Three (3) dimensional one-phase fluid (oil) reservoir simulator using a partially implicit method with lagging coefficients. Skin damage after drilling the production well was expected; the impact of the skin damage and the necessity of well stimulation was assessed
Numerical Simulation of 2-D Subsurface ReservoirJeffrey Daniels
This document provides a detailed summary of a reservoir simulation project, including:
- The properties of the rectangular oil reservoir being modeled, including dimensions, porosity, permeability, and fluid properties.
- The discretization of the reservoir into 165 grid blocks for simulation.
- The partial differential equations and initial/boundary conditions used in the reservoir simulation.
- How well production is incorporated and calculated using Peaceman's method.
- Details on running a "leakproof test" to validate that pressures and saturations remain constant with no well flow.
The lattice Boltzmann equation: background, boundary conditions, and Burnett-...Tim Reis
An overview of the lattice Boltzmann equation and a discussion of moment-based boundary conditions. Includes applications to the slip flow regime and the Burnett stress. Some analysis sheds insight into the physical and numerical behaviour of the algorithm.
Numerical Simulation Study of a 1-D Subsurface ReservoirJeffrey Daniels
The document describes a reservoir simulation study involving the discretization of a rectangular oil reservoir into 21 simulation cells. Equations describing fluid flow are presented for cells with and without wells. A system of implicit equations is developed by applying the block-centered grid discretization technique to each cell. The equations account for the presence of 5 wells - 2 production wells and 3 injection wells. Boundary and initial conditions are also specified. Discretized equations are written out for each cell to set up the system of equations to be solved.
The document provides solutions to several exercises related to slurry transport. For Exercise 4.1, the solution analyzes shear stress and shear rate data for a phosphate slurry and determines it follows a power-law relationship with a flow index of 0.15 and consistency index of 23.4 Ns0.15/m2. Exercise 4.2 verifies an equation for pressure drop in pipe flow of a power-law fluid. Exercise 4.3 similarly verifies an equation incorporating a yield stress. Subsequent exercises provide solutions for pressure drop, slurry concentration, and rheological properties calculations using data given.
The document describes several theoretical physics problems involving mechanics, thermodynamics, and radioactivity dating.
Problem 1 describes a bungee jumper attached to an elastic rope, deriving expressions for the distance dropped before coming to rest, maximum speed, and time taken.
Problem B involves a heat engine operating between two bodies at different temperatures, deriving an expression for the final temperature if maximum work is extracted, and using this to find the maximum work.
Problem C uses radioactive decay of uranium isotopes to date the age of the Earth, deriving equations relating isotope ratios to time and obtaining an approximate age of 5.38 billion years.
Poster to be presented at Stochastic Numerics and Statistical Learning: Theory and Applications Workshop 2024, Kaust, Saudi Arabia, https://cemse.kaust.edu.sa/stochnum/events/event/snsl-workshop-2024.
In this work we have considered a setting that mimics the Henry problem \cite{Simpson2003,Simpson04_Henry}, modeling seawater intrusion into a 2D coastal aquifer. The pure water recharge from the ``land side'' resists the salinisation of the aquifer due to the influx of saline water through the ``sea side'', thereby achieving some equilibrium in the salt concentration. In our setting, following \cite{GRILLO2010}, we consider a fracture on the sea side that significantly increases the permeability of the porous medium.
The flow and transport essentially depend on the geological parameters of the porous medium, including the fracture. We investigated the effects of various uncertainties on saltwater intrusion. We assumed uncertainties in the fracture width, the porosity of the bulk medium, its permeability and the pure water recharge from the land side. The porosity and permeability were modeled by random fields, the recharge by a random but periodic intensity and the thickness by a random variable. We calculated the mean and variance of the salt mass fraction, which is also uncertain.
The main question we investigated in this work was how well the MLMC method can be used to compute statistics of different QoIs. We found that the answer depends on the choice of the QoI. First, not every QoI requires a hierarchy of meshes and MLMC. Second, MLMC requires stable convergence rates for $\EXP{g_{\ell} - g_{\ell-1}}$ and $\Var{g_{\ell} - g_{\ell-1}}$. These rates should be independent of $\ell$. If these convergence rates vary for different $\ell$, then it will be hard to estimate $L$ and $m_{\ell}$, and MLMC will either not work or be suboptimal. We were not able to get stable convergence rates for all levels $\ell=1,\ldots,5$ when the QoI was an integral as in \eqref{eq:integral_box}. We found that for $\ell=1,\ldots 4$ and $\ell=5$ the rate $\alpha$ was different. Further investigation is needed to find the reason for this. Another difficulty is the dependence on time, i.e. the number of levels $L$ and the number of sums $m_{\ell}$ depend on $t$. At the beginning the variability is small, then it increases, and after the process of mixing salt and fresh water has stopped, the variance decreases again.
The number of random samples required at each level was estimated by calculating the decay of the variances and the computational cost for each level. These estimates depend on the minimisation function in the MLMC algorithm.
To achieve the efficiency of the MLMC approach presented in this work, it is essential that the complexity of the numerical solution of each random realisation is proportional to the number of grid vertices on the grid levels.
