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Quantifying Induced Polarization of Conductive Inclusions in Porous Media and Implications for Geophysical Measurements

Feb 3, 2020

Subjects Abstract Induced polarization (IP) mapping has gained increasing attention in the past decades, as electrical induced polarization has been shown to provide interesting signatures for detecting the presence of geological materials such as clay, ore, pyrite, and potentially, hydrocarbons. However, efforts to relate complex conductivities associated with IP to intrinsic physical properties of the corresponding materials have been largely empirical. Here we present a quantitative interpretation of induced polarization signatures from brine-filled rock formations with conductive inclusions and show that new opportunities in geophysical exploration and characterization could arise. Initially tested with model systems with solid conductive inclusions, this theory is then extended and experimentally tested with nanoporous conductors that are shown to have a distinctive spectral IP response. Several of the tests were conducted with nano-porous sulfides (pyrite) produced by sulfate-reducing bacteria grown in the lab in the presence of a hydrocarbon source, as well as with field samples from sapropel formations. Our discoveries and fundamental understanding of the electrode polarization mechanism with solid and porous conductive inclusions suggest a rigorous new approach in geophysical exploration for mineral deposits. Moreover, we show how induced polarization of biologically generated mineral deposits can yield a new paradigm for basin scale hydrocarbon exploration. Introduction As a key electromagnetic-based geophysical exploration method 1 , 2 , 3 , 4 , 5 , induced polarization (IP) has been widely utilized in field-scale geophysical surveys for many decades 6 , 7 , 8 , 9 to provide information about the complex conductivity (chargeability) of subsurface formations filled with ionically conducting fluids. These surveys supplement controlled-source electromagnetic 2 and magnetotelluric 1 , 5 surveys, which generate subsurface resistivity maps. Frequencies of interest in these geophysical surveys are usually orders of magnitude below the Debye relaxation frequency (ωD) so ionic conduction dominates the electromagnetic response. There are two distinct, commonly recognized mechanisms that give rise to complex conductivity in this low-frequency regime 4 : (1) diffusive relaxation of neutral modes associated with ionic concentrations, usually called “membrane polarization”, and (2) capacitive charging of the electric double layer on the surfaces of non-ionic conductors, usually called “electrode polarization”. Regardless of the dominant mechanism, frequency dependence of complex conductivity is typically fitted to a Cole-Cole type model 10 as the basis of various geophysical exploration methods 6 , 7 , 8 , 9 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , where the effective complex conductivity of the porous media σeff has the form 16 : $${\sigma }_{eff}(\omega )={\sigma }_{\infty }\left(1-\frac{m}{1+{(i\omega \tau )}^{c}\,}\right).$$ Here, ω is the angular frequency of the applied electric field and σ∞ is the high frequency limit of σeff. The “chargeability” m quantifies the relative change in conductivity between the low- and high-frequency limits, whereas τ is a characteristic relaxation time and c is the Cole-Cole exponent. This model has originated from dielectric spectroscopy and has been successfully developed to understand metal-dielectric systems 19 , 20 , 21 , although its application to model observed IP response 16 , 22 , 23 has been largely empirical. Membrane polarization 4 , 9 , 12 usually ties to IP effects observed in clay and shaley sand 3 , 9 , 24 , 25 , in glass materials such as silica sands with a Stern layer (also known as “charge polarization” 26 , 27 ), and ion-selective membranes in electrochemical cells (also known as “concentration polarization” 28 ). This type of polarization originates from inhomogeneities in the ionic transport properties, which drive ionic concentration gradients that ultimately relax through diffusion, therefore it leads to a relaxation time τ ∼ a2/D, where a is a characteristic length scale such as pore or grain size, and D is a characteristic diffusivity of dominant ions. In geophysical settings, this mechanism is more pronounced at lower pore fluid salinity (e.g., 1 mM NaCl or lower). Chargeability, which directly relates to the magnitude of the signal in IP surveys, diminishes at the higher salinities that are typically