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...partial differential equations (PDEs) are needed to describe realistic multiphase, multidimensional **flow** in a reservoir. As a rule, these equations cannot be solved analytically; they must be solved with ...**numerical methods**. This article provides an overview of these ...**methods**.
To illustrate the mathematics, we discuss the ...

Systems of nonlinear partial differential equations (PDEs) are needed to describe realistic multiphase, multidimensional flow in a reservoir. As a rule, these equations cannot be solved analytically; they must be solved with numerical methods. This article provides an overview of these methods. As a reminder, v is velocity, D is dispersion, and C is concentration. Eq. 1 is a good example to use because it illustrates many useful numerical methods that can be compared with the analytical solution given by Eq. 2. We first introduce the concept of finite differences to convert Eq. 1 to an equation that can be solved numerically.

Petrowiki

characteristic root, column vector, concentration, conjugate transpose, coordinate system, determinant, eigenvector, equation, finite difference, **fluid** **flow**, identity matrix, inverse matrix, knowledge management, matrix, null matrix, **numerical** method **analysis**, **numerical** solution, square matrix, transpose

Technology:

- Information Technology > Knowledge Management (0.40)
- Information Technology > Communications > Collaboration (0.40)

...**Fluid flow** equations in two and three dimensions can be compactly represented using concepts from vector ...**analysis**. This page reviews the mathematics involved.
The continuity equation in three space dimensions for...s, is
...(3)
The divergence operator • is an example of an operator from vector **analysis** that determines the spatial variation of a vector or scalar field. Following Fanchi, [1] we first r...

Fluid flow equations in two and three dimensions can be compactly represented using concepts from vector analysis. This page reviews the mathematics involved. The flux terms (Jy) and (Jz) have meanings analogous to (Jx) for flux in the y and z directions, respectively. If we write the components of flux as the flux vector {Jx, Jy, Jz}, Eq. 1 can be written in vector notation as The divergence operator • is an example of an operator from vector analysis that determines the spatial variation of a vector or scalar field. Following Fanchi, [1] we first review the concepts of scalar and vector fields and then define gradient (grad), divergence (div), and curl operators.

Petrowiki

Cartesian coordinate, Cartesian coordinate system, continuity equation, convection dispersion equation, coordinate system, dimension, Divergence, equation, **fluid** **flow**, flux, gradient, incompressible **fluid** **flow**, knowledge management, operator, PetroWiki, Reservoir Characterization, scalar field, spatial variation, Upstream Oil & Gas, vector **analysis**, Vector Field

SPE Disciplines:

Technology:

- Information Technology > Knowledge Management (0.41)
- Information Technology > Communications > Collaboration (0.41)

...The purpose of this page is to review the mathematics of **fluid flow**. We limit our review to essential aspects of partial differential equations, vector ...**numerical methods**, matrices, and linear algebra. These topics are discussed in the context of two ...**fluid flow** applications: ...

The purpose of this page is to review the mathematics of fluid flow. We limit our review to essential aspects of partial differential equations, vector analysis, numerical methods, matrices, and linear algebra. These topics are discussed in the context of two fluid flow applications: analysis of the convection/dispersion equation and diagonalization of the permeability tensor. For more details about the mathematics presented here, consult the literature.[1][2][3][4] Partial differential equations (PDEs) are frequently encountered in petroleum engineering.

Petrowiki

Artificial Intelligence, boundary condition, concentration, continuity equation, convection dispersion equation, convection term, dispersion term, equation, **flow** in porous media, **Fluid** Dynamics, **fluid** **flow**, flux, independent variable, initial condition, knowledge management, mathematics, partial derivative, partial differential equation, reservoir simulation, second-order pde, solute, Upstream Oil & Gas

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Technology:

... John R. Fanchi, Editor
Copyright 2007, Society of Petroleum Engineers
Chapter 2 – Mathematics of **Fluid Flow**
John R. Fanchi, Colorado School of Mines
ISBN 978-1-55563-108-6 Get permission for reuse
The pur...pose of this chapter is to review the mathematics of **fluid flow**. We limit our review to essential aspects of partial differential equations, vector ...**numerical methods**, matrices, and linear algebra. These topics are discussed in the context of two ...

