Horner plot aids analysis in stress-sensitive reservoirs

T.A. Jelmert
Norwegian University of Science & Technology
Trondheim

H. Selseng
Saga Petroleum ASA
Oslo

• Equations [221,891 bytes]

To add insight into the behavior of stress-sensitive reservoirs, a plot of permeability vs. Horner time may be well-suited to visualize permeability development as a function of time.

Stress-sensitive reservoirs occur in many places in the world. The Rotliegendes sandstone group in Northwest Europe and some North Sea chalks are examples.

Theory

Many papers have discussed well testing of stress-sensitive reservoirs. A study by Kikani and Pedrosa ¹ is notable since they apply their method to real field data. Their method was based on the constant modulus to permeability assumption. Equation 1 (see equation box) shows the Matthews and Russell² definition of the permeability modulus, g, which characterizes the sensitivity of the permeability to pressure.

From this definition, a nonlinear diffusion equation with a quadratic gradient term was obtained. This term may be removed by the Cole-Hopf transformation.

An integration of Equation 1 under the assumption of a constant permeability modulus yields Equation 2, where Equation 3 defines k_n.

The inverse relationship of Equation 2 is the Cole-Hopf transformation (Equation 4).

Substitution of the Cole-Hopf transformation into the Matthews and Russell equation gives a diffusion equation in terms of normalized permeability change. It is still nonlinear because diffusivity depends on pressure.

Usually this term may be linearized by neglecting the pressure dependency.³ The diffusivity is then evaluated at the initial condition.

This simplification may be justified for many engineering calculations. Any solution that is based on this simplification is of zero-order accuracy. Such a solution may be improved by adding higher-order perturbations.¹

Pedrosa⁴ proposed a variation of the Cole-Hopf transformation, which is ideally suited for the method of perturbations.

The dimensionless permeability modulus (Equation 6) was used as a perturbation parameter.

Comparison of Equations 4 and 5 shows that the Pedrosa substitution is proportional to the normalized permeability change (Equation 7).

Pedrosa found that the zero-order, partial-differential equation (in terms U) and the boundary conditions were equivalent to the corresponding equations for a reservoir without stress sensitivity. As a consequence, an approximate solution for the transformed variable U is given by the line-source solution.

Based on Equation 7, the equivalent solution in terms of the normalized permeability change, Dk_n, becomes Equation 8.

The skin factor, S_k, is consistent with steady-state flow in a stress-sensitive reservoir. It may be used if the permeability modulus of the altered zone is the same as for the reservoir.⁵

The exact solution of the skin effect is calculated by Equation 8 because the steady-state diffusion equation in terms of Dk_n is linear.

An approximate buildup solution may be found by superposition.⁴

Note that the normalized permeability is independent of a possible steady-state skin factor.

Superposition of solutions that derives from a nonlinear equation is questionable. Simulation studies, however, show that this approach is warranted.⁷

Kikani and Pedrosa¹ showed that the permeability modulus might be obtained by type-curve matching. We use the same procedure.

Substitution of Equation 9 into Equation 4 yields Equation 10.

Equation 10 depends only on one parameter, g_D; hence it is convenient for graphical type-curve matching. For larger values of shut-in time, Dt_D, Equation 9 will simplify to a Horner-type equation.

Substitution of Equation 3 into Equation 11 yields the relationship between the normalized permeability and time, which is given by Equation 12.

This equation will be a straight line on a Horner plot. If the permeability modulus is known, this may be a convenient way of displaying the development of the normalized well bore permeability during a build-up test.

The straight line has a slope 1.15 gD and it will extrapolate to k_n = 1 when the Horner ratio is one.

Assumptions

This analysis depends on the assumption that the permeability is the dominant nonlinear term, which may be described by an exponential function. The last assumption has experimental support. ^{3 6} Reference 3 presents data that illustrate the point ( Table 1 [52,092 bytes]). The permeability data in Table 1 correlate with an exponential function of pressure. From Equation 4, one obtains Equation 13, which will plot as a straight line on D p vs. ln k graph. Hence, linear regression is possible. The quality of a linear regression is measured by the correlation coefficient that in this case is r = -0.99. The calculation may be verified on a hand calculator.

