Robust evaluation of flow front data for in-plane permeability characterization by radial flow experiments

Abstract A novel approach is presented for modeling the temporally advancing fluid flow front in radial flow experiments for in-plane permeability characterization of reinforcing fabrics. The method is based on fitting an elliptic paraboloid to the flow front data collected throughout such an experiment. This “paraboloid” approach is compared to the conventional “ellipse” method and validated by means of data sets of optically tracked experiments from two different research institutions. A detailed discussion of the results reveals the benefits of the “paraboloid” method in terms of numerical efficiency as well robustness against temporal or local data variations. The “paraboloid” method is tested on temporally and spatially limited data sets from a testrig involving linear capacitive sensors. There, the method shows advantages over the conventional approach as it incorporates the entirety of available measurement data, particularly in the last stages of the experiments which are most characteristic for the material under test.


Introduction
In liquid composite molding, dry preforms of reinforcing fabrics are placed in a mold and then impregnated with the liquid polymer matrix material. The impregnation process plays a key role as insufficiently saturated regions directly affect the mechanical properties of the final component. In order to avoid elaborate and expensive impregnation trials, filling simulations can be conducted. These simulations strongly rely on accurate and reliable information on the permeability of the fibrous reinforcement. In the late 1990s, Weitzenböck et al. [1][2][3] introduced a framework for experimental permeability characterization techniques. For in-plane permeability characterization, the "rectilinear flow" (or "channel flow") method and the "radial flow" method were distinguished. The channel flow method is based on onedimensional fluid flow through the reinforcing structure placed in a specifically designed characterization cell. In general, this method has two major disadvantages: (1) Race tracking 4 is very likely to occur in gaps along the side edges of the material, deteriorating the rectilinear flow front advancement. Thus, specific care is required, typically by sealing the preform material at the side walls of the characterization cell. 5,6 (2) At least three experiments are required to fully characterize the in-plane permeability tensor of the material under test. This, by nature, causes

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increased experimental effort compared to the radial flow method. Application of "multi-cavity parallel flow cells", which enable the simultaneous characterization of samples cut along different material directions, was proposed by various authors [7][8][9] in order to reduce experimental efforts.
In contrast to the linear flow technique, the radial flow method allows for a full characterization of the in-plane permeability tensor from a single experiment and avoids race tracking effects on principle. The technique is based on the observation of radial flow experiments as introduced by Adams et al., [10][11][12][13] which enable strictly planar fluid flow as a result of a circular injection opening punched into the preform stack under test. The experiments then comprise three major aspects: (1) Flow front tracking: The radially advancing flow front is typically tracked by means of a camera system as originally proposed by Adams et al. 10,14 For this, the optically transparent mold half has to be stiffened, e.g. by bars or cross-beam structures, in order to avoid extensive mold deflection during the experiments. Alternatively, two metal mold halves can be used, asking for alternative flow front tracking techniques. Liu et al. 15 presented a cell with a star-like arrangement of line arrays of electrically sensitive point sensors. These are used to derive trigger signals as soon as the flow front reaches a sensor position. Morren et al. 16 and Hoes et al. 17 showed works with a similar approach based on arrays of dielectric sensors. Kissinger et al. 18 presented a system involving linear capacitive sensors in a star-like setup around the central injection opening. These sensors are calibrated in such manner that a linear relationship of sensor signal and saturated sensor area is obtained. (2) Flow front modeling: Typically, the overall shape of the flow front at a specific point in time during the experiment is reconstructed with an elliptic geometric model fitted to the flow front data. Evaluating the major and minor ellipse axes lengths throughout the experiment results in the temporal flow front characteristics required for the subsequent calculation of in-plane permeability values. Surprisingly few information is available in existing literature on the specific ellipse fitting approaches used. This might be due to the common availability of routines for fitting elliptic geometry models to noisy measurement data in standard data evaluation software. However, the choice of elliptic geometry model fitted to the data can easily have an impact on the resulting in-plane permeability values as reported by Fauster et al. 19 (3) Computation of in-plane permeability data: this step involves evaluation strategies in combination with specifically developed mathematical algorithms. Adams and Rebenfeld 11 presented an algorithm, which is based on transforming the problem to an elliptical coordinate system and uses an iterative numerical solution for the degree of anisotropy, i.e. the ratio of the two principal planar permeability values. Chan and Hwang 20 presented an algorithm which introduces a transformation of the anisotropic problem into an equivalent isotropic system (EIS) following principles originally introduced by Bear et al. 21,22 A modification of this algorithm was presented by Fauster et al. 19 which avoids a violation of the isotropic data properties in the EIS inherent to the original method. Finally, a number of evaluation strategies can be distinguished as discussed by Ferland et al. 23 for processing the data acquired during the experiments.
The paper at hand focusses on the aspect of flow front modeling by introducing a novel approach for processing flow front data acquired during radial flow experiments. In particular, the approach: (1) ensures increased numerical efficiency as the entirety of flow front data is modelled by means of an elliptic paraboloid in a single step approach, (2) provides a direct method for deriving a global value for the orientation angle of the principal permeability directions with respect to a testrig-specific coordinate frame and (3) is particularly beneficial for modelling data, which lack completeness in terms of temporal and/or spatial resolution:

