Patent Description:
Embodiments of the invention further relate to geological restoration in which the tomographic images of the present day geology are transformed into images of the past geology, as it was configured at an intermediate restoration time in the past τ before the present day and after the start of deposition of the oldest subsurface layer being imaged. New techniques are proposed herein to improve both the accuracy and computational speed of generating those images of the past restored geology. Improved images may aid geoscientists exploring the subsurface geology for applications such as predicting tectonic motion or earthquakes, or by engineers in the mining or oil and gas industries.

The accuracy of a geological model of the present day configuration of the subsurface of the Earth may be improved by "restoring" the model to a past intermediate time τ and checking model consistency at that time in the past. However, restoring geological models is a complex task and current methods are typically inefficient, requiring extensive processing resources and time, as well as inaccurate, relying on over-simplifications that induce errors to moderate the complexity of the task.

There is a longstanding need in the art to efficiently and accurately restore geological models from their present day geology to their past geology at restored past time τ.

<CIT> et al, discloses a geometric method for 3D structural restoration of a subsurface model including receiving data representative of a subsurface volume of interest including one or more chronohorizons and the geometry and topology of any faults of relevance; developing a fault framework model of the subsurface volume of interest; selecting a horizon, the deposition of which represents the geologic time to which the structural model should be restored; developing coordinate transformation constrained by a single datum horizon and, optionally, additional geologic constraints; applying the 3D transformation to all geologic features below and, optionally, above the datum surface; and scaling the vertical coordinates to accurately relate vertical and horizontal dimensions.

"<NPL>, discloses a new approach to petroleum systems analysis which allows full integration of tectonic and palinspastic restoration with three-dimensional (3D), PVT-controlled, multi-component, three-phase petroleum migration analysis through time.

Some embodiments, not forming part of the present invention, are directed to modeling restored geological models with τ-active and τ-inactive faults. In an embodiment of the invention, a system and method is provided for restoring a 3D model of the subsurface geology of the Earth from a present day geometry measured at a present time to a predicted past geometry at a past restoration time. The 3D model of the present day measured geometry comprising a network of faults may be received, wherein a fault is a discontinuity that divides fault blocks that slide in opposite directions tangential to the surface of the fault as time approaches a modeled time. A past restoration time τ may be selected that is prior to the present time and after a time when an oldest horizon surface in the 3D model was originally deposited. The network of faults may be divided into a subset of τ-active faults and a subset of τ-inactive faults, wherein a τ-active fault is a fault that is active at the past restoration time τ and a τ-inactive fault is a fault that is inactive at the past restoration time τ. A fault may be determined to be τ-active when the fault intersects a horizon Hτ that was originally deposited at the past restoration time τ and a fault may be determined to be τ-inactive when the fault does not intersect the horizon Hτ that was originally deposited at the past restoration time τ. The 3D model may be restored from the present day measured geometry to the predicted past geometry at the past restoration time τ by modeling each τ-active and τ-inactive fault differently. Each τ-active fault may be modeled to join end points of a horizon Hτ separated on opposite sides of the fault in the present day model to merge into the same position in the restored model by sliding the end points towards each other in a direction tangential to the surface of the τ-active fault. Each τ-inactive fault may be modeled to keep collocated points on opposite sides of the fault together.

Some embodiments, not forming part of the present invention, are directed to modeling restored geological models with new restoration coordinates uτ, vτ, tτ. In an embodiment of the invention, a system and method is provided for restoring a 3D model of the subsurface geology of the Earth from a present day measured geometry to a predicted past geometry at a restoration time in the past τ. The 3D model of the present day geometry of the subsurface may be received, including one or more folded geological horizon surfaces. A value may be selected of a restoration time in the past τ before the present day and after a time an oldest horizon surface in the 3D model of the subsurface was deposited. The 3D model may be restored from the present day measured geometry to the predicted past geometry at the restoration time in the past τ using a 3D transformation. The vertical component of the 3D transformation may restore the geometry to the vertical coordinate tτ such that: points along a horizon surface Hτ modeling sediment that was deposited at the selected restoration time in the past τ have a substantially constant value for the restored vertical coordinate tτ; and at any location in the 3D model, the restored vertical coordinate tτ is equal to a sum of a first approximation t'τ of the vertical coordinate and an error correction term ετ, wherein the error correction term ετ is computed by solving a linear relationship in which a variation in the sum of the first approximation t'τ of the vertical coordinate and the error correction term ετ between any two points separated by an infinitesimal difference in the direction of maximal variation of the sum is approximately equal to the distance between the points in the direction of maximal variation; and displaying an image of the restored 3D model of the subsurface geology of the Earth such that each point in the 3D model is positioned at the restored vertical coordinate tτ as it was configured at the restoration time in the past τ.

Some embodiments of the invention are directed to modeling restored geological models taking compaction into account at an intermediate restoration time in the past τ. In an embodiment of the invention, a system and computer-implemented method is provided for decompacting a 3D model of the subsurface geology of the Earth at an intermediate restoration time in the past τ. According to the invention, a method comprises receiving a 3D model of present-day geometry of the subsurface geology and a measure of present-day porosity experimentally measured within the subsurface geology of the Earth. A value of a restoration time in the past τ is selected before the present day and after a time an oldest horizon surface in the 3D model of the subsurface was deposited. The 3D model from the present day measured geometry is restored to the predicted past geometry at the restoration time in the past τ using a 3D transformation. The vertical dimension of the restored 3D model is decompacted to elongate vertical lengths of geological layers below a horizon layer deposited at the restoration time in the past τ. The vertical lengths are elongated based on a relationship between a depositional porosity of the geological layers at the time sediment in those layers was deposited, restoration porosity of the geological layers at the restoration time in the past τ, and the present-day porosity of the geological layers experimentally measured in the present-day.

The principles and operation of the system, apparatus, and method according to embodiments of the present invention may be better understood with reference to the drawings, and the following description, it being understood that these drawings are given for illustrative purposes only and are not meant to be limiting.

For simplicity and clarity of illustration, elements shown in the drawings have not necessarily been drawn to scale. Further, where considered appropriate, reference numerals may be repeated among the drawings to indicate corresponding or analogous elements throughout the serial views.

Embodiments of the invention improve conventional restoration techniques for imaging restored geological models as follows:.

In conventional restoration models, all faults are active (as discontinuous surfaces) at all times. However, in reality, certain faults have not yet formed or activated at various intermediate restoration times τ. Accordingly, conventional restoration models generate false or "phantom" faults that erroneously divide geology that has not yet fractured, leading to geological inaccuracies in subsurface images.

Embodiments of the invention solve this problem by selectively activating and deactivating individual fault surfaces to be discontinuous or continuous, respectively, depending on the specific restoration geological-time τ. For each intermediate restoration time in the past τ, embodiments of the invention split faults into two complementary subsets of "τ-active" faults and "τ-inactive" faults. τ-active faults are activated at restoration time τ (e.g., a discontinuous fault surface along which fault blocks slide tangentially), whereas τ-inactive faults are deactivated at restoration time τ (e.g., a continuous surface that does not behave as a fault).

As faults form and evolve over time, they behave differently at different geological times in the past. For example, a fault that forms at an intermediate geological-time τ, where τ<NUM><τ< τ<NUM>, is τ-active in a restored model at later time τ<NUM> (after the fault has formed), but τ-inactive in a restored model at earlier time τ<NUM> (before the fault has formed). This fault classification allows faults to be modelled differently at each restoration time τ in a geologically consistent way, thereby preventing unrealistic deformations from being generated in the neighborhood of these faults.

<FIG> shows the problem of a fault <NUM> being erroneously considered active at a restoration time before it formed (top image of <FIG>) and the solution of modeling the fault as a τ-inactive fault to correctly deactivate the fault at restoration time τ according to embodiments of the invention (bottom image of <FIG>). In the top image of <FIG>, when a horizon Hτ <NUM> is restored using uτ vτ tτ -transform <NUM>, fault block <NUM> (shaded region in the top-left image of <FIG>) is bounded by an active fault <NUM> and an inactive fault <NUM>. If however fault <NUM> is erroneously considered as an active fault then, after applying restoration <NUM>:.

It is clear that, if d(a, b) denotes the distance between any arbitrary pair of points (a, b), then: <MAT>.

This observation shows that erroneously considering fault <NUM> as a τ-active fault inevitably generates unrealistic deformations.

This problem is solved according to embodiments of the invention, e.g., as shown in the bottom image of <FIG>. In this image, fault <NUM> is modeled as a τ-active fault (activating the fault), but fault <NUM> is modeled as a τ-inactive fault (deactivating the fault). Accordingly, when uτ vτ tτ -transform <NUM> is applied to fault block <NUM> (shaded region in the bottom-left image of <FIG>), restored fault block <NUM> (shaded region in the bottom-right image of <FIG>) is no longer bounded by an active fault (τ-inactive fault <NUM> is inactive at restored time τ). Accordingly, the restored fault block <NUM> preserves volume and stays within (and does not cross) τ-active fault <NUM> (because the deactivated boundary transformed from τ-inactive fault <NUM> may shift to accommodate a shift in the restored τ-active fault <NUM>).

Contrary to conventional methods, the use of τ-active and τ-inactive faults produces more accurate results, e.g., even if there is no continuous path between (no way to continuously connect) a given fault block (e.g., <NUM>) and the horizon Hτ (e.g., <NUM>) deposited at geological time τ, which typically requires additional processing that may induce errors. By selectively activating and inactivating faults at the various restoration times according to when they form, embodiments of the invention eliminate erroneous phantom faults and more accurately represent the faulted geology.

Reference is made to <FIG>, which is a flowchart of a method to restore a geological model using τ-active and τ-inactive faults, according to an embodiment not forming part of the present invention.

In operation <NUM>, a processor may receive a 3D model of the present day measured geometry comprising a network of faults (e.g., present day model <NUM>). The present day model may be measured tomographically by scanning the Earth's subsurface e.g., as described in reference to <FIG> and <FIG>. In the present day geology, all faults in the model have already formed and so, represent active discontinuities that divide fault blocks which slide in opposite directions tangential to the surface of the fault as time approaches a modeled time.

In operation <NUM>, a processor may select or receive a past restoration time τ that is "intermediate" or prior to the present time and after the start of the subsurface's deposition (the time period when an oldest horizon surface in the 3D model was originally deposited).

In operation <NUM>, a processor may divide the network of faults into a subset of τ-active faults and a subset of τ-inactive faults, τ-active faults may be faults that are active at the past restoration time τ and τ-inactive faults are faults that are inactive at the past restoration time τ. A fault is determined to be τ-active when the fault intersects a horizon Hτ that was originally deposited at the past restoration time τ (e.g., see τ-active faults <NUM> of <FIG>) and a fault is determined to be τ-inactive when the fault does not intersect the horizon Hτ that was originally deposited at the past restoration time τ (e.g., see τ-inactive faults <NUM> of <FIG>). Because different faults activate to fracture the subsurface at different geological times, the processor may divide the fault network differently at different geological times. Thus, a fault may be τ-active at a first restoration time τ' (e.g., a time period during which the fault has formed) and τ-inactive at a second restoration time τ" (e.g., a time period different than that during which the fault has formed). In one embodiment, iso-value surfaces (e.g., <NUM> of <FIG>) of each restoration coordinate (e.g., uτ, vτ, and tτ) are continuous across τ-inactive faults (e.g., <NUM> of <FIG>) and discontinuous across τ-active faults (e.g., <NUM> of <FIG>).

