Morphology Decoder to Predict Heterogeneous Rock Permeability with Machine Learning Guided 3D Vision

Permeability has a dominant inﬂuence on the ﬂow behavior of a natural ﬂuid, and without proper quantiﬁcation, biological ﬂuids (Hydrocarbons) and water resources become waste. During the ﬁrst decades of the 21st century, permeability quantiﬁcation from nano-micro porous media images emerged, aided by 3D pore network ﬂow simulation, primarily using the Lattice Boltzmann simulator. Earth scientists realized that the simulation process holds millions of ﬂow dynamics calculations with accumulated errors and high computing power consumption. Therefore, accuracy and eﬃciency challenges obstruct planetary exploration. To eﬃciently, consistently predict permeability with high quality, we propose the Morphology Decoder. It is a parallel and serial ﬂow reconstruction of machine learning-driven semantically segmented heterogeneous rock texture images of 3D X-Ray Micro Computerized Tomography ( μ CT) and Nuclear Magnetic Resonance (MRI). For 3D vision, we introduce controllable-measurable-volume as new supervised semantic segmentation, in which a unique set of voxel intensity corresponds to grain and pore throat sizes. The morphology decoder demarks and aggregates the morphologies’ boundaries in a novel way to quantify permeability. The morphology decoder method consists of ﬁve novel processes, which we describe in this paper, these novel processes are (1) Geometrical: 3D Permeability Governing Equation, (2) Machine Learning: Guided 3D Properties Recognition of Rock Morphology, (3) Analytical: 3D Image Properties Integration Model for Permeability, (4) Experimental: MRI Permeability Imager, and (5) Morphology Decoder (the process that integrates the other four novel processes).

Although CNN identified pore size distribution, we observed limited application to 1D nuclear magnetic resonance data or 2D μCT images.A 2D image analysis of heterogeneous Cretaceous morphology produces a localized description [28] and does not represent the whole 3D morphological structure.None of the vendors delivered an acceptable permeability value when testing four commercial computer vision models [29] for permeability determination using the 2D image.The use of CNN for pore size identification and permeability prediction from 3D μCT is progressing [30]; however, its focus is on homogenous sandstone [30,31].On the contrary, Cretaceous carbonate is heterogeneous due to fossils (we also use the term "bioclast" interchangeably in this paper) contents.At the same time, diagenesis impacts Cretaceous carbonate texture [32].Carbonate heterogeneity made it complicated for CNN to analyze its morphology compared to sandstone [31].
In our research, to overcome the challenges of pore network simulation and solve heterogeneous porous media properties quantification challenges, we introduce Morphology Decoder (MorphD).It is an interdisciplinary computer vision-guided physics for fluid flow properties quantification.MorphD is a standalone technique -non-simulation based -predicts permeability using 3D vision [21].MorphD ensembles image processing, machine learning, 3D printing, 3D μCT, and MRI vision.Rather than the conventional interpore connectivity [33][34][35] of the pore network, MorphD builds a pore throat network (PorThN) by segmenting 3D μCT intensity of comparable morphology using machine learning algorithms [36,37].Then MorphD identifies -with MRI intensity [38,39] -the impact of pore throat size (PorTS) on the control volume boundary of the 3D segmented section [40][41][42].

