Infinite Acting Reservoir Solutions for Pressure Transient Analysis

Overview

The infinite acting reservoir model is the foundation of pressure transient analysis. It describes a well producing from a homogeneous reservoir that is large enough (or tested for short enough time) that boundary effects have not yet influenced the pressure response. This is the baseline against which all bounded reservoir solutions are compared.

The infinite acting period is characterized by:

• Radial flow around the wellbore
• Flat derivative at 0.5 on a log-log plot (after wellbore storage effects diminish)
• Semi-log straight line during the infinite acting radial flow period

Key analytical solutions include:

• Line Source Solution - The classical exponential integral solution
• Finite Wellbore Solution - Accounts for actual wellbore geometry
• Storage and Skin Solution - Includes wellbore storage and skin effects

Historical Context

The mathematical foundation was established by Van Everdingen and Hurst (1949), who applied Laplace transforms to solve the diffusivity equation. Stehfest (1970) later provided a practical numerical inversion algorithm that enabled computerized solutions. The introduction of pressure derivatives by Bourdet et al. (1983) revolutionized well test interpretation by enabling clear flow regime identification.

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Theory

The Diffusivity Equation

The radial diffusivity equation in dimensionless form governs pressure behavior:

$$\frac{\partial^2 p_D}{\partial r_D^2} + \frac{1}{r_D}\frac{\partial p_D}{\partial r_D} = \frac{\partial p_D}{\partial t_D}$$

For an infinite acting reservoir, the outer boundary condition is:

$$\lim_{r_D \to \infty} [p_D(r_D, t_D)] = 0$$

Line Source Solution

For large dimensionless time ($t_D > 25$), the line source solution provides an excellent approximation:

$$p_D(r_D, t_D) = -\frac{1}{2} \text{Ei}\left(-\frac{r_D^2}{4t_D}\right)$$

where Ei is the exponential integral function:

$$-\text{Ei}(-x) = \int_x^{\infty} \frac{e^{-u}}{u} du$$

At the wellbore ($r_D = 1$):

$$p_D(t_D) = -\frac{1}{2} \text{Ei}\left(-\frac{1}{4t_D}\right)$$

For $t_D > 100$, this can be approximated as:

$$p_D(t_D) \approx \frac{1}{2}\left[\ln(t_D) + 0.80907\right]$$

Exponential Integral Function

The exponential integral Ei(x) requires numerical evaluation. Petroleum Office uses the polynomial approximations from Abramowitz and Stegun:

For $0 < x \leq 1$:

$$\text{Ei}(x) + \ln(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + a_5 x^5$$

where:

• $a_0 = -0.57721566$
• $a_1 = 0.99999193$
• $a_2 = -0.24991055$
• $a_3 = 0.05519968$
• $a_4 = -0.00976004$
• $a_5 = 0.00107857$

For $x > 1$:

$$x \cdot e^x \cdot \text{Ei}(x) = \frac{x^4 + a_1 x^3 + a_2 x^2 + a_3 x + a_4}{x^4 + b_1 x^3 + b_2 x^2 + b_3 x + b_4}$$

where:

• $a_1 = 8.5733287401$, $a_2 = 18.0590169730$, $a_3 = 8.6347608925$, $a_4 = 0.2677737343$
• $b_1 = 9.5733223454$, $b_2 = 25.6329561486$, $b_3 = 21.0996530827$, $b_4 = 3.9584969228$

Wellbore Storage and Skin Effects

Real wells exhibit two additional phenomena:

Skin Effect: A thin zone around the wellbore with altered permeability causes an additional pressure drop:

$$p_{wD} = \left[p_D - S\left(\frac{\partial p_D}{\partial r_D}\right)\right]_{r_D=1}$$

where $S$ is the skin factor (positive for damage, negative for stimulation).

Wellbore Storage: Fluid stored in the wellbore causes early-time distortion:

$$C_D\frac{dp_{wD}}{dt_D} - \left[r_D\frac{\partial p_D}{\partial r_D}\right]_{r_D=1} = 1$$

Solution in Laplace Space

The complete solution including wellbore storage and skin is obtained in Laplace space:

With skin factor S:

$$\bar{p}_{wD}(u) = \frac{K_0(\sqrt{u}) + S\sqrt{u}K_1(\sqrt{u})}{u\left\{C_D u\left[K_0(\sqrt{u}) + S\sqrt{u}K_1(\sqrt{u})\right] + \sqrt{u}K_1(\sqrt{u})\right\}}$$

Without skin (S = 0):

$$\bar{p}_{wD}(u) = \frac{K_0(\sqrt{u/C_D})}{u\left\{\sqrt{u/C_D}K_1(\sqrt{u/C_D}) + uK_0(\sqrt{u/C_D})\right\}}$$

where $K_0$ and $K_1$ are modified Bessel functions of the second kind.