In operation <NUM>, a processor may restore the 3D model from the present day measured geometry to the predicted past geometry at the past restoration time τ. During restoration, the processor may flatten a horizon Hτ (e.g., <NUM> of <FIG>) that was originally deposited at time τ to a substantially planar surface of approximately constant depth. For horizons older (e.g., deposited deeper in the subsurface) than horizon Hτ, the processor may restore the horizons to non-planar surfaces, e.g., when the thickness of the layers is not constant. Because the region of the subsurface deposited after the restoration time τ (e.g., deposited shallower in the subsurface) did not yet exist at the time of the restored model, restoring the 3D model to a past restoration time τ may eliminate (e.g., removing or not displaying) all relatively shallower horizon surfaces that were originally deposited after the past restoration time τ. During restoration, the processor may treat τ-active and τ-inactive faults differently in operations <NUM> and <NUM>, respectively.

In operation <NUM>, for each τ-active fault, a processor may model the τ-active fault as an active discontinuous fault surface and restore the horizon surface by removing or omitting the fault surface at the time of restoration. The processor may eliminate the τ-active fault during restoration by sliding its adjacent fault blocks together. This may join end points of a horizon Hτ separated on opposite sides of the fault in the present day model to merge into the same position in the restored model by sliding the end points towards each other in a direction tangential to the surface of the τ-active fault.

In operation <NUM>, for each τ-inactive fault, a processor may model the τ-inactive fault, not as a discontinuous fault surface, but as a continuous non-fault surface in the restoration transformation. The τ-inactive fault may be modeled as a surface in which the discontinuity induced by the fault has been deactivated to prevent fault blocks from sliding in directions tangential to the surface of the fault as time approaches the restoration time τ. The processor may model the τ-inactive fault during restoration by keeping collocated points on opposite sides of the fault in the present day model together in the restored model.

After the geological model has been restored for a first past restoration time τ (operations <NUM>-<NUM>), the process may repeat to restore the model for a second different past restoration time τ'. In some embodiments, the geological model may be sequentially restored to a sequence of multiple past restoration times τ<NUM>, τ<NUM>,. In multiple (all or not all) of the past restoration times τ<NUM>, τ<NUM>,. , τn, the fault network may be divided into a different subset of τ-active and τ-inactive faults, e.g., because different faults fracture the subsurface at different geological times. In some embodiments, a processor may play a moving image sequence in which the 3D model is iteratively restored in a forward or reverse order of the sequence of past restoration times τ<NUM>, τ<NUM>,. , τn to visualize changes in the subsurface geology over the passage of time.

In operation <NUM>, a processor may display a visualization of an image of the subsurface geology of the Earth overlaid with τ-active faults and τ-inactive faults in the restored model at past restoration time τ. The processor may display the τ-active faults and the τ-inactive faults with different visual identifiers, such as, different levels of translucency, different colors, different patterns, etc..

A restoration transformation may transform a geological image of the subsurface of the Earth from a present day space (e.g., x,y,z coordinates) to a restoration space (e.g., uτ, vτ, and tτ coordinates) as it was formed at an intermediate restoration time in the past τ (before the present-day but after the start of the subsurface deposition). An ideal restoration should transform the vertical coordinate tτ in a manner that strictly honors the thickness of layers, to preserve areas and volumes of the Earth, so that terrains are not stretched or squeezed over time in the vertical dimension. However, conventional restoration transformations typically deform the vertical coordinates, forcing terrains to stretch and squeeze, resulting in errors in the restoration model.

Embodiments of the invention improve the accuracy of the restoration model by establishing a vertical restoration coordinate tτ that preserves layer thickness. This may be achieved by implementing a thickness-preserving constraint that sets a variation in the vertical restoration coordinate tτ between any two points separated by an infinitesimal difference in the direction of maximal variation of the vertical coordinate tτ to be approximately equal to the distance between the points in the direction of maximal variation. An example of this constraint may be modeled by ∥grad tτ(x,y,z)∥ = <NUM>. This constraint, however, is non-linear and highly complex and time-consuming to solve. Due to its complexity, this constraint is rarely used in conventional restoration models, and instead replaced by over-simplifications, such as equations (<NUM>) and (<NUM>), that result in model errors as shown in histograms <NUM> and <NUM> of <FIG>, and histograms <NUM> and <NUM> of <FIG>, respectively.

Embodiments of the invention improve the accuracy of the restored model by establishing a new thickness-preserving constraint that introduces an error correction term ετ. The new thickness-preserving constraint sets the restored vertical coordinate tτ to be equal to a sum of a first approximation t'τ of the vertical coordinate and an error correction term ετ, wherein the error correction term ετ is computed by solving a relationship in which a variation in the sum of the first approximation t'τ of the vertical coordinate and the error correction term ετ between any two points separated by an infinitesimal difference in the direction of maximal variation of the sum is approximately equal to the distance between the points in the direction of maximal variation. An example of this constraint may be modeled by ∥grad (t'τ + ετ)∥ = <NUM>. The new thickness-preserving constraint preserves layer thickness with greater accuracy as shown in histogram <NUM> of <FIG> as compared to conventional approximations shown in histograms <NUM> and <NUM> of <FIG> and minimizes volume variation with greater accuracy as shown in histogram <NUM> of <FIG> as compared to conventional approximations shown in histograms <NUM> and <NUM> of <FIG>, respectively.

Embodiments of the invention further improve the performance and computational speed of the computer generating the restored model by linearizing the new thickness-preserving constraint. As an example, the new thickness-preserving constraint may be linearized as follows. ∥grad (t'τ + ετ)∥ = <NUM> may be squared to obtain ∥grad t'τ∥<NUM> + ∥grad ετ∥<NUM> + ∥ <NUM> · grad t'τ · grad ετ ∥ = <NUM>. The error correction term ετ may be generated such that the square of its spatial variation, ∥grad ετ∥<NUM>, is negligible. Accordingly, the thickness-preserving constraint simplifies to a new linear thickness-preserving constraint of grad ετ · grad t'τ ≅ ½ {<NUM>-∥grad t'τ∥<NUM>} (eqn. This thickness-preserving constraint is linear because t'τ is already known, so the constraint is a relationship between the gradient of the error ετ and the gradient of the known first approximation of the vertical coordinate t'τ. The computer may therefore compute the new thickness-preserving constraint in linear time, which is significantly faster than computing the non-linear constraints ∥grad tτ∥ = <NUM> or ∥grad (t'τ + ετ)∥ = <NUM>.

Contrary to conventional methods, the computational complexity for performing the restoration transformation according to embodiments of the invention is significantly reduced compared to classical methods that are based on the mechanics of continuous media. As a consequence, the modeling computer uses significantly less computational time and storage space to generate the inventive restoration model.

Contrary to conventional methods that allow variations of geological volumes and deformations, embodiments of the invention implement a new set of geometrical constraints and boundary conditions that preserve geological volumes and deformations while adhering to geological boundaries.

Contrary to conventional methods, embodiments of the invention restore faults along fault striae (e.g., see <FIG>) induced by the twin points associated with the paleo-geographic coordinates of a depositional (e.g., GeoChron) model, given as input of the restoration method.

An ideal restoration should also transform the horizontal coordinates uτ and vτ in a manner that strictly honors lateral spatial distribution, to preserve areas and volumes of the Earth, so that terrains are not stretched or squeezed over time in the horizontal dimensions. However, conventional restoration transformations based on depositional coordinates (e.g., paleo-geographic coordinates u and v) typically deform the horizontal coordinates, forcing terrains to stretch and squeeze, resulting in errors in the restoration model.

Embodiments of the invention improve the accuracy of the restoration model at time τ by establishing horizontal restoration coordinates uτ and vτ that restore the horizon surface Hτ deposited at time τ consistently with horizontal depositional coordinates u and v whilst minimizing deformations. In one embodiment, on the horizon surface Hτ only, the horizontal restoration coordinates uτ and vτ are equal to the depositional coordinates u and v (see e.g., equation (<NUM>)) and the spatial variations of the horizontal restoration coordinates uτ and vτ are preserved with respect to the horizontal depositional coordinates u and v (see e.g., equation (<NUM>)). Thus, each restoration model at time τ, presents a horizon surface Hτ, as it was configured at that time τ when it was originally deposited. Additionally or alternatively, horizontal restoration coordinates uτ and vτ are modeled in a tectonic style (e.g., using constraints (<NUM>) or (<NUM>)) that is consistent with that of the horizontal coordinates u and v of the depositional model, which makes the restoration more accurate because the geological context is taken into account. Additionally or alternatively, horizontal restoration coordinates uτ and vτ are modeled to minimize deformations induced by the restoration of horizon Hτ, rather than minimizing deformations in the whole volume G. This may be achieved by implementing constraints (<NUM>) and (<NUM>) that only enforce orthogonality of gradients of uτ and vτ with local axes bτ and aτ, but which do not constrain the norm of grad uτ and grad vτ, as is typically constrained for horizontal depositional coordinates u and v consistent with the depositional time model. Horizontal restoration coordinates uτ and vτ may also be constrained only in Gτ, thereby only taking into account the part of the subsurface to be restored, not the entire model G. Additionally or alternatively, horizontal restoration coordinates uτ and vτ may be constrained to be equal on opposite sides of τ-active faults at twin point locations, where the twin points are computed from fault striae, which also ensures consistency with the depositional model (see e.g., equation (<NUM>)). Additionally or alternatively, horizontal restoration coordinates uτ and vτ are constrained to be equal on opposite sides of τ-inactive faults at mate point locations to cancel the effect of inactive faults on the restoration model (see e.g., equation (<NUM>)).

Reference is made to <FIG>, which is a flowchart of a method to restore a geological model with improved accuracy using a new thickness-preserving constraint, according to an embodiment not forming part of the present invention.

In operation <NUM>, a processor may receive a 3D model of the present day measured geometry (e.g., present day model <NUM>) comprising one or more folded (e.g., curvilinear or non-planar) geological horizon surfaces (e.g., <NUM>). The present day model may be measured tomographically by scanning the Earth's subsurface e.g., as described in reference to <FIG> and <FIG>.

In operation <NUM>, a processor may restore the 3D model from the present day measured geometry (e.g., present day model Gτ <NUM> in xyz-space G <NUM>) to the predicted past geometry at the restoration time in the past τ (e.g., restored model Gτ <NUM> in uτ vτ tτ - space <NUM>) using a 3D restoration transformation (e.g., uτ vτ tτ -transform <NUM>). At the restored time in the past τ, the geological layers above Hτ (e.g., HTτ+<NUM>. Hn) did not yet exist, so the subregion above Hτ in the present day space G <NUM> is eliminated or omitted, and only the subregion Gτ <NUM> below and aligned with Hτ (e.g., H<NUM>. Hτ) in the present day space G <NUM> is restored. The 3D restoration transformation includes a vertical component that restores the geometry to the vertical coordinate tτ and two lateral or horizontal components that restore the geometry to the horizontal coordinates uτ and vτ. The restored vertical coordinate tτ and horizontal coordinates uτ and vτ represent the predicted vertical and horizontal positions, respectively, where particles in the subsurface were located in the Earth at the restoration time in the past τ. Because the region of the subsurface deposited after the restoration time τ (e.g., deposited shallower in the subsurface than Hτ) did not yet exist at the time of the restored model, the processor may restore and compute coordinates for the part or subregion Gτ of the subsurface G that was deposited at a geological time of deposition t prior to or during the past restoration time τ (e.g., deposited deeper than, or at the same layer in the subsurface as, Hτ). Accordingly, the restored model eliminates or omits all relatively shallower or younger horizon surfaces or layers that were originally deposited after the past restoration time τ.

