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# Transient analysis mathematics

This page introduces 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 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. This page briefly discusses the statement of the IBVP for transient fluid flow in porous media.

## Equations of transient fluid flow in porous media

In essence, fluid motion in porous media can be specified by the knowledge of the velocity vector, , and the density of the fluid, *ρ*, as a function of the position (*x*, *y*, *z*) and time, *t*; that is, = (*x*, *y*, *z*, *t*) and *ρ*= *ρ* (*x*, *y*, *z*, *t*). Relative to the fixed Cartesian axes, the velocity vector can be written as

where *v*_{x}, *v*_{y}, and *v*_{z} are the velocity components, and , , and are the unit vectors in the *x*, *y*, and *z* directions, respectively.

The physical law governing the macroscopic fluid-flow phenomena in porous media is the conservation of mass, which states that mass is neither created nor destroyed. The mathematical formula of this rule is developed by considering the flow through a fixed arbitrary closed surface, Γ, lying entirely within a porous medium of porosity, *Φ*, which is filled with a fluid of viscosity *μ*. **Fig. 1** illustrates an arbitrary closed surface in porous medium.

The conservation of mass principle requires that the difference between the rates at which fluid enters and leaves the volume through its surface must equal the rate at which mass accumulates within the volume. The total mass within the volume at any time is given by

Then, the time rate of change of mass within Γ is

which, by the conservation of mass law, is equal to the rate at which mass enters *V* through the surface.

Consider the differential surface element, dΓ, shown in **Fig. 1**. The mass entering the volume through dΓ at the normal velocity, , in a time increment, Δ*t*, is , and the total mass of the fluid passing through Γ during Δ*t* is

The surface integral in **Eq. 4** accounts for both influx and outflux through the surface of the volume; that is, Δ*M*_{g} is the difference between the masses entering and leaving the control volume during the time increment, Δ*t*. Then, the mass rate entering the volume, *V*, through its surface, Γ, can be written as

By the principle of conservation of mass, equating the right sides of **Eqs. 3** and **5** yields

A more useful relation is found with the divergence theorem, which states that the flux of through the closed surface, Γ, is identical to the volume integral of (the divergence of ) taken throughout *V*; that is,

Here, ∇ is the gradient operator, which in 3D Cartesian and cylindrical coordinates is given, respectively, by

and

With the relation in **Eq. 7**, **Eq. 6** can be recast into

If the functions involved in the argument of the integral in **Eq. 10** are continuous, then the integral is identically zero if and only if its argument is zero (because the volume integral in **Eq. 10** is identically zero for any arbitrarily chosen volume). Then, the following continuity equation can be obtained.

**Eq. 11** is a PDE that is equivalent to the statement of the conservation of mass for fluid flow in porous media. For practical purposes, however, **Eq. 11** is expressed in terms of pressure because density and velocity cannot be measured directly. To express density, *ρ*, and velocity, , in terms of pressure, we use an equation of state and a flux law, known as Darcy’s law, respectively.

The following definition of isothermal fluid compressibility, *c*, is a useful equation of state that relates density to pressure.

If *c* is a constant (the compressibility of many reservoir liquids may be considered as constant), then **Eq. 12** can be integrated to yield

where subscript 0 indicates the conditions at the datum. Similarly, the compressibility of the porous rock, *c*_{f}, is defined by

and the total system compressibility, *c*_{t}, is given by

These definitions of compressibility help recast **Eq. 11** in terms of pressure.

Darcy’s law for fluid flow in porous media is a flux law. Neglecting the gravity effect, it is expressed by

In **Eq. 16**, *μ* is the viscosity of the fluid, and *k* is the permeability tensor of the formation given by

where *α*, *β*, and *γ* are the directions, and *k*_{ij} is the permeability in the *i* direction as a result of the pressure gradient in the *j* direction.

If **Eqs. 13** through **16** are used in **Eq. 11**, an alternative statement of the conservation of mass principle for fluid flow in porous media is obtained:

**Eq. 18** is the PDE that governs transient fluid flow in porous media. In the present form, **Eq. 18** is not very useful in obtaining practical solutions because of the nonlinearity displayed in the second term of the left side. For liquid flow, the viscosity, *μ*, is constant and **Eq. 18** can be linearized by assuming that the pressure gradients, ∇*p*, are small in the reservoir and the compressibility of the reservoir liquids, *c*, is on the order of 10^{−5} or smaller. Then, the second term of the left side of **Eq. 18** may be neglected compared with the remaining terms and the following linear expression is obtained:

**Eq. 19** (or **Eq. 18**) is known as the diffusivity equation. As an example in Cartesian coordinates, assuming that the coordinate axes can be chosen in the directions of the principal permeabilities, *k*, in **Eq. 19**, may be represented by the following diagonal tensor:

Then, **Eq. 19** may be written

If each coordinate, *j* = *x*, *y*, or *z*, is multiplied by , where *k* may be chosen arbitrarily (to preserve the material balance, *k* is usually chosen to be ), **Eq. 21** may be transformed into the diffusion equation for an isotropic domain:

where *η* is the diffusivity constant defined by

If the same transformation is also applied to the boundary conditions (see the "Initial and boundary conditions" section below), the problems in anisotropic reservoirs may be transformed into those in isotropic reservoirs provided that the system is infinite or bounded by planes perpendicular to the principal axes of permeability. In all other cases, this transformation distorts the bounding surfaces.