This document discusses the longest common subsequence problem and provides an example of how it can be solved using dynamic programming. It begins by defining the problem of finding the longest subsequence that is common to two input sequences. It then shows that this problem exhibits optimal substructure and can be solved recursively. However, a recursive solution is inefficient due to redundant subproblem computations. Instead, it presents an algorithm that uses dynamic programming to compute the length of the longest common subsequence in O(mn) time by filling out a 2D table in a bottom-up manner and returning the value at the last index. It also describes how to construct the actual longest common subsequence by tracing back through the table.
The document discusses the longest common subsequence (LCS) problem and presents a dynamic programming approach to solve it. It defines key terms like subsequence and common subsequence. It then presents a theorem that characterizes an LCS and shows it has optimal substructure. A recursive solution and algorithm to compute the length of an LCS are provided, with a running time of O(mn). The b table constructed enables constructing an LCS in O(m+n) time.
Numerical Simulation of 2-D Subsurface ReservoirJeffrey Daniels
This document provides a detailed summary of a reservoir simulation project, including:
- The properties of the rectangular oil reservoir being modeled, including dimensions, porosity, permeability, and fluid properties.
- The discretization of the reservoir into 165 grid blocks for simulation.
- The partial differential equations and initial/boundary conditions used in the reservoir simulation.
- How well production is incorporated and calculated using Peaceman's method.
- Details on running a "leakproof test" to validate that pressures and saturations remain constant with no well flow.
The lattice Boltzmann equation: background, boundary conditions, and Burnett-...Tim Reis
An overview of the lattice Boltzmann equation and a discussion of moment-based boundary conditions. Includes applications to the slip flow regime and the Burnett stress. Some analysis sheds insight into the physical and numerical behaviour of the algorithm.
Numerical Simulation Study of a 1-D Subsurface ReservoirJeffrey Daniels
The document describes a reservoir simulation study involving the discretization of a rectangular oil reservoir into 21 simulation cells. Equations describing fluid flow are presented for cells with and without wells. A system of implicit equations is developed by applying the block-centered grid discretization technique to each cell. The equations account for the presence of 5 wells - 2 production wells and 3 injection wells. Boundary and initial conditions are also specified. Discretized equations are written out for each cell to set up the system of equations to be solved.
The document provides solutions to several exercises related to slurry transport. For Exercise 4.1, the solution analyzes shear stress and shear rate data for a phosphate slurry and determines it follows a power-law relationship with a flow index of 0.15 and consistency index of 23.4 Ns0.15/m2. Exercise 4.2 verifies an equation for pressure drop in pipe flow of a power-law fluid. Exercise 4.3 similarly verifies an equation incorporating a yield stress. Subsequent exercises provide solutions for pressure drop, slurry concentration, and rheological properties calculations using data given.
The document describes several theoretical physics problems involving mechanics, thermodynamics, and radioactivity dating.
Problem 1 describes a bungee jumper attached to an elastic rope, deriving expressions for the distance dropped before coming to rest, maximum speed, and time taken.
Problem B involves a heat engine operating between two bodies at different temperatures, deriving an expression for the final temperature if maximum work is extracted, and using this to find the maximum work.
Problem C uses radioactive decay of uranium isotopes to date the age of the Earth, deriving equations relating isotope ratios to time and obtaining an approximate age of 5.38 billion years.
Poster to be presented at Stochastic Numerics and Statistical Learning: Theory and Applications Workshop 2024, Kaust, Saudi Arabia, https://cemse.kaust.edu.sa/stochnum/events/event/snsl-workshop-2024.
In this work we have considered a setting that mimics the Henry problem \cite{Simpson2003,Simpson04_Henry}, modeling seawater intrusion into a 2D coastal aquifer. The pure water recharge from the ``land side'' resists the salinisation of the aquifer due to the influx of saline water through the ``sea side'', thereby achieving some equilibrium in the salt concentration. In our setting, following \cite{GRILLO2010}, we consider a fracture on the sea side that significantly increases the permeability of the porous medium.
The flow and transport essentially depend on the geological parameters of the porous medium, including the fracture. We investigated the effects of various uncertainties on saltwater intrusion. We assumed uncertainties in the fracture width, the porosity of the bulk medium, its permeability and the pure water recharge from the land side. The porosity and permeability were modeled by random fields, the recharge by a random but periodic intensity and the thickness by a random variable. We calculated the mean and variance of the salt mass fraction, which is also uncertain.
The main question we investigated in this work was how well the MLMC method can be used to compute statistics of different QoIs. We found that the answer depends on the choice of the QoI. First, not every QoI requires a hierarchy of meshes and MLMC. Second, MLMC requires stable convergence rates for $\EXP{g_{\ell} - g_{\ell-1}}$ and $\Var{g_{\ell} - g_{\ell-1}}$. These rates should be independent of $\ell$. If these convergence rates vary for different $\ell$, then it will be hard to estimate $L$ and $m_{\ell}$, and MLMC will either not work or be suboptimal. We were not able to get stable convergence rates for all levels $\ell=1,\ldots,5$ when the QoI was an integral as in \eqref{eq:integral_box}. We found that for $\ell=1,\ldots 4$ and $\ell=5$ the rate $\alpha$ was different. Further investigation is needed to find the reason for this. Another difficulty is the dependence on time, i.e. the number of levels $L$ and the number of sums $m_{\ell}$ depend on $t$. At the beginning the variability is small, then it increases, and after the process of mixing salt and fresh water has stopped, the variance decreases again.
