Local Permeability Variations Research Articles

Fluid Migration Characterization of Full-Scale Annulus Cement Sections Using Pressure-Pulse-Decay Measurements

Abstract Fluid migration behind casings is a well integrity problem that can result in sustained casing pressure, undetected leaks to the environment, and potentially very challenging remediation attempts. Understanding the geometric dimensions and extent of annular migration paths is important for diagnosing and effectively treating fluid migration and sustained casing pressure problems in wells. In this study, permeability and micro-annuli sizes in two full-scale cemented annulus test sections are measured using transient pressure-pulse-decay and steady-state seepage measurements. One of the studied sections is a cemented 9 5/8-in. and 13 3/8-in. casing section from a 30 years old Norwegian North Sea production well. A model for predicting the transient pressure decay in annular sections with non-uniform permeability is presented and the permeabilities of the two sections are determined by fitting the transient model to pressure measurements at either side of the test sections and at selected axial positions. For both sections, measured micro-annulus sizes are within the range of effective wellbore permeabilities based on sustained casing pressure records and previous vertical interference tests from other wells. The test sections display measurable axial permeability variations with the bottom part of these vertical sections having the lower permeability. For the retrieved casing section, the axial permeability variation occurs close to the middle of the test section and is attributed to the top-of-cement. Increasing internal casing pressure is found to slightly reduce the equivalent micro-annulus size, indicative of fracture-like response of the migration paths. Using two independent test protocols, we have measured effective permeabilities as well as local permeability variations in full-scale test sections and found consistent results. The study suggests that the transient test procedure can be used to more effectively characterize low-permeable annular cement where it is otherwise time-consuming or difficult to establish steady-state flow conditions.

Impregnation quality diagnosis in Resin Transfer Moulding by machine learning

In recent years, several optimization strategies have been developed which reduce the overall defectiveness of the RTM manufactured part. RTM filling simulations showed that, even using optimized injection strategies, local variations in permeability could still generate local defects in the part. Such defects must be identified quickly and accurately right after the injection phase in order to repair or scrap the part. The present paper analyses the feasibility of diagnosing the quality after the injection stage of RTM parts by Machine Learning. The diagnosis was performed using a Supervised Learning binary classification model trained with synthetic data set. Among the different predictive models studied, Extreme Gradient Boosting and Light Gradient Boosting Machine were the most accurate models for predicting the filling quality of RTM, with an accuracy of 84.9% and 83.35%, respectively. In addition, a scaling of the ML model allowed not only to predict the quality of the part but also to locate the areas where the defect was generated. The computation time of the trained ML model versus the FEM model, both computed with a workstation (CPU Intel Xeon Processor E5.2690 with 2.9 GHz–8 core), were compared. In the specific case studied, a reduction from 360 s to 1 s was observed in the computation time utilizing the trained ML model. Thus, Supervised Learning predictive models can be used for diagnosing the composite quality. They are fast enough to be integrated in real processes and could be helpful for designing the monitoring system.

Bayesian inversion algorithm for estimating local variations in permeability and porosity of reinforcements using experimental data

A novel Regularising Ensemble Kalman filter Algorithm based on the Bayesian paradigm was applied to RTM processes to estimate local porosity and permeability of fibrous reinforcements using measured values of local resin pressure and flow front positions during resin injection. The algorithm allows to detect locations of defects in the preform. It was tested in virtual experiments with two geometries, a two-dimensional rectangular preform and a more complex 3D shape, as well as in laboratory experiments. In both the virtual and laboratory experiments, it was demonstrated that the proposed methodology is able to successfully discover defects and estimate local porosity and permeability with good accuracy. The algorithm also provides confidence intervals for the predictions and estimations of defect probabilities, which are valuable for analysis of the process.