1-Geometrical -Novel 3D Permeability
In homogenous texture, permeability depends on PorTS, while in heterogeneous one [38], permeability depends on the pore throat network.We define a pore throat network (PorThN) as a heterogeneous texture with multi pore throat sizes distributed in a specific system to produce a unique morphology.In exploring research options for identifying pore throat sizes and PorThN, we identified five possible paths: analytical, machine learning, image processing, experimental, and simulation (Fig. 2).We focused on the geometrical analysis [61][62][63][64].We target understanding the physics of the main factors that derive permeability.Machine learning provides an efficient and consistent quality process in segmenting a mass of 3D image data [65,66].Image processing eases the interpretation of machine learning outcomes to produce usable quantitative results [67,68].While experiments provide a calibration and validation assurance of our analytical, machine learning, and image processing approaches.We omitted the use of simulation to free researchers from simulation dependency; instead, we quantify permeability with MorphD.The analytical path focuses on the 2D and 3D geometrical analysis of objects.It describes the grain and the texture to determine PorTS and network.We start first with a fundamental step of analyzing a homogenous geometry.Then we add further complexities to analyze heterogeneous morphology, which contains more than one pore geometry.Finally, we apply the learning to analyze natural Cretaceous carbonate.Intergranular and intragranular pores [69] are two different geometrical systems that Cretaceous carbonate comprises.
We start from the homogenous cubic structure of well-sorted spherical grains displayed in 2D projection [38,[70][71][72][73] (Fig. 3).In Fig. 3, we see that the edges of the circle are also the edges of the pore throat.We deduced Axiom-0; a pore throat is a plane with enclosure from all directions.The red square is in the X-Y plane, and the image analysis moves perpendicular to this plane.The red square also represents the largest plane of the pore.The pore size is the square lateral, equivalent to a circle diameter.
The yellow concaved diamond is the pore throat.Imagine the red square and yellow concaved diamond shape in 3D; it is an arched pyramidal structure.The arch pyramid has a square base, four sides that are one-eighth of a sphere, and a concaved diamond top.
We show the fundamental geometrical analysis in Eq. 1, determining the pore area in 2D at the largest plane.The corners of this plane are the four points of contact of the eight spheres: where,  !: The grain size (grain radius).
In Eq. 2, we show the area determination of PorTS in 2D (plane view of Fig. 3): The relation between the area of pore size and PorTS in 2D is the ratio between Eq. 1 and 2: The natural grain-based porous system is of 3D configuration, not 2D, and this requires calculating the volume rather than the area.In Fig. 4, A, we see the same porous system of Fig. 3, but in 3D.The full 3D representation of the pore and pore throat is in Fig. 4, B.
Porosity is the void volume ratio to an object's total bulk volume (void + solid), so it is dimensionless property (volume/volume).Porosity does not change with the change of grain size.Still, it varies only with the grain configuration (i.e., sorting and compaction), determining the ratio transformation between the void and the total bulk volume [38].The porosity of our Cretaceous sample ranges from 0.18 to 0.32.This range gives an average porosity of 0.25 for the pore structure of rhombohedral configuration, as shown in Fig. 4, C [38,61].It is also called rhombohedral-pyramidal [74], and in this paper, we call it rhombohedral.The porosity value does not change with grain size change if the grain configuration remains the same, Fig. 4, D.
The PorTS changes with grain size change, Fig.The grain size impacts the pore throat radius, Fig. 4, D, also shown in Eq. 2. We derive the equation for the rhombohedral configuration in Fig. 4, C, to determine PorTS.We start with a more straightforward structure than rhombohedral, triclinic, shown in Fig. 5, A, a 2D illustration of three spheres.We derived the equations that govern the pore size and PorTS below from Fig. 5, A, and B: We notice that Eq. 5 represents a 2D  "#$%&$'$% , while our desired geometrical configuration is a 3D rhombohedral configuration (Fig. 4, C), which holds more complexity than triclinic (Fig. 5, A).The 3D cubic configuration of eight spheres shown in Fig. 4, B consists of six faces: top, bottom, and four slides.A pore throat shape of a concaved diamond on each face, like the yellow area shown in Fig. 3. Therefore, the 3D pore throat area of cubic configuration is the sum of six concaved diamonds areas, Eq. 2, to be 5.148 ! ( . Then we calculate the Effective 3D Pore Throat Size of cubic configuration ( %)*$% +, -../%0$1/ ) as shown in Eq. 6 below: where, The area of all pore throats of cubic configuration,  The 3D triclinic configuration of eight spheres consists of six faces that hold two different shapes of pore throats; the top, bottom, and two sides have a pore throat shape of a concaved diamond.The other two sides hold a concaved triangle pore throat shape: two-pore throats per side.Therefore, the 3D pore throat area of triclinic configuration is the sum of four concaved diamonds and four concaved triangles, Eq. 2 and 5 to be 4.08 ! ( .Then we calculate the Effective 3D Pore Throat Size of triclinic configuration ( 0#$%&$'$% +, -../%0$1/ ) as shown in Eq. 7, below: where, The area of all pore throats for triclinic configuration.

Grain
Pore Throat

A B
The 3D rhombohedral configuration of eight spheres consists of six faces that hold two different shapes of pore throats; the top and bottom faces hold a pore throat shape of a concaved diamond.The four sides have a concaved triangle pore throat shape: two-pore throats per side.
We rewrite Eq. 8 in terms of grain surface area, as shown in Eq. 9 below: Permeability is a resultant of both grain size and grain configuration to form a proportional relation between permeability and grain surface area [58] " this physical aspect of permeability has been used to create empirical equations for prediction of permeability," as described below in Eq. 10: For 3D rhombohedral configuration, we determine the permeability by substituting Eq. 9 and Eq. 10 to produce Eq.11 as shown below: +, #/"01"/-2#3* =  #>3?*3>/@#A& +, -../%0$1/ = 0.0858 ! ( We display Eq. 11, with experimental data [39] on different grain sizes and permeability with poorly sorted grains (rhombohedral), as shown in Fig. 6.