The number of random samples required at each level was estimated by calculating the decay of the variances and the computational cost for each level. These estimates depend on the minimisation function in the MLMC algorithm.
To achieve the efficiency of the MLMC approach presented in this work, it is essential that the complexity of the numerical solution of each random realisation is proportional to the number of grid vertices on the grid levels.
This document discusses the longest common subsequence problem and provides an example of how it can be solved using dynamic programming. It begins by defining the problem of finding the longest subsequence that is common to two input sequences. It then shows that this problem exhibits optimal substructure and can be solved recursively. However, a recursive solution is inefficient due to redundant subproblem computations. Instead, it presents an algorithm that uses dynamic programming to compute the length of the longest common subsequence in O(mn) time by filling out a 2D table in a bottom-up manner and returning the value at the last index. It also describes how to construct the actual longest common subsequence by tracing back through the table.
The document discusses the longest common subsequence (LCS) problem and presents a dynamic programming approach to solve it. It defines key terms like subsequence and common subsequence. It then presents a theorem that characterizes an LCS and shows it has optimal substructure. A recursive solution and algorithm to compute the length of an LCS are provided, with a running time of O(mn). The b table constructed enables constructing an LCS in O(m+n) time.
This document describes the adaptive restore algorithm, a non-reversible Markov chain Monte Carlo method. It begins with an overview of the restore process, which takes regenerations from an underlying diffusion or jump process to construct a reversible Markov chain with a target distribution. The adaptive restore process enriches this by allowing the regeneration distribution to adapt over time. It converges almost surely to the minimal regeneration distribution. Parameters like the initial regeneration distribution and rates are discussed. Examples are provided for the adaptive Brownian restore algorithm and calibrating the parameters.
This document discusses cubic spline interpolation. Cubic splines are piecewise cubic polynomials that are continuously differentiable and match function values at sample points. They provide a smooth interpolation that avoids oscillations seen in higher-degree global polynomials. The document outlines the construction of cubic spline interpolation, including determining the coefficients for each cubic polynomial piece based on function values and derivatives at nodes. An example interpolates the function f(x)=x^4 on the interval [-1,1] using cubic splines.
The time scale Fibonacci sequences satisfy the Friedmann-Lema\^itre-Robertson-Walker (FLRW) dynamic equation on time scale, which are an exact solution of Einstein's field equations of general relativity for an expanding homogeneous and isotropic universe. We show that the equations of motion correspond to the one-dimensional motion of a particle of position $F(t)$ in an inverted harmonic potential. For the dynamic equations on time scale describing the Fibonacci numbers $F(t)$, we present the Lagrangian and Hamiltonian formalism. Identifying these with the equations that describe factor scales, we conclude that for a certain granulation, for both the continuous and the discrete universe, we have the same dynamics.
The document discusses heat transfer via fins. It defines fins as protrusions that increase the surface area in contact with a fluid to facilitate heat transfer. It presents the governing differential equation for one-dimensional heat transfer through a fin and describes the boundary conditions and solutions for different fin types, including infinitely long fins, short fins with insulated tips, and fins with convection at the tip. It also provides equations for calculating the heat transfer rate for different fin configurations.
The purpose of this note is to elaborate in how far a 4 factor affine model can generate an incomplete bond market together with the flexibility of a 3 factor flexible affine cascade structure model.
Computing the masses of hyperons and charmed baryons from Lattice QCDChristos Kallidonis
Poster presented at the Computational Sciences 2013 Conference (Winner of poster competition). We present results on the masses of all forty light, strange and charm baryons from Lattice QCD simulations, focusing particularly on the computational aspects and requirements of such calculations.
This document is a semester project report submitted by Preeti Sahu to the Physics Department at Syracuse University. It summarizes her mathematical modeling of cellular oscillations driven by contractility and turnover in the actomyosin cytoskeleton. The report introduces increasingly complex mechanical models of cell contractility and viscosity. It then presents a model incorporating actomyosin turnover, showing the fixed point becomes unstable with oscillations emerging around the equilibrium state.
This document discusses Frullani integrals, which are integrals of the form ∫01 f(ax)−f(bx)x dx = [f(0)−f(∞)]ln(b/a). It provides 11 examples of integrals from Gradshteyn and Ryzhik that can be reduced to this Frullani form by appropriate choice of the function f(x). It also lists 9 examples found in Ramanujan's notebooks. One example, involving logarithms of trigonometric functions, requires a more complex approach. The document concludes by deriving the solution to this more delicate example.
This document discusses various types of mechanical vibrations including undamped, underdamped, critically damped, and overdamped motion. It provides examples of solving differential equations describing simple harmonic motion under different damping conditions and applying the solutions to problems involving springs, pendulums, and other oscillating systems. Specific cases are worked through to determine characteristics like frequency, period, and damping behavior.
1. The document describes 5 experiments related to mechanical oscillations and vibrations. Experiment 1 verifies the relation between the period of a simple pendulum and its length. Experiment 2 determines the radius of gyration of a compound pendulum.
2. Experiment 3 uses a bifilar suspension to determine the radius of gyration of a bar. Experiment 4 studies longitudinal vibrations of a helical spring and determines the frequency theoretically and experimentally.