For the flow of gases, the assumptions of small fluid compressibility and pressure gradient may not be appropriate and the *c*(∇*p*)^{2} term in **Eq. 18** may not be negligible. In these cases, an expression similar to **Eq. 21** may be obtained from **Eq. 18** in terms of pseudopressure, *m*, as

Here, the pseudopressure is defined by^{[1]}

where *Z* is the compressibility factor. To define a complete physical problem, **Eq. 21** (or **24**) should be subject to the initial and boundary conditions discussed below.

### Initial and boundary conditions

The solution of the diffusivity equation (**Eq. 19**) should satisfy the initial condition in the porous medium. The initial condition is normally expressed in terms of a known pressure distribution at time zero; that is,

The most common initial condition is the uniform pressure distribution in the entire porous medium; that is, *f* (*x*, *y*, *z*) = *p*_{i}.

The boundary conditions are specified at the inner (wellbore) and outer boundaries of the reservoir. These are usually in the form of prescribed flux or pressure at the boundaries. The condition of prescribed flux can be formulated as

where Γ is the surface of the boundary, and *n* indicates the outward normal direction of the boundary surface. The prescribed flux condition may be used at the inner and outer boundaries of the reservoir. The most common use of the prescribed flux condition at the inner boundary is for the production at a constant rate. In this case, the function, *g*(*t*), is related to a constant production rate, *q*. At the outer boundary, the prescribed flux condition is usually used to indicate impermeable boundaries [*g*(*t*)=0] and leads to a pseudosteady state under the influence of boundaries.

For some applications, pressure may be specified at the inner and outer boundaries. In this case,

When used at the inner boundary, this condition represents production at a constant pressure, *p*_{wf}; that is, *h*(*t*) = *p*_{wf}. At the outer boundary, specified pressure, *p*_{e}, is usually a result of injection or influx from an adjacent aquifer, which usually leads to steady state in the reservoir.

It is also possible to have boundary conditions of mixed type. These usually correspond to interface conditions in porous media. Raghavan^{[2]} contains more details about the common boundary conditions for the diffusion equation.

### Assumptions and limits

Some assumptions have been made in the derivation of the diffusivity equation given by **Eq. 19**. These assumptions determine the limits of applicability of the solutions obtained from **Eq. 19**. One of the most important assumptions involved is the continuity of the properties involved in **Eq. 19**. (This was required to obtain **Eq. 19** from the more general integral form in **Eq. 10**.) Therefore, sharp changes in the properties of the reservoir rock and fluid (such as faults and fluid banks) should be incorporated in the form of boundary or interface conditions in the solution of **Eq. 19**.

The second important assumption is that Darcy’s law describes the flux in porous media. This assumption is valid at relatively low fluid velocities that may be appropriate to describe liquid flow. At high velocities (when Reynolds number based on average sand grain diameter approaches unity) such as those observed in gas reservoirs, Darcy’s law is not valid.^{[3]} In this case, Forchheimer’s equation,^{[4]} which accounts for the inertial effects, should be used. In petroleum engineering, it is a common practice to consider the additional pressure drop as a result of non-Darcy flow in the form of a pseudoskin because it is usually effective in a small vicinity of the wellbore.

## Mathematical tools used for solving transient analysis problems

These tools include:

- Bessel functions in transient analysis
- Laplace transformation for solving transient flow problems
- Green’s function for solving transient flow problems

## Nomenclature

## References

- ↑ Al-Hussainy, R., Ramey Jr., H.J., and Crawford, P.B. 1966. The Flow of Real Gases Through Porous Media. J Pet Technol 18 (5): 624–636. SPE-1243-A-PA. http://dx.doi.org/10.2118/1243-A-PA
- ↑ Raghavan, R. 1993. Well Test Analysis, 28–31, 336–435. Englewood Cliffs, New Jersey: Petroleum Engineering Series, Prentice-Hall.
- ↑ Fancher, G.H., Lewis, J.A., and Barnes, K.B. 1933. Some Physical Characteristics of Oil Sands. Bulletin 12, Mineral Industries Experimental Station, Pennsylvania State University, University Park, Pennsylvania, 65–167.
- ↑ Forchheimer, P.F. 1901. Wasserbewegung durch Boden. Zeitschrift des Vereines deutscher Ingenieure 45 (5): 1781–1788.

## Additional references

Matthews, C. S., & Russell. D. G. 1967. Pressure Buildup and Flow Tests in Wells, SPE Monograph Series Vol. 1, Society of Petroleum Engineers, Richardson, TX, 163 pp.

Van Everdingen, A. F., & Hurst, W. (1949, December 1). The Application of the Laplace Transformation to Flow Problems in Reservoirs. Society of Petroleum Engineers. doi:10.2118/949305-G

## External links

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## See also

Bessel functions in transient analysis

Laplace transformation for solving transient flow problems

Green’s function for solving transient flow problems

Source function solutions of the diffusion equation

Solving unsteady flow problems with Green's and source functions

Solving unsteady flow problems with Laplace transform and source functions