Patent Publication Number: US-2021165124-A1

Title: Modeling Geological Strata Using Weighted Parameters

Description:
TECHNICAL FIELD 
     The present disclosure relates generally to geological interpretation. More specifically, but not by way of limitation, this disclosure relates to modeling geological strata using weighted parameters. 
     BACKGROUND 
     A well system (e.g., for extracting oil or gas) can include multiple wellbores drilled through a subterranean formation. Each wellbore can have a well logging tool that provides data in the form of a well log back to a well operator. A well log can be a record indicative of the geologic formations that are penetrated by a wellbore. The well operator can manually review the well logs to identify strata or other features of interest in the subterranean formation. For example, the well operator can manually review multiple well logs in a two-dimensional (2D) cross-sectional view or a three-dimensional (3D) view to identify structures or features of interest. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a cross-sectional side view of an example of a well system for obtaining well logs according to some aspects. 
         FIG. 2  is a cross-sectional side view of an example of a model of strata according to some aspects. 
         FIG. 3  is a block diagram of an example of a computing device according to some aspects. 
         FIG. 4  is flow chart of an example of a process for modeling geological strata using weighted parameters according to some aspects. 
         FIG. 5  is a flow chart of an example of a process for generating a geological model according to some aspects. 
     
    
    
     DETAILED DESCRIPTION 
     Certain aspects and features of the present disclosure relate to generating a geological model of a subterranean formation by concurrently determining shapes of strata in the subterranean formation based on user-provided weights for parameters. The parameters can define requirements for the geological model. For example, the parameters can be geologic metrics usable for generating the geological model. And the user-provided weights can be values indicating the relative influences of the parameters on the geological model. In some examples, generating the geological model based on the user-provided weights for the parameters can enable the geological model to be customized in a manner that is most useful for the user. And because the shapes of strata are interrelated due to their layered positioning on one another, a geological model that accounts for these interrelationships by concurrently determining the shapes of the strata (as opposed to sequentially determining the shapes of the strata) can be more realistic and accurate. 
     In some examples, the parameters can include constraints for generating the model. Examples of the constraints can include that (i) a gradient in a thickness of a stratum is to be minimized, (ii) a minimum thickness of a stratum is to be maintained, (iii) a maximum thickness of a stratum is to be maintained, (iv) a curvature in the thickness of a stratum is to be minimized, (v) a user-provided indication of a feature of the stratum is to be respected, (vi) surfaces of the strata are not to cross each other, or (vii) any combination of these. One specific example of a constraint can include a regional thickness constraint that forces the thicknesses of the strata to remain substantially consistent over the region of the subterranean formation being modeled. This may be useful when modeling, for example, a topset region of the subterranean formation. The topset region can include a nonmarine-to-shallow marine region of a basin fill, and may exhibit consistent thicknesses over long distances as a result of the long-wavelength subsidence trends that control the thickness of the topset region. By invoking (or heavily weighting) the regional thickness constraint, a more realistic and accurate model of the topset region may be generated. 
     The shapes for the strata can be determined concurrently using a system of equations. For example, a computing device can include multiple quadratic functions to be solved concurrently to determine thicknesses, surface shapes, or both of the strata. The quadratic functions can also take into account the user-provided weights for the parameters. The computing device can solve the quadratic functions together to determine the shapes of the strata. The computing device can then form the geological model using the determined shapes of the strata. 
     In some examples, a user can set a priority of a first parameter over a second parameter, such that the first parameter is to be honored over the second parameter (e.g., if the two parameters conflict). The geological model can then be generated based on this prioritization. For example, a computing device can receive user input indicating that a parameter for minimizing a change in a thickness of a stratum is to be prioritized over another parameter for maintaining a minimum thickness for the stratum. Based on this prioritization, if these two parameters conflict while generating the geological model, the computing device will ensure that the change in the thickness of the stratum is minimized. As another example, the computing device can receive user input identifying a feature of a stratum, such as a surface of the stratum. The user input can be used as a parameter. The computing device can also receive user input indicating that this parameter is to be prioritized over another parameter for minimizing a change in the thickness of a stratum. Based on this prioritization, if these two parameters conflict while generating the geological model, the computing device will ensure that the user input is respected. 