3. Experiment 5 examines vibrations of a system with springs in series. Each experiment involves setting up the apparatus, collecting observations in tables, performing calculations, and drawing conclusions by comparing experimental and theoretical results.
1. This document contains solutions to problems from Chapter 1 of the textbook "The Physics of Vibrations and Waves".
2. It provides detailed calculations and explanations for problems related to simple harmonic motion, including determining restoring forces, stiffness, frequencies, and solving differential equations of motion.
3. Examples include a simple pendulum, a mass on a spring, and vibrations of strings, membranes, and gas columns.
This document discusses continuous-time Markov chains and provides examples of different types of Markov chains including birth-death processes and pure birth processes. It introduces key concepts such as transition rates and the Kolmogorov differential equations. Specifically, it derives the Kolmogorov backward and forward equations, which describe the time evolution of the transition probabilities of a Markov chain. Examples are provided to illustrate how to apply these equations to different Markov chain models.
The document discusses recursive algorithms and recurrence relations. It provides examples of solving recurrence relations for different algorithms like Towers of Hanoi, selection sort, and merge sort. Recurrence relations define algorithms recursively in terms of smaller inputs. They are solved to find closed-form formulas for the running time of algorithms.
This document summarizes a talk on solving density-driven groundwater flow problems with uncertain porosity and permeability coefficients. The major goal is to estimate pollution risks in subsurface flows. The presentation covers: (1) setting up the groundwater flow problem; (2) reviewing stochastic modeling methods; (3) modeling uncertainty in porosity and permeability; (4) numerical methods to solve deterministic problems; and (5) 2D and 3D numerical experiments. The experiments demonstrate computing statistics of contaminant concentration and its propagation under uncertain parameters.
The document provides a design example for a reinforced concrete retaining wall with the following conditions:
1. The wall must retain a backfill with a unit weight of 100 pcf and a surcharge of 400 psf.
2. The wall stem is designed as a vertical cantilever beam to resist lateral earth pressures.
3. The base thickness is selected as 16 inches and the stem thickness as 15 inches with #8 reinforcing bars at 6 inches.
4. The heel width is selected as 7.5 feet to prevent sliding failure based on resisting and driving forces.
The document discusses concepts related to curves, tangents, normals, and curvature. It provides formulas for calculating the equation of the tangent and normal to a curve at a given point, as well as the length of the subtangent and subnormal. It also discusses how to find the radius of curvature and center of curvature at a point on a curve. An example is worked out to find the center of curvature of the curve xy=16 at the point (4,4).
The document discusses decreasing hysteresis in shape memory alloys (SMAs). It first provides background on SMA phases and properties. Key points include SMAs having two stable phases at different temperatures, and the hysteresis size being determined by transition temperature ranges. The document then discusses conditions for minimizing hysteresis, including no volume change, an invariant plane between phases, and compatibility conditions allowing infinite interfaces. It presents an example transition layer between austenite and martensite, calculating its energy based on linear elasticity theory. The energy is minimized by choosing strain and variant volume fraction parameters, showing how hysteresis can be decreased for SMAs.
Variants of the Christ-Kiselev lemma and an application to the maximal Fourie...VjekoslavKovac1
1. The document discusses variants of the Christ-Kiselev lemma and its application to maximal Fourier restriction estimates.
2. The Christ-Kiselev lemma allows block-diagonal and block-triangular truncations of operators while controlling their operator norms.
3. These lemmas can be used to prove maximal and variational estimates for the restriction of the Fourier transform to surfaces, which has applications in harmonic analysis.
Two basic topics of heat transfer have been covered up by me based on the famous books of :-
1) John H. Lienhard (Professor Emeritus, University of Houston)
2) J.P. Holman (Professor, Southern Methodist University)
3) Prabal Talukdar (Associate Professor, IIT, India)
System overflow blocking-transients-for-queues-with-batch-arrivals-using-a-fa...Cemal Ardil
This document summarizes a research paper that analyzes the transient behavior of the overflow probability in a queuing system with fixed-size batch arrivals. It introduces a set of polynomials that generalize Chebyshev polynomials and can be used to assess the transient behavior. The key findings are:
1≤k ≤ B
k≥B+1
which is just the generating function of the Chebyshev
polynomials of the second kind.
Furthermore, if we consider the special case when B = 1 in
(9), and make the substitution x → 2x, we obtain
k =0
0≤k ≤ B
Pk −1
λ
μ
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System overflow blocking-transients-for-queues-with-batch-arrivals-using-a-fa...
Numerical Simulation of a 3-D Reservoir
1. PTE 7397 Midterm Project Report
Jeffrey Daniels
Tuesday 30th
March, 2021
1 Introduction
In this study, an operating company is working on a three (3) dimensional single phase
oil reservoir. Skin damage after drilling the well is expected and the company wants to
verify the impact of the skin damage and the necessity of well stimulation. Therefore,
this simulation study seeks to forecast oil production in three cases:
a. Standard case (skin = 0)
b. Damaged well case (skin = 3)
c. Stimulated well case (skin = -1)
2 Reservoir Description
2.1 Reservoir Properties
The oil reservoir under study is approximated by a rectangular prism The geometric
measurements of this prism is length Lx = 450 ft, width Ly = 450 ft and height
Lz = 150 ft. The reservoir is assumed to have a constant porosity of φ = 0.25, a hori-
zontal permeability in both the x and y-directions of kx = ky = 70 mD and a vertical
permeability of kz = 7 mD. The formation compressibility factor of cr = 3x10−6
psia−1
is assumed to be constant.