encountered below freshwater aquifers and in offshore marine settings. In contrast, electrode polarization 9 , 13 , 24 requires the presence of conductive grains with non-ionic charge carriers, for example electronic conductors or semiconductors, such as gold or sulfide ores like pyrite 23 , 29 , 30 , 31 , 32 . These grains act as short circuits in the high-frequency limit and insulators in the low-frequency limit, therefore the chargeability depends primarily on the effective volume fraction of the conductive grains, and does not diminish at higher salinities. Unlike membrane polarization, the electrode polarization relaxation time has strong dependence on the conductivity of the pore fluid 22 , 30 , 32 . There are several mechanistic and semi-empirical models for electrode polarization 29 , 31 , 32 , 33 with broad agreement on chargeability but very different explanations for the relaxation time. In this study, we show that capacitive charging of the Stern layer on the conductive grain is the primary mechanism for the observed electrode polarization relaxation time. This mechanism imparts a distinct and quantifiable spectral IP response linking directly to the intrinsic physical properties of the associated rocks, such as feature size, surface area, and electrical properties of the grains. In high salinity environments this mechanism dominates the observed IP signal and enables us to relate the response to the intrinsic physical properties of the conductive particulates. It also enables us to extend this mechanistic model to porous conducting grains in order to demonstrate the profound influence of nanoporous conductors on the characteristic relaxation time of the IP signal. We then discuss the geophysical implications of this insight in various geological settings. Treatment of IP response when membrane polarization effects are significant is beyond the scope of this study. Typically this occurs in low salinity environments. Induced polarization with solid conductive inclusions To isolate the fundamental mechanism for the electrode polarization relaxation time, we performed a series of frequency domain experiments with model systems. One of our model systems consists of natural cubic pyrite crystals (each side ~17 mm) embedded in a framework of glass beads (average particle diameter ~0.4 mm). The model systems are held in a custom-designed four-probe measurement cell as shown in Figs. 1(a) and S2(a) . We use a National Instruments system to generate sinusoidal waves and measure the response from the voltage electrodes, and obtained the phase shifts between the injected current and measured voltage in a frequency range from 0.1 Hz to 10000 Hz, with an uncertainty about 0.5mrad. (See Methods). Figure 1 Induced polarization in porous media with conductive inclusions, (a) Graphic view of four-probe experimental cell: current electrodes are meshes; voltage electrodes are rings; thin porous frits are used to hold glass bead pack in position. (b) Frequency dependent phase shift of glass bead pack with 17 mm cubic pyrites. Black stars: control sample with 3 wt% NaCl; Green Squares: 1 pyrite cube with 3 wt% NaCl; Red circles: 2 pyrite cubes with 3 wt% NaCl; Black Squares: 2 pyrite cubes with 15 wt% NaCl; Blue Triangles: 2 pyrite cubes with 0.3 wt% NaCl; Pink Inverted Triangles: 2 pyrite cubes with 0.03 wt% NaCl; (c) Characteristic frequencies versus brine conductivities (NaCl solution) for the samples containing 2 pyrite cubes. In subsea sediments, it is possible to detect IP features of different forms of pyrite 14 , but in the absence of massive ore deposits associated with significant hydrothermal or volcanic activities 52 , 53 , 54 , we think that nano-porous pyrite is the only form of pyrite that would both correlate with hydrocarbon presence and would give rise to a distinguishable signature with the long relaxation times required for detection in conventional IP surveys. To form such specific nano-porous conductive structures, it is necessary to have sulfate reducing bacteria that produce H2S with the consumption of sulfate ($${{{\rm{SO}}}_{4}}^{2-}$$) and hydrocarbon (CxHy). The in-situ H2S production from anaerobic bacterial activity then reacts with different iron species and eventually forms nano-porous pyrite. In anoxic water columns, sulfate species diffuse downward from the seafloor, and are consumed by anaerobic bacteria in the sulfate reduction zone 42 . This process can also occur in euxinic water columns. In addition, mobile and constant hydrocarbon feed is necessary to form large clusters of framboidal pyrites with nano-porous overgrowths 15 , 43 and