     Some examples of the present disclosure can provide enhanced geological models that reduce or eliminate the presence of crossing stratum-surfaces, strata that have rapidly changing thicknesses (e.g., where they become too thick or too thin), and other problems resulting from mismatched, disparate, or incongruous pieces of data. 
     These illustrative examples are given to introduce the reader to the general subject matter discussed here and are not intended to limit the scope of the disclosed concepts. The following sections describe various additional features and examples with reference to the drawings in which like numerals indicate like elements, and directional descriptions are used to describe the illustrative aspects but, like the illustrative aspects, should not be used to limit the present disclosure. 
       FIG. 1  is a cross-sectional side view of an example of a well system  100  for obtaining well logs according to some aspects. The well system  100  includes multiple wellbores  102   a - h  drilled through a subterranean formation  104 . The wellbores  102   a - h  extend from the well surface  108  into strata  106   a - c  of the subterranean formation  104 . The strata  106   a - c  can include different materials (e.g., rock, soil, oil, water, or gas) and vary in thickness and shape. 
     Some or all of the wellbores  102   a - h  can include well tools, such as logging tools  110   a - b , for generating the well logs. Each of the well tools can measure properties of the rocks, fluid, or other contents of the strata  106   a - c  and use the measured properties to generate a respective well log. For example, the logging tool  110   a  can measure the electrical, acoustic, radioactive, electromagnetic, or pressure properties of the strata regions proximate to wellbore  102   a . The logging tool  110   a  can then use the measurements to generate a well log. A separate well log can be generated for each of the wellbores  102   a - h.    
     The well tools can electronically communicate the well logs to a computing device  112 , which can be positioned onsite (as shown in  FIG. 1 ) or offsite. The well tools can electrically communicate with the computing device  112  via a wired or wireless interface. In some examples, the well tools can electronically communicate the well logs to the computing device  112  indirectly, such as over the Internet or another network. 
     The computing device  112  can use the well logs or other geological data to generate a geological model. The geological model can be a two-dimensional (2D) or three-dimensional (3D) representation of at least a portion of the subterranean formation  104 . One example of such a geological model is shown in  FIG. 2 . The geological model  200  includes a visual representation of a cross-section of the strata  106   a - c  in the subterranean formation  104 . Each of the strata  106   a - c  in the cross-section has a thickness and a surface. For example, stratum  106   a  has thickness  204  and stratal surface  202 . In some examples, a stratal surface can define a boundary between two of the strata  106   a - c . For example, stratal surface  206  can define a boundary between stratum  106   b  and stratum  106   c . The thicknesses and the surface shapes of the strata  106   a - c  can provide useful information about the subterranean formation  104 . 
     One example of the computing device  112  is shown in  FIG. 3 . The computing device  112  can include a processing device  302 , a bus  304 , a display device  306 , a memory device  308 , a user input device  316 , and a communication interface  318 . In some examples, some or all of the components shown in  FIG. 3  can be integrated into a single structure, such as a single housing. In other examples, some or all of the components shown in  FIG. 3  can be distributed (e.g., in separate housings) and in electrical communication with each other. 
     The processing device  302  can execute one or more operations for modeling geological strata using weighted parameters. The processing device  302  can execute instructions stored in the memory device  308  to perform the operations. The processing device  302  can include one processing device or multiple processing devices. Non-limiting examples of the processing device  302  include a Field-Programmable Gate Array (“FPGA”), an application-specific integrated circuit (“ASIC”), a microprocessing device, etc. 
