superficial tension between these two substances effectively disappears. Immiscible CO2 displacement   It occurs when reservoir pressure is two low and or oil gravity too dense. The injected gas remain physically distinct from the oil within the reservoir. The interfacial tension between these two substance effectively disappears.

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Properties of CO2 CO2 is a colorless and odorless gas.. It acts as an asphyxiate and an irritant. Carbon dioxide is used in enhanced oil recovery. It act as a pressurizing agent . Water solubility Density and Viscosity Diffusivity

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Instability phenomenon arising in double phase flow through porous media When a fluid contained in a porous medium is displaced by another of lesser viscosity, instead of regular displacement of whole front , Perturberance may occur which shoot through the porous medium at relatively greater speed .These perturberances are called fingers. Statement of the problem We consider here that there is a uniform water injection into oil saturated porous media of homogenous physical characteristics such that the injecting water outs through the oil formation and gives rise to pertuberance (fingers).This furnishes a well developed fingers flow. Since the entire oil at the initial boundary x = 0 (x being measured in the direction of displacement), is displaced through a small distance due to water injection. Therefore, it is further assumed that complete saturation exists at the initial boundary.

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Mathematical Formulation The filtration velocity of injected liquid (water) (VW) and native liquid (oil) (VO) Scheidegger[13] may be written as (by Darcy’s law) (3.1)    (3.2) The equation of continuity (phase densities are regarded as constant) are  (3.3) (3.4) PC = PO – PW (3.5)

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From the definition of phase saturation, it is evident that SO + SW = 1 (3.6) (3.7)   (3.8) Eliminating from equation (3.5) and (3.7) we have (3.9)

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(3.9) Combining equation (3.8),(3.9) and using (3.6) we have (3.10)  On integrating (3.10) with respect to x we get (3.11)   Where ’C’ is the constant of integration From(3.11) (3.12)

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By putting the value of in (3.9) it gives (3.13)   The value of the pressure of native liquid (PO) can be written as where , is the mean pressure and is constant Now , Substituting the value of in (11) it gives (3.14) On substituting the value of C we obtain in equation (3.13) (3.15)

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For the analytical relationship between the relative permeability, Phase saturation and capillary pressure phase saturation . , Where β is constant and –ve sign shows the direction of saturation of water opposite to capillary pressure. Substituting the value of PC in equation (3.15) (3.16) Changing equation (3.16) in to dimensionless form by substituting (3.17) This is the desired non-linear partial differential equation describing the instability phenomenon with capillary pressure. , we get

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Gas injection and fingering in porous media On average, two-thirds of the original oil in any reservoir remains unrecovered, even after water injection into the reservoir.  If we inject a gas into a reservoir which is completely or partially saturated with oil, and if the gas and the oil-in-place mix in all proportions, the gas and the oil are said to be first-contact miscible. The injected gas and the oil may form two different phases, that is, they may not be first contact miscible, but mass transfer between the two phases and long-time contact between the fluids may achieve miscibility. This is usually called multiple-contact or dynamic miscibility , One way of enhancing the recovery of oil is by injecting into a reservoir a fluid which is miscible with the oil in place.

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Factor affecting the efficiency of miscible displacement Mobility and mobility ratio It is defined as the ratio of the effective permeability of the porous medium and the fluid viscosity i.e. (3.19) Viscosity µm of mixed zone is estimated from the following empirical law is (3.20) When M >1 , it may lead to the formation of fingers which reduce the efficiency of a miscible displacement.

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Diffusion and dispersion Diffusion Dispersion  It is mixing of two miscible fluids flowing in a system, such as porous medium. Mixing by a dispersion process can decrease the viscosity and density contrast between the displacing gas and the displaced fluids. Types of dispersion  Longitudinal dispersion  The mixing which is in the direction of the macroscopic flow.   Transverse dispersion  The mixing occurs in the direction perpendicular to the direction of macroscopic flow. Dispersion in porous media is usually modelled base on convective diffusion(CD)equation. (3.21)

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DL is usually larger than DT by a factor that depending on the morphology of the porous medium. Since pore space heterogeneties strongly affects DL and DT implies that heterogeneities also affect miscible displacement. It also depends on the mean flow velocity. In principle the relation between the dispersion coefficient on the value of the peclet number Pe i.e. (3.22)

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The relationship between dispersion coefficient and Pe can be non-linear but in numerical simulation we assume that , (3.23)   Where αL is the longitudinal dispersivities and αT is the transverse dispersivities. C D equation is applicable to describing the dispersion if the length scale of the observation or measurement is larger than dispersivities and if it cannot be described by C D equation it is stated as “ Anamalous” dispersion.

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Phenomenon of fingering If the front is unstable and many fingers of the mixture of the gas and the displaced fluid develop leaving behind large amount of oil, which have very irregular shapes , reduces strongly the efficiency of the miscible displacement. Effect of mobility ratio on the formation and shape of fingers. This phenomenon is usually referred to as “ viscous fingering”. Fig(3.24) Displacement fronts for two values of the mobility ratio M. The injection point is the lower left corner .