aggregates 42 , as the conversion from localized organic carbon to pyrite is in some ways limited 55 and the resulting euhedral and framboidal pyrites are relatively small. The biomimetic experiment (Fig. 4 ) along with the fundamental physics discovered here provide the heretofore missing structure-property relationships that are necessary to infer the correct IP relaxation time scales of nano-porous aggregates, and also provide a potential mechanism to explain the long relaxation time 7 observed in the geological setting with sedimentary pyrite formation above a hydrocarbon reservoir. For large solid euhedral pyrite(a ∼ 1 cm) or conventional framboidal pyrite (< rp > ∼ 100 nm, a ∼ 10 μm) without any nano-porous pyrite overgrowth, we can use Eqs. ( 1 ) or ( 2 ), respectively, to calculate a relaxation time τ < 10 ms under representative conditions (σm ∼ 0.2 S/m and C0 ∼ 30 μF/cm2). Similar to the earlier results with euhedral pyrite in Fig. 1(b) , our initial testing with conventional framboidal pyrite aggregates isolated from organic-rich sapropel layers with some levels of biological activities indicate peak frequencies in the kHz range, consistent with the mechanism and morphology described here. (These samples of sapropel layers from the Mediterranean were acquired from British Ocean Sediment Core Research Facility (BOSCORF); See SI-Text- S4 and Fig. S7 for more detail ). In contrast, under the same conditions, pyrite with nano-porous morphologies demonstrated through our biomimetic experiments (Fig. 4 ) and natural samples 43 (< rp > ∼ 10 nm, a ∼ 10–200 μm 15 , 42 , 43 ) would experience IP relaxation times τ ranging from ∼10 ms to ∼2 s assuming that their metallic nano-porous structures are inter-connected and filled with electrolytic conductive materials such as brine or clay. This mechanism based on nano-porous pyrite provides a possible explanation of the long IP relaxation time 7 about 0.5–5 seconds observed in geological settings in the absence of massive ore deposits. It remains to be seen whether natural subsurface grains with the nanoporosity required to obtain large relaxation times form ubiquitously within hydrocarbon seeps, which would potentially allow IP surveys to help identify hydrocarbon charge, seep, or migration pathways 7 by locating these nano-porous pyrite aggregates. Implications for geophysical measurements While most of our discussions so far are primarily focused on offshore marine environments, our understanding and interpretation of induced polarization can be readily applied to many other geophysical applications utilizing both frequency-domain and time-domain IP methods (See SI-Text- S3 for our time-domain analysis). Based on our theory (Eqs. 1 and 2 and SI-Text) and experimental validations, in order to make meaningful interpretations from field as well as borehole induced polarization data across various geophysical applications, additional information are generally required such as mineralogy, morphology and size distribution of the conductive particles as well as the brine conductivity. Here we provide general perspectives on utilizing IP surveys to acquire additional subsurface and/or materials information with key examples of geophysical applications. In offshore marine settings, either with a basin-scale controlled-source electromagnetic survey 2 , 7 or an ocean-bottom shallow inductive method 56 , 57 , it is possible to acquire induced polarization or complex conductivity information 7 . In these scenarios, the real part of the subsurface conductivity can be acquired through an inversion algorithm with proper constraints 58 , which provides a map of the effective conductivity σm of the porous media (without conductive grains) in the subsurface. In such environments, salinity is typically high with σw > 1S/m, and as a result “membrane polarization” is less relevant leaving “electrode polarization” as the main signal for interpretation. For example, if the size distributions of the pyrite or other mineral deposits could be approximated from basin analysis, even just within an order of magnitude, it is possible to use Eq. (2) with the measured characteristic frequency and relaxation time scale to generate a subsurface map of surface-area-to-volume-ratio $${s}_{f} \sim \frac{{\sigma }_{m}}{\pi {C}_{0}}\frac{3}{{a}^{2}{f}_{c}}$$, thus providing key insights on the morphology of the pyrite grains or other mineral deposits in the depositional enviroments. Such information can then be used in conjunction with basin analysis and geo-bio-chemistry analysis to link to micro-biological activities and potential indirect connections to