The processor may restore the vertical coordinate tτ such that points along a horizon surface Hτ (e.g., <NUM>) modeling sediment that was deposited at the selected restoration time τ have a substantially constant value for the restored vertical coordinate tτ (see e.g., eqn. Further, the processor may restore the vertical coordinate tτ such that at any location in the 3D model, the restored vertical coordinate tτ is equal to a sum of a first approximation t'τ of the vertical coordinate and an error correction term ετ, wherein the error correction term ετ is computed by solving a relationship in which a variation in the sum of the first approximation t'τ of the vertical coordinate and the error correction term ετ between any two points separated by an infinitesimal difference in the direction of maximal variation of the sum is approximately equal to the distance between the points in the direction of maximal variation. The error correction term ετ may correct errors in the first approximation t'τ of the vertical coordinate. This constraint may be represented by a linear second order approximation (see e.g., eqn.

In some embodiments, the processor computes the first approximation t'τ of the vertical coordinate by solving a relationship in which the spatial variation of the vertical coordinate t'τ is locally approximately proportional to the spatial variation of a geological time of deposition t. In some embodiments, the coefficient of proportionality is locally equal to the inverse of the magnitude of the maximal spatial variation of the geological time of deposition (see e.g., eqn. (<NUM>)-(<NUM>)). This relationship may give the vertical restoration coordinate tτ the shape of the horizon Hτ because, on the horizon, the gradient of depositional time t is normal to the horizon surface. Thus, the ratio grad t/∥grad t|| follows the shape of the horizon.

In some embodiments, the processor computes the first approximation t'τ of the vertical coordinate by solving a relationship in which any infinitesimal displacement in the direction orthogonal to horizon surface Hτ results in a variation of the vertical coordinate t'τ approximately equal to the length of the infinitesimal displacement for points on the horizon surface Hτ (see e.g., eqn. (<NUM>)-(<NUM>)).

In some embodiments, the processor computes the restored vertical coordinate tτ in parts of the subsurface which are older than restoration time τ such that iso-value surfaces of the restored vertical coordinate tτ are parallel to the horizon surface Hτ and the difference in the restored vertical coordinate tτ between two arbitrary iso-values is equal to the distance between the corresponding iso-surfaces (see e.g., eqn. Parallel surfaces may be planar parallel in the restored model, and curved parallel (e.g., having parallel tangent surfaces) in present day model, such that the surfaces are non-intersecting at limits.

In some embodiments, the error correction term ετ is null at points along the horizon surface Hτ that was deposited at the selected restoration time in the past τ so that the restored horizon surface Hτ is flat (see e.g., eqn.

In some embodiments, the restored horizontal coordinates uτ and vτ are constrained such that for each point along the horizon surface Hτ that was deposited at the selected restoration time in the past τ: the restored horizontal coordinates uτ and vτ are equal to depositional horizontal coordinates u and v, respectively, and the spatial variations of the restored horizontal coordinates uτ and vτ are equal to the spatial variations of the depositional horizontal coordinates u and v, respectively (see e.g., eqns. (<NUM>)-(<NUM>)). On average, globally over the entire model, the processor may compute ∥grad u∥ = <NUM> and ∥grad v∥ = <NUM>. However, locally, this is not necessarily true e.g., on horizon Hτ. So, while the processor sets grad uτ = grad u and grad vτ = grad v on Hτ, the processor may not constrain ∥grad uτ∥ = <NUM> and ∥grad vτ∥ = <NUM> on Hτ. Moreover, the processor may not constrain grad uτ to be orthogonal to grad tτ. This results from the boundary condition on Hτ and propagation through its constant gradient.

In some embodiments, the restored horizontal coordinates uτ and vτ are constrained in parts of the subsurface which are older than restoration time τ such that directions of maximal change of the restored horizontal coordinates uτ and vτ are linearly constrained by a local co-axis vector bτ and a local axis vector aτ, respectively (see e.g., eqn.

In some embodiments, the local axis vector aτ is oriented approximately in the direction of maximal change of depositional horizontal coordinate u and orthogonal to the direction of maximal change of the vertical restoration coordinate tτ, and the local co-axis vector bτ is oriented orthogonal to the direction of the local axis vector aτ and orthogonal to the direction of maximal change of the vertical restoration coordinate tτ (see e.g., eqn.

In some embodiments, if the tectonic style of the 3D model is minimal deformation, the restored horizontal coordinates uτ and vτ are computed over the part of the 3D model of the subsurface which is older than restoration time τ such that the directions of maximal change of uτ and vτ are approximately orthogonal to the local co-axis vector bτ and the local axis vector aτ, respectively. For example, equation (<NUM>) constrains the local axis vector aτ to be parallel to the gradient of u and the local co-axis vector bτ to be orthogonal to the local axis vector aτ, which means that the gradient of u is orthogonal to the local co-axis vector bτ. Equation (<NUM>) further constrains the gradient of uτ to be approximately orthogonal to the local co-axis vector bτ. Accordingly, the gradient of uτ is approximately parallel to the gradient of u. The same logic implies the gradient of vτ is approximately parallel to the gradient of v.

In some embodiments, if the tectonic style of the 3D model is flexural slip, the restored horizontal coordinates uτ and vτ are computed over the part of the 3D model of the subsurface which is older than restoration time τ such that projections of their directions of maximal change over the iso-value surfaces of the restored vertical coordinate tτ are approximately orthogonal to local co-axis vector bτ and the local axis vector aτ, respectively (see e.g., eqn.

In some embodiments, the values of the restored horizontal coordinates uτ and vτ are constrained in parts of the subsurface which are older than the restoration time τ to be respectively equal on twin points on τ-active faults, wherein twin points are points on opposite sides of a τ-active fault that were collocated at the restoration time τ and are located on the same fault stria in the present day model, to merge the twin points into the same position in the restored model by sliding the twin points towards each other in a direction tangential to the surface of the τ-active fault (see e.g., eqn.

In some embodiments, the values of the restored horizontal coordinates uτ and vτ are constrained in parts of the subsurface which are older than the restoration time τ to be respectively equal on mate points on τ-inactive faults, wherein mate points are points on opposite sides of a τ-inactive fault that are collocated at present day time, to move mate points together on opposite sides of τ-inactive faults (see e.g., eqn.

In operation <NUM>, a processor may display an image of the restored 3D model of the subsurface geology of the Earth such that each point in the 3D model is positioned at the restored coordinates uτ, vτ, tτ defining the location that a piece of sediment represented by the point was located at the restoration time in the past τ.

In some embodiments, the processor may receive an increasing chronological sequence of past restoration times τ<NUM>, τ<NUM>,. For each restoration time τi in sequence τ<NUM>, τ<NUM>,. , τn, the processor may repeat operations <NUM>-<NUM> to compute a corresponding 3D restoration transformation Rτi. 3D restoration transformation Rτi restores the part of the subsurface older than horizon Hτi to its predicted past geometry at time τi, e.g., to 3D restored coordinates uτi, vτi, and tτi.

In operation <NUM>, in some embodiments, a processor may play a moving image sequence in which the 3D model is iteratively restored in a forward or reverse order of the sequence of past restoration times τ<NUM>, τ<NUM>,. , τn to visualize changes in the subsurface geology over the passage of time.

In some embodiments, the processor may edit the model in the restoration space and then reverse the restoration transformation to apply those edits in the present day space. For example, the processor may edit the depositional values u, v, and t associated with the restored 3D model, and then reverse transform the restored 3D model forward in time from the predicted past geometry at the restoration time in the past τ to the present day measured geometry using an inverse of the 3D restoration transformation <NUM> to incorporate the edits from the restored model into the present day model.

Compaction may refer to the pore space reduction in sediment within the Earth's subsurface. Compaction is typically caused by an increase in load weight of overlying geological layers as they are deposited over time. As sediment accumulates, compaction typically increases, as time and depth increase. Conversely, porosity typically decreases, as time and depth increase. For example, at a depositional time t<NUM> when a layer is deposited with no overlaying geology, the depositional model has minimal or no compaction and maximum depositional porosity ψ<NUM>. At an intermediate restoration time τ, when there is an intermediate load of overlying deposited layers, the restored model has an intermediate level of compaction and an intermediate level of porosity ψτ (or simply ψ). At the present-day time tp, when the present-day model has the most deposited layers, the present-day model typically has a maximal level of compaction and minimum porosity ψp. Accordingly, the depositional porosity ψ<NUM> is greater than the intermediate time porosity ψτ, which in turn is greater than the present-day porosity ψp, i.e., ψ<NUM>> ψτ >ψp. Further, because deeper layers are typically deposited at relatively earlier times than are shallower layers, within each model at the same time τ, a relatively deeper geological layer typically experiences a relatively greater load than does a relatively shallower geological layer, resulting in greater compaction and lesser porosity.

Whereas compaction is a result of deposition over the forward passage of time, the process of restoration reverses the passage of time to visualize geology at an intermediate time in the past τ (before the present day and after the start of deposition of the oldest subsurface layer). Accordingly, embodiments of the invention generate a restoration model by reversing the effects of compaction in a process referred to as "decompaction" to more accurately depict how the geometry of geological layers change as their depths increase. Whereas compaction compresses the geological layers, decompaction reverses those effects, decompressing and uplifting terrains, resulting in increased layer thicknesses and increased intermediate time porosity ψτ (or simply ψ) in the restored domain as compared with the compacted present-day domain ψp. Decompaction decompresses the geology by a greater amount the earlier the intermediate restoration time τ is in the past and the deeper the layer is underneath the Earth's surface.

Conventional decompaction techniques, however, are notoriously unreliable. Laboratory experiments on rock samples show that, during burial when sediments contained in a volume V(rτ) compact under their own weight, their porosity ψ(rτ) exponentially decreases according to Athy's law: <MAT> where V(rτ) represents an infinitely small volume of sediment centered on a point rτ ∈ Gτ underneath the sea floor Sτ(<NUM>) ≡ Hτ, δ(rτ) is the absolute distance, or depth, from point rτ ∈ Gτ to sea floor Sτ(<NUM>) measured at restoration time τ, and ψ<NUM>(rτ) < <NUM> and κ(rτ) are known non-negative coefficients which depend only on rock type at location rτ. ψ<NUM>(rτ) is the porosity of the rock type with approximately no (zero) compaction, i.e., the porosity at its depositional time t<NUM> before any layers were deposited to compress from above. κ is an experimental measurement derived from compression experiments of Athy's law performed in laboratory tests. As an example, assuming that geological depth δ(rτ) is expressed in meters, the following average coefficients for sedimentary terrains were observed in southern Morocco:.

Because, in the restored Gτ-space, -tτ(rτ) measures the vertical distance from point rτ to the sea floor Sτ(<NUM>), the depth δ(rτ) in equation (<NUM>) may be equivalently expressed as: <MAT>.

Accordingly, in the context of embodiments of the invention, Athy's law may be reformulated as: <MAT>.

