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...rvoirs.
Green's function for transient flow in a porous medium is defined as the pressure at M (x, **y**, z) at time t because of an instantaneous point source of unit strength generated at point M′(x′, ...**y**′, z′) at time τ t with the porous medium initially at zero pressure and the boundary of the mediu...reen's functions for the 1D and/or 2D systems. For example, an infinite line-source well at x x′, **y y**′, and z′ in an infinite reservoir may be visualized as the intersection of two infinite, ...

As discussed in Source function solutions of the diffusion equation, the conventional development of the source function solutions uses the instantaneous point-source solution as the building block with the appropriate integration (superposition) in space and time. In 1973, Gringarten and Ramey[1] introduced the use of the source and Green's function method to the petroleum engineering literature with a more efficient method of developing the source solutions. Specifically, they suggested the use of infinite-plane sources as the building block with Newman's product method.[2] In this page we discuss the use of Green's functions and source functions in solving unsteady-flow problems in reservoirs. Green's function for transient flow in a porous medium is defined as the pressure at M (x, y, z) at time t because of an instantaneous point source of unit strength generated at point M′(x′, y′, z′) at time τ t with the porous medium initially at zero pressure and the boundary of the medium kept at zero pressure or impermeable to flow.[1][3]

Petrowiki

approximation, boundary, diffusion equation, horizontal well, impermeable boundary, infinite reservoir, infinite-plane source, infinite-slab reservoir, infinite-slab source, instantaneous point source, instantaneous point-source solution, instantaneous source function, knowledge management, plane source, porous medium, pressure transient analysis, pressure transient testing, product method, reservoir, reservoir system, source function, Source function solution, Upstream Oil & Gas

SPE Disciplines: Reservoir Description and Dynamics > Formation Evaluation & Management > Pressure transient analysis (1.00)

Technology:

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

...field, offshore UK) After Woodhouse.[7]
Fig. 3 – Core-plug porosity vs. well-log bulk density with **y**-on-x and RMA line fits (South Morecambe gas field, offshore UK) After Woodhouse.[7]
The variance o...tandard.
The generally recommended method for obtaining a line-fit for porosity prediction is the "**y**-on-x" ordinary least-squares regression method. [7][8] The recommendation presumes that the calibra... respects of the environment of the equation's future use. The dependent-variable calibration data, **y**, the values wanted in the future (e.g., core porosity), are regressed against the selected independ...

The accurate calculation of porosity at the wellbore is essential for an accurate calculation of original oil in place (OOIP) or original gas in place (OGIP) throughout the reservoir. The porosity and its distribution also need to be calculated as accurately as possible because they are almost always directly used in the water saturation (Sw) and permeability calculations and, possibly, in the net pay calculations. In most OOIP and OGIP studies, only the gross-rock-volume uncertainties have a greater influence on the result than porosity does. Occasionally, where porosity estimates are difficult, porosity is the leading uncertainty. Fractured and clay-mineral-rich reservoirs remain a challenge. For this discussion, it is assumed that the core data have been properly adjusted to reservoir conditions, that the data from various logs have been reviewed and validated as needed, and that all of the required depth-alignment work has been completed. There are a few preliminary steps in the use of routine core porosity data over the reservoir interval.

Petrowiki

Artificial Intelligence, calculation, clay mineral, core data, core measurement, core porosity, correlation, density log, effective porosity, estimates of resource in place, gas field, knowledge management, machine learning, Offshore UK, permeability, porosity, porosity measurement, porosity value, reserves evaluation, reservoir, resource in place estimate, shaly sand, total porosity, Upstream Oil & Gas, vhcp, water saturation

Oilfield Places:

- Europe > United Kingdom > Irish Sea > East Irish Sea > Morecambe Bay > Morecambe Bay > East Irish Sea Basin > Block 110/8a > South Morecambe Field (0.99)
- Europe > United Kingdom > Irish Sea > East Irish Sea > Morecambe Bay > Morecambe Bay > East Irish Sea Basin > Block 110/3a > South Morecambe Field (0.99)
- Europe > United Kingdom > Irish Sea > East Irish Sea > Morecambe Bay > Morecambe Bay > East Irish Sea Basin > Block 110/2a > South Morecambe Field (0.99)

SPE Disciplines:

Technology:

... the equation.
...(7)
for a function (ψ) of two or more independent variables {x,**y**, ...}. A PDE is linear if it is first order in the unknown function and its partial derivatives, an...fficients of the partial derivatives, are either constant or depend on the independent variables {x,**y**, ...}. We illustrate these concepts by considering the continuity equation for flow of a fluid with...(13)
where the functions {A, B, C, G} are known functions of two independent variables {x,**y**} and the first-order partial derivatives ψ(x,...