hydrocarbon seeps, potentially assisted with structural constraints from seismic stratigraphy and joint seismic-CSEM inversion. In an electromagnetic-based land survey 3 , 25 , 26 , 27 involving near surface fresh water, both “membrane polarization” 4 , 9 , 27 from clay, sand, and other ion-selective materials and “electrode polarization” from conductive inclusions play a role in this low salinity environment. As a result meaningful interpretations of induced polarization signals would typically require additional information such as clay content distribution so that IP response from “electrode polarization” can be decoupled from “membrane polarization” with a correct mechanistic model. However, in near surface geological settings with a known type of massive mineral/ore deposit 52 , 53 , 54 , “electrode polarization” could potentially be the primary source of the measured IP signals. This electrode polarization response can easily be quantified when the mineral grain size distribution is relatively narrow. In this case the grain size can be estimated from $$a=\frac{{\sigma }_{m}}{\pi {f}_{c}{C}_{0}}$$ with an inverted σm and mineral information (C0). The abundance or volume fractions of the conducting grains can also be directly tied to the maximum phase shift of IP response as in $${\phi }_{c}=\frac{9}{4}\frac{{V}_{cond}}{1+3{V}_{cond}}$$ regardless of the brine conductivity. If the size distribution is wide, the convoluted spectral IP response can also be used to estimate the size range and abundance of conductive grains based on equation- S2 in the SI. In principle, the above mentioned methods can also be used in offshore marine settings with the appropriate understanding of geological settings and sedimentary structures. Like all electromagnetic-based geophysical surveys, the electromagnetic and IP signal strength will be limited by energy dissipation in conductive media, as electromagnetic waves in subsurface attenuate exponentially with the travelling distance. This attenuation length or skin depth is of the form of $$\delta =\sqrt{\frac{1}{\pi \,f\,\mu {\sigma }_{m}}}$$ (σm is the effective conductivity of the porous media and μ is the magnetic permeability). Using this attenuation length we provide some perspectives to outline the detectability limitations of some specific materials (e.g. euhedral, framboidal and nanoporous pyrites) in different types of geophysical applications (e.g. offshore, on-land and borehole) For conventional framboidal pyrite typically existing in subsea organic-rich layers (SI-Text- S4 ) or oil/gas reservoirs 42 , the detectability from a basin-scale or borehole IP survey depends highly on their sedimentary environment and in particular the salinity. We have demonstrated that the characteristic frequency of their IP response is about 5000 Hz (SI-Text- S4 ) in high salinity (3 wt% NaCl) translating to a skin depth of about 7 meters. The small skin depth in high salinity environments makes such conventional framboidal pyrites difficult to detect with a basin scale offshore or land survey (km-scale or larger). In this scenario, proper IP or complex conductivity interpretation in borehole logging 59 can potentially take advantage of such a frequency range and skin depth and could be particularly useful in identifying the content and morphology of pyrite and other minerals in organic layers or oil/gas reservoirs. To contrast, in low salinity or freshwater 58 (similar to 0.03 wt% NaCl), the characteristic frequency of the same pyrites would be about 50 Hz leading to a skin depth of 700 meters due to both the frequency and conductivity changes. This large skin depth in the shallow freshwater condition could enable the acquisition and interpretation of meaningful IP response for identifying the abundance and potentially morphology of the shallow mineral species, assuming the minerals such as pyrite are relatively abundant in a local region. Unlike the conventional framboidal pyrites, pyrites with nanoporous morphologies are unique in the sense that even in marine environments with high salinity, these electronically conductive nanoporous structures can produce a uniquely low IP characteristic frequency (fc ∼ 1 Hz) resulting in a large skin depth over 1000 meters under representative conditions such as σm ∼ 0.2 S/m for brine saturated porous media. In addition, to our knowledge, the nanoporous layers demonstrated through natural samples 43 and our biomimetic experiments (Fig. 4 ) link directly to the anaerobic bacteria activity. Fed by hydrocarbons, these anaerobic bacteria respire sulfate and reduce it to H2S 51 leading to the formation of