A correlation coefficient, p =± 1, corresponds to a perfect correlation.

The data set in Table 1 has been used to compare analytical against numerical solutions.^{3 7} It was found that the analytical solution in terms of the transformed variable correlates with the solution of a slightly compressible fluid.

References 3 and 7 use the Kirchhoff transformation. The resulting differential equation is of zero-order accuracy. It may be shown that the Kirchhoff and Cole-Hopf transformation are equivalent when the permeability is described by an exponential function of pressure.⁵

For convenience, the equation box contains the argument (Equations 14-16).

This implies that conclusions from simulation studies on the properties of the Kirchhoff transformation are valid also for the Cole-Hopf transformation.

Samaniego and Cinco-Ley⁷ conducted extensive simulation studies based on the Raghavan, et al., data. Their study supports the assumption that superposition is acceptable.

Case study

The field case published by Kikani and Pedrosa ¹ provides the test case. The results obtained by our zero-order model is shown in Fig. 1 [28,893 bytes]. The dimensionless field curve obtained by Kikani and Pedrosa was digitized and plotted on the same graph. As can be seen, there is a reasonably good match for a dimensionless permeability modulus of value gD = 0.25, which is the same value as found by Kikani and Pedrosa using their second-order perturbation technique.

Their technique is slightly more accurate.

The pressure data from the well test were converted to the equivalent permeability data with Equation 2, and Fig. 2 [40,367 bytes] plots the results.

The graph shows that the permeability at the well bore has increased three times during this particular build-up test.

The same plot shows the theoretical Horner equation for permeability (Equation 12).

This plot is a convenient way for visualizing the behavior of stress-sensitive reservoirs. It is certainly more instructive than the pressure plot shown in Fig. 1.

In the present analysis, we used a simplified analytical solution of zero-order accuracy to derive the type curve to determine the dimensionless permeability modulus. A more complex solution technique may be used, for instance, the second-order perturbation technique of Kikani and Pedrosa¹ or a numerical model.

Horner time vs. permeability may then be plotted as a result of the more-refined analysis. But there is a possibility that the theoretical response could deviate slightly from a straight line. This has minor importance because the purpose of the plot is to display the development of the normalized permeability at the well bore as a function of time.

References

1. Kikani, J., and Pedrosa, O.A.,"Perturbation Analysis of Stress-Sensitive Reservoirs," SPEFE, September 1991, pp. 379-86.
2. Matthews, C.S., and Russell, D.G., Pressure Buildup and Flow Tests in Wells, SPE, Dallas, 1967.
3. Raghavan, R., Scorer, J.D.T., and Miller, F.G., "An Investigation by Numerical Methods of the Effect of Pressure-Dependent Rock and Fluid Properties on Well Flow Tests," SPEJ, June 1972, pp. 267-75.
4. Pedrosa, O.A., "Pressure Transient Response in Stress-Sensitive Formations," Paper No. SPE 15115, SPE California Regional Meeting, Apr. 2-4, 1986.
5. Jelmert, T.A., and Selseng, H., "Pressure Transient Behavior of Stress-Sensitive Reservoirs," Paper No. SPE 38970, 5th Latin American and Carebbian Conference and Exhibition, Aug. 30-Sept. 3, 1997 Rio de Janeiro.
6. Wyble, D.O., "Effect of Applied Pressure on the Conductivity, Porosity and Permeability of Sandstones," T.N. 2022, Trans AIME 213, 1958, pp. 431-32.
7. Samaniego, F., and Cinco-Ley, H., "On the Determination of the Pressure-Dependent Characteristics of a Reservoir through Transient Pressure Testing," Paper No. SPE 19774, 64th SPE Annual Technical Conference and Exhibition, San Antonio, Oct. 8-11, 1989.

The Authors