Athy's law alone, however, incorrectly models porosity ψ and therefore often models decompaction inaccurately. Under Athy's law, restoration porosity ψ depends only on predictions extrapolated based on rock properties (ψ<NUM> and κ), but does not actually measure real-world porosity. Because Athy's law is not rooted in the real-world geology, it often leads to inaccurate overestimated or underestimated compaction. Further, Athy's law models compaction based on porosity only at the time of earliest deposition, ψ<NUM>, but not porosity that occurs in the present-day, ψp. Once the model is transformed from the present-day to restored time τ, but prior to decompaction, the restored model still erroneously exhibits present-day compaction ψp. Because Athy's law does not eliminate present-day compaction, which erroneously over-compresses terrains compared to restoration porosity, the resulting model is incorrectly decompacted at the restoration time τ.

Embodiments of the invention improve decompaction techniques by modeling decompaction at an intermediate restoration time in the past τ based on real-world measurements of present-day compaction ψp experimentally observed within the subsurface geology of the Earth. Modeling decompaction based on present-day compaction measurements accounts for the many real-world geological variables, such as those in the above example scenarios, that Athy's law misses.

Some embodiments accurately decompact the restoration model by simultaneously (<NUM>) removing the impact of present-day compaction affecting terrains in (incorrectly) restored version at time τ (e.g., "total" decompaction, such as, defined in equations (<NUM>)); and (<NUM>) recompacting these terrains according to their depth in the restored model (e.g., "partial" recompaction, such as, defined in equations (<NUM>)). Embodiments of the invention solve the difficult problem of performing these two operations (decompaction and recompaction) simultaneously.

Reference is made to <FIG>, which schematically illustrates an example 3D geological volume of a compacted model <NUM> representing the porosity of a subsurface region before decompaction (right image) and a corresponding decompacted model <NUM> representing the porosity of the region after decompaction (left image) in the restored Gτ-space at an intermediate restoration time in the past τ, according to an embodiment of the invention. Embodiments of the invention replace original restoration coordinates of the compacted model <NUM> with new restoration coordinates {uτ, vτ, tτ}rτ of the decompacted model <NUM>. Decompacted model <NUM> may represent a new uτ, vτ, tτ-transform from the present-day model in Gτ-space to the restored Gτ-space that restores the terrains and induces thickness variations as a consequence of decompaction. This decompaction transformation is modeled to be the inverse ("reversing time") of the compaction that occurred over the forward passage of time between geological-time τ and the present geological-time. Some embodiments may start with a region of the compacted model <NUM> under the horizon Hτ (geology deposited before time τ with a present-day level of compaction) and restore the region to the decompacted model <NUM> in Gτ-space (geology deposited before time τ with a level of compaction at intermediate time τ). Because the compacted model <NUM> has not yet been decompacted, its low porosity is similar to the present-day porosity, yielding vertical lengths dh(rτ) that are too short and compressed for the restored time τ. Accordingly, decompaction vertically stretches the lengths dh⊕(rτ) of the decompacted model <NUM> to yield a greater porosity predicted at the time in the past τ. This process may repeat iteratively, layer-by-layer, starting at the top horizon Hτ deposited at the restoration time τ and ending at the bottom horizon deposited at the depositional time t<NUM>.

Elasto-plastic mechanical frameworks developed to model compaction rely on a number of input parameters which may be difficult for a geologist or geomodeler to assess and are solved using a complex system of equations. Isostasic approaches are typically simpler to parameterize and still provide useful information on basin evolution. Therefore, compaction may be considered a primarily one-dimensional vertical compression induced by gravity which mainly occurs in the early stages of sediment burial when horizons are still roughly horizontal surfaces close to the sea floor.

At any point rτ ∈ Gτ within a geological layer, the decompacted thickness dh⊕(rτ) e.g., of a vertical probe of infinitely small volume V(rτ) comprising an infinitely short column of sediment roughly orthogonal to the restored horizon passing through rτ is linked to the thickness dh(rτ) of the shorter, compacted vertical column by, for example, the following relationship: <MAT>.

In this equation, φ(rτ) denotes the "compaction coefficient" which characterizes the vertical shortening of the probe at restored location rτ ∈ Gτ. As an example, <FIG> shows the same infinitely short vertical column of sediment where average porosity is equal to (ψ<NUM> = <NUM>/<NUM>) before compaction and (ψ = <NUM>/<NUM>) after compaction. The compaction coefficient (ψ<NUM> - ψ) is then equal to (φ = <NUM>/<NUM>) and column shortening (<NUM> - φ) is (<NUM>/<NUM>).

Compacted model <NUM>, built assuming there is no compaction, incorrectly ignores the compaction characterized by present-day porosity ψp(rτ). Compacted model <NUM> thus results in geology with greater compaction and smaller porosity than occurred at intermediate restoration time τ. Embodiments of the invention correct the restored model by decompacting compacted model <NUM>. The decompaction process involves decompressing the vertical dimension's compacted height dh(rτ) or compacted time dt(rτ) (relatively shorter) to elongate the vertical dimension with a decompacted height dh⊕(rτ) or decompacted time dt⊕(rτ) (relatively longer) (see e.g., equation (<NUM>) and/or (<NUM>)). This decompaction of height (e.g., in equation (<NUM>)) or time (e.g., in equation (<NUM>)) is elongated based on compaction coefficient <MAT>, which is a function of the present-day porosity ψp(rτ) (see e.g., equation (<NUM>)). Because the present-day porosity ψp(rτ) is less than the restoration porosity ψτ(rτ) the ratio term in equations (<NUM>) and (<NUM>) is > <NUM>. Accordingly, the decompacted length dh⊕(rτ) and time dt⊕(rτ) are greater than the compacted length dh(rτ) and time dt(rτ), respectively, resulting in an elongation of the vertical dimension after decompaction. This elongation is thus defined based on real-world measurements of the present-day porosity ψp(rτ), which yields more accurate decompaction than conventional simulations that ignore real-world porosity and compaction, such as Athy's law.

Present-day porosity ψp(rτ) is measured by direct inspection of the Earth's subsurface material composition. In one example, porosity may be measured by directly analyzing core samples of the Earth's subsurface, for example, using a variety of methods to compare bulk rock volume and total sample volume. In one example, porosity may be derived from well logs, which are measurements performed on rock inside wells. Samples may be collected and porosity measured at regularly or irregularly spaced intervals within the Earth (e.g., bored into the Earth or along well paths). After porosity measurements are taken at those discrete locations, porosity may be extrapolated throughout the entire studied domain. In one example, at least one (and preferably multiple) samples are collected at each distinct depositional layer or depth (e.g., deposited at each distinct period of time).

Let <MAT> be a total compaction coefficient (representing a total compaction as a difference between the minimum present-day porosity and maximum depositional porosity) and let <MAT> be an intermediate compaction coefficient (representing a partial compaction as a difference between the intermediate restoration porosity and maximum depositional porosity). The pair of compaction coefficients, <MAT> and <MAT>, may be defined, for example, as: <MAT>.

Because compaction typically increases over time, the present day porosity ψp(rτ) may be assumed to be less than the restored time porosity ψ(rτ): <MAT>.

This inequality implies that intermediate compaction coefficient <MAT> ≤ total compaction coefficient <MAT>, and so, the ratio in equations (<NUM>) and (<NUM>) is greater than <NUM>, resulting in a vertical elongation in height to dh⊕(rτ) and/or time to dt⊕(rτ) in the decompacted model <NUM> relative to the compacted model <NUM>.

Considering once again the vertical probe introduced above in restored space Gτ, decompaction may proceed by using equation (<NUM>) twice, once in a forward and then in a backward transformation, for example, as follows:.

After this second operation, the probe porosity is equal to the intermediate time restoration porosity ψ(rτ).

Therefore, to take present-day compaction into account, equation (<NUM>) may be replaced, for example, by: <MAT> where compaction coefficient <MAT> is based on the measured present-day porosity ψp(rτ), for example, as defined in equation (<NUM>). Accordingly, the decompacted vertical thickness dh⊕(rτ) at intermediate restoration time τ is elongated based on real-world measurements of the present-day compaction ψp(rτ) experimentally observed within the subsurface of the Earth.

In the restored Gτ-space, the geological time of deposition tτ(rτ) may be interpreted as an arc-length abscissa s(rτ) along the vertical straight line passing through rτ oriented in the same direction as the vertical unit frame vector {rtτ = rz}. Therefore, in the Gτ-space, <MAT> may represent the height of an infinitely short vertical column of restored sediment located at point rτ∈Gτ, subject to present-day compaction. As a consequence, to take compaction into account in the restored Gτ-space, according to equations (<NUM>) and (<NUM>), geological-time tτ(rτ) may be replaced by a "decompacted" geological-time <MAT> such that, for example: <MAT>.

Assuming that {rtτ = rz} is the unit vertical frame vector of the Gτ-space, it follows, for example, that: <MAT>.

From this, it can be concluded that the compacted geological-time tτ(rτ) of point rτ ∈ Gτ should be transformed into a decompacted geological-time <MAT>, for example, honoring the following differential equation: <MAT> with : <MAT>.

Due to the vertical nature of compaction, on the top restored horizon {S(<NUM>)≡ Hτ}, geological-time <MAT> should vanish or reduce to zero and its gradient should be vertical. In other words, in addition to the constraint of equation (<NUM>), geological-time <MAT> may also honor the following example boundary conditions where ruτ and rvτ may represent the unit horizontal frame vectors of the Gτ-space: <MAT>.

Boundary condition (<NUM>)(<NUM>) may ensure that the top restored horizon Hτ is flat and planar at intermediate restoration time τ when it was deposited. Boundary conditions (<NUM>)(<NUM>) and (<NUM>)(<NUM>) may ensure that the direction of change (gradient) of the geological-time tτ(rτ) is vertical in the Gτ-space.

As compaction is a continuous process, geological-time <MAT> may be continuous (e.g., C<NUM>-continuous) across all faults affecting Gτ. As a consequence, in addition to the constraints in equations (<NUM>) and (<NUM>), for any fault F in Gτ, geological-time <MAT> may also honor the following boundary conditions where ( <MAT>) are pairs of "τ-mate-points" defined as collocated points respectively lying on the positive face F + and negative face F- (opposite sides of fault F) at geological time τ: <MAT> <MAT>.

Boundary condition (<NUM>) may ensure that, for any pair of collocated points on opposite sides of the fault, the two points have the same decompacted geological-time coordinate <MAT>. This ensures there are no (or reduced) gaps or overlaps along the fault in the restored Gτ-space.

Using an appropriate numerical method, <MAT> may be computed in Gτ whilst ensuring that differential equation (<NUM>) and boundary conditions (<NUM>) and (<NUM>) are honored. To ensure smoothness and uniqueness of <MAT>, the following constraint may also be added: <MAT>.

In summary, the following GeoChron Based Restoration technique may be used to take compaction into account:.

This approach to decompaction may be seamlessly integrated into the GeoChron Based Restoration framework according to embodiments of the invention and is wholly dissimilar to the sequential decompaction following Athy's law along IPG-lines. In particular, embodiments of the invention perform decompaction based on real-world present-day porosity, a quantity that is accurately measured and extrapolated for any type of rock without having to make assumptions. Additionally, embodiments of the invention allow decompaction in the restored Gτ space representing the Earth's subsurface at an intermediate restoration time in the past τ, before the present day and after the start of deposition of the oldest subsurface layer being imaged.

In the general case, the system of equations (<NUM>), (<NUM>) and (<NUM>) is typically too complex to be solved analytically and may be approximated using numerical methods. However, in a specific case where κ(rτ), ψ<NUM>(rτ) and ψp(rτ) are all constant ∀ rτ ∈ Gτ, the compaction ratio may be integrated at once over the entire domain, and there is no need to iteratively and independently decompact one layer at a time. This special case allows an analytical solution to the system of equations, for example, as follows.