Basics of flow in porous media and geometric considerations
Fluid flow in fibrous, i.e. porous, structures is commonly modeled according to the fundamental work of Henry Darcy, 24 whose empirical findings are commonly represented as a relation between the flow velocity v and the driving pressure gradient ∇p. It can be reproduced as: with the dynamic fluid viscosity as well as the permeability tensor . While Neuman 25 analytically verified the correspondence of Darcy's law with the Navier-Stokes equations, Liakopoulos 26 proved the permeability tensor to be symmetric and of second order. For planar fluid flow in anisotropic porous media, such as in an in-plane permeability characterization cell, the tensor is: However, by transformation to a coordinate frame aligned with the principal axes of fluid flow, the tensor simplifies to orthotropic shape: with = This second order partial differential equation does not show a closed form solution, however for flow in isotropic porous media, i.e. = 1, Equation (4) simplifies to the well-known Laplace differential equation, which can be solved analytically.

Flow in isotropic porous media
For fluid flow in isotropic porous media, Adams et al. 11 reported an analytical solution for the resulting radially symmetric pressure distribution. This in turn enables the derivation of an analytic solution for the temporally advancing radial flow front extent: with the radius r 0 of the injection opening and the pressure difference Δp between the injection point and the flow front. Furthermore, the following three material anisotr. = k x k xy k yx k y , with: k yx = k xy .
(3) orthotr. = properties are included: fluid viscosity , fabric porosity ε and isotropic fabric permeability k. Although widely covered in literature, the mathematical derivations are briefly reproduced in Appendix 1 for the sake of completeness and consistency. Equation (5) can be reorganized to: with the nonlinear function f (r f ) = 2 ln r f and the constant coefficients: c 3 = −2 ln r 0 − 1, c 4 = − 2kΔp and c 6 = r 2 0 (the reason for the unordinary choice of coefficient indices c k will be clarified at the end of this section). Studying the two terms involving r 2 f in Equation (6) in a range relevant for typical in-plane permeability characterization cells (0 < r f < 0.25 m) reveals that the constant coefficient term, c 3 r 2 f , dominates over the nonlinear function term, f (r f )r 2 f , as is shown in Figure 1. This finding is of general validity, as (a) the coefficient c 3 as well as the function f (r f ) are independent of the material properties and (b) the only experimental parameter contained, the radius r 0 of the injection opening, shows insignificant variations in laboratory practice, typically being in the range: 5 < r 0 < 10 mm. Figure 1 additionally illustrates that in the given range of r f , the sum of the two terms can be reasonably well approximated by a single, constant coefficient term: with the empirical scaling constant, 0 < s < 1. This indicates that the temporal advancement of the radially symmetric flow front can be well approximated by a simple equation of second order: Comparison with the well-known general equation of quadratic forms, 27 now reveals the reason for the choice of coefficients indices c k and more importantly, the fundamental nature of Equation (5): The temporal advancement of the fluid flow front in isotropic porous media exhibits the characteristics of a parabola, symmetric to the time axis. Obviously, the approximation of the flow front advancement described by Equation (7) could be improved by fitting even-degree polynomials of higher order to the sum of the two functional terms. However, this would add computational costs to the fitting task and raise questions for the subsequent steps of data processing, which require further investigations.
(9) c 1 x 2 + 2c 2 xy + c 3 y 2 + 2c 4 x + 2c 5 y + c 6 = 0, However, some deviations can be observed especially in the early stages of the experiments. These are due to the parabolic fit being dominated by the large number of data points available in the later stages of the experiments (due to increasing flow ellipse circumference), which show rather low curvature characteristics. From a strictly mathematical point of view, this effect could easily be avoided, e.g. by weighting techniques. From a flow dynamics point of view however, the early stages of radial flow experiments are dominated by starting effects, where the flow front migrates from the shape of the circular injection opening to an elliptic shape specific to the material under test. Hence, this stage does actually not reflect the true material behavior. Thus, the deviations of the fitted parabolae from the measurement data