The purpose of this chapter is to review the mathematics of fluid flow. We limit our review to essential aspects of partial differential equations, vector analysis, numerical methods, matrices, and linear algebra. These topics are discussed in the context of two fluid flow applications: analysis of the convection/dispersion equation and diagonalization of the permeability tensor. For more details about the mathematics presented here, consult the literature.[1][2][3][4] Partial differential equations (PDEs) are frequently encountered in petroleum engineering. We review basic concepts of PDEs by considering the relevant mathematical properties of the continuity equation. Fluid flow equations in two and three dimensions can be compactly represented using concepts from vector analysis.

Petrowiki

Artificial Intelligence, column vector, continuity equation, coordinate system, determinant, eigenvector, equation, **flow** in porous media, **Fluid** Dynamics, **fluid** **flow**, **fluid** **flow** equation, identity matrix, knowledge management, mathematics, matrix, permeability tensor, reservoir simulation, square matrix, transformation, transpose, Upstream Oil & Gas, vector, Vector Field

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Technology:

...trophysics
Edward D. Holstein, Editor
Copyright 2007, Society of Petroleum Engineers
Chapter 8 – **Fluid Flow** Through Permeable Media
John Lee, Texas A&M U.
ISBN 978-1-55563-120-8 Get permission for reuse
T...his chapter discusses **fluid flow** in petroleum reservoirs. Basic concepts, which include ...**flow** equations for unsteady-state, pseudosteady-state, and steady-state ...

Basic concepts, which include flow equations for unsteady-state, pseudosteady-state, and steady-state flow of fluids, are discussed first. Various flow geometries are treated, including radial, linear, and spherical flow. The pseudosteady-state equations provide the basis for a brief discussion of oil well productivity, and the unsteady-state equations provide the basis for a lengthy discussion of pressure-transient test analysis. For pressure-transient test analysis, semilog techniques, type curves, damage and stimulation, modifications for gases and multiphase flow, the diagnostic plot, bounded reservoirs, average pressure in the drainage area, hydraulically fractured wells, and naturally fractured reservoirs are included. The chapter also discusses transient and stabilized flow in horizontal wells and gas-well deliverability tests. It concludes with considerations of coning in vertical and horizontal wells. Many important applications of fluid flow in permeable media involve 1D, ...

Petrowiki

boundary, buildup test, coefficient, diagnostic plot, diffusivity equation, Directional Drilling, drainage area, drilling operation, Drillstem Testing, drillstem/well testing, equation, **flow** period, **flow** regime, isochronal test, knowledge management, permeability, pressure drop, radial **flow**, reservoir, skin factor, straight line, test data, type curve, Upstream Oil & Gas, wellbore

SPE Disciplines:

Technology:

- Information Technology > Knowledge Management (0.40)
- Information Technology > Communications > Collaboration (0.40)

...The three primary functions of a drilling **fluid** depend on the ...**flow** of drilling fluids and the pressures associated with that ...**flow**. These functions includes: The transport of cuttings out of the wellbore, prevention of ...

The three primary functions of a drilling fluid depend on the flow of drilling fluids and the pressures associated with that flow. These functions includes: The transport of cuttings out of the wellbore, prevention of fluid influx, and the maintenance of wellbore stability. If the wellbore pressure exceeds the fracture pressure, fluids will be lost to the formation. If the wellbore pressure falls below the pore pressure, fluids will flow into the wellbore, perhaps causing a blowout. It is clear that accurate wellbore pressure prediction is necessary. To properly engineer a drilling fluid system, it is necessary to be able to predict pressures and flows of fluids in the wellbore.