2.2 Production Well
One well is drilled in the reservoir first, and its location on the grid is at xw = 435 ft,
yw = 15 ft, zw = 6900 ft. The well serves as a producer in this reservoir simulation
study and partially penetrates the reservoir. The penetration thickness is zt = 30 ft.
It has a wellbore radius of rw = 0.35 ft. The well is produced at a constant pressure of
Pwf = 2700 psia
2.3 Reservoir Fluid Properties
The fluid flow is one-phase which is oil. At a reference pressure of 3000 psia, the fluid
properties of oil are viscosity µo
= 0.99 cp, oil compressibility co = 10−5
psia−1
, forma-
tion volume factor Bo
= 1 rcf/scf and density ρo
= 45 lbm
ft3 .
1
2. The oil properties are defined as:
Density (lbm/ft3
) : ρ(P) = 45 exp[Co(P − 3000)]
Formation Volume Factor (rcf/scf) : B(P) = 1.0 exp[−Co(P − 3000)]
Viscosity (cP) : µ(P) = 6.00961538 ∗ 10−9
P2
− 9.13324176 ∗ 10−5
P + 1.21
3 Creation of Simulation Grid Blocks
In this study, the reservoir is three dimensional and was discretized into 1125 simula-
tion cells (i.e characterized by 15x15x5 grid blocks). Each grid block has a length of
∆X = 30 ft, a width of ∆Y = 30 ft and a height of ∆Z = 30 ft. The volume of a grid
block is 27000 ft3
.
4 Partial Differential Equations (PDEs) With Initial and Bound-
ary Conditions
We start with the oil material balance equation which is as follows:
Rate of mass accumulation = Net rate of mass flow in − Sink
In mathematical terms this is expressed with a partial differential equation (PDE) as
follows:
∂(φρ)
∂t
= ∇[
ρ
µ
k(∇P + ρg∇Z)] − Q
The density and formation volume factor at reference pressure (Po
) are defined as ρo
and Bo
respectively. Hence density can be computed at any pressure as follows:
ρ =
ρo
Bo
B
Substituting this into the second equation listed, we obtain
∂
∂t
(
φ
B
) = ∇[
k
Bµ
(∇P + ρg∇Z)] −
Q
ρoBo
where
Mobility in x,y and z directions : λx =
kx
Bµ
λy =
ky
Bµ
λz =
kz
Bµ
Flow Rate : e
q =
Q
ρoBo
Pressure gradient : γ = ρg however for this project γ(psi/ft) = ρ/144
2
3. Therefore in terms of mobility and pressure gradient our partial differential equation
becomes
∂
∂t
(
φ
B
) = ∇[λ(∇P + γ∇Z)] − e
q
Since we are using a three (3) dimensional simulation grid, the PDE to be discretized is
as follows:
∂
∂t
(
φ
B
) =
∂
∂x
[λx(
∂P
∂x
+ γx
∂z
∂x
)] +
∂
∂y
[λy(
∂P
∂y
+ γy
∂z
∂y
)] +
∂
∂z
[λz(
∂P
∂z
+ γz
∂z
∂z
)] − e
q
We have no flow boundary conditions which are mathematically represented as follows:
∂P
∂x
|x=0ft = 0, ∂P
∂x
|x=450ft = 0
∂P
∂y
|y=0ft = 0, ∂P
∂y
|y=450ft = 0
∂P
∂y
|z=6825ft = 0, ∂P
∂z
|z=6975ft = 0
Initial condition: P(z = 6900ft)|t=0 = 3000 psia
5 Implicit Block-Centered Grid System Discretization with
Lagging Coefficients
We will derive the finite difference equation for the lefthand side (LHS) and right hand
side (RHS) of the PDE separately, starting with the RHS which deals with spatial dis-
cretization. The Finite Difference Equation (FDE) for the PDE derived is as follows:
∂
∂t
(
φ
B
)|i,j,k =
λn
i+1,j,k
∆x
(
Pn+1
i+1,j,k − Pn+1
i,j,k
∆xi+1,j,k
+ γn
i+1,j,k
zi+1,j,k − zi,j,k
∆xi+1,j,k
)
+
λn
i−1,j,k
∆x
(