Effects of post-infusion dwell on vacuum infusion of thermoset composites toughened by thermoplastic interlaminar veils

The effects of post-infusion dwell on vacuum infusion of thermoset composites toughened by non-woven thermoplastic interlaminar veils were investigated. Permeability measurements and simulation of the resin infusion process demonstrated that the toughening interlayers can effectively act as interlaminar flow distribution media. Local variations in permeability induced non-uniform flow fronts, resulting in high porosity. However, introduction of a low-temperature post-infusion dwell allowed more time for the resin to equilibrate pressure and redistribute during the post-filling stage, achieving full saturation of dry regions. The process parameters of the post-infusion dwell were determined using cure kinetics and viscosity models, while in situ process adjustments were implemented using dielectric cure monitoring system in conjunction with resin cure maps. Laminates fabricated with the modified cycle exhibited reduced porosity and greater peak load during impact testing. This work highlights potential advantages of the post-infusion dwell, which can similarly be applied to other vacuum infusion processes requiring a protracted post-filling stage.

Explicit tracking of CO2-flow at the core scale using micro-Positron Emission Tomography (μPET)

Safe subsurface sequestration of carbon dioxide (CO2) is becoming increasingly important to meet climate goals and curb atmospheric CO2 concentrations. The world-wide CO2 storage capacity in carbonate formations is significant; within deep, saline aquifers and through several CO2-enhanced oil recovery projects, with associated CO2 storage. Carbonates are complex, both in terms of heterogeneity and reactivity, and improved core scale and sub core-scale analysis of CO2 flow phenomena is necessary input to simulators, aiming to establish large-scale behavior. This paper presents a recent advancement in in-situ imaging of CO2 flow, utilizing high-resolution micro-Positron Emission Tomography and radioactive tracer [11C]arbon dioxide to explicitly track CO2 during dynamic flow and subsequent trapping at the core scale. Unsteady state water injection (imbibition) and CO2 injection (drainage) were performed in a low-permeable chalk core at elevated pressure conditions. Short-lived radioisotopes were used to label water and CO2, respectively, and facilitated explicit tracking of each phase separately during single phase injection. Local flow patterns and dynamic spatial fluid saturations were determined from in-situ imaging during each experimental step. Initial miscible displacement revealed displacement heterogeneities in the chalk core, and dynamic image data was used to disclose and quantify local permeability variations. Radial permeability variations influenced subsequent flow patterns, where CO2 predominantly flooded the higher-permeability outer part of the core, leaving a higher water saturation in the inner core volume. Injection of water after CO2 flooding is proposed to be the most rapid and effective way to ensure safe storage, by promoting capillary trapping of CO2. PET imaging showed that presence of CO2 reduced the flow of water in higher-permeability areas, improving sweep efficiency and promoting a nearly ideal core-scale displacement. Alternate injections of water and gas is also expected to improve sweep efficiency and contribute to improved oil recovery and CO2 storage on larger scales. Sub-core analysis showed that residually trapped CO2 was evenly distributed in the chalk core, occupying 40% of the pore volume after ended water injection. Micro-Positron Emission Tomography yielded excellent small-scale resolution of both water and CO2 flow, and may contribute to unlocking fluid flow dynamics and determining mechanisms on the millimeter scale; presenting a unique opportunity in experimental core-scale evaluations of CO2 storage and security.

Prediction of gas hydrate saturation throughout the seismic section in Krishna Godavari basin using multivariate linear regression and multi-layer feed forward neural network approach

Stepwise linear regression, multi-layer feed forward neural (MLFN) network method was used to predict the 2D distribution of P-wave velocity, resistivity, porosity, and gas hydrate saturation throughout seismic section NGHP-01 in the Krishna-Godavari basin. Log prediction process, with uncertainties based on root mean square error properties, was implemented by way of a multi-layer feed forward neural network. The log properties were merged with seismic data by applying a non-linear transform to the seismic attributes. Gas hydrate saturation estimates show an average saturation of 41 % between common depth point (CDP) 600 and 700 and an average saturation of 35 % for CDP 300–400 and 700–800, respectively. High gas hydrate saturation corresponds to high P-wave velocity and high resistivity except in a few spots, which could be due to local variation of permeability, temperature, fractures, etc.