Petrowiki

area change, assumption, calculation, coefficient, complexity, cross-sectional area, differential equation, drilling **fluid** management & disposal, drilling fluids and materials, energy equation, equation, **flow** stream, **fluid** mechanics, friction factor, increment, knowledge management, nozzle, pressure drop, production control, production logging, production monitoring, Reservoir Surveillance, standpipe pressure, steady **flow**, Upstream Oil & Gas, variation, Wellbore Design, wellbore integrity

SPE Disciplines:

Technology:

- Information Technology > Knowledge Management (0.40)
- Information Technology > Communications > Collaboration (0.40)

...This page discusses the implementation and **analysis** of ...**flow-after-flow** testing for gas well deliverability assessment. Both the Rawlins and Schellhardt and Houpeurt ...**analysis** techniques are presented in terms of pseudopressures.
...

Both the Rawlins and Schellhardt and Houpeurt analysis techniques are presented in terms of pseudopressures. Flow-after-flow tests, sometimes called gas backpressure or four-point tests, are conducted by producing the well at a series of different stabilized flow rates and measuring the stabilized BHFP at the sandface. Each different flow rate is established in succession either with or without a very short intermediate shut-in period. Conventional flow-after-flow tests often are conducted with a sequence of increasing flow rates; however, if stabilized flow rates are attained, the rate sequence does not affect the test. Fig 1 illustrates a flow-after-flow test.

Petrowiki

SPE Disciplines:

Technology:

- Information Technology > Knowledge Management (0.41)
- Information Technology > Communications > Collaboration (0.41)

...ngineering
Robert F. Mitchell, Editor
Copyright 2006, Society of Petroleum Engineers
Chapter 3 - **Fluid** Mechanics for Drilling
By R. F. Mitchell, Landmark Graphics and Kris Ravi, Halliburton
ISBN 978-1...-55563-114-7 Get permission for reuse
Introduction The three primary functions of a drilling **fluid**--the transport of cuttings out of the wellbore, prevention of ...**fluid** influx, and the maintenance of wellbore stability--depend on the ...

Introduction The three primary functions of a drilling fluid--the transport of cuttings out of the wellbore, prevention of fluid influx, and the maintenance of wellbore stability--depend on the flow of drilling fluids and the pressures associated with that flow. For example, if the wellbore pressure exceeds the fracture pressure, fluids will be lost to the formation. If the wellbore pressure falls below the pore pressure, fluids will flow into the wellbore, perhaps causing a blowout. It is clear that accurate wellbore pressure prediction is necessary. To properly engineer a drilling fluid system, it is necessary to be able to predict pressures and flows of fluids in the wellbore. The purpose of this chapter is to describe in detail the calculations necessary to predict the flow performance of various drilling fluids for the variety of operations used in drilling and completing a well. Overview Drilling fluids range from relatively incompressible fluids, such as water and brines, to ...

Petrowiki

annular pressure drilling, application, assumption, calculation, cross-sectional area, Directional Drilling, drilling **fluid** chemistry, drilling **fluid** formulation, drilling **fluid** management & disposal, drilling **fluid** property, drilling **fluid** selection and formulation, drilling fluids and materials, drilling operation, Drillstem Testing, drillstem/well testing, equation of state, **flow** in porous media, **flow** rate, **Fluid** Dynamics, **fluid** loss control, **fluid** modeling, foam, friction factor, frictional pressure drop, knowledge management, laminar **flow**, pipe, pressure drop, production control, production logging, production monitoring, Reservoir Surveillance, rheology, shear stress, trajectory design, turbulent **flow**, Upstream Oil & Gas, viscosity, well control, well planning, Wellbore Design, wellbore integrity

Country:

- North America > United States > Texas (0.67)
- North America > Canada (0.67)
- North America > United States > Oklahoma (0.47)

Oilfield Places:

- North America > United States > Texas > Permian Basin (0.99)
- North America > United States > Texas > Anadarko Basin (0.99)
- North America > United States > Oklahoma > Anadarko Basin (0.99)
- (2 more...)

SPE Disciplines:

- Well Drilling > Wellbore Design > Wellbore integrity (1.00)
- Well Drilling > Well Planning > Trajectory design (1.00)
- Well Drilling > Pressure Management > Well control (1.00)
- (7 more...)