a nanoporous conductive layer made of FeS or pyrite (FeS2). Given sufficient abundance, in principle such pyrite with nanoporous morphologies can be detected through offshore marine surveys and land surveys. Furthermore in borehole logging, along with resistivity, density, neutron porosity and other data, induced polarization data can potentially shed light on the nanoporous pyrite content and morphology and differentiate them from conventional framboidal pyrite from the unique frequency response. Conclusion This study allows us to progress the interpretation of induced polarization measurements from the empirical and qualitative to the mechanistic and quantitative. We are now able to relate the measured spectral IP response to the abundance (volume fraction), morphology (size, and surface-area-to-volume-ratio) and intrinsic material properties (specific capacitance) of the conducting grains. The broad agreements between theory and experiments, and the applicability to time-domain analysis (See SI-Text- S3 ), suggest that our understanding can be readily applied to field-scale electromagnetic surveys for advanced IP mapping and interpretation. In geological settings, basin analysis can provide reliable insights into parameters such as porosity and brine concentration, and further analysis based on our IP method can be used to directly predict and map out useful mineralogical information, e.g., characteristic size and volume fraction of solid conductive inclusions. The formulation of the IP response from first principles presented here will enable development of structure-property relationships that can be used in partial-differential-equation based forward models and inversions, without resorting to empirical formulations that have made interpretation of induced polarization parameters difficult and subjective 1 . Furthermore, our discovery on the scaling of porous conductor suggests that a long relaxation time (~1 s) is a unique feature in sedimentary rocks containing nano-porous sulfides such as pyrite, which strongly correlates with the activity of sulfate reducing bacteria and hydrocarbon occurrence 6 , 7 , 14 , 15 , 43 . In addition to the broad geophysical applications, our discovery also leads to new strategies for surface area characterization of porous materials, super-capacitor design from biomimetic processes, and bio-inspired materials 47 . Methods Custom-designed four-probe measurement cell As shown in Figs. 1(a) and S2 , the gap between the bead pack with pyrite inclusions and the current electrodes is approximately 5 cm. The probe electrodes are made of gold coated copper with different geometries as in Fig. 1(a) : the two current electrodes are square meshes with spacing about 3 mm to guarantee the uniformity of an electric field in the bead pack while preventing the accumulation of small bubbles; the two voltage electrodes are ring-shaped and only touching the cylindrical cell on the rim, with minimal metal in the current path. Electronics National Instruments cards PXI-4461 and PXI-4462 (rates up to 200k Samples/s) are used respectively for signal generation and data acquisition along with National Instruments chassis (PXIe-1078). Two current electrodes of the measurement cell are connected to two symmetric 1000 Ω resistors that balance the circuits. We obtain the injected current by measuring the voltages across the 1000 Ω resistors, and all the voltages are measured differentially through SR560 voltage pre-amplifiers with 100 MΩ/25 pF input impedance to minimize the capacitance coupling signals. Materials Glass beads are purchased through Sigma-Aldrich; Cubic pyrites are purchased through Ward’s science; 304 and 316 stainless steel spheres are purchased through McMaster Carr. Sodium Chloride is purchased through Sigma-Aldrich. Conductor coatings 50 nm of Gold and platinum coating on various sizes of spheres are achieved by plasma sputtering deposition (Denton sputter coater Hummer X) with gold and platinum targets (99.99% pure from Anatech USA). The uniformity of coating on spherical objects is achieved by a wireless controlled shaking device inside the vacuum chamber during the coating process. Our theoretical derivation suggests that the polarization effect of a hollow metal sphere would be the same as the solid spherical metal, since the solid shell behaves as a Faraday cage and the polarization charge is present only at the outer metal surface. Surface-area-to-volume ratios for porous conductors with specific geometries For solid spheres, sf = 3/a and from Equation-2 we recover the earlier result with solid a conductive sphere. For