Effects of Permeability Variations on CO2 Convection in Anisotropic and Heterogeneous Saline Formations

This study simulated the natural convection of dissolved carbon dioxide (CO_2) in a small-scale heterogeneous saline formation using the state module ECO2N equation in the TOUGHREACT model. A one-way downscaling approach that involves using a series of sub-models in simulation procedures was proposed to efficiently simulate problems with high-scale discrepancies. This study evaluated the effects of different degrees of small-scale permeability variations on the vertical migration of dissolved CO_2. The sequential Gaussian simulation model was used to generate unconditional random permeability fields for different natural logarithm of permeability (lnk) variations (i.e., lnk variances and correlations in x and z directions). The results showed an identical transition zone of dissolved CO_2 near the top boundary, where a constant CO_2 gas saturation was specified. The local permeability variations can trigger fingerings and enhance the vertical convection of the dissolved CO_2. The number of fingerings depends on the variations in permeability near the front interface of the dissolved CO_2 (i.e., the bottom edge of the transition zone for the dissolved CO_2). However, the fingering patterns and developments are constrained by the permeability variations along the fingering paths. At the same mean lnk permeability the convection fluxes increase with an increase in lnk variances. However, an increase in lateral correlations (i.e., increase in the correlation lengths in the x direction) can slightly reduce the convection fluxes at the same lnk variance. The highly variable flux rates of the dissolved CO_2 occur early and the variations in the flux rate decrease with time.

A Finite Element-based Heuristic Estimation of Local Preform Permeability for Resin Transfer Molding

Uniformity of fabrics significantly affects the resin flow behavior in the resin transfer molding (RTM) process. Due to fabric defects or improper fiber preform preparation/loading, non-uniformity in fabric structure frequently occurs in RTM processing and creates local permeability variations. Such variations often lead to unbalanced resin flow patterns and thus result in defects of finished composite parts. In RTM process modeling, an accurate estimation of the whole field permeability profile of the fiber preform is critical for predicting resin flow pattern correctly. In this article, a finite element-based heuristic computing method is introduced for estimating the in situ whole-field isotropic permeability profile of the preform using a steady flow of gas. Compared with conventional approaches, this method is effective in measuring local permeability variations and applicable to molds with complex 2-D geometries, as well as diverse injection strategies. Several case studies were presented with experimental designs and numerical computations to illustrate the effectiveness and efficiency of the method.

Influence of the fabric architecture on the variations in experimentally determined in-plane permeability values

Fabric non-uniformities can be described by the geometric parameters R p / a 0, which refers to the average ratio of in-plane dimension of fibre tows R p and fibre tow spacing a 0 and characterises the fibre-void distribution in the fabric, and the standard deviation of the inter-fibre-angle σ α . For three different woven fabrics and a stitch-bonded non-crimp fabric, it was found that in general the higher R p / a 0, the lower is σ α , with an approximately linear relation between R p / a 0 and σ α . Woven fabrics show mainly macro-scale non-uniformities determined by the fibre tow mobility, which is limited by the void space between the tows. The mainly mesoscopic non-uniformities in non-crimp fabrics are imposed by the stitching pattern, which again determines the values of R p / a 0 and σ α . The principal values and the orientation of the principal axes for the global in-plane permeabilities were determined based on series of experiments that were evaluated statistically. All relative variations (standard deviations/mean values) of the principal permeability values were found to be in the range between 9.1% and 29.3%. The variation in the angle describing the orientation of the principal permeability axes increases with increasing relative permeability variations. Analysis of the relation between permeability variations and geometrical fabric parameters suggests that the more homogeneous the fibre-void distribution in the fabric, i.e., the higher R p / a 0 and the lower σ α , respectively, the lower the local permeability variations and the more similar the flow front shape to a perfect ellipse. The global permeability variations, on the other hand, increase with the local variations until they reach a maximum and then decrease again. Quantitative comparison of measured permeability variations and the results of stochastic simulations indicates that, while the results are in the same order of magnitude, the stochastic model is not detailed enough for quantitatively accurate prediction of permeability variations.