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, differential equation, 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

SPE Disciplines:

Technology:

...ustrates the concept of a complex seismic trace in which x(t) represents the real seismic trace and **y**(t) is the Hilbert transform of x(t). In this discussion, we ignore what a Hilbert transform is and ...how the function **y**(t) is calculated. Most modern seismic data-processing software packages provide Hilbert transform a...lgorithms and allow processors to create the function **y**(t) shown in Fig. 2 easily. These two data vectors are displayed in a 3D (x, ...

Any procedure that extracts and displays these seismic parameters in a convenient, understandable format is an invaluable interpretation tool. Taner and Sheriff[1] and Taner et al.[2] began using the Hilbert transform to calculate seismic amplitude, phase, and frequency instantaneously, meaning that a value of amplitude, phase, and frequency is calculated for each time sample of a seismic trace. Since that introduction, numerous Hilbert transform algorithms have been implemented to calculate these useful seismic attributes (e.g., Hardage[3]). Figure 1 illustrates the concept of a complex seismic trace in which x(t) represents the real seismic trace and y(t) is the Hilbert transform of x(t). In this discussion, we ignore what a Hilbert transform is and how the function y(t) is calculated.

Petrowiki

amplitude, complex seismic trace, complex trace, equation, frequency, frequency value, Hilbert transform, hilbert transform algorithm, instantaneous amplitude, instantaneous frequency, instantaneous phase, instantaneous phase function, knowledge management, phase function, plane, Reservoir Characterization, stratigraphic discontinuity, time axis, Upstream Oil & Gas, vector

SPE Disciplines: Reservoir Description and Dynamics > Reservoir Characterization > Seismic processing and interpretation (1.00)

Technology:

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

...very, Tulsa, 19–23 April. SPE-113997-MS. http://dx.doi.org/10.2118/113997-MS
Bai, B., Li, L., Liu, **Y**. et al. 2004. Preformed Particle Gel for Conformance Control: Factors Affecting Its Properties and ...ium on Improved Oil Recovery, Tulsa, 17–21 April. http://dx.doi.org/10.2118/89389-MS
Bai, B., Liu, **Y**., Coste, J.-P. et al. 2007. Preformed Particle Gel for Conformance Control: Transport Mechanism Thr...SPE Res Eval & Eng 10 (2): 176–184. SPE-89468-PA. http://dx.doi.org/10.2118/89468-PA
Bai, B., Liu, **Y**., Coste, J.-P. et al. 2007. Preformed Particle Gel for Conformance Control: Transport Mechanism Thr...

Petrowiki

application, Bai, Conformance Control, conformance control treatment, conformance improvement, enhanced recovery, Exhibition, improved oil recovery, knowledge management, porous media, preformed particle gel, presented, SPE DOE Symposium, Symposium, technical conference, Tulsa, Upstream Oil & Gas, Vasquez

Country:

- North America > United States (1.00)
- Asia (1.00)

SPE Disciplines: Reservoir Description and Dynamics > Improved and Enhanced Recovery > Conformance improvement (1.00)

Technology:

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

...int source of strength acting at t 0:
...(31)
If the source is located at x′D, **y**′D, z′D, then, by translation, we can write
...(32)
...(33)
...dure to obtain the solution is fairly easy, the use of the method of images in three directions (x, **y**, z) yields triple infinite Fourier series, which may pose computational inconveniences. As an examp...le, the solution for a continuous point source located at x′, **y**′, z′ in a rectangular porous medium occupying the region 0 x xe, 0 ...

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, closed cylindrical reservoir, complex reservoir, Computation, 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:

Technology:

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

... 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...ervoirs. Green's function for transient flow in a porous medium is defined as the pressure at M (x, **y**, z) at time t because of an instantaneous point source of unit strength generated at point M′(x′, ...**y**′, z′) at time τ t with the porous medium initially at zero pressure and the boundary of the mediu...

