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The salient significance of the solution of radial diffusivity equation to well testing analysis done in oil and gas industry cannot be over-emphasized. Varieties of solutions have been proposed to the radial diffusivity equation, of which the Van Everdingen-Hurst constant terminal rate solution is the most widely accepted and the others are approximate solution having their respective limitations. The main objective of this project, being its first application to oil and gas industry, is to use a new mathematical technique, the homotopy analysis method (HAM) to solve the radial diffusivity equation for slightly compressible fluid. In Using HAM, the Boltzmann transformation method was used to transform the radial PDE to ODE, then a homotopy series was then constructed for the new equation with the linear boundary condition from the original radial diffusivity equation of slightly compressible fluid and the final equation then solved using computation software Maple. The result gotten reveals that the homotopy analysis method gives good results compared to the Van Everdingen and Hurst Solution (Exact solution) and thus proves to be very effective, simple, and accurate when compared to other form of solutions. Hence from the results gotten, Homotopy Analysis Method can therefore be applied in solving other non-linear equations in the petroleum engineering field since it is simple and accurate.

Nonlinear partial differential equations are useful in describing the various phenomena in many disciplines. Apart from a limited number of these problems, most of them do not have a precise analytical solution, so these nonlinear equations should be solved using approximate methods. In 1992, Shijun Liao employed the basic ideas of the homotopy in topology to propose a general analytic method for nonlinear problems, namely homotopy analysis method (HAM) and then modified it [

The Radial Diffusivity Equation is considered one of the most salient and widely used mathematical expressions in the Oil and Gas Industry. The equation is particularly applied to the analysis of well testing data. The use of this equation is to determine salient information or parameters during the testing analysis of a well and they include; Initial pressure (p), permeability (k), mechanical skin factor (s), flow efficiency, productivity index etc. The equation is also used to investigate reservoir productivity.

To develop the radial diffusivity equation, some assumptions are assumed. These are:

・ Formation is a homogenous and isotropic porous media of uniform thickness.

・ Negligible gravity effect.

・ Applicability of Darcy’s law.

・ Pressure independent rock properties.

・ Central well is perforated across the entire formation thickness. Hence; radial flow.

・ Single phase flow is present in the reservoir.

・ Small pressure gradient.

These assumptions are needed to combine the three (3) governing equations needed in deriving the radial diffusivity equation:

・ The law of conservation of mass.

・ Darcy’s law.

・ Equation of state.

We consider a radial flow of fluid towards a well in a circular reservoir. Combining the conservation of mass and Darcy’s law for the isothermal flow of fluids of small and constant compressibility, a partial differential equation is obtained that in field units simplifies to

where it is assumed that compressibility, c, is small and independent of pressure, permeability, k, is constant and isotropic; viscosity, µ, is independent of pressure, porosity ɸ, is constant. The above equation is called the

non-linear radial diffusivity equation and the term

quently denoted by η.

Re-writing Equation (1) in terms of dimensionless variables, the following salient points are established:

1) The dimensionless radius,

2) The dimensionless pressure,

3) The initial condition:

4) The constant rate inner boundary condition:

As stated above, the definition of the dimensionless radius is given as

and

And dimensionless time is given as:

Therefore, the Dimensionless radial diffusivity equation is given as:

When

The above Equation (6) is the linear dimensionless radial diffusivity equation.

There are four solutions to Equation (6) (when

1) Van Everdingen-Hurst constant terminal rate solution for bounded cylindrical reservoir.

2) The solution for an infinite reservoir with a well-considered to be a line source with zero wellbore radius.

3) The pseudo-steady state solution.

4) The solution that includes wellbore storage for a well in an infinite reservoir.

a) Van Everdingen-Hurst Constant Terminal rate solution for Bounded cylindrical reservoir

Van Everdingen and Hurst developed two set of solution for the linear radial diffusivity equation namely, “the constant terminal pressure case solution” and “the constant terminal rate case”. In the constant terminal pressure case, the pressure at the terminal boundary is lowered by unity at zero time, kept constant thereafter, and the cumulative amount of fluid flowing across the boundary is computed as a function of the time. The constant terminal rate solution is explained explicitly below.

Constant Terminal Rate SolutionIt is solutions of the radial diffusivity Equation (1) when

In summary, Van Everdingen and Hurst solution to the Equation (6) requires two boundary conditions and an initial condition. A realistic and practical solution is obtained if we assume that

1) A well produces a constant rate, qB, into the wellbore (q refers to flow rate in STB/D at surface conditions, and B is the formation volume factor in RB/STB);

2) The well, with wellbore radius,

3) Before production begins, the reservoir is at uniform pressure

The most useful form of the desired solution relates flowing pressure,

The solution after applying the above initial and boundary conditions is expressed as follows;

This solution is considered the exact solution to the radial diffusivity equation, but approximate solutions are used instead of this exact solution because of the cumbersomeness of the equation.

b) Pseudo-steady state solution for Bounded Cylindrical Reservoir

The solution here is simply a limiting form of the Van Everdingen-Hurst constant terminal rate solution (the exact solution), where the summation involving exponentials and Bessel functions is considered negligible.

The solution proposed here to the linear radial diffusivity equation is expressed as;

c) Solution for an infinite Reservoir with Line source well

Assumptions made here are that;

1) A well produces at a constant rate, qB;

2) The well has zero radius;

3) The reservoir is at uniform pressure, pi, before production begins; and

4) The well drains an infinite area (i.e., that p → p_{i} as r → ∞).