In situ measurement and monitoring of whole-field permeability profile of fiber preform for liquid composite molding processes

Abstract For liquid composite molding (LCM) processes, such as resin transfer molding (RTM), the quality of final parts is heavily dependent on the uniformity of the fiber preform. However, the conventional permeability measurement method, which uses liquid (oil or resin) as its working fluid, only measures the average preform permeability in an off-line mode. This method cannot be used to create an in situ permeability profile because of fiber pollution. Further, the conventional method cannot be used to reveal preform's local permeability variations. This paper introduces a new permeability characterization method that uses gas flow to detect and measure preform permeability variations in a closed mold assembly before resin injection. This method is based upon two research findings: (1) resin permeability is highly correlated with air permeability for the same fiber preform with well-controlled gas flow, and (2) the whole-field air permeability profile of a preform can be obtained through measuring the pressure field of gas flow. In this study, first the validity of the gas-assisted, in situ permeability measurement technique was established. Then the technique was demonstrated as effective by qualitatively detecting non-uniformities and permeability variations in fiber performs. Finally, a two-dimensional flow model, based on the finite difference scheme, was developed to quantitatively estimate the whole-field preform permeability profile using predetermined pressure distribution. The efficacy of the new method was illustrated through experimental results.

Enhancement of flow in VARTM using localized induction heating

A critical step in the vacuum assisted resin transfer molding (VARTM) process is the permeation of a porous preform by the reactive resin. A real time flow control is often imperative to achieve complete preform saturation and void-free fill. Local permeability variations in the preform, however, challenge the flow control endeavor. Control strategies that use manipulation of the inlet port parameters are shown in the literature to be limited in the controllability of flow in regions of localized preform variability, particularly at locations away from the controlled injection ports. This paper explores an innovative scheme of using induction heating as a method of locally reducing the resin viscosity to counteract the effects of such localized low permeability regions within the preform. Toward this end, the paper presents a process model for nonisothermal flow during the VARTM process, in the presence of induction heating. The process model, validated with experiments, is used to conduct process simulations to investigate the effects of processing parameters such as induction heating location, induction heating power level, and vacuum level on three heterogeneous preform geometries with varying permeability ratios between the low permeability and high permeability regions. Results of these studies are presented in the form of processing windows and processing maps, which show that induction heating is capable of reducing void and dry spot formation during the VARTM process.

Rheological effects of synovial fluid on nutritional transport

Synovial fluid is an excellent source of nutrients for the cells of the cartilage, through which water and other solutes like glucose are permeable. It has long been established that the tissue imbibes and exudes fluid when deformed, metabolities are believed to move to and from the cells through the ground substance by mechanical effects and by diffusion. Local variation of the permeability within the articular cartilage plays an important role in nutritional transport. To account for the effects of structural configuration of tissue, rate, depth and amount of solute penetration, we have modelled the cartilage by mixture of two distinct constituents, i.e., an incompressible fluid phase and an incompressible porous solid phase. It is observed that when local permeability decreases, the concentration decreases. When fluid flows into the cartilage due to metabolism, the solid portion of the porous matrix increases. Due to the increased solidity of the cartilage matrix, less fluid enters into the cartilage and nutritional transport decreases. In the case of diseased joints the nutritional transport is very difficult, owing to increased rigidity or local variation of permeability within the cartilage. The concentration distribution at the same depth in articular cartilage for low-molecular-weight solutes is less than that for high-molecular-weight solutes. Thus, for low-molecular-weight solutes, the phenomenon of nutrition transport is diffusion dominated whereas for high-molecular-weight solutes, it is dominated by mechanical pumping action. The paper further concludes that in the process of imbibition and exudation, the cells of the middle area of the cartilage surface get more nutrition as compared to the cells at the periphery, so the earliest signs of cartilage degeneration appear in the unstressed areas. Therefore joint motion is assumed necessary for cartilage nutrition. It also concludes that as the viscoelastic parameter increases, the concentration decreases in the articular cartilage so that the cells of the cartilage get less nutrition and can deteriorate.