     The processing device  302  can be communicatively coupled to the memory device  308  via the bus  304 . The non-volatile memory device  308  may include any type of memory device that retains stored information when powered off. Non-limiting examples of the memory device  308  include electrically erasable and programmable read-only memory (“EEPROM”), flash memory, or any other type of non-volatile memory. In some examples, at least some of the memory device  308  can include a medium from which the processing device  302  can read instructions. A computer-readable medium can include electronic, optical, magnetic, or other storage devices capable of providing the processing device  302  with computer-readable instructions or other program code. Non-limiting examples of a computer-readable medium include (but are not limited to) magnetic disk(s), memory chip(s), read-only memory (ROM), random-access memory (“RAM”), an ASIC, a configured processing device, optical storage, or any other medium from which a computer processing device can read instructions. The instructions can include processing device-specific instructions generated by a compiler or an interpreter from code written in any suitable computer-programming language, including, for example, C, C++, C#, etc. 
     In some examples, the memory device  308  can include geological data  310 , such as well logs or seismic data). The geological data  310  can be communicated to the computing device  112  from one or more well tools positioned in one or more wellbores. 
     In some examples, the memory device  308  can include a modelling engine  312 . The modelling engine  312  can be a software application for modeling geological strata using weighted parameters. More specifically, the modelling engine  312  may be able to receive or generate weights  322  for parameters. Some or all of the parameters can be constraints. The modelling engine  312  can then use the weights  322  for the parameters to determine shapes (e.g., cross-sectional shapes, surface shapes, thicknesses, or any combination of these) of strata in a subterranean formation. In some examples, the modelling engine  312  can determine the surface shapes and thicknesses of all the strata substantially simultaneously to one another. The modelling engine  312  can generate a geological model  314  based on the surface shapes and thicknesses for the strata. 
     The memory device  308  can also include equations  320  for generating the geological model  314 . Although the equations  320  are shown separately from the modelling engine  312  in this example, in other examples the modelling engine  312  can include the equations  320 . The equations  320  can form a system of equations that may be solved concurrently. For example, the equations  320  can depend on one another so that each equation affects the entire system of equations. Constraints placed on the equations  320 , and parameter values, may also affect the entire system of equations. 
     The computing device  112  can include a user input device  316 . The user input device  316  can represent one or more components used to input data. Examples of the user input device  316  can include a keyboard, mouse, touchpad, button, or touch-screen display, etc. 
     The computing device  112  can include a display device  306 . The display device  306  can represent one or more components used to output data. Examples of the display device  306  can include a liquid-crystal display (LCD), a television, a computer monitor, a touch-screen display, etc. In some examples, the user input device  316  and the display device  306  can be a single device, such as a touch-screen display. 
     The computing device  112  can include a communication interface  318 . The communication interface  318  can represent one or more components that facilitate a network connection or otherwise facilitate communication between electronic devices. Examples include, but are not limited to, wired interfaces such as Ethernet, USB, IEEE 1394, and/or wireless interfaces such as IEEE 802.11, Bluetooth, near-field communication (NFC) interfaces, RFID interfaces, or radio interfaces for accessing cellular telephone networks (e.g., transceiver/antenna for accessing a CDMA, GSM, UMTS, or other mobile communications network). 
     In some examples, the computing device  112  can implement one or more of the steps shown in  FIG. 4  for modeling geological strata using weighted parameters. The computing device  112  may alternatively implement a process that has more steps, fewer steps, different steps, or a different order of the steps depicted in  FIG. 4 . The steps below are described with reference to the components of  FIG. 3  discussed above. 
     In block  402 , the computing device  112  receives geological data  310  representative of the strata in a subterranean formation. The computing device  112  can receive the data from a well tool, a remote database, or a local memory device  208 . The geological data  310  can be a well log, seismic data, or other data associated with strata in the subterranean formation  104 . 
     In block  404 , the computing device  112  determines one or more weights  322  for one or more parameters that are to be used to generate a geological model  315 . In some examples, the weights  322  for the parameters are provided as user input via user input device  316 . In other examples, the weights  322  can be determined using other methods. For example, the weights  322  for the parameters can be determined (e.g., automatically) by a software application that is local to, or remote from, the computing device  112 . An example of the software application can be the modelling engine  312 . The computing device  112  can determine any number and combination of weights  322  for any number and combination of parameters. 