In this special case, in Gτ, terrain porosity is homogeneous and characterized as for example follows where κ, ψ<NUM> and ψp are known constants: <MAT>.

Due to its homogeneity, Gτ may be considered continuous and the intrinsic, vertical nature of compaction implies that any function ϕ(rτ) defined in Gτ associated to compaction may only depend on the vertical component tτ(rτ) of rτ. Therefore, it follows, for example, that: <MAT>.

Let constants A and B be defined, for example, by: <MAT>.

Let the following example functions be derived from Athy's law in equation (<NUM>) and equations (<NUM>) and (<NUM>): <MAT>.

On the one hand, the following example indefinite integral holds true: <MAT>.

On the other hand, according to equation (<NUM>): <MAT>.

Therefore, for any {tτ ≤ <NUM>}, the decompacted restoration function <MAT> may be analytically defined, for example, by: <MAT> with : <MAT>.

Other equations or permutations of these equations or terms may also be used.

Reference is made to <FIG>, which is a flowchart of a method for decompacting a 3D model of the subsurface geology of the Earth at an intermediate restoration time in the past τ, according to an embodiment of the invention.

In operation <NUM>, a processor receives a 3D model of present-day geometry of the subsurface geology and a measure of present-day porosity experimentally measured within the subsurface geology of the Earth. The present day model may be measured tomographically by scanning the Earth's subsurface e.g., as described in reference to <FIG> and <FIG>. To obtain the measure of present-day porosity, a probe may burrow into the Earth's subsurface or into one or more wells to collect and/or analyze material from within the subsurface geology of the Earth. Samples of subsurface materials are collected at spaced intervals, from which porosity is extrapolated throughout the studied domain.

In operation <NUM>, a processor selects a past restoration time τ that is "intermediate" or prior to the present time and after the start of the subsurface's deposition (the time period when an oldest horizon surface in the 3D model was originally deposited).

In operation <NUM>, a processor restores the 3D model from the present day measured geometry (e.g., present day model Gτ <NUM> in xyz-space G <NUM>) to the predicted past geometry at the restoration time in the past τ (e.g., restored model Gτ <NUM> in uτ vτ tτ - space <NUM>) using a 3D restoration transformation (e.g., uτ vτ tτ -transform <NUM>). The 3D model may be restored, for example, as described in reference to <FIG>. Prior to decompaction, the restored model may be a compacted model (e.g., <NUM> of <FIG>).

In operation <NUM>, a processor decompacts the vertical dimension of the restored 3D model. This elongates compacted vertical lengths in the compacted model (e.g., <NUM> of <FIG>) to relatively longer vertical lengths in a decompacted model (e.g., <NUM> of <FIG>). In one embodiment, the vertical length may be a measure of height and the vertical dimension may be expanded from relatively shorter heights dh(rτ) in the compacted model to relatively longer heights dh⊕(rτ) in a decompacted model (e.g., as defined in equation (<NUM>)). In another embodiment, the vertical length may be a measure of geological-time when the particles of sediment were originally deposited on the Earth's surface and the vertical dimension may be expanded from relatively shorter times dt(rτ) in the compacted model to relatively longer times dt⊕(rτ) in a decompacted model (e.g., as defined in equation (<NUM>)). The vertical lengths are elongated based on a relationship between a depositional porosity (e.g., ψ<NUM>(rτ)) of the geological layers at the time sediment in those layers was deposited, restoration porosity (e.g., ψ(rτ)) of the geological layers at the restoration time in the past τ, and the present-day porosity (e.g., ψp(rτ)) of the geological layers experimentally measured in the present-day. In some embodiments, the relationship between the depositional porosity, the restoration porosity, and the present-day porosity may be, for example: <MAT>, where compaction coefficients <MAT> and <MAT> for all points in the restored 3D model, e.g., as defined in equations (<NUM>), (<NUM>) and (<NUM>). Since porosity decreases over time, the restoration porosity is typically greater than the present-day porosity (e.g., equation (<NUM>)) and typically less than the depositional porosity. Accordingly, the compaction coefficients have a relationship <MAT>, and the relationship between the depositional, restoration, and present-day porosities, e.g., <MAT>, is greater than <NUM>, resulting in a stretching or elongating effect to increase the vertical lengths when they are decompacted.

In some embodiments, a processor may decompact the vertical dimension of the restored 3D model by a combination (e.g., equation (<NUM>)) of total decompaction corresponding to an increase in porosity from the present day porosity to the depositional porosity (e.g., equation (<NUM>)) and partial recompaction corresponding to a partial decrease in the porosity from the depositional porosity to the restored porosity (e.g., equation (<NUM>)).

At the restored time in the past τ, the geological layers above Hτ (e.g., Hτ+<NUM>. Hn) did not yet exist, so decompaction may elongate lengths of geological layers below the horizon layer Hτ (e.g., H<NUM>. Hτ) deposited at the restoration time in the past τ. In some embodiments, decompaction may be performed by iteratively decompacting the subsurface layer-by-layer, starting at the top horizon Hτ deposited at the restoration time τ and ending at the bottom horizon H<NUM> deposited at the depositional time. In some embodiments, the depositional porosity and the present-day porosity may be independently determined for each geological layer of the subsurface. In other embodiments, when the depositional porosity and the present-day porosity are substantially constant throughout the subsurface geology, decompaction may occur in one operation over the entire domain of the restored 3D model (e.g., as in equation (<NUM>)).

Some embodiments may implement a boundary condition that ensures that a top horizon Hτ deposited at the restoration time τ is a horizontal plane in the restored 3D model (e.g., equation (<NUM>)(<NUM>)). Additionally or alternatively, some embodiments may implement a boundary condition that ensures that a direction of change of geological-time when the particles of sediment were originally deposited on the Earth's surface is vertical in the restored 3D model (e.g., equation (<NUM>)(<NUM>) and (<NUM>)(<NUM>)). Additionally or alternatively, some embodiments may implement a boundary condition that ensures that, for any pair of collocated points on opposite sides of a fault, the two collocated points are decompacted to have the same coordinate (e.g., equation (<NUM>)).

In some embodiments, for example, implemented in a past-time model, such as the GeoChron model, a processor may decompact the vertical dimension of the restored 3D model by: computing an elongated geological-time (e.g., dt⊕(rτ)) in the restored 3D model (e.g., by solving equation (<NUM>)), transforming the elongated geological-time from the restored 3D model to generate a 1D geological-time (e.g., tτ(rτ)) in the present-day 3D model (e.g., equation (<NUM>)), computing 2D paleo-depositional coordinates (e.g., uτ(rτ) and vτ(rτ)) based on the transformed geological-time (e.g., tτ(rτ)) in the present-day 3D model, and performing a 3D transformation (e.g., a uτ, vτ, tτ-transformation)) comprising the 1D geological-time and 2D paleo-depositional coordinates from the present-day 3D model (e.g., Gτ) to the restored 3D model (e.g., Gτ) that is decompacted based on the elongated geological-time.

Operations of <FIG>, <FIG> and <FIG> may be performed for example using system <NUM> of <FIG>, e.g., by one or more processor(s) <NUM> of <FIG>, or another suitable computing system. The embodiments disclosed in reference to <FIG>, <FIG> and <FIG> may be performed using other operations or orders of the operations, and the exact set of steps shown in the figures may be varied.

In the past <NUM> years, many methods have been proposed to build geological models of sedimentary terrains having layers that are both folded and faulted. For any given geological-time τ, checking geological model consistency is considered both simpler and more accurate if terrains have previously been "restored" to their pre-deformational, unfolded and unfaulted state, as they were at geological-time τ.

Embodiments of the invention provide a new, purely geometrical 3D restoration method based on the input of a depositional (e.g., GeoChron model). Embodiments of the invention are able to handle depositional models of any degree of geometrical and topological complexity, with both small and large deformations, do not assume elastic mechanical behavior, and do not require any prior knowledge of geo-mechanical properties. Embodiments of the invention further reduce or eliminate gaps and overlaps along faults as part of the restoration transformation and do not resort to any post-processing to minimize such gaps and overlaps. Compared to other conventional methods, embodiments of the invention minimize deformations and volume variations induced by geological restoration with a higher degree of precision, unequaled so far (see e.g., <FIG> and <FIG>). Embodiments of the invention further ensure that 2D deformations of horizon surfaces induced by the uvt-transform are kept coherent with 3D deformations of volumes induced by the new proposed 3D restoration method.

Referring to <FIG>, for a given restoration time τ, the set of faults is split into τ-active and τ-inactive subsets. Such a distinction allows:.

Embodiments of the invention input a 3D model of sedimentary terrains in the subsurface. In one example, the input model may be the GeoChron™ model generated by SKUA® software for use in mining and oil and gas industries. Embodiments of the invention may build a 3D restoration transformation of this model in such a way that, after transformation, the new model represents terrains as they were at a given intermediate restoration-time τ (where τ<NUM> < τ < τ<NUM>, before the present day τ<NUM> and after the time of the deposition of the oldest layer τ<NUM>).

For example, G may represent the present day 3D geological domain of the region of the subsurface being modeled and Gτ <NUM> may represent the subset of G containing particles of sediment that were deposited at a time prior to or equal to τ. In some embodiments, for all points r ∈ G, a geologic restoration transformation may move a particle of sediment observed today at location r to a new restored location rτ(r), e.g., defined as follows: <MAT> where Rτ(r) represents a 3D field of restoration vectors, e.g., generated to minimize deformations in Gτ.

A depositional model may be generated by inputting a tomographic model of the present day subsurface geology of the Earth and transforming that geology to a past depositional time as each particle was configured when originally deposited in the Earth. Sedimentary particles are deposited over time in layers from deepest to shallowest from the earliest to the most recent geological time periods. Since various layers of terrain are deposited at different geological times, a depositional model does not image the geology at any one particular time period, but across many times periods, each layer modeled at the geological time when the layer was deposited. Accordingly, the vertical axis or depth in the depositional model may be a time dimension representing the time period of deposition, progressing from oldest to newest geological time as the model progresses vertically from deepest to shallowest layers.

In one embodiment, the depositional model may be the GeoChron™ model, which is generated by SKUA™ software, that is routinely used by many oil & gas companies to build models of geologic reservoirs which help optimize hydrocarbon production and exploration. An example implementation of the GeoChron model is discussed in <CIT>. The depositional model is described in reference to the GeoChron model only for example, though any other depositional model may be used.

Reference is made to <FIG>, which schematically illustrates an example transformation from a present day model (upper-left image) to a depositional GeoChron model (bottom-right image), according to an embodiment of the invention. The transformation may be referred to as a "uvt-transform" <NUM> that transforms a particle of sediment observed today at location r = r(x,y,z) in the present day geological domain G (also referred to as "G-space") <NUM> to be moved to a new depositional location r(r) = r(u,v,t) in the depositional geological domain G (also referred to as "G-space"). The new depositional location r has a vertical coordinate that is the geological time t(r) when the particle at location r was deposited and has horizontal or paleo-geographic coordinates {u(r), v(r)} equal to the lateral spatial location where the particle at r was located at its depositional time t(r). The paleo-geographic coordinates {u(r), v(r)} may be linked to the vertical time coordinate t(r) by different relationships (e.g., constrained by different systems of differential equations) depending on the structural style of their deposition (e.g., minimal deformation or flexural slip).