An application of ultrasonic tomographic imaging to study smoldering combustion

An ultrasonic imaging technique has been developed and applied to examine smoldering combustion within a permeable medium. The technique provides information about local permeability variations within a smoldering sample, which can, in turn, be interpreted to visualize the propagation of the smolder reaction. The method utilizes the observation that transmission of an ultrasonic signal through a porous material increases with increasing permeability. Since a propagating smolder reaction leaves behind a char that is higher in permeability than the original (unburnt) material, ultrasonic transmission can be employed to monitor smolder progress. Additionally, the technique allows observation of the evolution of the char (i.e., material left by the smolder reaction), which, in certain circumstances, can continue to increase in permeability, due to secondary reactions (either oxidative or pyrolytic in nature). Experiments have been conducted, applying the technique to smoldering combustion in a two-dimensional geometry with line-of-sight imaging. For axisymmetric configurations, tomographic techniques have been implemented, providing three-dimensional mappings of the smolder front, as well as visualization of the smolder process itself. The results have furthered the understanding of two-dimensional smolder, and have been especially informative in identifying the controlling mechanisms leading to the transition from smoldering to flaming combustion.

Early diagenetic partial oxidation of organic matter and sulfides in the Middle Pennsylvanian (Desmoinesian) Excello Shale member of the Fort Scott Limestone and equivalents, northern Midcontinent region, USA

A process of early diagenetic partial oxidation of organic matter and sulfides has altered the chemical composition of the Middle Pennsylvanian Excello Shale Member of the Fort Scott Limestone and equivalents in the northern Midcontinent region. This process was identified by comparison of organic carbon contents, Rock-Eval hydrogen indices, organic carbon δ 13C and element compositions of core and surface mine samples of the Excello Shale Member with analyses of three other underlying and overlying organic-matter-rich marine shales (offshore shale lithofacies) from southern Iowa, northern Missouri, eastern Kansas and northeastern Oklahoma. The end product of the partial oxidation process is shale with relatively low contents of hydrogen-poor, C 13-enriched organic matter, lower contents of sulfur and sulfide-forming elements, and relatively unchanged contents of phosphorus and many trace elements (e.g. Cr, Ni, and V). However, because of lower organic carbon contents, element/organic carbon ratios are greatly increased. The partial oxidation process apparently took place during subaerial exposure of the overlying marine carbonate member (Blackjack Creek Member of the Fort Scott Limestone) following a marine regression when meteoric waters percolated down to the level of the Excello muds allowing oxidation of organic matter and sulfides. This hypothesis is supported by earlier workers, who have identified meteoric carbonate cements within, and soil horizons at the top of the Blackjack Creek Member. The period of oxidation is constrained in that organic matter and sulfides in the Little Osage Shale Member of the Fort Scott Limestone and equivalents (immediately overlying the Blackjack Creek Member) appear unaltered. Similar alteration of other shales in the Middle and Upper Pennsylvanian sections may be local to regional in extent and would depend on the extent and duration of the marine regression and be influenced by local variations in permeability and topography. The partial oxidation process has likely led to a redistribution of sulfur and sulfide-forming elements into other organic-rich lithologies in the section. The altered/oxidized shales are nongenerative with respect to hydrocarbon generation.