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Factor affecting fingering phenomenon Tip splitting In this the tip of a finger become unstable and split into two branches that compete with each other for further growth. Spreading It is usually occur when flow velocity is not large in porous medium. Here transverse dispersion causes lateral spreading of the fingers which help them to join and therefore increase the efficiency of the displacement process. Shielding It occurs when one finger spread out and grow much faster than other fingers and therefore shield them. Fig 3.25 shows the phenomenon of tip splitting and finger shielding.

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FIG 3.25 Tip splitting (top) and shielding (bottom) in viscous fingering

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Displacement rate Miscible displacements are much less sensitive to the displacement rate in comparison to Immiscible displacements. Smaller and more numerous fingers are formed in immiscible displacement at high flow rates comparatively at low rates. Heterogeneity characteristics It plays a fundamental role in finger formation is the heterogeneity of the porous medium. Permeability variations have been found to play an important role in finger initiation and growth. The spatial variation of the permeability is usually described by two parameters. One is the coefficient of permeability variation CK defined by   (3.26)

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Viscosity ratio At low viscosity ratio, there were no fingers that grows dramatically faster than the rest. It results in finger merging and growth suppression occur to a much lesser extent and form a large number of active fingers with growth rates that do not vary much. At high viscosity ratio, the active fingers start to outgrow the others earlier and more finger merging and suppression of growth occur . It results in a smaller number of long active fingers.

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Aspect ratio and boundary condition The aspect ratio of a porous medium as the ratio of its length and width as (3.27) If KO increases the initial fingers are close to one another and their interaction is stronger it results in suppression of growth of the smaller fingers and their merging at the initial stages of the displacement yielding a small number of active fingers at early times. At large aspect ratio only one active finger will form. Gravity segregation Gravity influences vertical sweeps in miscible displacement .To study the effects of gravity on miscible displacement, two dimensionless number are introduced. (3.28) (3.29)

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If NG is small, gravity is unimportant and viscous fingering dominates the flooding behaviour . If increases, viscous fingering can still occur but gravity influences the growth rate of individual fingers Averaged continuum models of miscible displacements Two popular one dimensional and semi-empirical continuum models of miscible displacement are due to Koval (1963) and Todd and Longstaff (1972). Koval recognized that the central features of physics of viscous fingers is linear growth of the fingers length with time. He cast the problem as a hyperbolic transport equation similar to more familiar Buckley-leverett equation of two phase flow. In Koval model the displacing fluid is assumed to travel at a constant velocity ν .He made an analogy between miscible and immiscible displacement

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In Koval model the displacing fluid is assumed to travel at a constant velocity . If viscous fingering is dominant phenomenon we write (3.30) Where µes and µeo are effective viscosities of solvent and oil. ‘H’ is heterogeneity index that characterizes the inhomogeneity of porous medium. Koval defined that for a homogeneous porous medium with H=1 for which oil recovery is 99% and other porous medium for which the recovery is less than 99% is characterized as H >1. Based on experiment data Kovel suggested the following expression (3.31) .

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The second model is due to Todd and Longstaff Here the average concentration of the solvent is described by (3.32) This equation is a limiting case of a C D equation in which dispersion has been neglected. Todd and Longstaff assumed that With a similar expression for and where is given by equation (8.1) in which instead of and , is given by = (3.33) Leading edge of the finger moves at a speed of Trailing edge of the finger moves at a speed of Despite of some shortcomings, because of their simplicity these model used heavily in petroleum industry.

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Dynamics of fluid in porous media , Dover Publications ,Jacob Bear ISBN 0486656756,1988-9-1 Principles of applied reservoir simulation ,Gulf professional publishing, Dr John R ,ISBN 0750679336 third edition by John R Fanchi. Madhav M Kulkarni “Thesis on immiscible and miscible gas oil displacement in porous media” at Graduate faculty of the Lousiana state university and Agricultural and Mechanical College. Experimental and Computational studies of fluid flow Phenomena in Carbon dioxide Sequestration in brine and oil field, Duana H Smith National energy technology laboratory, Department of energy Morgantown. Gas transport in porous media by Clifford k.Ho and Stephen W. Webb, Sandia National Laboratory USA

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Mathematical Model and Simulation of gas flow through a porous medium in high breaking capacity fuses, D.Rochette,S.clain ,11 september 2003 FRANCE. Fluid Dynamics of Carbon dioxide disposal into saline aquifers by Julio Enrique Garcia, University Of Los Andes Columbia (1991) and University Of Colorado (1995) Essential Of Multiphase Flow and transport in porous media by George F.Pinder and William G.Gray, University Of Vermont ,John Wiley and Sons ,INC., Publication. Yusuf ziya Pamukcu “Simulating oil recovery during CO2 sequestration into a mature oil reservoir” A thesis submitted to the Graduate school of Natural and Applied Sciences of middle east technical university in August 2006.

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Physical modelling of CO2 Sequestration, California Energy Commision , Stanford University 2008. Reservoir Simulation-Mathematical techniques in oil recovery, Zhangxin Chen ISBN 0898716403 ,Society for industrial and Applied Mathematic ,March 2008. Ferer, M., Grant S. Bromhal, and Duane H. Smith. “Pore-level modeling of carbon dioxide sequestration in brine fields.” Proceedings of the First National Conference on Carbon Sequestration.” Washington, DC May 15-17, 2001.

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