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)

...h two or more ESP systems are installed concurrently in the same wellbore. One configuration uses a **Y**-tool with the first ESP attached, as described in ESP optional components, and a second ESP system ...attached to the bottom of the bypass tube or to another **Y**-tool bypass head (Fig. 4). For a triple system, another ...**Y**-tool is attached to the bottom of the first bypass tube, allowing for a third unit to be incorporat...

Electrical submersible pumps focuses on the standard ESP configuration. It has the pump, seal chamber section, and motor attached to the production tubing, in this order from top down. In some wellbore completions and unique ESP applications, the arrangement and configuration of the system is modified. For a bottom-intake design, the production fluid is drawn in the intake ports located at the very bottom of the ESP system and discharged out of ports located just below the connection to the seal-chamber section. Because the discharged production fluid cannot flow through the seal-chamber section and motor, it has to exit into the casing or liner annulus and flow past these units.

Petrowiki

application, artificial lift system, bypass tube, centrilift graphic, Claremore, configuration, disposal zone, ESP configuration, ESP system, flow rate, individual unit, knowledge management, Oklahoma, Petroleum Engineer, PetroWiki, pressure increase, production tubing, seal-chamber section, Upstream Oil & Gas, **y**-tool

SPE Disciplines:

Technology:

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

...ccording to their impact on a specified (target) output.
Rank correlation sensitivity analysis Let **Y** be an output and X an input for the model. The rank correlation coefficient, rr, between ...**Y** and X is a number between –1 and 1. (See the definition and discussion in The tools of the trade.)... The closer rr is to 1 or –1, the more influence X has on **Y**. Positive correlation indicates that as X increases, ...

A Monte Carlo model is, in principle, just a worksheet in which some cells contain probability distributions rather than values. Thus, one can build a Monte Carlo model by converting a deterministic worksheet with the help of commercial add-in software. Practitioners, however, soon find that some of their deterministic models were constructed in a way that makes this transition difficult. Redundancy, hidden formulas, and contorted logic are common features of deterministic models that encumber the resulting Monte Carlo model. Likewise, presentation of results from probabilistic analysis might seem no different from any other engineering presentation (problem statement, summary and conclusions, key results, method, and details).

Petrowiki

Artificial Intelligence, audience, cash-flow model, decision tree learning, deterministic model, input distribution, machine learning, Monte Carlo model, PetroWiki, presentation, rank correlation coefficient, reserve model, reservoir simulation, risk management, sensitivity analysis, software, Standard Deviation, total cost, uncertainty model

SPE Disciplines:

Technology: Information Technology > Artificial Intelligence > Machine Learning > Decision Tree Learning (0.36)

...(psia/ft). The form of Darcy's law with full permeability tensor in Cartesian coordinate system {x, **y**, z} is
...(3)
where we have treated the cross-sectional area, A, as a constant w... Eq. 12 is the same form as Eq. 5.
It is mathematically possible to find a coordinate system {x′, **y**′, z′} in which the permeability tensor has the diagonal form . We diagonalize the matrix by finding... for finding a matrix that diagonalizes an n n matrix is as follows:[4]
The coordinate axes {x′, **y**′, z′} are the principal axes of the diagonalized tensor, and the diagonal form of the permeability ...

The form of Darcy's law that is most widely used in formulating fluid flow equations in reservoir simulators assumes that the coordinate system is aligned with the principal axes of the permeability tensor. The simplified equations are easier to code and can be solved with less computation time than fluid flow equations that include the full permeability tensor. Research in naturally-fractured-reservoir modeling,[1] geomechanics,[2] and upscaling[3] has demonstrated that the full permeability tensor is needed to correctly solve fluid flow problems in a variety of realistic settings. The mathematical procedure for diagonalizing the permeability tensor is presented here as an illustration of the mathematics discussed in matrices and linear algebra. An understanding of the relationship between grid orientation and the permeability tensor can help us decide how to orient a fluid flow grid to most accurately represent the permeability distribution in a reservoir.

Petrowiki

coordinate system, coordinate transformation, dependence, equation, flow in porous media, flow rate, Fluid Dynamics, fluid flow equation, formulation, grid orientation, knowledge management, matrix, off-diagonal element, permeability tensor, principal axis, reservoir simulation, similarity transformation, square matrix, transformation, transformation matrix, Upstream Oil & Gas

SPE Disciplines: Reservoir Description and Dynamics > Reservoir Fluid Dynamics > Flow in porous media (1.00)

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

Thank you!