isotropic-shaped porous objects (radius a) composed of electronically-connected small conductive spheres (radius rp) with a porosity of φ, $${s}_{f}=\frac{3(1-\varphi )}{{r}_{p}}$$. This consideration is also consistent with an increased surface capacitance $${C}_{s} \sim \frac{(1-\varphi ){a}^{3}}{{r}_{p}}{C}_{0}$$ for a porous conductor as compared to Cs ∼ a2C0 for a solid metal sphere, and therefore leads to a peak frequency $${f}_{c} \sim \frac{{\sigma }_{m}}{\pi {C}_{0}}\frac{{r}_{p}}{(1-\varphi ){a}^{2}}$$ for porous conductors. For a thin nano-porous conductive layer (average pore size < rp >, layer thickness h) on a solid metal bead (radius a), simple geometrical considerations suggest that the surface-area-to-volume ratio should be $${s}_{f} \sim \frac{9(1-\varphi )h}{ < {r}_{p} > a}$$ which leads to a peak frequency of $${f}_{c} \sim \frac{{\sigma }_{m}}{\pi a{C}_{0}}\frac{ < {r}_{p} > }{3(1-\varphi )h}$$. Nanoporous FeS scale from sulfate reduced bacteria coating The wild-type Desulfovibrio vulgaris (D. vulgaris) was obtained from Judy Wall at the University of Missouri. Freezer stocks of D. vulgaris containing 10% glycerol were used to inoculate overnight cultures. The growth media contains 30 mM lactate, 30 mM sulfate, 8 mM MgCl2, 20 mM NH4Cl, 2.2 mM phosphate buffer, 0.6 mM CaCl2, 24 mM NaCO3, 0.02% resazurin, 0.06 mM FeCl2, trace elements and Thauer’s vitamins, with pH adjusted to 7.2. Media was bubbled with 15% CO2 balance N2 and sodium dithionite was added immediately before inoculation to a final concentration of 1.5 mM. Overnight cultures of D. vulgaris were subcultured into serum bottles containing 80 ml of media and low-carbon steel balls (McMaster-Carr) and incubated at 30 °C overnight. After incubation, one set of carbon steel balls were transferred to 3 wt% NaCl solution for induced polarization measurement, while we process the other set with critical point drying and then use it for helium ion microscopy imaging. Correlating IP results to nanoporous FeS scale From our earlier theoretical calculation $${f}_{c} \sim \frac{{\sigma }_{m}}{\pi a{C}_{0}}$$, independently measured parameters (fc ∼ 3 Hz, h ∼ 5 μm, a = 0.8 mm, and C0 ∼ 30 μF/cm2 for pyrite), and by assuming a reasonable range of porosity (0.1 < φ < 0.5), we infer an average pore size < rp > ∼ 25–50 nm, demonstrating the consistency between induced polarization measurements and the helium ion images in Fig. 4(b) . References 1. Key, K., Constable, S., Liu, L. & Pommier, A. Electrical image of passive mantle upwelling beneath the northern East Pacific Rise. Nature 495, 499–502 (2013). 3. Vacquier, V., Holmes, C. R., Kintzinger, P. R. & Lavergne, M. Prospecting for ground water by induced electrical polarization. Geophysics 22, 660–687 (1957). 4. Marshall, D. J. & Madden, T. R. Induced polarization, a study of its causes. Geophysics 24, 790–816 (1959). Acknowledgements We would like to thank Mike B. Ray, Eric Herbolzheimer, Amy Herhold, Steven Meier, Paul Chaikin, Clifford Walters, John Valenza, Ken Desmond, Heather Elsen, William Horn, William Lamberti, Peter Ravikovitch, Simon Weston, Zara Summers, Giovanni Pilloni, Jack Johnson and Peter Jacobs for fruitful discussions. We would like to also thank British Ocean Sediment Core Research Facility (BOSCORF) for providing core samples from MD81 and JC77 research cruises, and Anna Lichtschlag at National Oceanography Centre, Southampton for the reference data from JC77 research cruise. In addition, we would like to thank Suzanne MacLachlan (National Oceanography Centre, Southampton) for fruitful discussion and assistance on handling core samples. Author information Competing interests We have a US patent application (US20180149020A1) on an engineering application of the concept disclosed here. We cited this patent application as a reference49 with a one-sentence description in the manuscript. We do not believe this is a competing interest, but do want to disclose such information to ensure full transparency. Additional information Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Supplementary information Rights and permissions Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ .

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## Porous Media Patents

Porous Media has filed 10 patents.

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11/24/2015

Marxian economics, Engine technology, National accounts, Gas technologies, Piping

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Application Date 7/23/2012 11/24/2015 Toroidal filter vessel system Marxian economics, Engine technology, National accounts, Gas technologies, Piping Grant