2-Machine Learning -Novel 3D Properties Recognition of Carbonate Morphology
The main morphology decoder deliverable is the permeability of heterogeneous Cretaceous rock (kHeC).One of the cornerstones of morphology decoder for determining kHeC is Eq.11.
Another critical cornerstone is differentiating heterogeneity zones (DHZ), where machine learning-based computer vision plays the role of deciding DHZ for classifying minerals [75] with the 3D μCT [21,76] images.In Cretaceous carbonate, more than one mineral exists, and the volumes of these minerals are quantifiable [77], in our case, calcite, and pyrite [21].Fig. 7, A, shows the original μCT image with 28 um resolution.To identify DHZ, we run machine learning with the Random Forest (RF) algorithm to perform image recognition of different rock sections [21].After conducting comparisons between several algorithms, we chose the RF algorithm because RF proved to be the most suitable one with the highest accuracy [78].Fig. 7, B, shows the training image with four critical features -DHZs.We display the classification results in Fig. 7, C; with Pyrite (red color), Open Vugs (Green color), Intergranular-1 (Pink color), and Intergranular-2 Bioclast (Yellow color).We separate each DHZ geometrically as a discrete block to create the controllable-measurable-volume (CMV) as shown in Fig. 7, D -G.

3-Analytical -Novel 3D Vision Property Integration Model for Permeability
We innovated a new permeability aggregation process for the 3D image stack.We aggregate the permeability using parallel and serial permeability equations [58] shown in Eq. 12 and 13 (Fig. 8, A) to produce the general 3D permeability equation of heterogeneous rock, kHeC.We call this aggregation; 3D Property Integration Model (3DPIM).The steps for achieving 3DPIM starts keeping the flow direction (the arrows in Fig. 8, B) perpendicular to the Control Volume ( ∀ ) boundary.In the next step of 3DPIM, we integrate the Permeability of each 2D slice (x-y plane), using the parallel permeability equation, Eq. 13, to calculate the slice permeability.The last step of 3DPIM is integrating the z-axis direction for the 3D stack using the serial equation, Eq. 12, Fig. 8, C.  To validate our novel 3DPIM model, we designed a 3D porous media (3DPM) conceptual model, shown in Fig. 9 top (Conceptual Design).We draw five different 3DPM configurations, utilizing two mesh sizes (mesh inner laterals): 2000 um (the dark blue color mesh shown in Fig. 9) and 4000 um (the green color mesh shown in Fig. 9).These configurations reflect homogeneity and heterogeneity with two types of homogenous rocks -Sample 1 and 2 -shown in Fig. 9, one serial configuration of two homogenous rocks -Sample 3 -shown in Fig. 9, one parallel configuration of two homogenous rocks -Sample 4 -shown in Fig. 9, and heterogeneous rock of arbitrary distribution of two rock types -Sample 5 -shown in Fig. 9.
The inner laterals sizes demonstrate a double difference in size; this helps us accurately differentiate the flow effects.The outer dimensions of 3DPM cylinders are 7.8 cm x 3.8 cm (length x Diameter), a size that can fit the flooding apparatus we use for permeability measurement.The conceptual model had its final 3D drawing engineered with Computer-Aided Design (CAD) software, as shown in Fig. 9, the second from the top (3D Computer Design).Then we 3D-printed 3DPM five cylinders with a polymer material to have the final physical look of the cylinders shown in Fig. 9, the second from the bottom (3D Printed).Then we measured the five 3DPM for determining permeability by equipping each cylinder with a rubber sleeve and injecting air at 35 psi (~241 KPa) sleeve conformance pressure.The conformance pressure we used is the maximum pressure the 3DPM can hold before deformation occurs to the cylinders.We also calculated the permeability using Eq. 12 and 13.
Then we validated the calculated permeability with the measured permeability, as shown in Fig. 9, the bottom (Samples Measured and Predicted), to prove our novel 3DPIM model's ability to quantify permeability with high confidence.

Fig. 9. The 3D aggregation model validation with 3D printed samples of various mesh sizes and
configurations.We compare the predicted (calculated using our novel 3DPIM model) and the physically measured permeability.