Validity for flow in anisotropic porous media
The derivation given in the previous section refers to fluid flow in isotropic porous media, however, the major finding of the parabolic nature inherent to the temporally advancing flow front extent is equally valid for fluid flow in anisotropic porous media. Although an analytic proof cannot be given here (the differential equation for the pressure distribution in anisotropic media does not show a closed form solution), empirical evidence is possible. As visualized in Figure 2 by means of exemplarily chosen experimental data, parabolic geometry models (Appendix 2 briefly reflects the mathematical basics) can be fitted to major and minor radial extent data r 1 (t) and r 2 (t), respectively, with a high level of accuracy.  In order to derive affine paraboloid parameters from the set of homogeneous quadric coefficients, principal component analysis (PCA, 29 ) is applied. The smallest eigenvalue 1 of the upper-left 3 × 3 submatrix Q 33 turns out to be zero as a result of the infinite length of the primary paraboloid axis. The corresponding eigenvector e 1 = [0 0 1] T indicates its direction, i.e. the z-axis. Eigenvector e 2 = [e 21 e 22 0] T corresponding to the second-smallest eigenvalue 2 of Q 33 is pointing towards the major vertex of the elliptic paraboloid cross-section. Thus, the orientation angle of the paraboloid with respect to the global x-axis can be directly derived from its components: Rotating the paraboloid around the z-axis by aligns its axes with the coordinate frame and thus, diagonalizes Q 33 . This in turn results in the following implicit quadric equation: in the early stages of the radial flow experiments can be considered uncritical as the data characteristics as well as the quality of the parabola fit in the later stages of the experiments are of higher relevance for characterizing the in-plane permeability of the material under test. Note that the parabolic models shown in Figure 2 are fitted to the radial extent data r 1 (t) and r 2 (t) while enforcing a common parabola vertex. This in turn is located at a time t V < 0 because the fit is dominated by the low curvature data present in the later stages of the experiments. The fitting is mathematically realized by coupling the geometry models according to a principle proposed by O'Leary et al. 28 Due to this choice of specifically coupled parabolae, the fitting implicitly presumes the initiation of the fluid flow from an infinitely small injection point. In addition, the resulting coupled parabolae are in line with a newly proposed data evaluation approach as presented in the subsequent section.

Global analysis of temporally advancing radial extend data
Applying the approach presented above to a three-dimensional coordinate frame with the x-and y-axes adopted from the testrig running the radial flow experiments and the experimental time chosen along the z-axis, the entirety of flow front data points collected throughout the experiment can be modeled by an elliptic paraboloid as schematically illustrated in Figure 3.
Effectively, the fitted elliptic paraboloid is chosen to show the following properties: • the primary axis of the paraboloid is aligned with the z-axis, • the vertex is located on the z-axis at a distance z V from the origin and • the major axis of the elliptic cross-section of the paraboloid is oriented at an angle with respect to the x-z-plane.
Note that the vertex distance z V is a result of the paraboloid fit being dominated by the low curvature data in the later stages of the radial flow experiments. Forcing the paraboloid through the boundary of the circular injection hole would be an alternative option as it would reflect the initial condition for the flow into the fibrous preform. However, this would lead to a circular ellipsoid, which is inadequate to cover the radially asymmetric characteristics of planar fluid flow in anisotropic media. Type-specific fitting of an elliptic paraboloid (Appendix 3 briefly reproduces the mathematical basics) to flow front data points collected throughout a radial flow experiment enables the derivation of global fabric properties in a single-step approach, which is termed as "global" data evaluation method for the remainder of this work. The fitting result is a quadric represented as a matrix with five non-zero quadric coefficients: This result is of major importance, as it reveals that the degree of anisotropy: (1) is independent of time and (2) can be analytically computed from the ratio of parabola coefficients.
Hence, a direct and analytical method for computing the degree of anisotropy is available which avoids the need for an iterative optimization procedure as proposed in the original work of Adams and Rebenfeld. 12 This results in a significant simplification of the algorithm and a reduction of computational costs.

