Net-Stress/Permeability Effect on Gas Non-Darcy Flow
Authors: Kelly M. , Jan B. Mohamed , Lee R.L. 
Published: December 2007

Abstract
The correct explanation for the non-Darcy behavior (or effect) on gas flow through porous media has been debated for decades. Non-Darcy behavior (i.e. extra pressure drop) has been more logically ascribed to fluid inertia (caused when gas under high flow rate is forced through tortuous rock) than to turbulence. This high flow rate, non-Darcy concept has been adopted and extended to explain the concave upwards nature of the back-pressure plot. However, two anomalies arise:the laboratory determined values are much lower than the field observations, in other words, the field gas velocity is much less than the gas velocity used in laboratory;there are numerous experimental data showing that the rock permeability is a function of the net-stress (mainly, overburden pressure minus pore pressure) regardless of the gas flow rates.
We define this permeability reduction due to net-stress decrease as the net-stress effect. During the gas production, we believe that the non-Darcy behavior should be caused by both the effect and the net-stress effect. By combining these two effects, the scale of the non-Darcy observed in fields is in the range of laboratory experimental values. This paper also shows the applications of using these two effects to analyze the back-pressure test data.

Introduction
While measuring gas permeability in the laboratory at atmospheric conditions, researchers have found that gas no longer follows Darcy's law in the high gas flow regime. An extra term is added to the Darcy equation to account for this non-Darcy behavior. In this paper, we refer the extra pressure drop due to high velocity as the effect.
Most states in the US require a back-pressure test for all gas wells to estimate the deliverability of the wells. The results of back-pressure tests are plotted in the form of p 2 vs. Qsc. Usually, the slope of the back-pressure plot is greater than one. This means that the well exhibits non-Darcy behavior. The logical explanation is the high velocity effect experienced in the laboratory.
The effect was originally mistaken as the turbulent effect. Later, it was recognized and accepted as the effect of inertia. It is almost impossible to have turbulent flow in a consolidated rock. Laboratory experimental values of k, the product of permeability and coefficient, are in the range of 104 to 106 darcy/cm. In many gas wells, even at low production and flow velocity, their corresponding back-pressure curves still give slopes greater than one. Rarely does a back-pressure curve give an unit slope. To match the field data, sometimes, the value must be increased 100 times or more. Compared to laboratory estimates, the factor may increase in high pressure environments such as in gas fields. Only one study (Warpinski et al.) shows the measurement of the factor under high pressures. The interpreted data indicates that the k values are still in the same range of the data observed under atmospheric conditions. Actually, their data shows the factor increases when the net-stress increases, however, their k values were relatively insensitive to the net-stress.
It is well known that the gas permeability may reduce during drawdown. This is particularly true for naturally fractured reservoirs. The reduction of pore pressure increases net-stress. If no new fractures are induced during this stress alternation process, the increase of the net-stress may restrict the flow path, therefore, reduces the gas effective permeability. This gas permeability reduction is referred to as the net-stress effect in this paper. This net-stress effect has not been incorporated in many petroleum engineering practices including in analyzing the back-pressure data.

Back-Pressure Test
In many states of the US, the back-pressure test is required for a gas well and is documented by the regulatory agency. P. 191^