In the example uvt-transform <NUM> shown in <FIG>, when the geological time coordinate t(r) is equal to the curvilinear distance to the top horizon Hτ <NUM> along curvilinear axis <NUM>, the uvt-transform is a valid technique for imaging the depositional model. In other words, the uvt-transform is a valid depositional rendering technique if the module of its gradient grad t(r) honors the following constraint: <MAT>.

Embodiments of the invention observe that when ∥grad t(r)∥ differs from "<NUM>," replacing the depositional coordinates {u(r), v(r), t(r)} of the uvt-transform <NUM> by new restoration coordinates {uτ(r), vτ(r), tτ(r)} where ∥grad tτ∥ = <NUM> allows the uvt- transform to be replaced by a uτ vτ tτ -transform that generates a valid restoration model at restoration time τ.

In some embodiments, the depositional (e.g., GeoChron) model includes the following data structures stored in a memory (e.g., memory <NUM> of <FIG>) (see <FIG>, <FIG>, <FIG>, and <FIG>):.

Moreover, referring to <FIG> and <FIG>, the depositional model may have the following properties:.

It would be appreciated by a person of ordinary skill in the art that the GeoChron model and its features described herein are discussed only as an example of a depositional model and that these elements may differ in other models or implementations.

Referring to the volume deformation of <FIG>, the restoration time τ may be a given geological time in the past and subdomain Gτ <NUM> may be a part of a 3D present day geological domain G that has terrains older than (deposited at a time prior to) or equal to restoration time τ and defined by a depositional model. Embodiments of the invention provide a new unfolding technique that replaces the uvt-transform (converting the present day model to a depositional model rendering all layers at their many respective times of deposition) by a uτ vτ tτ -transform <NUM> (converting the present day model to a restored model at a single restoration time τ before the present day but after the earliest times of deposition of the deepest model layer): <MAT>.

Accordingly, present day geological space Gτ <NUM> is transformed into a restored geological space Gτ <NUM>, such that:.

Compaction may be handled in pre and post-restoration stages, as is known in the art. Thus, the model may be restored without taking compaction into account.

Some embodiments of the invention provide an inventive volume deformation with a new set of inventive geometric constraints on the depositional model to allow geologic layers to be restored at a given geological time τ with a precision that has never before been reached. As shown in <FIG>, in this volume deformation, paleo-geographic coordinates {u(r) and v(r)} and the geological time coordinate t(r) are replaced by new restoration coordinates respectively denoted {uτ(r), vτ(r)} and tτ(r).

For simplicity and without loss of generality, the coordinate frame unit vectors {ruτ, rvτ, rtτ} <NUM> of the Gτ -space and its origin Ouτ vτ tτ may be equal to the coordinate frame unit vectors {rx, ry, rz} <NUM> of the G-space and its origin Oxyz: <MAT>.

Referring to <FIG>, the following notation is used:.

Equivalently to equations (<NUM>) and in accordance with equation (<NUM>), during restoration of Gτ, a particle of sediment observed today at location r <NUM> is moved to a new location r(r) <NUM> defined e.g., as follows, where Rτ(r) is a restoration vector field: <MAT> with, in matrix notation: <MAT>.

Referring to <FIG>, surface Sτ(<NUM>) <NUM> is located at an altitude (tτ = zτO) with respect to the vertical unit vector rtτ oriented upward. In the frame of the presentation of the volume deformation and without loss of generality, zτO may be assumed to be constant, e.g., equal to zero.

Referring to <FIG>, surface Sτ(d) <NUM> is located at a distance (d) from Sτ(<NUM>) <NUM>, implying that: <MAT> such that: <MAT>.

<FIG> shows the folded present-day volume Gτ <NUM> resulting from the deformation of restored volume Gτ <NUM> under tectonic forces following either a "minimal deformation" or a "flexural slip" tectonic style: <MAT>.

As shown in <FIG>, the part of the subsurface observed today stratigraphically below Hτ <NUM> may be identified with the deformed volume Gτ <NUM>, e.g., such that:.

With compaction handled separately in pre and post restoration steps, leaving aside the very particular case of clay and salt layers, tectonic forces generally induce no or negligible variations in volume. Therefore, restoration coordinates {uτ(r), vτ(r), tτ(r)} may be chosen in such a way that the uτ vτ tτ -transform <NUM> of the present-day volume Gτ <NUM> into the restored volume Gτ <NUM> minimizes deformations and volume variations. This is achieved by constraining restoration coordinates {uτ(r), vτ(r), tτ(r)} to honor the two following conditions in the present day Gτ domain:.

So as not to conflict with equations (<NUM>) and (<NUM>), and contrary to conventional depositional coordinates u and v (e.g., in the GeoChron model), new constraints (<NUM>) and (<NUM>) do not constrain ∥grad uτ∥, ∥grad vτ∥, ∥gradS uτ∥, or ∥gradS vτ∥ to be equal to "<NUM>".

Referring to <FIG> and <FIG>, at geological-time τ, the horizon Hτ <NUM> to be restored was coincident with a given surface Sτ(<NUM>) (<NUM>=<NUM>) considered as the sea-floor. The task of restoration includes:.

At geological time τ, the sea floor Sτ(<NUM>) (<NUM>) is assumed to be a continuous, unfaulted surface whose altitude zτO is a given function zτO(u, v). In practice, Sτ(<NUM>) (<NUM>=<NUM>) is typically a flat, horizontal plane whose altitude zτO(u, v) at geological time τ is constant. Accordingly, for concision, zτO may refer to a given function zτO(u(r), v(r)) which may or may not be constant: <MAT>.

Deformation of sedimentary terrains is typically induced both by tectonic forces and terrain compaction. In order to model separately the effects of these phenomena, the restoration process may proceed as follows:.

As an input to the restoration process, a given depositional (e.g., GeoChron) model may be received from storage in a digital device (e.g., from memory <NUM> of <FIG>).

Referring to <FIG>, a geological time τ may be selected that is associated with the given horizon Hτ <NUM> to be restored and the given altitude zτO of the surface S τ(<NUM>) <NUM> onto which the horizon Hτ should be restored.

The region Gτ <NUM> may be retrieved as the part of the depositional model where geological time of deposition t(r) is less than or equal to τ (subsurface regions deposited in a layer deeper than or equal to the layer deposited at time τ).

The set of faults may be split into a subset of r-active faults cutting Hτ <NUM> and a subset of τ-inactive faults which do not cut Hτ,.

A geologist or other user may decide to manually transfer some faults from the τ-inactive fault set to the τ-active set, or vice versa, which typically causes greater restoration deformations. For example, manually altering the set of automatically computed τ-active and τ-inactive faults typically makes the restoration less accurate.

Four new 3D piecewise continuous discrete functions {uτ, vτ, tτ, ετ}r may be created that are defined on grid Γ <NUM> whose discontinuities occur only across τ-active faults;.

Referring to <FIG>, to remove discontinuities of discrete functions {uτ, vτ, tτ, ετ}r across τ-inactive faults, for all τ-inactive faults F <NUM>, one or more of the following inventive (e.g., DSI) constraints may be installed on Γ <NUM>, e.g., as: <MAT> <MAT> <MAT> where <MAT> (<NUM>,<NUM>) represents a pair of "mate-points" collocated on both sides of F <NUM> and assigned to F + <NUM> and F - <NUM>, respectively, and ετ(r) represents an error correction constraint. Constraints (<NUM>), (<NUM>), (<NUM>) and (<NUM>) may be referred to collectively as "fault transparency constraints.

Assuming that THmin > <NUM> is a given threshold chosen by a geologist or other user, fault transparency constraints (<NUM>), (<NUM>), (<NUM>) and (<NUM>) may be locally installed at any point rF on a τ-active fault F where fault throw is lower than THmin. As a non-limitative example, THmin may be equal to <NUM> meter.

Two new discrete vector fields r* and Rτ may be defined on 3D grid Γ <NUM>.

Referring to <FIG> and <FIG>, the depositional uvt-transform <NUM> of Gτ <NUM> is typically correct when equation (<NUM>) is honored. Based on this observation, embodiments of the present invention adapt equation (<NUM>) for the inventive restoration technique, replacing the vertical depositional coordinate t(r) by a vertical restoration coordinate tτ(r) and replacing equation (<NUM>) by the following inventive thickness-preserving constraint to ensure layer thickness is preserved and surfaces {Sτ(d) : d ≥ <NUM>} are parallel: <MAT>.

In addition, to allow Hτ <NUM> to be restored on surface Sτ(<NUM>) <NUM>, the vertical restoration coordinate tτ(r) may honor the following boundary condition, e.g., as a DSI constraint on grid Γ <NUM>, referred to as the "paleo-sea-floor constraint": <MAT>.

Due to its non-linearity, thickness-preserving equation (<NUM>) cannot be implemented as a DSI constraint, which must be linear. In order to incorporate the thickness-preserving equation into the restoration model using the DSI method, various linear surrogates of equation (<NUM>) may be used to approximate tτ(r) as follows:.

Constraints (<NUM>) and (<NUM>) are only examples of possible surrogate-thickness-preserving constraints. Other examples of such surrogate thickness-preserving constraints may be used.

Referring to <FIG>, contrary to constraint (<NUM>)-(<NUM>), new inventive constraint (<NUM>)-(<NUM>) benefits from the geologic observation that, throughout the entire domain Gτ <NUM>, surfaces {Sτ(d) : d ≥ <NUM>} <NUM> generally have a shape roughly similar to the level sets of the geologic time of deposition t(r);.

Assuming that constraints (<NUM>) and (<NUM>) or (<NUM>) are installed on grid Γ <NUM>, a first approximation of vertical restoration coordinate t'τ(r) may be computed by running the DSI method on grid Γ <NUM>.

Honoring constraint (<NUM>) significantly increases the accuracy of the restoration model and a violation of this constraint not only degrades the accuracy of the vertical restoration coordinate tτ(r) but also the horizontal restoration coordinates {uτ(r), vτ(r)} as they are linked to tτ(r) (e.g., by equations (<NUM>) and (<NUM>)). Accordingly, there is a great need for validating any approximation technique used to compute tτ(r).

To test the accuracy of the various approximations of tτ(r), an example geological terrain is provided in <FIG>. Despite the apparent simplicity of this terrain, because the thicknesses of the layers vary, this test example is challenging and useful in comparing the accuracy of inventive embodiments with other conventional techniques.

<FIG> shows histograms <NUM> and <NUM> of the distributions of ∥grad tτ∥, where tτ is approximated using constraints (<NUM>) or (<NUM>), respectively, in the example geological terrain Gτ <NUM> shown in <FIG>. <FIG> shows that when tτ is approximated by constraints (<NUM>) or (<NUM>), ∥grad tτ∥ significantly differs from "<NUM>". Therefore, while constraints (<NUM>) provide a better approximation of the thickness-preserving equation (<NUM>) than constraints (<NUM>), both of these approximations are inaccurate.

Similarly, <FIG> shows histograms <NUM> and <NUM> of relative variations of volume ΔV /V induced by the restoration of Hτ <NUM> over Gτ <NUM> shown in <FIG>, where tτ is approximated using constraints (<NUM>) or (<NUM>), respectively. Ideally, a restoration transformation should minimize variations in volume ΔV /V from the present day to the restored model. <FIG> however shows that a restoration based on constraints (<NUM>) or (<NUM>) results in a volume variation ΔV /V that significantly differs from the ideal value of "<NUM>". While constraints (<NUM>) result in a smaller volume variation ΔV /V than constraints (<NUM>), both of these approximations induce a significant volume variation ΔV and are inaccurate.