Experimental work
Following the approach presented in the previous section, experimental data from three different test rigs for in-plane permeability characterization following the radial flow method was evaluated. Due to space limitations, the particular testrigs are not described here in detail. However, the interested reader is referred to relevant literature previously published by the authors: (1) optical permeameter at LVV, [30][31][32][33][34][35] (2) optical permeameter available at PuK 36,37 and (3) capacitive permeameter systems at IVW and LVV. 18,38

Validation of the evaluation method on optically tracked radial flow experiments
In-plane permeability characterization through optical observation of radial flow experiments involves at least one optically transparent mold half. In order to avoid extensive mold deflection, specific arrangements for increased structural stiffness are required. However, these elements cause partial occlusions of the fluid flow Intersecting ̂ with the x-z-plane yields the equation of the major intersecting parabola: Analogously, the minor intersecting parabola is obtained by intersecting ̂ with the y-z-plane: Figure 4 visualizes a schematic of an elliptic paraboloid and its intersections with the x-zand y-z-planes, respectively, together with the major and minor parabolae as intersecting curves. These intersecting parabolae can be further processed towards fabric in-plane permeability as will be shown in the subsequent section.

In-plane permeability calculation
Adams and Rebenfeld 12 introduced an algorithm for calculating in-plane permeability values from given sets of radial extent data, which is based on an iterative numerical solution for the degree of anisotropy .
Considering the analytic expressions for the major and minor parabolae in Equation (13) and (14) with x, y and z corresponding to r 1 , r 2 and t, respectively, we find: and thus: (12) :q 1 x 2 +q 3 y 2 + 2q 9 z + q 10 = 0.
(14) q 3 y 2 + 2q 9 z + q 10 = 0.  compared as determined according to the following two methods: (1) the conventional "ellipse" method, where elliptic geometry models are individually fitted to the flow front data extracted from the particular sequence images, and (2) the newly proposed "paraboloid" method, where a single elliptic paraboloid model is fitted to the entirety of available flow front data points as visualized in Figure 8.

Comparison of radial extent data
In Figure 9, the temporal characteristics of radial extent data r 1 (t) and r 2 (t) are compared. The data obtained with the "ellipse" method indicate the major and minor semi-axis length of the ellipse fitted to flow front data at a particular point of time. By contrast, the data of the "paraboloid" method are to be understood as major and minor elliptic extent of the paraboloid fitted to the data points available up to a specific point of time. The data characteristics are more or less perfectly coinciding.
front as it advances during the radial flow experiments. These occlusions have to be handled with digital image processing techniques such as application of mask images. 39,40 The sequence images acquired at the test rig of LVV ( Figure 5, left) show a geometrically simple occluded region in terms of a single stiffening bar, vertically oriented in the central portion of the images. As a result, the amount of data points available to each side of the stiffening frame is well balanced, but obviously increasing over experimental time as depicted in Figure 6.
The images acquired with the test rig of PuK ( Figure 5, right) show variations in the data point balance as the location and degree of flow front occlusion is varying during the experiment as visualized in the sequence images shown in Figure 7.
In order to validate the data evaluation approach proposed in this paper, two exemplarily chosen data sets were evaluated from optically tracked radial flow experiments conducted independently at LVV and PuK, respectively. In particular (a) radial extent data, (b) orientation angle and (c) in-plane permeability values are   experiments as shown in Figure 8, its orientation angle (which corresponds to the very last value of the temporal characteristics of "paraboloid" fits depicted in Figure 10) is very well in line with the final trend of the temporal characteristics derived with the "ellipse" method. This is in distinct contrast to the average of orientation angle characteristics derived with the "ellipse" method, which is clearly impacted by the strong variations in the early stages of the experiment. Although both approaches involve some kind of averaging effects, these are clearly different: The elliptic paraboloid fit involves the entirety of data points found along the temporally advancing fluid flow front and thus, inherently adds a stronger weight to the data in the later stages of the experiment as a result of the increasing amount of available data points. By contrast, averaging the temporal characteristics of orientation angles from the "ellipse" method provides equal weight to all of the values obtained throughout the experiment. Consequently, the results not only support the validity of the newly proposed data evaluation method, they also indicate a clear advantage: Figure 10 shows a comparison of orientation angle data (t) as determined with the two methods. The temporal characteristics obtained with the "ellipse" method exhibit a higher degree of scatter compared to those of the "paraboloid" method. This is a result of the amount of data points involved in the particular fitting tasks: while the number of data points is slightly increasing over time in the "ellipse" method, it is rising in a cumulative manner in the "paraboloid" method.

