Petroleum Reservoir Simulation

Non-Darcy behavior is important for describing fluid flow in porous media in situations where high velocity occurs. A criterion to identify the beginning of non-Darcy flow is needed. Two types of criteria, the Reynolds number and the Forchheimer number, have been used in the past for identifying the beginning of non-Darcy flow. Because each of these criteria has different versions of definitions, consistent results cannot be achieved. Based on a review of previous work, the Forchheimer number is revised and recommended here as a criterion for identifying non-Darcy flow in porous media. Physically, this revised Forchheimer number has the advantage of clear meaning and wide applicability. It equals the ratio of pressure drop caused by liquid–solid interactions to that by viscous resistance. It is directly related to the non-Darcy effect. Forchheimer numbers are experimentally determined for nitrogen flow in Dakota sandstone, Indiana limestone and Berea sandstone at flowrates varying four orders of magnitude. These results indicate that superficial velocity in the rocks increases non-linearly with the Forchheimer number. The critical Forchheimer number for non-Darcy flow is expressed in terms of the critical non-Darcy effect. Considering a 10% non-Darcy effect, the critical Forchheimer number would be 0.11.

A Criterion for Non-Darcy Flow in Porous Media

In this paper, a well test interpretation method based on the analysis of the time rate of change of pressure, together with the actual pressure response, is discussed. A differentiation algorithm is proposed and several field examples are provided to illustrate how the method simplifies the analysis process. Interpretation of well tests is therefore easier and more accurate.

Use of Pressure Derivative in Well Test Interpretation

During the last few years, there has been an explosion of information in the field of well-test analysis. Because of increased physical understanding of transient fluid flow, it is possible to analyze t.l:ie entire pressure history of a well test not just long-time data as in conventional analysis.i It is now often possible to specify the time of beginning of the correct semilog straight line and determine whether the correct straight line has been properly identified. It is also possible to identify well bore storage effects, and the nature of wellbore stimulation as to permeability improvement, or fractU"ring, and to quantitatively analyze. those effects. Such accomplishments have been augmented by attempts to understand the short-time pressure data from well testing data that were often classified as too complex for analysis. One recent study of short-time pressure behavior2 showed that it was important to specify the physical nature of the stimulation in considering the behavior of a stimulated well. That is, stating that the van Everdingen Hurst infinitesimal skin effect was negati ve was not sufficient to define short-time well behavior. For instance, acidized (but not acid-fractured) and hydraulically fractured wells might not necessarily exhibit the same behavior at early times, even though they could possess the same value of negative skin effect. In the same manner, hydraulic fracturing leading to horizontal or vertical fractures could produce the same skin effect, but with possibly different shorttime pressure data. This could then provide a way to determine the orientation of fractures created by this type of well stimulation. In fact, it is generally agreed that hydraulic fracturing usually results in one vertical fracture, the plane of which includes

Unsteady-State Pressure Distributions Created by a Well With a Single Infinite-Conductivity Vertical Fracture

A process for recovering methane from a coal bed and/or sequestering a fluid more strongly adsorbing than methane (“SAG”) in a coal bed involves fracturing an injection well and cyclically injecting SAG with intervening shut-in periods. Even though the absolute permeability of the coal bed decreases with continued SAG sorption, the SAG injectivity, surprisingly and unexpectedly, increases while injecting SAG at pressures greater than reservoir pressure, but less than fracture pressure or fracture extension pressure. The increased injectivity is thought to be due to a reduction in near-well flow resistance.

Process for recovering methane and/or sequestering fluids

An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: I. Analytical Treatment

The purpose of this paper is to provide solutions for analyzing pressure data for constant-rate wells crossed by finite-conductivity vertical fractures producing at high rates to cause non-Darcy flow effects in the fracture. The following conclusions can be drawn. 1. To describe the non-Darcy effect, two physical parameters are required: the dimensionless fracture conductivity and the dimensionless flow rate constant. 2. The effect of non-Darcy flow in the fracture is to cause an apparent fracture conductivity significantly less than the true fracture conductivity. 3. The bilinear flow period also is observed in high-rate wells that have the non-Darcy effect. Graphs of log P /SUB D/vs. log (t /SUB Dxf/) exhibit the one-fourth slope for apparent fracture conductivities less than 50. 4. Techniques used for analyzing finite-conductivity fractures also apply to the non-Darcy case, except the fracture conductivities obtained are apparent values. 5. A correlation between the flow rate constant, the true fracture conductivity, and the apparent fracture conductivity is developed in this study. 6. Two methods are presented for determining the true fracture conductivity of a well that has two sets of drawdown data obtained at two different high flow rates.

Non-Darcy Flow in Wells With Finite-Conductivity Vertical Fractures

Equipment was constructed to perform dynamic displacement experiments on small core samples under conditions of elevated temperature. Oil-water flowing fraction and pressure drop were recorded continuously for calculation of both the relative permeability ratio and the individual relative permeabilities. Imbibition relative permeabilities were measured for five samples of Boise sandstone at room temperature and at 175/sup 0/F. The fluids used were distilled water and a white mineral oil. The effect of temperature on absolute permeability was investigated for six Boise sandstone samples and two Berea sandstone samples. Results for all samples were similar. The irreducible water saturation increased significantly, while the residual oil saturation decreased significantly with temperature increase. The individual relaive permeability to oil increased for all water saturations. The individual relative permeability to water decreased with temperature increase for water saturations below the room-temperature residual oil saturation, but the relative permeability to water at flood-out increased with temperature increase. Absolute permeability decreased with temperature increase.

The Effect of Temperature on Relative and Absolute Permeability of Sandstones

Over the last 20 years, a number of studies have reported temperature effects on two-phase relative permeabilities in porous media. However, some of the reported results have been contradictory. Also, observed effects have not been explained in terms of fundamental properties known to govern two-phase flow. The purpose of this study was to attempt to isolate the fundamental properties affecting two-phase relative permeabilities at elevated temperature. Laboratory dynamic displacement relative permeability measurements were made on unconsolidated and consolidated sand cores, using water and a refined white mineral oil. Experiments were run on 2 in. (51 mm) diameter, 20 in. (510 mm) long cores from room temperature to 300/sup 0/F (149/sup 0/C). Unlike the results of previous researchers, essentially no changes with temperatures were observed in either residual saturations or relative permeability relationships. It was concluded that previous results may have been affected by viscous instabilities, capillary end-effects, and/or difficulties in maintaining material balances.

H.J. Ramey

Papers

An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: I. Analytical Treatment

Non-Darcy Flow in Wells With Finite-Conductivity Vertical Fractures

The Effect of Temperature on Relative and Absolute Permeability of Sandstones

Effect of temperature on oil/water relative permeabilities of unconsolidated and consolidated sands

Gas Well Testing With Turbulence, Damage and Wellbore Storage