An approximation of the vertical restoration coordinate t'τ(r) may be improved by a "tτ-incremental improvement" constraint, e.g., as follows: <MAT> where ετ(r) is an error correction term, e.g., as characterized below.

Accordingly, assuming that an initial approximation t'τ(r) has already been obtained, to compute an improved version of tτ(r), the following inventive incremental procedure may be executed:.

This constraint specifies that, after applying correction constraint (<NUM>), in the close neighborhood of τ-active faults, the shape of level sets of tτ(r) remains roughly unchanged.

Referring to <FIG>, with respect to surfaces {Sτ(d) : d ≥ <NUM>} <NUM>, horizontal restoration coordinates {uτ(r), vτ(r)} play a role similar to the one played by paleo-geographic coordinates {u(r), v(r)} with respect to horizons {Ht : t ≥ <NUM>} <NUM> of the depositional model provided as input. Based on this similarity, horizontal restoration coordinates {uτ(r), vτ(r)} may be generated as follows:.

The τ-axe and τ-coaxe vector fields aτ(r) and bτ(r) differ considerably from the local axe and co-axe vectors fields a(r) and b(r), e.g., as discussed in <CIT>.

These new τ-axe and τ-coaxe vectors aτ(r) and bτ(r) strongly depend on the new vertical restoration coordinate tτ(r) (e.g., already computed as above) and also take into account the gradient of the paleo-geographic coordinate u(r) (e.g., associated to the depositional model provided as input).

The restoration vector field Rτ(r) represents the field of deformation vectors from the present day (e.g., xyz) space to the restoration (e.g., uτ vτ tτ) space, e.g., computed from the uτ vτ tτ - transform.

Referring to <FIG>, for each node α <NUM> of 3D grid Γ <NUM>, move α to restored location r(α), e.g., defined as follows: <MAT>.

For each node α <NUM> of 3D grid Γ <NUM>:.

Consider a series of geological restoration times {τ<NUM> < τ<NUM> < < τn} associated with reference horizons Hτ<NUM>, Hτ<NUM>,. , Hτn, respectively. Using the restoration method described herein, for each (τi = τ), build and store on a digital device a restoration vector field Rτi(r) = Rτ(r), e.g., as: <MAT>.

In addition to these reference restoration times, an additional restoration time τn+<NUM> may be added to be associated with the horizontal plane Htn+<NUM> located at a constant altitude <MAT> of the sea level. Time τn+<NUM> may be the present day geological time and, provided that τn+<NUM> is greater than τn, any arbitrary value may be chosen for τn+<NUM>. As a non-limitative example, τn+<NUM> may be defined as: <MAT>.

Because τn+<NUM> is the present day, applying the restoration vector field Rτn+<NUM>(r) to the present day horizon Htn+<NUM> should leave Htn+<NUM> unchanged e.g., as follows: <MAT>.

To explore subsurface evolution throughout geological times, a process may proceed as follows:.

Geological models are generated using geological or seismic tomography technology. Geological tomography generates an image of the interior subsurface of the Earth based on geological data collected by transmitting a series of incident waves and receiving reflections of those waves across discontinuities in the subsurface. A transmitter may transmit signals, for example, acoustic waves, compression waves or other energy rays or waves, that may travel through subsurface structures. The transmitted signals may become incident signals that are incident to subsurface structures. The incident signals may reflect at various transition zones or geological discontinuities throughout the subsurface structures, such as, faults or horizons. The reflected signals may include seismic events. A receiver may collect data, for example, reflected seismic events. The data may be sent to a modeling mechanism that may include, for example, a data processing mechanism and an imaging mechanism.

Reference is made to <FIG>, which is a schematic illustration of a geological tomography technique in which a series of incident rays <NUM> and reflected rays <NUM> are propagated through a subsurface region of the Earth <NUM> to image the subsurface, according to an embodiment of the invention.

One or more transmitter(s) (e.g., <NUM> of <FIG>) located at incident location(s) <NUM> may emit a series of incident rays <NUM>. Incident rays <NUM> may include for example a plurality of energy rays related to signal waves, e.g., sonic waves, seismic waves, compression waves, etc. Incident rays <NUM> may be incident on, and reflect off of, a subsurface structure or surface <NUM> at a reflection point <NUM>. Multiple reflection points <NUM> may be identified or imaged or displayed in conjunction to display, for example, a horizon.

One or more receiver(s) (e.g., <NUM> of <FIG>) located at reflected location(s) <NUM> may receive the reflection rays <NUM>. Reflection rays <NUM> may be the reflected images of incident rays <NUM>, for example, after reflecting off of image surface <NUM> at target point <NUM>. The angle of reflection <NUM> may be the angle between corresponding incident rays <NUM> and reflected rays <NUM> at reflection point <NUM>. An incident rays <NUM> and a corresponding reflected rays <NUM> may propagate through a cross-section of a subsurface structure <NUM>. Incident rays <NUM> may reflect off of a subsurface feature <NUM> at a reflection point <NUM>, for example, a point on an underground horizon, the seafloor, an underground aquifer, etc..

One or more processor(s) (e.g., <NUM> of <FIG>) may reconstitute incident and reflected rays <NUM> and <NUM> to generate an image the subsurface <NUM> using an imaging mechanism. For example, a common reflection angle migration (CRAM) imaging mechanism may image reflection points <NUM> by aggregating all reflected signals that may correspond to a reflection point, for example, reflected signals that may have the same reflection angle. In other examples, imaging mechanisms may aggregate reflected signals that may have the same reflection offset (distance between transmitter and receiver), travel time, or other suitable conditions.

The processor(s) may compose all of the reflection points <NUM> to generate an image or model of the present day underground subsurface of the Earth <NUM>. The processor(s) may execute a restoration transformation (e.g., uτ vτ tτ - transform <NUM>) to transform the present day model of subsurface <NUM> to a restored subsurface image <NUM> at a restoration time τ. One or more display(s) (e.g., <NUM> of <FIG>) may visualize the present day subsurface image <NUM> and/or the restored subsurface image <NUM>.

Reference is made to <FIG>, which schematically illustrates a system including one or more transmitter(s), one or more receiver(s) and a computing system in accordance with an embodiment of the present invention. Methods disclosed herein may be performed using a system <NUM> of <FIG>.

System <NUM> may include one or more transmitter(s) <NUM>, one or more receiver(s) <NUM>, a computing system <NUM>, and a display <NUM>. The aforementioned data, e.g., seismic data used to form intermediate data and finally to model subsurface regions, may be ascertained by processing data generated by transmitter <NUM> and received by receiver <NUM>. Intermediate data may be stored in memory <NUM> or other storage units. The aforementioned processes described herein may be performed by software <NUM> being executed by processor <NUM> manipulating the data.

Transmitter <NUM> may transmit signals, for example, acoustic waves, compression waves or other energy rays or waves, that may travel through subsurface (e.g., below land or sea level) structures. The transmitted signals may become incident signals that are incident to subsurface structures. The incident signals may reflect at various transition zones or geological discontinuities throughout the subsurface structures. The reflected signals may include seismic data.

Receiver <NUM> may accept reflected signal(s) that correspond or relate to incident signals, sent by transmitter <NUM>. Transmitter <NUM> may transmit output signals. The output of the seismic signals by transmitter <NUM> may be controlled by a computing system, e.g., computing system <NUM> or another computing system separate from or internal to transmitter <NUM>. An instruction or command in a computing system may cause transmitter <NUM> to transmit output signals. The instruction may include directions for signal properties of the transmitted output signals (e.g., such as wavelength and intensity). The instruction to control the output of the seismic signals may be programmed in an external device or program, for example, a computing system, or into transmitter <NUM> itself.

Computing system <NUM> may include, for example, any suitable processing system, computing system, computing device, processing device, computer, processor, or the like, and may be implemented using any suitable combination of hardware and/or software. Computing system <NUM> may include for example one or more processor(s) <NUM>, memory <NUM> and software <NUM>. Data <NUM> generated by reflected signals, received by receiver <NUM>, may be transferred, for example, to computing system <NUM>. The data may be stored in the receiver <NUM> as for example digital information and transferred to computing system <NUM> by uploading, copying or transmitting the digital information. Processor <NUM> may communicate with computing system <NUM> via wired or wireless command and execution signals.

Memory <NUM> may include cache memory, long term memory such as a hard drive, and/or external memory, for example, including random access memory (RAM), read only memory (ROM), dynamic RAM (DRAM), synchronous DRAM (SD-RAM), flash memory, volatile memory, non-volatile memory, cache memory, buffer, short term memory unit, long term memory unit, or other suitable memory units or storage units. Memory <NUM> may store instructions (e.g., software <NUM>) and data <NUM> to execute embodiments of the aforementioned methods, steps and functionality (e.g., in long term memory, such as a hard drive). Data <NUM> may include, for example, raw seismic data collected by receiver <NUM>, instructions for building a mesh (e.g., <NUM>), instructions for partitioning a mesh, and instructions for processing the collected data to generate a model, or other instructions or data. Memory <NUM> may also store instructions to divide and model τ-active faults and τ-inactive faults. Memory <NUM> may generate and store the aforementioned constraints, restoration transformation (e.g., uτ vτ tτ-transform <NUM>), restoration coordinates (e.g., uτ, vτ, tτ), a geological-time and paleo-geographic coordinates (e.g., u, v, t), a model representing a structure when it was originally deposited (e.g., in uvt-space), a model representing a structure at an intermediate restoration time (e.g., in uτ, vτ, tτ -space), and/or a model representing the corresponding present day structure in a current time period (e.g., in xyz-space). Memory <NUM> may store cells, nodes, voxels, etc., associated with the model and the model mesh. Memory <NUM> may also store forward and/or reverse uτ, vτ, tτ -transformations to restore present day models (e.g., in xyz-space) to restored models (e.g., in uτ, vτ, tτ -space), and vice versa. Memory <NUM> may also store the three-dimensional restoration vector fields, which when applied to the nodes of the initial present day model, move the nodes of the initial model to generate one of the plurality of restored models. Applying a restoration vector field to corresponding nodes of the present day model may cause the nodes to "move", "slide", or "rotate", thereby transforming modeled geological features represented by nodes and cells of the initial model. Data <NUM> may also include intermediate data generated by these processes and data to be visualized, such as data representing graphical models to be displayed to a user. Memory <NUM> may store intermediate data. System <NUM> may include cache memory which may include data duplicating original values stored elsewhere or computed earlier, where the original data may be relatively more expensive to fetch (e.g., due to longer access time) or to compute, compared to the cost of reading the cache memory. Cache memory may include pages, memory lines, or other suitable structures. Additional or other suitable memory may be used.

Computing system <NUM> may include a computing module having machine-executable instructions. The instructions may include, for example, a data processing mechanism (including, for example, embodiments of methods described herein) and a modeling mechanism. These instructions may be used to cause processor <NUM> using associated software <NUM> modules programmed with the instructions to perform the operations described. Alternatively, the operations may be performed by specific hardware that may contain hardwired logic for performing the operations, or by any combination of programmed computer components and custom hardware components.

Embodiments of the invention may include an article such as a non-transitory computer or processor readable medium, or a computer or processor storage medium, such as for example a memory, a disk drive, or a USB flash memory, encoding, including or storing instructions, e.g., computer-executable instructions, which when executed by a processor or controller, carry out methods disclosed herein.