Comparison of orientation angles
Furthermore, it can be clearly seen that the characteristics of the orientation angles are poorly defined in the early stages of the radial flow experiments and thus, show particularly strong variations. However, the characteristics trend towards a constant level in the later stages of the experiments. This transition reflects the flow front migration from an initially circular boundary to the elliptic shape characteristic for the material under test.
Considering the elliptic paraboloid fitted to the entirety of flow front data points collected during the  start, i.e. t single,k = {t 1 , t k } with k = {2 … n} and the total number of n time steps.
(2) The "elementary" method: Again, a multi-step approach is followed, whereas each evaluation step involves measurement data from a pair of consecutive time steps: t elem,k = {t k−1 , t k } with k = {2 … n}. (3) The "interpolation" method: For each evaluation step, the entire set of measurement data acquired up to this point of time is considered, i.e. t interp,k = {t 1 , t 2 , … , t k } with k = {2 … n}. Figure 11 shows a comparison of temporal characteristics of major and minor in-plane permeability data computed according to the "interpolation" strategy based on radial extent data with the "ellipse" and the "paraboloid" method, respectively. The characteristics are more or less perfectly coinciding, which is not surprising as a result of the matching characteristics of radial extend data (see Figure 9).
While specific care is required when averaging the temporal characteristics of orientation angles found with the "ellipse" method, the "paraboloid" method robustly provides a representative value as a direct output.

Comparison of in-plane permeability values
The validity of the newly proposed "paraboloid" method is further demonstrated by computing major and minor in-plane permeability values k 1 and k 2 from the temporal characteristics of radial extent data r 1 (t) and r 2 (t), respectively, and by comparison with results obtained from the "ellipse" method. Following the systematics introduced by Ferland et al. 23 for the computation of in-plane permeability data from channel flow experiments, three different strategies can be distinguished: (1) The "single point" strategy: The entire set of measurement data is evaluated by relating data from each point of time of the experiment to its  Following the "ellipse" evaluation method, the shape of the flow front can be reconstructed at each point of time during the experiment by fitting an elliptic geometry model to the sensor data. However, this requires to restrict the measurement data to a portion showing valid data on at least three or five sensors, depending on the choice of ellipse model being fitted. 19 In other words, the measurement data needs to be cut when the required minimum of valid data points is reached and thus, a significant portion of measurement data from the lastand thus most accurate -stage of the radial flow experiment is excluded from being processed towards in-plane permeability data. By contrast, the "paraboloid" method proposed in this work allows for making full use of the available measurement data as the elliptic paraboloid can be fitted to the entirety of flow front data collected throughout the experiment without any restrictions as shown in Figure 13, right.

Comparison of radial extent data and orientation angle
Similar to the comparative analysis presented for the optically tracked experiments, the radial extent data r 1 (t) and r 2 (t), respectively, as well as the orientation angle (t) obtained with the "ellipse" and "paraboloid" methods are compared as shown in Figure 14.
The characteristics of major and minor radial extent data clearly reveal the impact of the available number of data points on the radial extent values found with the "ellipse" method. As the number of valid sensor data reduces from eight to six at t ≈ 60 s, the data characteristics (particularly the major radial extent) show significant changes in the overall trend and the scatter associated with the radial extent data increases significantly as well. This is because the two data points located Table 1 shows a quantitative comparison of the final values of the temporal characteristics of major and minor in-plane permeability values computed according to the three strategies listed above. The maximum relative difference value found is 1.64%, which is very small compared to uncertainty values typically reported for in-plane permeability data 41 and thus, further supports the validity of the newly developed "paraboloid" data evaluation approach.

