Drupal How to fix notice unserialize() and Undefined index?

I am getting below mention notice on /admin/reports/status page.

1. Notice: unserialize(): Error at offset 0 of 728 bytes in field_read_fields() (line 374 of /my_website/modules/field/field.crud.inc).
2. Notice: Undefined index: settings in text_field_schema() (line 17 of /my_website/modules/field/modules/text/text.install).

How to fix the notice?

The function unserialize() fails to convert a serialized value back into a PHP value because of an invalid length (or a length mismatch) in the stored representation of the data.

This means that a field record was not stored correctly in its field table.

The second notice tells us the type of the incriminated field : text_field_schema(), invoked via the hook_field_schema, exposes three field types : text, text_long, text_with_summary.

// line 392 of modules/field/field.crud.inc
$schema = (array) module_invoke($field['module'], 'field_schema', $field);

Just above, line 388, you got this :

module_invoke_all('field_read_field', $field);

By implementing this hook you should be able to point out which record is broken :

function yourmodule_field_read_field($field) {
  if ($field['module'] === 'text' && !isset($field['settings'])) {
    dpm($field);
  }
}

Once identified, you will probably have to "repair" the field structure before saving it as usual via FieldAPI, e.g. :

$field += array(
  'settings' => array(
    'max_length' => $somelength
  )
);
field_update_field($field);

The structure should be as defined in field_create_field().

---

[EDIT]

If you can't use field API, still you should be able to see which record is broken for that field.

Call field_read_field($field_name) and debug :

// line 370 of modules/field/field.crud.inc
$fields = array();
$results = $query->execute();
foreach ($results as $record) {
  dpm($record); // check $record['data']
  $field = unserialize($record['data']); 
  // ...
}

The broken record should be printed just before the first notice (if you display it).

Check the serialized value ($record['data']), and compare it with other records to see what's wrong. In the end you may have to update the field using SQL statements if you can't use field API.