Reconstruction of palaeothermal conditions in a passive margin using illite-smectite mixed-layer series (BA1 scientific deep drill-hole, Ardeche, France)

AbstractThe comparison between measured and predicted smectite percent (%Sm) in an illite-smectite (I-S) mixed-layer series was used to estimate the thermal conditions of diagenesis in the Ardèche sedimentary series. Measurements were performed using the decomposition procedure (Lanson & Besson, 1992) of XRD patterns obtained from oriented preparations (<0.2 μm fractions). Three particle populations were identified: I-S, poorly crystallized illite (PCI), and well crystallized illite (WCI); the %Sm was measured in the I-S population by comparison with calculated XRD patterns using the NEWMOD software (Reynolds, 1985). The theoretical evolution of %Sm with depth obtained with the two first-order kinetic reaction processes described by Velde & Vasseur (1992) was compared to the measured one at Balazuc. The best fit was obtained for the following conditions: 35°C km–1and between 1500 and 2000 m for the thermal gradient and the eroded piles, respectively. The predicted values for I-S compositions in the 10–0% Sm range are overestimated by the model because the reaction controlling the composition is no longer the dissolution of I-S but the ripening of illitic particles (PCI and WCI).Two zones can be distinguished in the sedimentary series: (1) down to 55 m, the %Sm of I-S particles is highly variable and seems to be controlled by local variations in permeability and chemical composition; (2) down to 550 m, the %Sm of I-S particles is more strictly dependent on the general time-temperature conditions. An anomalously high %Sm value is located in the vicinity of faults at 1669 m due to local change of the water/rock ratio.

Method to reduce local permeability variations in magnets

Since the achieved magnetic flux density in steel depends on the excitation history of the steel, precision magnets often suffer from local changes in permeability in the steel of the pole pieces. These effects can be reduced by once fully saturating the pole pieces of the magnet. For high field magnets this can easily be done, but also in low field magnets such a saturation can be achieved by temporarily introducing a bar of steel into the magnet air gap.

Paleomagnetism and the evolution of fluids in the Proterozoic Athabasca Basin, northern Saskatchewan, Canada

In the Athabasca Basin, diagenetic hematite of variable paragenesis occurs throughout the sandstones and underlying paleoregolith. This hematite carries three distinct, single-component magnetizations: A (D = 158°, I = 62°, α95 = 5°, n = 21); B (D = 11°, I = −36°, α95 = 7°, n = 6); and C (D = 18°, I = 79°, α95 = 3°, n = 27). In some areas of the sandstones, such as near reactivated fault zones, the diagenetic hematite has been altered to goethite which yields a very low-intensity, incoherent D magnetization. Ages for the A, B, and C magnetizations, inferred from comparisons with paleomagnetic directions in Precambrian rocks whose ages are known approximately, are 1750–1600, 1600–1450, and about 900 Ma, respectively. The A magnetization is carried by the earliest formed hematite, and its estimated age compares well with U–Pb ages of 1650–1700 Ma for early diagenetic apatite. U–Pb and Rb–Sr ages of approximately 1500 and 900 Ma for uraninite and illite coeval with hematite that carries the B and C magnetizations compare well with their ages estimated from paleomagnetism. The development of B magnetization appears to be coeval with high-grade, unconformity-type uranium mineralization.Petrographic and field relationships indicate that the A magnetization is carried by hematite formed during initial diagenesis of the Athabasca sandstones, the B magnetization is carried by hematite formed during peak diagenesis, and the C magnetization is carried by hematite formed during subsequent high-temperature hydrothermal alteration. The incoherent D magnetizations have resulted from degradation of hematite to goethite as a result of incursion of low-temperature meteoric waters along fault zones that have been continuously reactivated since the late Precambrian. δ18O values of clay minerals and of the coeval hematite which carries the B and C magnetization indicate that they were formed from a fluid having temperatures of 150–200 °C and δ18O values near 1.0‰. Fluids that deposited the early formed hematite carrying the A magnetism are relatively 18O depleted, with values of approximately 0.8‰ and somewhat lower temperatures of 120–160 °C. Intermingling of A, B, and C magnetizations indicates either that hematite may be deposited by one fluid and reprecipitated by a subsequent fluid, or that fluid flow was controlled by local variations in permeability. Evidently, fluid flow has been episodic and basin wide and has occurred over a time span on the order of 108 years. It is suggested that the stratigraphy of the sandstones controlled the basin-wide lateral migration of the basinal fluids and that faults facilitated interformational fluid flow.