4-Experimental -Novel MRI Permeability Imager
We proposed MRI as a grain radius quantifier.MRI can sense the fluid (containing hydrogen) rather than anything else.Nuclear magnetic relaxation time (T2) value differs for different pore sizes.In MRI measurement, the larger the pore, the slower the T2 [79].Despite all the progress in NMR and MRI technology, there has not been any published work that shows a direct relation between MRI Image Intensity (MRIII) and grain size with rhombohedral configuration until writing this paper.Earlier in this paper, we proved the direct relation between pore size and throat size.We also demonstrated the direct link between pore throat size and grain size of rhombohedral configuration.Then we established the direct relation between grain size and permeability of rhombohedral configuration.Therefore, we targeted the ability to measure the grain size using MRIII for the first time and human history in this research part.We built a new laboratory apparatus specifically for this purpose.We used an NMR system of 0.5 Tesla.The MRI instrument setup consists of three main parts, the magnetic field with the core holder, the NMR radio frequency and temperature controller, and the 3-phase pressurized flooding system, as shown in Fig. 10.We then calibrate MRIII quantification of grain size with rhombohedral configuration.This step is vital to building a novel model that enables humanity to measure grain size using MRIII.
We select a narrower range of MRIII measurement (x-y axis) for three distant sizes of glass beads: 1500 um, 400 um, and 50 um, (Fig. 12).Then we plug the grain diameter value of 68.82 um on Eq. 11 to calculate  +, #/"01"/-2#3* for this natural rock sample to find permeability value to be 113 mD.To validate the 113 mD permeability value, which is the result of our innovative process of Morphology Decoder, we have measured the natural rock sample at an independent industry laboratory.The laboratory measurement confirms 126 mD, which is considered a close match to the MorphD permeability value.The oil and gas industry's accepted practice is to have a matching accuracy of less than 0.5 of a logarithmic decade scale.We achieved less than 0.013 of a logarithmic decade scale in our case.We map the Morphology Decoder algorithm (Fig. 15) as a summary road map from imaging to machine learning to CMV to the physics analytical model to experimental calibration and validation.

Conclusion
This research has targeted delivering a novel quantification of simulation-free permeability.
An innovative interdisciplinary method, the Morphology Decoder, is the outcome of this research.For developing the Morphology Decoder, we established several interlinks between physical and chemical properties of heterogeneous carbonate rock, supported by geometrical analysis, machine learning, 3D printed calibrator experiments, and analytical derivations, some of which we conclude below: 1-This paper proved several new interlinks between various physical properties: a. We have established the link between grain configuration and pore throat size, Eq. 9.
b.We have established the link between pore throat size and permeability, Eq. 11.
2-Use these established links of (1) above to interpret the images acquired by μCT and MRI to deliver a novel quantification method of natural rock permeability.
3-We created a novel computer vision semantic segmentation representation, Controllable Measurable Volume CMV, guided with calibration models, to have a novel 3D image flow property aggregation, leading to quantifying permeability.

4-Morphology
Decoder is an efficient method for developing and interpreting new generation measuring devices.Morphology Decoder-based logging tools (wireline, logging while drilling, logging while coring, permanent sensors, and macro-micro-nano sensing robots [80]) is a solution for autonomous planetary exploration in Earth, Moon, and Mars.

Fig. 2 .
Fig. 2. Pore throat identification options.Green highlighted boxes are research approaches we considered,

Fig. 3 .
Fig. 3. Handcrafted 2D projection of four spherical grains representing one of the most homogeneous sphere-

Fig. 4 .
Fig. 4. Different 3D configurations of spherical grains.(A) Four spherical grains in 3D represent homogeneous 23#": The number of pore throats in a 3D configuration, 4 ∀ $'/05: The number of outlets of the fluid flow control volume.

Fig. 5 .
Fig. 5. Triclinic configuration of spherical grains.(A)The red triangle shows the pore size, while the yellow

Fig. 6 .
Fig. 6.Experimental and analytical permeability determination.The validation of permeability determination

Fig 10 .
Fig 10.Bench Top NMR and MRI Analyzer with Pressurized Flooding System.

Fig. 12 .
Fig. 12. MRI Image Intensity (MRIII) calibration for quantifying grain size using three sizes of Glass Beads.

Fig. 15 .
Fig. 15.Morphology Decoder.A road map shows the novel protocol for quantifying natural rock permeability