Petroleum Reservoir Engineering II Lecture 6: Fundamentals Of Reservoir .
Tishk International University Engineering Faculty Petroleum and Mining Engineering Department Petroleum Reservoir Engineering II Lecture 6: Fundamentals of Reservoir Fluid Flow (I) Third Grade- Spring Semester 2020-2021 Instructor: Sheida Mostafa Sheikheh
Content: Flow in Porous Media Reservoir Characteristics: o Reservoir Fluid Types according to Compressibility o Types of Flow Regimes o Types of Reservoir Geometries o Number of flowing fluids in the reservoir Fluid Flow Equations
Flow in Porous Media Flow in porous media is a very complex phenomenon and as such cannot be described as explicitly as flow through pipes of conduits. It is rather easy to measure the length and diameter of a pipe and compute its flow capacity as a function of pressure. In porous media, however, flow is different in that there are no clearcut flow paths that lend themselves to measurement.
Flow in Porous Media The analysis of fluid flow in porous media has evolved throughout the years along two fronts- the experimental and the analytical. Physicists, engineers, hydrologists, and the like have examined experimentally the behavior of various fluids as they flow through porous media ranging from sand packs to fused Pyrex glass. On the basis of their analyses, they have attempted to formulate laws and correlations that can then be utilized to make analytical predictions for similar systems.
Flow in Porous Media The main objective of this lecture is to present the mathematical relationships that are designed to describe the flow behavior of the reservoir fluids. The mathematical forms of these relationships will vary depending upon the characteristics of the reservoir. The primary reservoir characteristics that must be considered include: o Reservoir Fluid Types according to Compressibility o Types of Flow Regimes o Types of Reservoir Geometries o Number of flowing fluids in the reservoir
Reservoir Characteristics Reservoir Fluid Types according to Compressibility: The isothermal compressibility coefficient is essentially the controlling factor in identifying the type of the reservoir fluid. Isothermal compressibility coefficient is defined as the change in volume per unit volume for a unit change in pressure. In general, reservoir fluids are classified into three groups: Incompressible fluids Slightly compressible fluids Compressible fluids
Reservoir Characteristics Reservoir Fluid Types according to Compressibility: The isothermal compressibility coefficient, c, is described mathematically by the following two equivalent expressions: In terms of fluid volume: π 1 π 1 π π In terms of fluid density: 1 Ο π 2 Ο π Where V and p are the volume and density of the fluid, respectively.
Reservoir Characteristics Reservoir Fluid Types according to Compressibility: Homework (Part I): Explain the reason why the minus sign (-) is present in equation (1) and disappeared in equation (2)
Reservoir Characteristics Reservoir Fluid Types according to Compressibility: Incompressible Fluids: An incompressible fluid is defined as the fluid whose volume (or density) does not change with pressure, i.e.: π 0 π Ο 0 π Incompressible fluids do not exist; this behavior, however, may be assumed in some cases to simplify the derivation and the final form or many flow equations.
Reservoir Characteristics Reservoir Fluid Types according to Compressibility: Slightly Compressible Fluids: These βslightlyβ compressible fluids exhibit small changes in volume, or density, with changes in pressure. Knowing the volume ππππ of a slightly compressible liquid at a reference (initial) pressure ππππ , the changes in the volumetric behavior of this fluid as a function of pressure, p, can be mathematically described by integrating equation (1): 1 π π π π ππ π ππ π By integrating above equation: π π ππ π ΰΆ± ππ ΰΆ± ππππ π ππππ
Reservoir Characteristics Reservoir Fluid Types according to Compressibility: Slightly Compressible Fluids: After integrating: π π ππππ π ππππ π π(ππππ π) (3) π(ππππ π) Where p pressure, psia V volume at pressure p, ππ‘ 3 ππππ initial (reference) pressure, psia ππππ fluid volume at initial (reference) pressure, ππ‘ 3
Reservoir Characteristics Reservoir Fluid Types according to Compressibility: Slightly Compressible Fluids: The π π₯ may be represented by a series expansion as: 2 3 π π₯ π₯ π₯ π π₯ 1 π₯ (4) 2! 3! π! Because the exponent x [which represents the term π(ππππ π)] is very small, the π π₯ term can be approximated by truncating equation (4) to: π π₯ 1 π₯ 5 Combining equation (5) with equation (3), gives: π ππππ 1 π ππππ π (6) It should be pointed out that crude oil and water systems fit into this category.
Reservoir Characteristics Reservoir Fluid Types according to Compressibility: Slightly Compressible Fluids: Homework (Part II): Derive the equation of isothermal compressibility coefficient for density and pressure relationship (equation 2)
Reservoir Characteristics Reservoir Fluid Types according to Compressibility: Compressible Fluids: These are fluids that experience large changes in volume as a function of pressure. All gases are considered compressible fluids. The isothermal compressibility of any compressible fluid is described by the following expression: 1 1 π§ ππ π π§ π (7) π
Reservoir Characteristics Reservoir Fluid Types according to Compressibility: Following figures show schematic illustrations of the volume and density changes as a function of pressure for the three types of fluids.
Reservoir Characteristics Types of Flow Regimes: There are basically three types of flow regimes that must be recognized in order to describe the fluid flow behavior and reservoir pressure distribution as a function of time. There are three flow regimes: Steady-state flow Unsteady-state flow Pseudosteady-state flow
Reservoir Characteristics Types of Flow Regimes: Steady-state flow: The flow regime is identified as a steady-state flow if the pressure at every location in the reservoir remains constant, i.e., does not change with time. Mathematically, this condition is expressed as: π ( )π 0 8 π‘ The above equation states that the rate of change of pressure p with respect to time t at any location I is zero. In reservoirs, the steady-state flow condition can only occur when the reservoir is completely recharged and supported by strong aquifer or pressure maintenance operations.