A study of nutritional transport in a synovial joint

Aged articular cartilage has no other means of nutritional transport to the cartilage cells except the inbuilt mechanism of interstitial fluid exchange in between the cartilage and synovial fluid. The secreted nutrition from blood mixes well in the synovial fluid and during articulation it enters into the cartilage where it meets with the cartilage cells. The local variation of the permeability of the cartilage, therefore, plays a very important role in the whole mechanism. Analytically the problem is formulated as a two region diffusion and flow model: flow and diffusion in between two approaching cartilage surfaces and within the porous cartilages. The solution of the coupled mixed boundary value problem could only be obtained for a few particular cases. It has been observed that the increased permeability at the surface does not allow nutrition to the cells in the deeper region and they die out. In case of diseased joints the nutritional transport is very difficult owing to increased rigidity or local variation of permeability within the cartilage. The paper further concludes that for low molecular weight solutes, the phenomenon of nutritional transport is diffusion dominated whereas for large molecular weight solutes, it is dominated by mechanical pumping action.

Analysis of Interference Data in a Highly Heterogeneous Naturally Fractured Geothermal Reservoir

Summary An approach consisting of a combination of composite and naturally fractured reservoir models has been proposed to interpret pressure-transient data from an 11-well interference test in a highly heterogeneous geothermal reservoir. The approach leads to an integrated interpretation that accounts for all of the salient features of the data. An initial appraisal of the data indicates excellent agreement with theoretical results for double-porosity reservoirs. For some of the wells, however, the bulk storativity, calculated with the double-porosity model, differs significantly from the average value of this parameter. Furthermore, it increases systematically as the distance between the active and observation wells decreases. To explain this peculiar result, an alternative method of analyzing data from composite reservoirs is presented. The method demonstrates that local permeability variations can result in enormous variations in the apparent storativity without causing variations in the apparent permeability-thickness of the reservoir. A simple graphical technique, based on analyzing systematic variations in the apparent storativity, has been developed to determine the size of the region and the mobility contrast between different regions of the reservoir. By using this technique, we show that the observed variations in the storativity are the result of a small, high-permeability region that surrounds the pumped well and is penetrated by several of the observation wells.

Flow Through a Porous Medium With Periodic Barriers or Fractures

Abstract Apparent permeability of a reservoir may be affected by local variations in permeability whose extent and location are not known in detail. One way to approximate this condition is to postulate a regular pattern of typically shaped and oriented inclusions. An example is the presence of a large number of very thin, relatively impermeable shale streaks distributed throughout an otherwise homogeneous reservoir. If these barriers to flow generally are oriented alike, then at a scale that is large compared to the individual barriers, apparent permeability will be anisotropic, being higher parallel to the barriers than perpendicular to them. Discontinuous horizontal shale streaks can be responsible for reduced vertical oil drainage to the pumping well1; therefore, a reasonable prediction of vertical permeability in such settings can be of value. This paper examines the special case of a periodic array of infinitely thin, impermeable barriers opposed to single-phase, two-dimensional (2D) flow. Results provide a quantitative picture of the importance of barrier geometry and arrangement to the apparent anisotropy of permeability. The problem setting, formulated previously,2,3 allows a closed-form analytical expression to be obtained for anisotropy as a function of barrier geometry. However, a previously published solution3 is incorrect.2,4 The purpose of this paper is to present a correct solution and to discuss some of its consequences. Analytical results for this and other geometries, in two and three dimensions, have been developed previously.2 Numerical results for this problem are also available for both single-phase5 and two-phase (gas/oil)1 flow.