Under these conditions, the solution to the radial diffusivity equation is;

To describe the main idea of Homotopy Analysis Method, the following differential equations are considered:

Let U be a function of homotopy parameter q. Then,

where N is a non-linear operator, and

The Construction of homotopy takes the form:

where

In deriving the zeroth order deformation equation we set the homotopy Equation (10) equal to zero,

whose solution transforms continuously with respect to q.

When q = 0,

Similarly when q = 1,

Thus, as q varies from 0 to 1, the solution

The homotopy series is gotten by expanding

where

If the auxiliary linear operator, the initial guess, auxiliary parameter and auxiliary function are properly chosen, the series Equation (16) converges at q = 1 and we get homotopy series solution;

which must be one of the solution of the original non linear equation

For

Differentiating Equation (11) m times with respect to the embedding parameter q and then setting q = 0 and finally dividing them by m!, we have the so-called mth-order deformtion equation:

where

and

It is of paramount importance to emphasize that

Also, transforming Equation (5) using Boltzmann transformation to ordinary differential equation for ease in solving, we have:

Equation (5) then becomes;

Initial condition

Boundary conditions

The Construction of homotopy for Equation (25) takes the form:

where

In deriving the zeroth order deformation equation we set the homotopy Equation (28) equal to zero,

whose solution transforms continuously with respect to q.

When q = 0 ,

Similarly when q = 1,

Thus, as q varies from 0 to 1, the solution

The homotopy series is gotten by expanding

where

If the auxiliary linear operator, the initial guess, auxiliary parameter and auxiliary function are properly chosen, the series Equation (3.13) converges at q = 1 and we get homotopy series solution;

which must be one of the solution of the original non linear equation

For

Therefore from Equation (7), the mth order deformation equation is derived as;

The solution of the mth-order deformation Equation (37) is given as:

According to the rule of solution expression Equation (39), the auxilliary function

In this way, we derive

The analytical solution of the nonlinear diffusivity Equation (25) is obtained by using the Homotopy Analysis Method (HAM). The convergence parameter values for which the solution converges is shown in

As long as we choose h in this horizontal region

Evaluating _{D} = 1).

Homotopy Analysis Method solution was also used in solving for dimensionless pressure P_{D} for finite radial

system with closed exterior boundary, constant rate at inner boundary (

t_{D} | Van Everdigen and Hurst Exact Solution for (P_{D}) | Homotopy Analysis Method (P_{D}) | Absolute Error |
---|---|---|---|

1500 | 4.061 | 4.000 | 0.061 |

2000 | 4.205 | 4.200 | 0.005 |

2500 | 4.317 | 4.300 | 0.017 |

3500 | 4.498 | 4.478 | 0.020 |

4000 | 4.552 | 4.548 | 0.004 |

5000 | 4.663 | 4.655 | 0.008 |

6000 | 4.754 | 4.750 | 0.004 |

7000 | 4.829 | 4.820 | 0.009 |

9000 | 4.949 | 5.042 | 0.093 |

10,000 | 4.996 | 5.100 | 0.104 |

12,000 | 5.072 | 5.180 | 0.108 |

From _{D} = 200.

Furthermore, Homotopy Analysis Method solution gotten for the radial diffusivity equation was applied in getting the values of P_{D} for respective Dmensionless time t_{D} values for r_{D} = 300 for finite radial system with closed exteriorboundary and constant rate at inner boundary.

Van Everdigen and Hurst solutions against various dimensionless time t_{D} values at r_{D} = 300 and their respective absolute errors.

Also, from _{D} = 300.

It is important to note that

t_{D} | Van Everdigen and Hurst Exact Solution for (P_{D}) | Homotopy Analysis Method (P_{D}) | Absolute Error |
---|---|---|---|

6000 | 4.754 | 4.632 | 0.122 |

8000 | 4.898 | 4.784 | 0.114 |

10,000 | 5.010 | 4.970 | 0.040 |

12,000 | 5.101 | 5.070 | 0.031 |

14,000 | 5.177 | 5.175 | 0.002 |

16,000 | 5.242 | 5.223 | 0.019 |

18,000 | 5.299 | 5.282 | 0.017 |

20,000 | 5.348 | 5.350 | 0.002 |

24,000 | 5.429 | 5.422 | 0.007 |

28,000 | 5.491 | 5.500 | 0.009 |

30,000 | 5.517 | 5.540 | 0.023 |

40,000 | 5.606 | 5.841 | 0.235 |

50,000 | 5.652 | 5.952 | 0.300 |

of the HAM in solving approximately the non-linear flow problem in the reservoir is evident. Also, Figures 2-4 show the accuracy of the Homotopy Analysis method when compared to the Van Everdingen and Hurst solution when the radial diffusivity equation solution used was linearized.

From

In this work, an approximate but very accurate solution of a non-linear reservoir model with coefficient of square of pressure gradient Co was successfully obtained using a powerful analytic method called the homotopy analysis method. It was shown that:

1) The pressure distribution in the reservoir with coefficient of square of pressure gradient Co, can be accurately predicted with the HAM solution.

2) The freedom of choice of the auxiliary parameter h, gives way to adjust and control the convergence of the solution series, which can be considered as a fundamental difference between the homotopy analysis method and other existing methods such as the adomian decomposition method, homotopy perturbation method and variational iteration method.

3) The Homotopy Analysis method is a mathematical model that can be used to solve the non-linear radial diffusivity equation effectively.

It is then believed that the homotopy analysis method is an efficient and capable technique in handling a wide variety of engineering problems.

Olugbenga Adebanjo Falode,Victor Sunday Chukwunagolu, (2016) Homotopy Analysis Solution to Radial Diffusivity Equation of Slightly Compressible Fluid. Applied Mathematics,07,993-1004. doi: 10.4236/am.2016.79087