     Examples of the parameters can include (i) a first parameter for reducing a change in a thickness of a stratum, (ii) a second parameter for maintaining a minimum thickness of the stratum, (iii) a third parameter for maintaining a maximum thickness of the stratum, (iv) a fourth parameter for reducing a curvature in the thickness of the stratum, (v) a fifth parameter for a user-provided indication of a feature of the stratum, (vi) a sixth parameter for preventing stratal surfaces from crossing, or (vii) any combination of these. Examples of weights  322  for the parameters can be 0.25, 0.5, 0.75, and 1.0. Examples of the feature of the stratum can include a top surface of the stratum, a bottom surface of the stratum, a length of the stratum, a curvature in the stratum, a point within the stratum, or any combination of these. 
     The computing device  112  may additionally or alternatively determine priorities between two or more parameters. For example, the computing device  112  may output a graphical user interface through which a user can order parameters according to a desired priority. As another example, the priorities for the parameters can be determined (e.g., automatically) by a software application that is local to, or remote from, the computing device  112 . The software application can prioritize the parameters according to predetermined rules provided by a manufacturer or distributor of the software application. 
     In block  406 , the computing device  112  generates the geological model  314  based on the one or more weights  322  for the one or more parameters and the geological data  310 . For example, the computing device  112  can generate the geological model by concurrently determining cross-sectional shapes of the strata by solving a system of equations using the weight for the parameter and the geological data. In some examples, the computing device  112  can execute the modelling engine  312  to generate the geological model  314 . One method for generating the geological model  314  will be described in greater detail below with reference to  FIG. 5 . 
     In block  408 , the computing device  112  displays the geological model  314 . For example, the computing device  112  can output the geological model  314  using display device  306 . The geological model  314  can visually represent various properties of the strata in the subterranean formation. The geological model  315  can include a 2D or 3D representation of the strata. 
     Other operations can additionally or alternatively be performed after generating the geological model  314  in block  406 . For example, the computing device  112  or a well operator can control operation of a well tool, such as a drilling tool, based on the geological model  314 . In one particular example, the computing device  112  can transmit electronic communications for steering a drilling tool to avoid a stratum having certain undesirable properties or to target a stratum having certain desirable properties. In some examples, the computing device  112  can communicate the geological model  314  to a remote computing device to control the remote computing device or for analysis by a remote user. 
       FIG. 5  is a flow chart of an example of a process for generating a geological model according to some aspects. Some examples can be implemented using more steps, fewer steps, different steps, or a different order of the steps depicted in  FIG. 5 . The steps below are described with reference to the components of  FIG. 3  discussed above. 
     In block  502 , the computing device  112  determines equations  320  to be solved. For example, the computing device  112  can receive some or all of the equations  320  from a local memory device  308 , a remote memory device of a remote computing device, via user input, or any combination of these. 
     In some examples, the equations  320  include quadratic functions, or quadratic equations, and corresponding parameters. For example, the equations  320  can include a quadratic function representing the smoothness (e.g., curvature) of a surface of a stratum. Additionally or alternatively, the equations  320  can include another quadratic function representing a gradient of a thickness between two stratal surfaces. Each of the equations  320  can have one or more of the parameters associated with it. 
     In block  504 , the computing device  112  solves the equations  320  using the parameters (e.g., using the weights  322  for the parameters). The computing device  112  can use any number and combination of techniques for solving the equations  320  using the parameters. 
     In some examples, the equations  320  can include a system of equations, such as a system of quadratic functions. The computing device  112  may solve the system of equations by optimizing (e.g., find a maximum value or a minimum value of) the system of equations. For example, the system of equations can be solved simultaneously as a global optimization problem. The global optimization problem may have the following form:
         Minimize ½x T Qx+c T x subject to Ex≥d
 