Pn+1
i−1,j,k − Pn+1
i,j,k
∆xi−1,j,k
+ γn
i−1,j,k
zi−1,j,k − zi,j,k
∆xi−1,j,k
)
+
λn
i,j+1,k
∆y
(
Pn+1
i,j+1,k − Pn+1
i,j,k
∆yi,j+1,k
+ γn
i,j+1,k
zi,j+1,k − zi,j,k
∆yi,j+1,k
)
+
λn
i,j−1,k
∆y
(
Pn+1
i,j−1,k − Pn+1
i,j,k
∆yi,j−1,k
+ γn
i,j−1,k
zi,j−1,k − zi,j,k
∆yi,j−1,k
)
+
λn
i,j,k+1
∆z
(
Pn+1
i,j,k+1 − Pn+1
i,j,k
∆zi,j,k+1
+ γn
i,j,k+1
zi,j,k+1 − zi,j,k
∆zi,j,k+1
)
+
λn
i,j,k−1
∆z
(
Pn+1
i,j,k−1 − Pn+1
i,j,k
∆zi,j,k−1
+ γn
i,j,k−1
zi,j,k−1 − zi,j,k
∆zi,j,k−1
) − e
qi,j,k
We then multiply each side by the volume of a grid block, Vi,j,k, where
3
4. Volume of each grid block (ft3
) : Vi,j,k = ∆x∆y∆z
Flow rate : qi,j,k = e
qi,j,kVi,j,k
The FDE then becomes
Vi,j,k
∂
∂t
(
φ
B
)|i,j,k = ∆y∆zλn
i+1,j,k(
Pn+1
i+1,j,k − Pn+1
i,j,k
∆xi+1,j,k
+ γn
i+1,j,k
zi+1,j,k − zi,j,k
∆xi+1,j,k
)
+ ∆y∆zλn
i−1,j,k(
Pn+1
i−1,j,k − Pn+1
i,j,k
∆xi−1,j,k
+ γn
i−1,j,k
zi−1,j,k − zi,j,k
∆xi−1,j,k
)
+ ∆x∆zλn
i,j+1,k(
Pn+1
i,j+1,k − Pn+1
i,j,k
∆yi,j+1,k
+ γn
i,j+1,k
zi,j+1,k − zi,j,k
∆xi,j+1,k
)
+ ∆x∆zλn
i,j−1,k(
Pn+1
i,j−1,k − Pn+1
i,j,k
∆yi,j−1,k
+ γn
i,j−1,k
zi,j−1,k − zi,j,k
∆yi,j−1,k
)
+ ∆x∆yλn
i,j,k+1(
Pn+1
i,j,k+1 − Pn+1
i,j,k
∆zi,j,k+1
+ γn
i,j,k+1
zi,j,k+1 − zi,j,k
∆zi,j,k+1
)
+ ∆x∆yλn
i,j,k−1(
Pn+1
i,j,k−1 − Pn+1
i,j,k
∆zi,j,k−1
+ γn
i,j,k−1
zi,j,k−1 − zi,j,k
∆zi,j,k−1
) − qi,j,k
We will now define our total transmissibility term, T. The total transmissibility is made
up of two terms; geometric transmissibility (Tg) and flow transmissibility (Tf ). This is
expressed as follows:
T = 0.00633TgTf
1. In x-direction
Ti+1,j,k = 0.00633
∆y∆z
∆xi+1,j,k
λi+1,j,k
Ti−1,j,k = 0.00633
∆y∆z
∆xi−1,j,k
λi−1,j,k
2. In y-direction
Ti,j+1,k = 0.00633
∆x∆z
∆yi,j+1,k
λi,j+1,k
Ti,j−1,k = 0.00633
∆x∆z
∆yi,j−1,k
λi,j−1,k
3. In z-direction
Ti,j,k+1 = 0.00633
∆x∆y
∆zi,j,k+1
λi,j,k+1
4
5. Ti,j,k−1 = 0.00633
∆x∆y
∆zi,j,k−1
λi,j,k−1
The flow transmissibility term is defined by the mobility (λ) while the geometric term
is defined by the other grid block dimensions.
The harmonic average of the flow transmissibility for each grid block is used, in order
to obtain more accurate results. This is given by:
1. In x- direction
Tf i+1,j,k =
Tf i+1,j,kTf i,j,k(
∆xi,j,k
2
+
∆xi+1,j,k
2
)
Tf i+1,j,k
∆xi,j,k
2
+
Tf i,j,k
∆xi+1,j,k
2
Tf i+1,j,k =
Tf i+1,j,kTf i,j,k(∆xi,j,k + ∆xi+1,j,k)
Tf i+1,j,k∆xi,j,k + Tf i,j,k∆xi+1,j,k
Tf i−1,j,k =
Tf i−1,j,kTf i,j,k(
∆xi,j,k
2
+
∆xi−1,j,k
2
)
Tf i−1,j,k
∆xi,j,k
2
+
Tf i,j,k
∆xi−1,j,k
2
Tf i−1,j,k =
Tf i−1,j,kTf i,j,k(∆xi,j,k + ∆xi−1,j,k)
Tf i−1,j,k∆xi,j,k + Tf i,j,k∆xi−1,j,k
2. In y-direction
Tf i,j+1,k =
Tf i,j+1,kTf i,j,k(
∆yi,j,k
2
+
∆yi,j+1,k
2
)
Tf i,j+1,k
∆yi,j,k
2
+
Tf i,j,k
∆yi,j+1,k
2
Tf i,j+1,k =
Tf i,j+1,kTf i,j,k(∆yi,j,k + ∆yi,j+1,k)
Tf i,j+1,k∆yi,j,k + Tf i,j,k∆yi,j+1,k
Tf i,j−1,k =
Tf i,j−1,kTf i,j,k(
∆yi,j,k
2
+
∆yi,j−1,k
2
)
Tf i,j−1,k
∆yi,j,k
2
+
Tf i,j,k
∆yi,j−1,k
2
Tf i,j−1,k =
Tf i,j−1,kTf i,j,k(∆yi,j,k + ∆yi,j−1,k)
Tf i,j−1,k∆yi,j,k + Tf i,j,k∆yi,j−1,k
3. In z-direction
Tf i,j,k+1 =
Tf i,j,k+1Tf i,j,k(
∆zi,j,k
2
+
∆zi,j,k+1
2
)
Tf i,j,k+1
∆zi,j,k
2
+
Tf i,j,k
∆zi,j,k+1
2
Tf i,j,k+1 =
Tf i,j,k+1Tf i,j,k(∆zi,j,k + ∆zi,j,k+1)