- Information Technology > Knowledge Management (0.40)
- Information Technology > Communications > Collaboration (0.40)

...anchi, Editor
Copyright 2007, Society of Petroleum Engineers
Chapter 3 – Mathematics of Transient **Analysis**
Erdal Ozkan, Colorado School of Mines
ISBN 978-1-55563-108-6 Get permission for reuse
This chapt...er explains how **fluid flow** in porous media can be translated into a mathematical statement and how mathematical ...**analysis** can be used to answer transient-...

This chapter explains how fluid flow in porous media can be translated into a mathematical statement and how mathematical analysis can be used to answer transient-flow problems. This broad area is common to many other disciplines, such as heat conduction in solids and groundwater hydrology. The objective of this chapter is to introduce the fundamentals of transient analysis, present examples, and guide the interested reader to relevant references. Most physical phenomena in the domain of transient fluid flow in porous media can be described generally by partial differential equations (PDEs). With appropriate boundary conditions and sometimes with simplifying assumptions, the PDE leads to an initial boundary value problem (IBVP) that is solved to find a mathematical statement of the resulting flow in the porous medium. Figure 1.1 – Arbitrary closed surface Γ in porous medium. The initial condition is normally expressed in terms of a known pressure distribution at time zero; that is, ....................(3.26) The most common initial condition is the uniform pressure distribution in the entire porous medium; that is, f (x, y, z) pi. The boundary conditions are specified at the inner (wellbore) and outer boundaries of the reservoir.

Petrowiki

approximation, asymptotic approximation, Bessel function, boundary, boundary condition, Computation, differential equation, diffusion equation, equation, example 3, **flow** in porous media, **Fluid** Dynamics, fracture, horizontal well, infinite-slab reservoir, inverse Laplace transform, knowledge management, Laplace transform, long-time approximation, point-source solution, porous medium, pressure transient **analysis**, pressure transient testing, relation, reservoir, Upstream Oil & Gas

SPE Disciplines:

- Information Technology > Knowledge Management (0.40)
- Information Technology > Communications > Collaboration (0.40)

...There are many advantages of developing transient **flow** solutions in the Laplace transform domain. For example, in the Laplace transform domain, Duhamel's ...theorem[1] provides a convenient means of developing transient **flow** solutions for variable rate production problems using the solutions for the corresponding constant ...(2)
The simplicity of the expression given in Eq. 2 explains our interest in obtaining transient-**flow** solutions in the Laplace transform domain.
Another example to explain the convenience of the Lapla...

There are many advantages of developing transient flow solutions in the Laplace transform domain. For example, in the Laplace transform domain, Duhamel's theorem[1] provides a convenient means of developing transient flow solutions for variable rate production problems using the solutions for the corresponding constant rate production problem. Applying the Laplace transform converts the convolution integral in Eq. 1 to an algebraic expression, and Duhamel's theorem is given in the Laplace transform domain as The simplicity of the expression given in Eq. 2 explains our interest in obtaining transient-flow solutions in the Laplace transform domain. Another example to explain the convenience of the Laplace domain solutions is for the naturally fractured reservoirs. Common transient flow models of naturally fractured reservoirs lead to the following differential equation in radial coordinates in the Laplace transform domain: [2] The naturally fractured reservoir function, f (s), is a function of matrix and fracture properties and depends on the model chosen to represent the naturally fractured reservoir.[2] The general solutions for Eqs. 3 and 4 are given, respectively, by This discussion demonstrates that it is possible to derive transient flow solutions for naturally fractured reservoirs by following the same lines as those for the homogeneous reservoirs.

Petrowiki

approximation, asymptotic approximation, boundary condition, complex reservoir, Computation, cylindrical reservoir, expression, fracture, geometry, infinite-slab reservoir, inverse Laplace transform, knowledge management, Laplace transform, Laplace transform domain, long-time approximation, point source, point-source solution, pressure transient **analysis**, pressure transient testing, relation, reservoir, right side, Upstream Oil & Gas

SPE Disciplines:

- Information Technology > Knowledge Management (0.40)
- Information Technology > Communications > Collaboration (0.40)

Thank you!