Reservoir Characteristics Types of Flow Regimes: Unsteady-state flow: The unsteady-state flow (frequently called transient flow) is defined as the fluid flowing condition at which the rate of change of pressure with respect to time at any position in the reservoir is not zero or constant. This definition suggests that the pressure derivative with respect to time is essentially a function of both position i and time t. Thus, π π π, π‘ (9) π‘
Reservoir Characteristics Types of Flow Regimes: Pseudosteady-state flow: When the pressure at different locations in the reservoir is declining linearly as a function of time, i.e., at a constant declining rate, the flowing condition is characterized as the psedosteady-state flow. Mathematically, this definition states that the rate of change of pressure with respect to time at every position is constant, or π ( )π ππππ π‘πππ‘ (10) π‘ It should be pointed out that the pseudosteady-state flow is commonly referred to as semisteady-state flow and quasisteady-state flow.
Reservoir Characteristics Types of Flow Regimes: Following figure shows a schematic comparison of the pressure declines as a function of time on the three flow regimes.
Reservoir Characteristics Types of Reservoir Geometries: The shape of a reservoir has a significant effect on its flow behavior. Most reservoirs have irregular boundaries and a rigorous mathematical description of geometry is often possible only with the use of numerical simulators. For many engineering purposes, however, the actual flow geometry may be represented by one of the following flow geometries: Radial flow Linear flow Spherical and hemispherical flow
Reservoir Characteristics Types of Reservoir Geometries: Radial Flow: In the absence of severe reservoir heterogeneities, flow into or away from a wellbore will follow radial flow lines from a substantial distance from the wellbore. Because fluids move toward the well from all directions and coverage at the wellbore, the term radial flow is given to characterize the flow of fluid into the wellbore.
Reservoir Characteristics Types of Reservoir Geometries: Radial Flow: Figure 4 shows idealized flow lines and iso-potential lines for a radial flow system.
Reservoir Characteristics Types of Reservoir Geometries: Linear Flow: Linear flow occurs when flow paths are parallel and the fluid flows in a singe direction. In addition, the cross-sectional area to flow must be constant. Figure 5 shows an idealized linear flow system.
Reservoir Characteristics Types of Reservoir Geometries: Linear Flow: A common application of linear flow equations is the fluid flow into vertical hydraulic fractures as illustrated in figure 6.
Reservoir Characteristics Types of Reservoir Geometries: Spherical and Hemispherical Flow: Depending upon the type of wellbore completion configuration, it is possible to have a spherical or hemispherical flow near the wellbore. A well with a limited perforated interval could result in spherical flow in the vicinity of the perforations as illustrated in figure 7.
Reservoir Characteristics Types of Reservoir Geometries: Spherical and Hemispherical Flow: Depending upon the type of wellbore completion configuration, it is possible to have a spherical or hemispherical flow near the wellbore. A well that only partially penetrates the pay zone, as shown in figure 8, could result in hemispherical flow. The condition could arise where coning of bottom water is important.
Reservoir Characteristics Number of Flowing Fluids in the Reservoir: The mathematical expressions that are used to predict the volumetric performance and pressure behavior of the reservoir vary in forms and complexity depending upon the number of mobile fluids in the reservoir. There are generally three cases of flowing systems: Single-phase flow (oil, water, or gas) Two-phase flow (oil-water, oil-gas, or gas-water) Three phase flow (oil, water, and gas) The description of fluid flow and subsequent analysis of pressure data becomes more difficult as the number of mobile fluids increases.
Fluid Flow Equations The fluid flow equations that are used to describe the flow behavior in a reservoir can take many forms depending upon the combination of variables presented previously (i.e., types of flow, types of fluids, etc.). Since all flow equations to be considered depend on Darcyβs Law, it is important to consider this transport relationship first. In the next lecture, the Darcyβs Law will be introduced and derived for various mentioned reservoir characteristics.
Reservoir Characteristics Number of Flowing Fluids in the Reservoir: The mathematical expressions that are used to predict the volumetric performance and pressure behavior of the reservoir vary in forms and complexity depending upon the number of mobile fluids in the reservoir. There are generally three cases of flowing systems:
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1. Fundamentals of Reservoir Engineering by Dake, L P, Elsevier 2. The Practice of Reservoir Engineering, by Dake, L.P., Elsevier 3. Reservoir Engineering Handbook, by Tarek Ahmed, Gulf Professional Publishing,Elsevier 4. Applied Petroleum Reservoir Engineering
1. Fundamentals of Reservoir Engineering by Dake, L P, Elsevier 2. The Practice of Reservoir Engineering by Dake, L.P., Elsevier 3. Reservoir Engineering Handbook by Tarek Ahmed, Gulf Professional Publishing,Elsevier 4. Applied Petroleum Reservoir Engineering
Reservoir characterization is a combined technology associated with geostatistics, geophsics, petrophysics, geology and reservoir engineering and the main goals of reservoir characterization research are to aid field de velopment and reservoir management teams in describing the reservoir in sufficient detail, developing 3D/4D data for reservoir
The Reservoir Engineering & Petroleum Department's strategy is to follow the entire life-cycle of any given asset from exploration to abandonment. Integrated reservoir models and proper reservoir management practices are at the heart of this process. Asset value is maximized by optimizing production on the short-term
Fundamentals of Reservoir Engineering, Dake (1978). The Practice of Reservoir Engineering, Dake (1994). Applied Petroleum Reservoir Engineering, Craft and Hawkins (rev. Terry) (1991). Advanced Reservoir Management and Engineering (2nd Edition), Ahmed (2011) 3.
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