Display <NUM> may display data from transmitter <NUM>, receiver <NUM>, or computing system <NUM> or any other suitable systems, devices, or programs, for example, an imaging program or a transmitter or receiver tracking device. Display <NUM> may include one or more inputs or outputs for displaying data from multiple data sources or to multiple displays. For example, display <NUM> may display visualizations of subsurface models including subsurface features, such as faults, horizons and unconformities, as a present day subsurface image (e.g., <NUM>), a restored subsurface image (e.g., <NUM>) and/or a depositional model (e.g., <NUM>). Display <NUM> may display one or more present day model(s), depositional model(s), restoration model(s), as well as a series of chronologically sequential restoration models associated with a sequence of respective restoration times (e.g., τ<NUM> < τ<NUM> < τ<NUM> < τ<NUM>, as shown in <FIG>). The models may be displayed one at a time, two at a time, or many at a time (e.g., the number selected by a user or automatically based on the difference between models or the total number of models). Display <NUM> may display the models in a sequence of adjacent models, through which a user may scan (e.g., by clicking a 'next' or 'previous' button with a pointing device such as a mouse or by scrolling through the models).

Input device(s) <NUM> may include a keyboard, pointing device (e.g., mouse, trackball, pen, touch screen), or cursor direction keys, for communicating information and command selections to processor <NUM>. Input device <NUM> may communicate user direction information and command selections to the processor <NUM>. For example, a user may use input device <NUM> to select one or more preferred models from among the plurality of perturbed models, recategorize faults as τ-active faults and τ-inactive, or edit, add or delete subsurface structures.

Processor <NUM> may include, for example, one or more processors, controllers or central processing units ("CPUs"). Software <NUM> may be stored, for example, in memory <NUM>. Software <NUM> may include any suitable software, for example, DSI software.

Processor <NUM> may generate a present day subsurface image (e.g., <NUM>), a restored subsurface image (e.g., <NUM>) and/or a depositional model (e.g., <NUM>), for example, using data <NUM> from memory <NUM>. In one embodiment, a model may simulate structural, spatial or geological properties of a subsurface region, such as, porosity or permeability through geological terrains.

Processor <NUM> may initially generate a three dimensional mesh, lattice, grid or collection of nodes (e.g., <NUM>) that spans or covers a domain of interest. The domain may cover a portion or entirety of the three-dimensional subsurface region being modeled. Processor <NUM> may automatically compute the domain to be modeled and the corresponding mesh based on the collected seismic data so that the mesh covers a portion or the entirety of the three-dimensional subsurface region from which geological data is collected (e.g., the studied subsurface region). Alternatively or additionally, the domain or mesh may be selected or modified by a user, for example, entering coordinates or highlighting regions of a simulated optional domain or mesh. For example, the user may select a domain or mesh to model a region of the Earth that is greater than a user-selected subsurface distance (e.g., <NUM> meters) below the Earth's surface, a domain that occurs relative to geological features (e.g., to one side of a known fault or riverbed), or a domain that occurs relative to modeled structures (e.g., between modeled horizons H(t<NUM>) and H(t<NUM>)). Processor <NUM> may execute software <NUM> to partition the mesh or domain into a plurality of three-dimensional (3D) cells, columns, or other modeled data (e.g., represented by voxels, pixels, data points, bits and bytes, computer code or functions stored in memory <NUM>). The cells or voxels may have hexahedral, tetrahedral, or any other polygonal shapes, and preferably three-dimensional shapes. Alternatively, data may include zero-dimensional nodes, one-dimensional segments, two-dimensional facet and three-dimensional elements of volume, staggered in a three-dimensional space to form three-dimensional data structures, such as cells, columns or voxels. The cells preferably conform to and approximate the orientation of faults and unconformities. Each cell may include faces, edges and/or vertices. Each cell or node may correspond to one or more particles of sediment in the Earth (e.g., a node may include many cubic meters of earth, and thus many particles).

Data collected by receiver <NUM> after the time of deposition in a current or present time period, include faults and unconformities that have developed since the original time of deposition, e.g., based on tectonic motion, erosion, or other environmental factors, may disrupt the regular structure of the geological domain. Accordingly, an irregular mesh may be used to model current geological structures, for example, so that at least some faces, edges, or surfaces of cells are oriented parallel to faults and unconformities, and are not intersected thereby. In one embodiment, a mesh may be generated based on data collected by receiver <NUM>, alternatively, a generic mesh may be generated to span the domain and the data collected by receiver <NUM> may be used to modify the structure thereof. For example, the data collected may be used to generate a set of point values at "sampling point". The values at these points may reorient the nodes or cells of the mesh to generate a model that spatially or otherwise represents the geological data collected from the Earth. Other or different structures, data points, or sequences of steps may be used to process collected geological data to generate a model. The various processes described herein (e.g., restoring a geological model using τ-active and τ-inactive faults, or restoring a geological model using a new thickness-preserving constraint) may be performed by manipulating such modeling data.

Restoration coordinates may be defined at a finite number of nodes or sampling points based on real data corresponding to a subsurface structure, e.g., one or more particles or a volume of particles of Earth. Restoration coordinates may be approximated between nodes to continuously represent the subsurface structure, or alternatively, depending on the resolution in which the data is modeled may represent discrete or periodic subsurface structures, e.g., particles or volumes of Earth that are spaced from each other.

The computing system of <FIG> may accept the data used in the operations of <FIG>, <FIG> and <FIG> as for example a set of data generated by tomographic scanning of a subsurface geological region of the Earth as disclosed in reference to <FIG>, or such data augmented by another process. The computing system may accept one or more of seismic and well data. The computing device may generate one or more of seismic and well data.

"Restoration" or "intermediate" time τ may refer to a time in the past before the present day and after a time when an oldest or deepest horizon surface in the 3D model was deposited. "Restoration" or "intermediate" transformation or model may refer to a model or image of the surface as it was configured at the "intermediate" time in the past τ. An intermediate horizon may refer to a horizon that was deposited at the "intermediate" time τ, which is located above the deepest horizon and below the shallowest horizon.

"Time" including the present-day, current or present time, the past restoration time τ, and/or the depositional time t, may refer to geological time periods that span a duration of time, such as, periods of thousands or millions of years.

"Geological-time" t(r) may refer to the time of deposition when a particle of sediment represented by point r was originally deposited in the Earth. In some embodiments, the geological-time of the deposition may be replaced, e.g., by any arbitrary monotonic increasing function of the actual geological-time. It is a convention to use an monotonically increasing function, but similarly an arbitrary monotonic decreasing function may be used. The monotonic function may be referred to as the "pseudo-geological-time".

The geological-time of the deposition and restoration time of particles are predicted approximate positions since past configurations can not typically be verified.

"Current" or "present day" location for a particle (or data structure representing one or more particles) or subsurface feature may refer to the location of the item in the present time, as it is measured.

In stratified terrain, layers, horizons, faults and unconformities may be curvilinear surfaces which may be for example characterized as follows.

Terrain deformed in the neighborhood of a point r in the G-space may occur according to a "minimal deformation" tectonic style when, in this neighborhood:.

Terrain deformed in the neighborhood of a point r in the G-space may occur according to a "flexural slip" tectonic style when, in this neighborhood:.

Discrete-Smooth-Interpolation (DSI) is a method for interpolating or approximating values of a function f(x,y,z) at nodes of a 3D grid or mesh Γ (e.g., <NUM>), while honoring a given set of constraints. The DSI method allows properties of structures to be modeled by embedding data associated therewith in a (e.g., 3D Euclidean) modeled space. The function f(x,y,z) may be defined by values at the nodes of the mesh, Γ. The DSI method allows the values of f(x,y,z) to be computed at the nodes of the mesh, Γ, so that a set of one or more (e.g., linear) constraints are satisfied. DSI generally only applies linear constraints on the model.

In some embodiments, bold symbols represent vectors or multi-dimensional (e.g., 3D) functions or data structures.

In some embodiments, the "simeq" symbol "≃" or "≅" may mean approximately equal to, e.g., within <NUM>-<NUM>% of, or in a least squares sense. In some embodiments, the symbol "≡" may mean identical to, or defined to be equal to.

While embodiments of the invention describe the input depositional model as the GeoChron model, any other depositional model visualizing the predicted configuration of each particle, region or layer at its respective time of depositional may be used.

While embodiments of the invention describe the present day coordinates as xyz, the restoration coordinates as uτvτtτ, the depositional coordinates as uvt, the restoration transformation as a uτvτtτ -transform, and the depositional transformation as a uvt-transform, any other coordinates or transformations may be used.

In the foregoing description, various aspects of the present invention have been described. For purposes of explanation, specific configurations and details have been set forth in order to provide a thorough understanding of the present invention. However, it will also be apparent to one skilled in the art that the present invention may be practiced without the specific details presented herein. Furthermore, well known features may have been omitted or simplified in order not to obscure the present invention. Unless specifically stated otherwise, as apparent from the following discussions, it is appreciated that throughout the specification discussions utilizing terms such as "processing," "computing," "calculating," "determining," or the like, refer to the action and/or processes of a computer or computing system, or similar electronic computing device, that manipulates and/or transforms data represented as physical, such as electronic, quantities within the computing system's registers and/or memories into other data similarly represented as physical quantities within the computing system's memories, registers or other such information storage, transmission or display devices. In addition, the term "plurality" may be used throughout the specification to describe two or more components, devices, elements, parameters and the like.

Embodiments of the invention may manipulate data representations of real-world objects and entities such as underground geological features, including faults and other features. The data may be generated by tomographic scanning, as discussed in reference to <FIG>, e.g., received by for example a receiver receiving waves generated e.g., by an air gun or explosives, that may be manipulated and stored, e.g., in memory <NUM> of <FIG>, and data such as images representing underground features may be presented to a user, e.g., as a visualization on display <NUM> of <FIG>.

When used herein, a subsurface image or model may refer to a computer-representation or visualization of actual geological features such as horizons and faults that exist in the real world. Some features when represented in a computing device may be approximations or estimates of a real world feature, or a virtual or idealized feature, such as an idealized horizon as produced in a uτ vτ tτ-transform. A model, or a model representing subsurface features or the location of those features, is typically an estimate or a "model", which may approximate or estimate the physical subsurface structure being modeled with more or less accuracy.

It will thus be seen that the objects set forth above, among those made apparent from the preceding description, are efficiently attained and it is intended that all matter contained in the above description and shown in the accompanying drawings shall be interpreted as illustrative and not in a limiting sense.

Claim 1:
A computer-implemented method for decompacting a 3D model of the subsurface geology of the Earth at an intermediate restoration time in the past τ, the method comprising:
receiving a 3D model of present-day geometry of the subsurface geology ;
selecting a value of a restoration time in the past τ before the present day and after a time an oldest horizon surface in the 3D model of the subsurface was deposited;
restoring the 3D model from the present day measured geometry to the predicted past geometry at the restoration time in the past τ using a 3D transformation; and
decompacting the vertical dimension of the restored 3D model to elongate vertical lengths of geological layers below a horizon layer deposited at the restoration time in the past τ, characterized in that
the method further comprises receiving a measure of present-day porosity experimentally measured within the subsurface geology of the Earth and
wherein the vertical lengths are elongated based on a relationship between a depositional porosity of the geological layers at the time sediment in those layers was deposited, restoration porosity of the geological layers at the restoration time in the past τ, and the present-day porosity of the geological layers experimentally measured in the present-day.