Tf i,j,k+1∆zi,j,k + Tf i,j,k∆zi,j,k+1
5
6. Tf i,j,k−1 =
Tf i,j,k−1Tf i,j,k(
∆zi,j,k
2
+
∆zi,j,k−1
2
)
Tf i,j,k−1
∆zi,j,k
2
+
Tf i,j,k
∆zi,j,k−1
2
Tf i,j,k−1 =
Tf i,j,k−1Tf i,j,k(∆zi,j,k + ∆zi,j,k−1)
Tf i,j,k−1∆zi,j,k + Tf i,j,k∆zi,j,k−1
So now we define the RHS of our FDE in terms of transmissibility which becomes:
Vi,j,k
∂
∂t
(
φ
B
)|i,j,k = Tn
i+1,j,k(Pn+1
i+1,j,k − Pn+1
i,j,k ) + γn
i+1,j,kTn
i+1,j,k(zi+1,j,k − zi,j,k)
+ Tn
i−1,j,k(Pn+1
i−1,j,k − Pn+1
i,j,k ) + γn
i−1,j,kTn
i−1,j,k(zi−1,j,k − zi,j,k)
+ Tn
i,j+1,k(Pn+1
i,j+1,k − Pn+1
i,j,k ) + γn
i,j+1,kTn
i,j+1,k(zi,j+1,k − zi,j,k)
+ Tn
i,j−1,k(Pn+1
i,j−1,k − Pn+1
i,j,k ) + γn
i,j−1,kTn
i,j−1,k(zi,j−1,k − zi,j,k)
+ Tn
i,j,k+1(Pn+1
i,j,k+1 − Pn+1
i,j,k ) + γn
i,j,k+1Tn
i,j,k+1(zi,j,k+1 − zi,j,k)
+ Tn
i,j,k−1(Pn+1
i,j,k−1 − Pn+1
i,j,k ) + γn
i,j,k−1Tn
i,j,k−1(zi,j,k−1 − zi,j,k) − qi,j,k
Now we tackle the LHS of the FDE which deals with temporal discretization. Since oil
is a slightly compressible fluid, the rate of accumulation term containing our formation
volume factor and porosity becomes
φ
B
=
Bo
exp[−Co(P − Po
)]
φoexp(−Cr(P − Po)]
=
φo
Bo
exp[Cr(P − Po
) + Co(P − Po
)]
Using a taylor expansion series, the above equation simplifies to
φ
B
=
φo
Bo
exp[1 + Cr(P − Po
) + Co(P − Po
)]
∂
∂t
(
φ
B
) =
φo
Bo
(Cr
∂P
∂t
+ Co
∂P
∂t
) =
φo
Bo
(Ct
∂P
∂t
)
Now our FDE becomes
6
7. Vi,j,k
φo
Bo
(Ct
∂P
∂t
) = Tn
i+1,j,k(Pn+1
i+1,j,k − Pn+1
i,j,k ) + γn
i+1,j,kTn
i+1,j,k(zi+1,j,k − zi,j,k)
+ Tn
i−1,j,k(Pn+1
i−1,j,k − Pn+1
i,j,k ) + γn
i−1,j,kTn
i−1,j,k(zi−1,j,k − zi,j,k)
+ Tn
i,j+1,k(Pn+1
i,j+1,k − Pn+1
i,j,k ) + γn
i,j+1,kTn
i,j+1,k(zi,j+1,k − zi,j,k)
+ Tn
i,j−1,k(Pn+1
i,j−1,k − Pn+1
i,j,k ) + γn
i,j−1,kTn
i,j−1,k(zi,j−1,k − zi,j,k)
+ Tn
i,j,k+1(Pn+1
i,j,k+1 − Pn+1
i,j,k ) + γn
i,j,k+1Tn
i,j,k+1(zi,j,k+1 − zi,j,k)
+ Tn
i,j,k−1(Pn+1
i,j,k−1 − Pn+1
i,j,k ) + γn
i,j,k−1Tn
i,j,k−1(zi,j,k−1 − zi,j,k) − qi,j,k
Now the LHS is ready to be discretized which becomes
Vi,j,k
φo
Bo
(Ct
Pn+1
i,j,k − Pn
i,j,k
∆t
) = Tn
i+1,j,k(Pn+1
i+1,j,k − Pn+1
i,j,k ) + γn
i+1,j,kTn
i+1,j,k(zi+1,j,k − zi,j,k)
+ Tn
i−1,j,k(Pn+1
i−1,j,k − Pn+1
i,j,k ) + γn
i−1,j,kTn
i−1,j,k(zi−1,j,k − zi,j,k)
+ Tn
i,j+1,k(Pn+1
i,j+1,k − Pn+1
i,j,k ) + γn
i,j+1,kTn
i,j+1,k(zi,j+1,k − zi,j,k)
+ Tn
i,j−1,k(Pn+1
i,j−1,k − Pn+1
i,j,k ) + γn
i,j−1,kTn
i,j−1,k(zi,j−1,k − zi,j,k)
+ Tn
i,j,k+1(Pn+1
i,j,k+1 − Pn+1
i,j,k ) + γn
i,j,k+1Tn
i,j,k+1(zi,j,k+1 − zi,j,k)
+ Tn
i,j,k−1(Pn+1
i,j,k−1 − Pn+1
i,j,k ) + γn
i,j,k−1Tn
i,j,k−1(zi,j,k−1 − zi,j,k) − qi,j,k
In the x and y direction, ∆Z = 0, therefore
Vi,j,k
φo
Bo
(Ct
Pn+1
i,j,k − Pn
i,j,k
∆t
) = Tn
i+1,j,k(Pn+1
i+1,j,k − Pn+1
i,j,k ) + Tn
i−1,j,k(Pn+1
i−1,j,k − Pn+1
i,j,k )
+ Tn
i,j+1,k(Pn+1
i,j+1,k − Pn+1
i,j,k ) + Tn
i,j−1,k(Pn+1
i,j−1,k − Pn+1
i,j,k )
+ Tn
i,j,k+1(Pn+1
i,j,k+1 − Pn+1
i,j,k ) + γn
i,j,k+1Tn
i,j,k+1(zi,j,k+1 − zi,j,k)
+ Tn
i,j,k−1(Pn+1
i,j,k−1 − Pn+1
i,j,k ) + γn
i,j,k−1Tn
i,j,k−1(zi,j,k−1 − zi,j,k) − qi,j,k
where
Dn
= Vi,j,k
φo
Bo
Ct
Gn
= Σγn
Tn
∆Z
For the pressure gradient for each block we use an arithmetic average for more accurate
results. Therefore
γi,j,k−1 =
γi,j,k−1 + γi,j,k