where Q can represent a matrix into which the weights  322  for the parameters are encoded; c can represent possible linear terms (e.g., constants or a constant vector); E can represent another matrix into which non-penetration constraints are encoded; T can indicate a transposition of the related variable; x can represent a final configuration of the strata for the geological model; and d can represent a vector of thicknesses. Each row r in matrix E can have a corresponding thickness in the r th  entry of d. In some examples, a non-penetration constraint can be a thickness constraint between two nodes i and j of adjacent strata. The entry for this thickness constraint in a row of E can be in the form [1 . . . −1 . . . ], and the row in d can have the desired thickness. Then, the equation Ex≥d can indicate that x i &gt;x j +d (e.g., a nonpenetration or distance of at least d). In some examples, a result for the abovementioned global optimization problem can be determined by solving the system of equations using undetermined Lagrange multipliers, λ, as follows:
       

     
       
         
           
             
               
                 [ 
                 
                   
                     
                       Q 
                     
                     
                       
                         E 
                         T 
                       
                     
                   
                   
                     
                       E 
                     
                     
                       0 
                     
                   
                 
                 ] 
               
                
               
                 [ 
                 
                   
                     
                       x 
                     
                   
                   
                     
                       λ 
                     
                   
                 
                 ] 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       - 
                       c 
                     
                   
                 
                 
                   
                     d 
                   
                 
               
               ] 
             
           
         
       
     
     Additionally or alternatively, the system of equations can be solved simultaneously as a global dynamics problem. The system of equations can be quasi-static. In some examples, the system of equations can be solved by integrating a first-order ordinary differential equation for forces f, velicities v, and to determine the final configuration: 
     
       
         
           
             
               
                 d 
                 
                   d 
                    
                   t 
                 
               
                
               
                 ( 
                 
                   
                     
                       x 
                     
                   
                   
                     
                       v 
                     
                   
                 
                 ) 
               
             
             = 
             
               ( 
               
                 
                   
                     v 
                   
                 
                 
                   
                     
                       
                         M 
                         
                           - 
                           1 
                         
                       
                        
                       
                         f 
                          
                         
                           ( 
                           
                             x 
                             , 
                             v 
                           
                           ) 
                         
                       
                     
                   
                 
               
               ) 
             
           
         
       
     
     where M can represent a mass matrix. In some examples, M can be an identity matrix, as inertial effects can be ignored for the quasi-static case. The forces f can be a first derivative of energy (x′*Q*x), or f=Q*x. In some examples, the system of equations can be augmented using Lagrange multipliers to force the system of equations to respect hard constraints: 
     
       
         
           
             
               
                 ( 
                 
                   
                     
                       ∂ 
                       f 
                     
                     
                       ∂ 
                       x 
                     
                   
                    
                   
                     M 
                     
                       - 
                       1 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                       ∂ 
                       
                         f 
                         T 
                       
                     
                     
                       ∂ 
                       x 
                     
                   
                 
                 ) 
               
                
               
                   
               
                
               λ 
             
             = 
             
               
                 - 
                 h 
               
                
               
                 M 
                 
                   - 
                   1 
                 
               
                
               f 
             
           
         
       
     
