(169g) Accessment of Non-Linearities on the Transport Properties of Compressible Fluids

Traditional permeability measurements on unconventional reservoirs involve the grinding of samples. This is done to allow rapid assessment of permeability and porosity. The measurements are however unstressed and the disturbance in the sample properties is due to grinding is unpredictable.

We have used COMSOL Multiphysics^TMsoftware to model four different measurement protocols for nonlinear transport properties on a plug scale. They include steady state, unsteady state, pulse decay and sinusoidal pressure measurements, assessing the strength and weakness of each technique. These measurement protocols differ due to the number of reference chambers and the pressure boundary conditions applied. The COMSOL models have all been validated by one-dimensional numerical solutions of the nonlinear equation and also been checked for consistency and stability by forward modeling the transport and then inverting to obtain the original modeled parameters. We include the effect of gas slippage (Klinkenberg corrections), pressure dependent density, but neglect the effects of pressure dependent viscosity and rock compressibility. Over the pressure ranges we model these effects are assumed to be minor. Forchheimer terms are also assumed negligible due to the low flow rates encountered.

Significant pressure drops must be used to obtain large enough flow rates to perform the measurements. This implies significant pressure dependent density effects, which reduce the flow rates compared to linear models (low pressure drops). In contrast, gas slippage increases the transport of gas by an increasing amount with reduced pressure. The result is that nearly identical fits are obtained for any value of the mass flow rate by varying the magnitudes of permeability and gas slippage. In order to separate these two effects, measurements must be made at distinctly different mean pressures.

We introduce a new analytical technique to calculate the pressure profile for steady state measurements once the transients have dissipated and constant mass flow is reached. This calculation allows large pressure difference measurements for steady state data to be interpreted without the use of these complex models. The steady state results are validated against finite element and finite difference calculations. The effects of the nonlinearities predicted by the models on the measured chamber pressures are presented for each of the measurement protocols (other than steady state).

In particular for sinusoidal permeability a rise in the average pressure in the downstream chamber pressure is predicted. This effect has not been discussed in the literature and is key to interpreting the transport properties.

The modeling indicates that plug scale measurements are practical but multiple mean pressures must be used to separate the competing effects of permeability and gas slippage. Typical measurement times are on the order of days and grinding of the samples used for the standard GRI technique is therefore not necessary. We recommend using unsteady state measurements at reservoir stress, supplemented with sinusoidal pressure and pulse decay to calibrate the nonlinear effects, i.e. the impact of diffusion, adsorption and absorption.

We have developed models for a variety of geometries and experimental protocols:

1. Steady State: Historically the most widely used technique for measuring permeability. For low permeability samples, there are nonlinearities that must be taken into account to accurately analyze the data for k₀and b. To acquire data that allows a linearized version of the equation, the flow rates are too low to be reliably measured.

2. Unsteady State: This technique takes the least time to perform the measurement over a wide range of flow rates. At low permeability, the full nonlinear equation must be used to extract the correct value of k0 and b. Measurements at two or more mean stresses are required to allow unambiguous extraction of the transport parameters. The measurement time is comparable to many other petro-physical measurements (days). This technique should become the standard for low permeability plug measurements.

3. Pulse Decay: The measurement provides mass balance which allows the effects of absorption and adsorption to be included in models. This makes it the most robust measurement for determining all the physical mechanisms, but small leaks would negate these advantages. Similar to unsteady state measurements at several means stresses should be made.

4. Sinusoidal: Nonlinearities are easily quantified through the use of Fourier analysis. The pressure dependent density introduces harmonic distortion. Addition of gas slippage reduces this effect. In particular the average pressure in the outlet chamber is higher than the average pressure in the inlet chamber due to the pressure dependent density. Fourier analysis of the data in the downstream chamber allows the nonlinear effects to be quantified.

The necessity of using the full nonlinear flow equation for the interpretation of the data is demonstrated in these models. From the results of this modeling, we believe plug scale measurements are practical. We recommend using unsteady state measurements, supplemented with sinusoidal pressure and pulse decay to calibrate the magnitude of the nonlinear effects, and the impact of diffusion and absorption.

Future work will include modeling the effects of anisotropic samples, gas sorption and diffusion effects, and comparing models to transport data. We will also include pressure dependent viscosities to model the effect of flowing gas above the critical point.