2
γi,j,k+1 =
γi,j,k+1 + γi,j,k
2
The final general form of our FDE with the lagging coefficients becomes
ΣTn
(∆Pn+1
) = Dn
Pn+1
i,j,k − Pn
i,j,k
∆t
+ Gn
− qi,j,k
7
8. The lagging coefficients are Dn
, Tn
and Gn
because they are computed using the cur-
rent pressure values (Pn
) at a particular time step and serve as coefficients for calculat-
ing our unknown pressure values (Pn+1
). In this study we consider Dn
to be constant
since compressibility and porosity are not pressure dependent.
For a grid block containing from which a well is not producing from, q = 0, however,
for a grid block from which a well is producing from, the flow rate for constant well
pressure production as is the case in this simulation study is defined as follows:
q(
ft3
day
) = 0.00633WIi,j,kλn
i,j,k ∗ (Pn+1
i,j,k − Pwf )
Well Index : WI =
2π∆Z
ln( re
rw
) + skin
For an isotropic reservoir, drainage radius is defined as:
re = 0.208∆X
Since this is radial flow to a vertical well,
λn
i,j,k = λn
xi,j,k
6 Simulation Studies Conducted
We conduct three simulation studies which variable skin values to assess skin damage
on oil production and the necessity for well stimulation
6.1 Standard Case: No Damage or Stimulation
In the first simulation study, we consider the case where there is no damage (skin = 0).
In the figure below we have our production rate vs time.
8
9. 0 20 40 60 80 100 120
Time,days
10-2
10-1
100
101
102
103
104
Production
Rate,
ft
3
/day
Standard Case: Oil Production Rate vs. Time
We also have the reservoir pressure distribution at 30, 60, 90 and 120 days of produc-
tion in the following figures.
9
12. 6.2 Damaged Well Case
In the second simulation study, we consider the case where there is damage (skin = 3).
In the figure below we have our production rate vs time.
0 20 40 60 80 100 120
Time,days
10-1
100
101
102
103
104
Production
Rate,
ft
3
/day
Damaged Well Case: Oil Production Rate vs. Time
We also have the reservoir pressure distribution at 30, 60, 90 and 120 days of produc-
tion in the following figures.
12
14. 6.3 Stimulated Well Case
In the third simulation study, we consider the case where with well stimulation (skin =
-1).
In the figure below we have our production rate vs time.
0 20 40 60 80 100 120
Time,days
10-2
10-1
100
101
102
103
104
105
Production
Rate,
ft
3
/day
Stimulated Well Case: Oil Production Rate vs. Time
14
15. We also have the reservoir pressure distribution at 30, 60, 90 and 120 days of produc-
tion in the following figures.
15
17. 7 Conclusion
The stimulated well has the highest initial production rate of 13456 ft3
day
followed by the
well with no damage, with an initial production rate of 8785 ft3
day
and then the damaged
well, with an initial production rate of 4303 ft3
day
. The order is in line with our expecta-
tion.
With the initial production rate of the stimulated well approximately 53% greater than
the well with no damage and approximately 212% greater than the well with damage,
well stimulation is justified. Also we note that skin damage can have a significant im-
pact on our well production.
17