     In some examples, equations in the system of equations can depend on one another. For example, a result from solving a first equation in the system of equations can be used as a value for a variable in a second equation in the system of equations. The computing device  112  can solve the second equation by solving the first equation. As a particular example, the system of equations can include at least two different equations, which can collectively define shapes for the strata. The at least two different equations can depend on each other. Each equation may, for example, define a different shape-characteristic of the strata. A shape-characteristic can be a characteristic of a shape of a stratum. The computing device  112  can solve the system of equations by first determining the result of the first equation, and then inserting the result into the second equation to solve the second equation. 
     The computing device  112  can solve the equations  320  to generate a dataset that includes data points defining characteristics of the strata. For example, the dataset can include data points that define the locations and shapes of stratal surfaces in the subterranean formation. The data points can additionally or alternatively define thicknesses of the strata in the subterranean formation. The data points can include X, Y, and Z coordinates. 
     In block  506 , the computing device  112  generates the geological model  314  based on the dataset. For example, the computing device  112  can graphically render the dataset to visually represent the strata in as layers positioned adjacent to one another, with each layer having a defined thickness and surface shape. The geological model  314  may help a well operator or geologist visualize the physical characteristics of the subterranean formation. 
     In some examples, the computing device  112  can update the geological model  314  in response to new data. For example, the computing device  112  can receive new geological data representative of the strata in the subterranean formation. The new geological data can be received, for example, while a wellbore is being drilled or after a wellbore has been drilled. The computing device  112  can generate an updated version of the geological model by solving the equations  320  using the weights  322  for the parameters and the new geological data. For example, the computing device  112  can concurrently re-determine the shapes (e.g., surface shapes and thicknesses) for strata by solving a system of quadratic functions. The computing device  112  can then display the updated version of the geological model via the display device  306 . 
     In some aspects, geological strata can be modeled using weighted parameters according to one or more of the following examples: 
     Example #1: A method for generating a geological model of strata in a subterranean formation can include receiving geological data representative of the strata in the subterranean formation. The method can include determining a weight for a parameter to be used to generate the geological model. The parameter can include (i) a first parameter for reducing a change in a thickness of a stratum, (ii) a second parameter for maintaining a minimum thickness of the stratum, (iii) a third parameter for maintaining a maximum thickness of the stratum, (iv) a fourth parameter for reducing a curvature in the thickness of the stratum, or (v) a fifth parameter for a user-provided indication of a feature of the stratum. The method can include concurrently determining cross-sectional shapes of the strata by solving a system of quadratic functions using the weight for the parameter and the geological data to generate the geological model. The method can include displaying, by a display device, the geological model. The geological model can visually represent the cross-sectional shapes of the strata. 
     Example #2: The method of Example #1 may include receiving the weight for the parameter from a user via a user input device. The method may include generating the geological model based on the weight for the parameter while ensuring that surfaces of the strata do not intersect with one another. 
     Example #3: The method of any of Examples #1-2 may feature the parameter including at least two different parameters. The method can include receiving a priority among the at least two different parameters from a user via a user input device. The method can include generating the geological model based on the priority among the at least two different parameters. 
     Example #4: The method of any of Examples #1-3 may include receiving new geological data representative of the strata in the subterranean formation. The method can include generating an updated version of the geological model by solving the system of quadratic functions using the weight for the parameter and the new geological data to concurrently re-determine the cross-sectional shapes of the strata. The method can include displaying the updated version of the geological model via the display device. 
     Example #5: The method of any of Examples #1-4 may feature the system of quadratic functions including at least two different quadratic functions that collectively define the cross-sectional shapes of the strata and have associated parameters. The at least two different quadratic functions can be dependent on one another. Each quadratic function of the at least two different quadratic functions can have at least one associated parameter and define a different shape-characteristic of the strata. The method can include concurrently determining the cross-sectional shapes of the strata by solving the at least two different quadratic functions subject to the associated parameters using Lagrange multipliers. 
     Example #6: The method of any of Examples #1-5 may feature the parameter including the first parameter. 
     Example #7: The method of any of Examples #1-6 may feature the parameter including the second parameter or the third parameter. 
     Example #8: The method of any of Examples #1-7 may feature the parameter including the fourth parameter. The cross-sectional shapes of the strata can include surface shapes and thicknesses of the strata. 
     Example #9: The method of any of Examples #1-8 may feature the parameter including the fifth parameter. The feature can include a surface of the stratum. 