Stehfest Numerical Inversion

To convert from Laplace space back to real time, the Stehfest algorithm is used:

$$f(t) = \frac{\ln(2)}{t} \sum_{i=1}^{N} V_i \cdot \bar{f}(s_i)$$

where:

• $s_i = i \cdot \frac{\ln(2)}{t}$
• $N$ is typically 8, 10, or 12

The Stehfest coefficients $V_i$ are:

$$V_i = (-1)^{N/2+i} \sum_{k=\lfloor(i+1)/2\rfloor}^{\min(i, N/2)} \frac{k^{N/2} \cdot (2k)!}{(N/2-k)! \cdot k! \cdot (k-1)! \cdot (i-k)! \cdot (2k-i)!}$$

Modified Bessel Functions

The solutions require modified Bessel functions $K_0$, $K_1$, $I_0$, and $I_1$. These are computed using polynomial approximations from Abramowitz and Stegun.

Modified Bessel function $I_0(x)$:

For $|x| < 3.75$: $$I_0(x) = 1 + 3.5156229 y^2 + 3.0899424 y^4 + 1.2067492 y^6 + 0.2659732 y^8 + 0.0360768 y^{10} + 0.0045813 y^{12}$$

where $y = x/3.75$.

Modified Bessel function $K_0(x)$:

For $x \leq 2$: $$K_0(x) = -\ln(x/2) \cdot I_0(x) + (-0.57721566 + 0.4227842 y^2 + 0.23069756 y^4 + ...)$$

where $y = x/2$.

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Equations

Dimensionless Pressure at Any Radius

Line Source Solution (infinite homogeneous reservoir):

$$p_D(r_D, t_D) = -\frac{1}{2} \text{Ei}\left(-\frac{r_D^2}{4t_D}\right)$$

For practical purposes, this is valid when $t_D/r_D^2 > 25$.

Dimensionless Wellbore Pressure

Including wellbore storage ($C_D$) and skin ($S$):

The dimensionless wellbore pressure $p_{wD}$ is obtained by numerical inversion of the Laplace space solution using the Stehfest algorithm.

For infinite acting radial flow (after wellbore storage effects):

$$p_{wD} = \frac{1}{2}\left[\ln(t_D) + 0.80907 + 2S\right]$$

Dimensional Pressure

To convert from dimensionless to dimensional pressure:

$$\Delta p = p_i - p_{wf} = \frac{141.2 \, q \, B \, \mu}{k \, h} \cdot p_{wD}$$

At the wellbore after wellbore storage effects (oilfield units):

$$\Delta p = \frac{162.6 \, q \, B \, \mu}{k \, h}\left[\log(t) + \log\left(\frac{k}{\phi \mu c_t r_w^2}\right) - 3.23 + 0.87 S\right]$$

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Applicability & Limitations

Valid Range

Flow Regime Identification

End of Wellbore Storage

The end of pure wellbore storage effects occurs approximately at:

$$\left(\frac{t_D}{C_D}\right)_{end} \approx (60 + 3.5S) \cdot e^{0.14S}$$

For $S = 0$: wellbore storage ends at $t_D/C_D \approx 60$.

Onset of Boundary Effects

The infinite acting period ends when boundaries begin to affect the pressure response. This occurs approximately at:

$$t_D \approx 0.1 \left(\frac{L}{r_w}\right)^2$$

where $L$ is the distance to the nearest boundary.

Limitations

1. Homogeneous Reservoir: No variation in permeability, porosity, or thickness
2. Single-Phase Flow: Oil flow only; no gas or water mobility effects
3. Constant Properties: Fluid viscosity, compressibility, and FVF assumed constant
4. Radial Geometry: Vertical well, fully penetrating, radial flow
5. Infinite Boundaries: No boundary effects during the analysis period
6. Slightly Compressible Fluid: Small and constant compressibility

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Related Documentation

Prerequisite Concepts

• Dimensionless Variables - Definitions of $p_D$, $t_D$, $C_D$, $L_D$

Boundary Effects

• Bounded Reservoir Solutions - Single and multiple boundary effects
• Method of images for sealing and constant pressure boundaries

Type Curve Analysis

• Gringarten type curves for wellbore storage and skin
• Bourdet derivative type curves

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References

1. Van Everdingen, A.F. and Hurst, W. (1949). "The Application of the Laplace Transformation to Flow Problems in Reservoirs." Petroleum Transactions, AIME, 186: 305-324.

2. Stehfest, H. (1970). "Algorithm 368: Numerical Inversion of Laplace Transforms." Communications of the ACM, 13(1): 47-49.

3. Abramowitz, M. and Stegun, I.A. (1970). Handbook of Mathematical Functions. National Bureau of Standards, pp. 231, 378-379.

4. Bourdet, D., Whittle, T.M., Douglas, A.A., and Pirard, Y.M. (1983). "A New Set of Type Curves Simplifies Well Test Analysis." World Oil, May 1983, pp. 95-106.

5. Bourdet, D., Ayoub, J.A., and Pirard, Y.M. (1989). "Use of Pressure Derivative in Well-Test Interpretation." SPE Formation Evaluation, 4(2): 293-302. SPE-12777-PA.

6. Abass, E. and Song, C.L. (2012). "Computer Application on Well Test Mathematical Model Computation of Homogeneous and Multiple-Bounded Reservoirs." IJRRAS, 11(1): 41-52.

7. Horne, R.N. (1994). "Advances in Computer-Aided Well Test Interpretation." Journal of Petroleum Technology, 46(7): 599-605.

8. Lee, J. (1982). Well Testing. SPE Textbook Series, Vol. 1. Society of Petroleum Engineers.