Beneficial application for spatially limited flow front tracking
The newly developed "paraboloid" method can be used in a particularly beneficial way when the tracking of the fluid flow front in radial flow experiments is spatially limited, such as with systems based on electric or dielectric point or line sensors. [15][16][17][18] This will be demonstrated by means of an exemplarily chosen experiment from a capacitive permeameter system, which was originally developed and patented 42 by IVW. The system involves two metal mold halves mounted to a hydraulic press or mold carrier as shown in Figure 12, left. A set of eight linear capacitive sensors is embedded in the lower mold half in a star-like scheme around a centrally located pressure sensor (see Figure 12, right).
The sensors along the east and west directions, respectively, show a length of 185 mm, while all other sensors are 105 mm in length. The width is 5 mm for all of the eight sensors, which capture the change of dielectric properties of the material covering their surface. 18,43 The flow front position at a particular sensor is linearly related with the level of sensor saturation and thus, can be reconstructed from the sensor signal as visualized in Figure 13, left. Table 1. comparison of in-plane permeability values computed from radial extent data according to the "step-by-step" and "global" data evaluation method, respectively.  closest to the major apex of the fitted ellipse fall away at this point of time, which leads to a poor definition of the major radial extent in the subsequent stage. By contrast, the radial extent data obtained with the "paraboloid" method shows steady characteristics throughout the entire experiment. Similar observations can be made for the temporal characteristics of the orientation angle.

Comparison of in-plane permeability values
Unsurprisingly, these variations directly translate into the temporal characteristics of major and minor in-plane permeability values as shown in Figure 15: The significant variation in the characteristics of the major radial extent data results in a strong change of the major principal in-plane permeability. Moreover, the characteristics reveal that in the section of concurrent availability of measurement data from all of the eight linear capacitive sensors, the experiment has not reached the final trend, i.e. the most characteristic behavior of the material under test is not adequately captured. The problem can generally be avoided when following the "paraboloid"    data evaluation method as the entirety of available flow front data is directly used for the fitting task as well as the calculation of in-plane permeability values.

Summary and conclusion
The paper at hand introduces a novel approach for evaluating flow front data acquired during radial flow experiments for in-plane permeability characterization of reinforcing fabrics. Based on the fitting of an elliptic paraboloid model, a data evaluation method is proposed, which directly and robustly delivers an estimate of the orientation angle as well as principal in-plane permeability values characteristic for the reinforcing material under test. In particular, the work comprises the following findings: (1) A mathematical study is given for the temporal advancement of the radial flow front in isotropic porous media, revealing that its fundamental nature can be well approximated by means of a parabolic geometry model. The validity of this finding for flow in anisotropic media is given by empirical evidence. (2) The approach is further extended to fitting an elliptic paraboloid model to the entirety of flow front data acquired during radial flow experiments. The fitting routine is based on singular value decomposition of a design matrix, specifically set up for the geometric model and shows increased numerical efficiency compared to standard nonlinear, iterative approaches. The paraboloid model allows for robustly determining the orientation angle of the principal directions of fluid flow with respect to a testrig-based coordinate frame. Furthermore, the major and minor intersecting parabolae can be extracted and directly processed towards principal in-plane permeability values. (3) An analytical method for computing the degree of anisotropy from the equations of major and minor intersecting parabolae is presented, which avoids the need for a non-linear, iterative search inherent to the original work of Adams and Rebenfeld 12 for in-plane permeability calculation. This not only provides a simplification of the algorithm, it also ensures increased accuracy and reduced computational costs. (4) The applicability of the newly proposed "paraboloid" method is demonstrated by means of data from exemplarily chosen, optically tracked radial flow experiments conducted at two different research institutions. Comparison with the conventionally applied "ellipse" data evaluation technique reveals negligibly small deviations in the resulting in-plane permeability values for all of the three involved evaluation strategies, i.e. the "single step", the "elementary" as well as the "interpolation" method.
with the integration constant C 1 . Back-substitution, once again separation of variables and subsequent integration leads to the general solution of the pressure distribution: with a second integration constant C 2 . Involving the boundary conditions, i.e.: p(r = r 0 ) = p 0 and p(r = r f ) = p f , i.e. the fluid pressure at the boundary of the injection opening r 0 and the fluid flow front r f are given by p 0 and p f , respectively, finally results in:

Temporal advancement of radial flow front
Due to the change to the polar coordinate system, Darcy's law simplifies to: Thus, we need to compute the first derivative of Equation (21) at first: and then substitute in Equation (22): to finally obtain the solution for the volume average flow velocity in the porous medium: Separation of variables yields: and subsequent integration by parts results in: The integration constant C is found by means of the initial condition: r f (t = 0) = r 0 , i.e. the radial extent at the experimental start equals the radius of the injection opening. Thus, C = −r 2 0 , and: As the moving flow front is radially symmetric in this situation, the use of a polar coordinate system is beneficial as the second-order partial differential equation further simplifies to the following ordinary differential equation: 44 Substitution of: q(r) = dp(r) dr , leads to the following first order differential equation: which is easily solved by separation of variables and subsequent integration to:

Fitting of a parabola with off-origin vertex to measurement data
Type-specific fitting of such a parabola model to a given set of n measurement data, {x k , y k }, k = 1 … n, results in a system of equations which can be formulated by matrix multiplication as: with the set of modified conic coefficients: c 3 = c 3,c4 = 2c 4 and c 6 = −2c 4 Δx as well as the vector r of algebraic residuals. The actual fitting task is mathematically set up as finding c such that the sum of squared residuals, i.e.: is minimized. This in turn is accomplished by singular value decomposition (SVD, 46 ) of the scatter matrix , i.e. finding the matrix decomposition, = T , with the nxs diagonal matrix holding the singular values j , j = 1 … s, of along its diagonal as well as the matrices of left and right singular vectors, and , respectively. Selecting the right singular vector corresponding to the smallest singular value yields the solution of the vector c of modified conic coefficients. The parabola parameters are finally found through back substitution.

Mathematical description of quadrics
The implicit equation of quadric surfaces, i.e. quadrics, in general form is given as: 45

Mathematical description of quadratic forms
The implicit equation of quadratic forms, i.e. conics, in general form is given as: 45 which can be reformulated as a matrix equation: with the vector of homogeneous two-dimensional point coordinates, x = [x y 1] T , and the symmetric matrix of homogeneous conic coefficients [ 27 ]:

Parabola of Normal Form
A parabola with parameter p in normal form, i.e. symmetric to the x-axis and the vertex at the point of origin is given by: with the conic coefficients: c 1 = c 2 = c 5 = c 6 = 0, c 3 = 1 and c 4 = −p.

Parabola with off-origin vertex
In order to investigate a parabola symmetric with the x-axis and the vertex shifted from the origin (see Figure 16), we simply apply a translating transformation of the parabola in normal form along the x-axis. Translation by Δx is realized by a matrix multiplication, Par,Δx = −T Δx Par

−1
Δx ,involving the inverse and inverse transpose of the following translatory transformation matrix: Thus, multiplication gives: and results in the following general equation for a parabola symmetric to the x-axis and the vertex at a distance Δx from the origin: (30) c 1 x 2 + 2c 2 xy + c 3 y 2 + 2c 4 x + 2c 5 y + c 6 = 0, (31) x T x = 0,  Thus, multiplication gives: and results in the following general equation for an elliptic paraboloid symmetric to the z-axis and the vertex at a distance Δz from the origin:

Elliptic paraboloid rotated around its axis
Equivalently, rotation of an elliptic paraboloid in normal form around its axis by a rotation angle z (see Figure 17, center), is realized by a matrix multiplication, Ell.Par, z = −T z Ell.Par −1 z , involving the inverse and inverse transpose of the following rotatory transformation matrix: Therein, c z = cos( z ) and s z = sin( z ), are introduced for the purpose of compact notation. Multiplication gives:
q 1 x 2 +q 2 xy +q 3 y 2 +q 9 z +q 10 = 0, which can be reformulated as a matrix equation: with the vector of homogeneous three-dimensional point coordinates, x = [x y z 1] T , and the symmetric matrix of homogeneous quadric coefficients: 27

Elliptic paraboloid in normal form
An elliptic paraboloid in normal form, i.e. showing the z-axis as primary axis, the vertex at the point of origin and the major apex aligned with the x-axis and is given by: x 2 a 2 + y 2 b 2 = 2cz, i.e. the quadric matrix is: with the set of homogeneous quadric coefficients: q 2 = q 4 = q 5 = q 6 = q 7 = q 8 = q 10 = 0, as well as: q 1 = 1 a 2 , q 3 = 1 b 2 and q 9 = −c.

Elliptic paraboloid with off-origin vertex
In order to investigate an elliptic paraboloid symmetric with the z-axis and the paraboloid vertex shifted from the origin (see Figure 17, left), we simply apply a translating transformation of the paraboloid in normal form along the z-axis. Translation by Δz is realized by a matrix multiplication, Ell.Par,Δz = −T Δz Ell.Par