     Example #10: A non-transitory computer-readable medium can include instructions for generating a geological model of strata in a subterranean formation. The instructions can be executable by a processing device. The instructions can cause the processing device to receive geological data representative of the strata in the subterranean formation. The instructions can cause the processing device to determine a weight for a parameter to be used to generate the geological model. The parameter can include (i) a first parameter for reducing a change in a thickness of a stratum, (ii) a second parameter for maintaining a minimum thickness of the stratum, (iii) a third parameter for maintaining a maximum thickness of the stratum, (iv) a fourth parameter for reducing a curvature in the thickness of the stratum, or (v) a fifth parameter for a user-provided indication of a feature of the stratum. The instructions can cause the processing device to concurrently determine cross-sectional shapes of the strata by solving a system of quadratic functions using the weight for the parameter and the geological data to generate the geological model. The instructions can cause the processing device to cause a display device to display the geological model. The geological model can visually represent the cross-sectional shapes of the strata. 
     Example #11: The non-transitory computer-readable medium of Example #10 may further include instructions that can cause the processing device to receive the weight for the parameter from a user via a user input device. The instructions can cause the processing device to generate the geological model based on the weight for the parameter while ensuring that surfaces of the strata do not intersect with one another. 
     Example #12: The non-transitory computer-readable medium of any of Examples #10-11 may further include instructions that can cause the processing device to receive a priority among at least two different parameters from a user via a user input device. The instructions can cause the processing device to generate the geological model based on the priority among the at least two different parameters. 
     Example #13: The non-transitory computer-readable medium of any of Examples #10-12 may feature the parameter including the first parameter and the fifth parameter. The feature can include a surface of the stratum. 
     Example #14: The non-transitory computer-readable medium of any of Examples #10-13 may further include instructions that can cause the processing device to solve the system of quadratic functions using a Lagrange multiplier corresponding to a constraint. 
     Example #15: The non-transitory computer-readable medium of any of Examples #10-14 may further include instructions that can cause the processing device to concurrently determine the cross-sectional shapes of the strata by determining a first surface shape of a first stratum in the strata substantially simultaneously to determining a second surface shape of a second stratum in the strata by solving the system of quadratic functions. 
     Example #16: The non-transitory computer-readable medium of any of Examples #10-15 may feature the system of quadratic functions including at least two different quadratic functions that collectively define the cross-sectional shapes of the strata. The non-transitory computer-readable medium can further include instructions that can cause the processing device to concurrently determine the cross-sectional shapes of the strata by solving the at least two different quadratic functions. 
     Example #17: A system for generating a geological model of strata in a subterranean formation can include a processing device and a memory device on which instructions executable by the processing device are stored. The instructions can cause the processing device to receive geological data representative of the strata in the subterranean formation. The instructions can cause the processing device to determine a weight for a parameter to be used to generate the geological model. The parameter can include (i) a first parameter for reducing a change in a thickness of a stratum, (ii) a second parameter for maintaining a minimum thickness of the stratum, (iii) a third parameter for maintaining a maximum thickness of the stratum, (iv) a fourth parameter for reducing a curvature in the thickness of the stratum, or (v) a fifth parameter for a user-provided indication of a feature of the stratum. The instructions can cause the processing device to concurrently determine cross-sectional shapes of the strata by solving a system of quadratic functions using the weight for the parameter and the geological data to generate the geological model. The instructions can cause the processing device to cause a display device to display the geological model. The geological model can visually represent the cross-sectional shapes of the strata. 
     Example #18: The system of Example #17 may feature the parameter including a plurality of parameters. The plurality of parameters can include the first parameter, the second parameter or the third parameter, and the fifth parameter. 
     Example #19: The system of any of Examples #17-18 may feature the memory device further including instructions executable by the processing device for causing the processing device to concurrently solving the system of quadratic functions using Lagrange multipliers corresponding to a plurality of constraints. 
     Example #20: The system of any of Examples #17-19 may feature the memory device further including instructions executable by the processing device for causing the processing device to concurrently solving the system of quadratic functions by: using a first quadratic function in the system of quadratic functions as a value for a variable in a second quadratic function in the system of quadratic functions, and solving the second quadratic function. 
     The foregoing description of certain examples, including illustrated examples, has been presented only for the purpose of illustration and description and is not intended to be exhaustive or to limit the disclosure to the precise forms disclosed. Numerous modifications, adaptations, and uses thereof will be apparent to those skilled in the art without departing from the scope of the disclosure.